It is known that the Poincaré inequality is equivalent to the quadratic transportation-variance inequality (namely $W_{2}^{2}(f\mu ,\mu ) \leqslant C_{V} \mathrm{Var} _{\mu }(f)$), see Jourdain [10] and most recently Ledoux [12]. We give two alternative proofs to this fact. In particular, we achieve a smaller $C_{V}$ than before, which equals the double of Poincaré constant. Applying the same argument leads to more characterizations of the Poincaré inequality. Our method also yields a by-product as the equivalence between the logarithmic Sobolev inequality and strict contraction of heat flow in Wasserstein space provided that the Bakry-Émery curvature has a lower bound (here the control constants may depend on the curvature bound).

Next, we present a comparison inequality between $W_{2}^{2}(f\mu ,\mu )$ and its centralization $W_{2}^{2}(f_{c}\mu ,\mu )$ for $f_{c} = \frac{|\sqrt {f} - \mu (\sqrt {f})|^{2}} {\mathrm{Var} _{\mu }(\sqrt{f} )}$, which may be viewed as some special counterpart of the Rothaus’ lemma for relative entropy. Then it yields some new bound of $W_{2}^{2}(f\mu ,\mu )$ associated to the variance of $\sqrt{f} $ rather than $f$. As a by-product, we have another proof to derive the quadratic transportation-information inequality from Lyapunov condition, avoiding the Bobkov-Götze’s characterization of the Talagrand’s inequality.

We use the theory of regularity structures to develop an Itô formula for $u$, the solution of the one-dimensional stochastic heat equation driven by space-time white noise with periodic boundary conditions. In particular, for any smooth enough function $\varphi $ we can express the random distribution $(\partial _{t}-\partial _{xx})\varphi (u)$ and the random field $\varphi (u)$ in terms of the reconstruction of some modelled distributions. The resulting objects are then identified with some classical constructions of Malliavin calculus.

Let $S$ be the random walk obtained from “coin turning” with some sequence $\{p_{n}\}_{n\ge 2}$, as introduced in [8]. In this paper we investigate the scaling limits of $S$ in the spirit of the classical Donsker invariance principle, both for the heating and for the cooling dynamics. We prove that an invariance principle, albeit with a non-classical scaling, holds for “not too small” sequences, the order const$\cdot n^{-1}$ (critical cooling regime) being the threshold. At and below this critical order, the scaling behavior is dramatically different from the one above it. The same order is also the critical one for the Weak Law of Large Numbers to hold. In the critical cooling regime, an interesting process emerges: it is a continuous, piecewise linear, recurrent process, for which the one-dimensional marginals are Beta-distributed. We also investigate the recurrence of the walk and its scaling limit, as well as the ergodicity and mixing of the $n$th step of the walk.

We consider a contact process on $\mathbb{Z} ^{d}$ with two species that interact in a symbiotic manner. Each site can either be vacant or occupied by individuals of species $A$ and/or $B$. Multiple occupancy by the same species at a single site is prohibited. The name symbiotic comes from the fact that if only one species is present at a site then that particle dies with rate 1 but if both species are present then the death rate is reduced to $\mu \le 1$ for each particle at that site. We show the critical birth rate $\lambda _{c}(\mu )$ for weak survival is of order $\sqrt{\mu } $ as $\mu \to 0$. Mean-field calculations predict that when $\mu < 1/2$ there is a discontinuous transition as $\lambda $ is varied. In contrast, we show that, in any dimension, the phase transition is continuous. To be fair to the physicists that introduced the model, [27], the authors say that the symbiotic contact process is in the directed percolation universality class and hence has a continuous transition. However, a 2018 paper, [30], asserts that the transition is discontinuous above the upper critical dimension, which is 4 for oriented percolation.

For every $R>0$, consider the stochastic heat equation \[ \partial _{t} u_{R}(t\,,x)=\tfrac {1}{2} \Delta _{S_{R}^{2}}u_{R}(t\,,x)+\sigma (u_{R}(t\,,x)) \xi _{R}(t\,,x) \] on $S_{R}^{2}$, where $\xi _{R}=\dot {W_{R}}$ are centered Gaussian noises with the covariance structure given by $\mathrm {E}[\dot {W_{R}}(t,x)\dot {W_{R}}(s,y)]=h_{R}(x,y)\delta _{0}(t-s)$, where $h_{R}$ is symmetric and semi-positive definite and there exist some fixed constants $-2< C_{h_{up}}< 2$ and $\tfrac {1}{2} C_{h_{up}}-1<C_{h_{down}} \leqslant C_{h_{up}}$ such that for all $R>0$ and $x\,,y \in S_{R}^{2}$, $(\log R)^{C_{h_{down}}/2}=h_{down}(R)\leqslant h_{R}(x,y) \leqslant h_{up}(R)=(\log R)^{C_{h_{up}}/2}$, $\Delta _{S_{R}^{2}}$ denotes the Laplace-Beltrami operator defined on $S_{R}^{2}$ and $\sigma :\mathbb {R}\mapsto \mathbb {R}$ is Lipschitz continuous, positive and uniformly bounded away from $0$ and $\infty $. Under the assumption that $u_{R,0}(x)=u_{R}(0\,,x)$ is a nonrandom continuous function on $x \in S_{R}^{2}$ and the initial condition that there exists a finite positive $U$ such that $\sup _{R>0}\sup _{x \in S_{R}^{2}}\vert u_{R,0}(x)\vert \leqslant U$, we prove that for every finite positive $t$, there exist finite positive constants $C_{down}(t)$ and $C_{up}(t)$ which only depend on $t$ such that as $R \to \infty $, $\sup _{x \in S_{R}^{2}}\vert u_{R}(t\,,x)\vert $ is asymptotically bounded below by $C_{down}(t)(\log R)^{1/4+C_{h_{down}}/4-C_{h_{up}}/8}$ and asymptotically bounded above by $C_{up}(t)(\log R)^{1/2+C_{h_{up}}/4}$ with high probability.

Markov intertwining is an important tool in stochastic processes: it enables to prove equalities in law, to assess convergence to equilibrium in a probabilistic way, to relate apparently distinct random models or to make links with wave equations, see Carmona, Petit and Yor [8], Aldous and Diaconis [2], Borodin and Olshanski [7] and Pal and Shkolnikov [23] for examples of applications in these domains. Unfortunately the basic construction of Diaconis and Fill [10] is not easy to manipulate. An alternative approach, where the underlying coupling is first constructed, is proposed here as an attempt to remedy to this drawback, via random mappings for measure-valued dual processes, first in a discrete time and finite state space setting. This construction is related to the evolving sets of Morris and Peres [22] and to the coupling-from-the-past algorithm of Propp and Wilson [27]. Extensions to continuous frameworks enable to recover, via a coalescing stochastic flow due to Le Jan and Raimond [16], the explicit coupling underlying the intertwining relation between the Brownian motion and the Bessel-3 process due to Pitman [25]. To generalize such a coupling to more general one-dimensional diffusions, new coalescing stochastic flows would be needed and the paper ends with challenging conjectures in this direction.

We establish exponential inequalities for a class of V-statistics under strong mixing conditions. Our theory is developed via a novel kernel expansion based on random Fourier features and the use of a probabilistic method. This type of expansion is new and useful for handling many notorious classes of kernels.

It has been shown by van den Berg and Steif [5] that the sub-critical Ising model on $\mathbb{Z} ^{d}$ is a finitary factor of a finite-valued i.i.d. process. We strengthen this by showing that the factor map can be made to have finite expected coding volume (in fact, stretched-exponential tails), answering a question of van den Berg and Steif. The result holds at any temperature above the critical temperature. An analogous result holds for Markov random fields satisfying a high-noise assumption and for proper colorings with a large number of colors.

For large $q$, does the (discrete) uniform distribution on the set of $q!$ permutations of the vector $\bar{\mathbf {x}} ^{q}=(1,2,\dots ,q)'$ closely approximate the (continuous) uniform distribution on the $(q-2)$-sphere that contains them? These permutations comprise the vertices of the regular permutohedron, a $(q-1)$-dimensional convex polyhedron. The answer is emphatically no: these permutations are confined to a negligible portion of the sphere, and the regular permutohedron occupies a negligible portion of the ball. However, $(1,2,\dots ,q)$ is not the most favorable configuration for spherical uniformity of permutations. A more favorable configuration $\hat{\mathbf {x}} ^{q}$ is found, namely that which minimizes the normalized surface area of the largest empty spherical cap among its $q!$ permutations. Unlike that for $\bar{\mathbf {x}} ^{q}$, the normalized surface area of the largest empty spherical cap among the permutations of $\hat{\mathbf {x}} ^{q}$ approaches $0$ as $q\to \infty $. Nonetheless the permutations of $\hat{\mathbf {x}} ^{q}$ do not approach spherical uniformity either. The existence of an asymptotically spherically uniform permutation sequence remains an open question.

In this work, we derive sufficient and necessary conditions for the existence of a weak and mild solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Lévy process. Our approach requires to establish a stochastic Fubini result for stochastic integrals with respect to cylindrical Lévy processes. This approach enables us to conclude that the solution process has almost surely scalarly square integrable paths. Further properties of the solution such as the Markov property and stochastic continuity are derived.

We give a construction of an infinite stable looptree, which we denote by $\mathcal{L} ^{\infty }_{\alpha }$, and prove that it arises both as a local limit of the compact stable looptrees of Curien and Kortchemski (2015), and as a scaling limit of the infinite discrete looptrees of Richier (2017), and Björnberg and Stefánsson (2015). As a consequence, we are able to prove various convergence results for volumes of small balls in compact stable looptrees, explored more deeply in a companion paper. We also establish the spectral dimension of $\mathcal{L} ^{\infty }_{\alpha }$, and show that it agrees with that of its discrete counterpart. Moreover, we show that Brownian motion on $\mathcal{L} ^{\infty }_{\alpha }$ arises as a scaling limit of random walks on discrete looptrees, and as a local limit of Brownian motion on compact stable looptrees, which has similar consequences for the limit of the heat kernel.

We study the Wasserstein distance $W_{2}$ for Gaussian samples. We establish the exact rate of convergence $\sqrt{\log \log n/n} $ of the expected value of the $W_{2}$ distance between the empirical and true $c.d.f.$’s for the normal distribution. We also show that the rate of weak convergence is unexpectedly $1/\sqrt{n} $ in the case of two correlated Gaussian samples.

We consider two-dimensional dependent dynamical site percolation where sites perform majority dynamics. We introduce the critical percolation function at time $t$ as the infimum density with which one needs to begin in order to obtain an infinite open component at time $t$. We prove that, for any fixed time $t$, there is no percolation at criticality and that the critical percolation function is continuous. We also prove that, for any positive time, the percolation threshold is strictly smaller than the critical probability for independent site percolation.

In this paper, we consider the addition of two matrices in generic position, namely $A+UBU^{*}$, where $U$ is drawn under the Haar measure on the unitary or the orthogonal group. We show that, under mild conditions on the empirical spectral measures of the deterministic matrices $A$ and $B$, the law of the largest eigenvalue satisfies a large deviation principle, in the scale $N$, with an explicit rate function involving the limit of spherical integrals. We cover in particular the case when $A$ and $B$ have no outliers.

In this paper, we study the cut-off phenomenon under the total variation distance of $d$-dimensional Ornstein-Uhlenbeck processes which are driven by Lévy processes. That is to say, under the total variation distance, there is an abrupt convergence of the aforementioned process to its equilibrium, i.e. limiting distribution. Despite that the limiting distribution is not explicit, its distributional properties allow us to deduce that a profile function always exists in the reversible cases and it may exist in the non-reversible cases under suitable conditions on the limiting distribution. The cut-off phenomena for the average and superposition processes are also determined.

We study the existence of (Markovian) solutions to the $(A,C_{c}^{\infty }(\mathbb{R} ^{d}))$-martingale problem associated with the Lévy-type operator $A$ with symbol $q(x,\xi )$. Firstly, we establish conditions which ensure the existence of a solution. The main contribution is that our existence result allows for discontinuity in $x \mapsto q(x,\xi )$. Applying the result, we obtain new insights on the existence of weak solutions to a class of Lévy-driven SDEs with Borel measurable coefficients and on the the existence of stable-like processes with discontinuous coefficients. Secondly, we prove a Markovian selection theorem which shows that – under mild assumptions – the $(A,C_{c}^{\infty }(\mathbb{R} ^{d}))$-martingale problem gives rise to a strong Markov process. The result applies, in particular, to Lévy-driven SDEs. We illustrate the Markovian selection theorem with applications in the theory of non-local operators and equations; in particular, we establish under weak regularity assumptions a Harnack inequality for non-local operators of variable order.

We prove a priori bounds for solutions of stochastic reaction diffusion equations with super-linear damping in the reaction term. These bounds provide a control on the supremum of solutions on any compact space-time set which only depends on the specific realisation of the noise on a slightly larger set and which holds uniformly over all possible space-time boundary values. This constitutes a space-time version of the so-called “coming down from infinity” property. Bounds of this type are very useful to control the large scale behaviour of solutions effectively and can be used, for example, to construct solutions on the full space even if the driving noise term has no decay at infinity. Our method shows the interplay of the large scale behaviour, dictated by the non-linearity, and the small scale oscillations, dictated by the rough driving noise. As a by-product we show that there is a close relation between the regularity of the driving noise term and the integrability of solutions.

For all $\kappa > 0$, we show that the support of SLE$_{\kappa }$ curves is the closure in the sup-norm of the set of Loewner curves driven by nice (e.g. smooth) functions. It follows that the support is the closure of the set of simple curves starting at $0$.

Let $\mathbb{M} $ be the space of finite measures on a locally compact Polish space, and let $\mathcal{G} $ be the Gamma distribution on $\mathbb{M} $ with intensity measure $\nu \in \mathbb{M} $. Let $\nabla ^{ext}$ be the extrinsic derivative with tangent bundle $T\mathbb{M} = \cup _{\eta \in \mathbb{M} } L^{2}(\eta )$, and let $\mathcal{A} : T\mathbb{M} \rightarrow T\mathbb{M} $ be measurable such that $\mathcal{A} _{\eta }$ is a positive definite linear operator on $L^{2}(\eta )$ for every $\eta \in \mathbb{M} $. Moreover, for a measurable function $V$ on $\mathbb{M} $, let ${\mathrm{{d}} }{\mathcal{G} }^{V}= {\mathrm{{e}} }^{V}{\mathrm{{d}} }{\mathcal{G} }$. We investigate the Poincaré, weak Poincaré and super Poincaré inequalities for the Dirichlet form \[ \mathcal{E} _{\mathcal{A} ,V}(F,G):= \int _{\mathbb{M} }\langle \mathcal{A} _{\eta }\nabla ^{ext}F(\eta ), \nabla ^{ext}G(\eta )\rangle _{L^{2}(\eta )}\, {\mathrm{{d}} }{\mathcal{G} }^{V}(\eta ), \] which characterize various properties of the associated Markov semigroup. The main results are extended to the space of finite signed measures.

We establish sharp exponential deviation estimates of the information content as well as a sharp bound on the varentropy for the class of convex measures on Euclidean spaces. This generalizes a similar development for log-concave measures in the recent work of Fradelizi, Madiman and Wang (2016). In particular, our results imply that convex measures in high dimension are concentrated in an annulus between two convex sets (as in the log-concave case) despite their possibly having much heavier tails. Various tools and consequences are developed, including a sharp comparison result for Rényi entropies, inequalities of Kahane-Khinchine type for convex measures that extend those of Koldobsky, Pajor and Yaskin (2008) for log-concave measures, and an extension of Berwald’s inequality (1947).

We provide in this work a robust solution theory for random rough differential equations of mean field type \[ dX_{t} = V\big ( X_{t},{\mathcal L}(X_{t})\big )dt + \textrm{F} \bigl ( X_{t},{\mathcal L}(X_{t})\bigr ) dW_{t}, \] where $W$ is a random rough path and ${\mathcal L}(X_{t})$ stands for the law of $X_{t}$, with mean field interaction in both the drift and diffusivity. We show that, in addition to the enhanced path of $W$, the underlying rough path-like setting should also comprise an infinite dimensional component obtained by regarding the collection of realizations of $W$ as a deterministic trajectory with values in some $L^{q}$ space. This advocates for a suitable notion of controlled path à la Gubinelli inspired from Lions’ approach to differential calculus on Wasserstein space, the systematic use of the latter playing a fundamental role in our study. Whilst elucidating the rough set-up is a key step in the analysis, solving the mean field rough equation requires another effort: the equation cannot be dealt with as a mere rough differential equation driven by a possibly infinite dimensional rough path. Because of the mean field component, the proof of existence and uniqueness indeed asks for a specific and quite elaborated localization-in-time argument.

This paper is devoted to uniform versions of the Hanson-Wright inequality for a random vector $X \in \mathbb{R} ^{n}$ with independent subgaussian components. The core technique of the paper is based on the entropy method combined with truncations of both gradients of functions of interest and of the components of $X$ itself. Our results recover, in particular, the classic uniform bound of Talagrand [28] for Rademacher chaoses and the more recent uniform result of Adamczak [2] which holds under certain rather strong assumptions on the distribution of $X$. We provide several applications of our techniques: we establish a version of the standard Hanson-Wright inequality, which is tighter in some regimes. Extending our results we show a version of the dimension-free matrix Bernstein inequality that holds for random matrices with a subexponential spectral norm. We apply the derived inequality to the problem of covariance estimation with missing observations and prove an almost optimal high probability version of the recent result of Lounici [21]. Finally, we show a uniform Hanson-Wright-type inequality in the Ising model under Dobrushin’s condition. A closely related question was posed by Marton [22].

We study the free energy of mixed $p$-spin spin glass models enriched with an additional magnetic field given by the canonical Gaussian field associated with a Ruelle probability cascade. We prove the conjecture in [15] that this free energy converges to the Hopf-Lax solution of a certain Hamilton-Jacobi equation. Using this result, we give a new representation of the free energy of mixed $p$-spin models with soft spins.

We provide a verification and characterization result of optimal maximal sub-solutions of BSDEs in terms of fully coupled forward backward stochastic differential equations. We illustrate the application thereof in utility optimization with random endowment under probability and discounting uncertainty. We show with explicit examples how to quantify the costs of incompleteness when using utility indifference pricing, as well as a way to find optimal solutions for recursive utilities.

Consider an epidemic model with a constant flux of susceptibles, in a situation where the corresponding deterministic epidemic model has a unique stable endemic equilibrium. For the associated stochastic model, whose law of large numbers limit is the deterministic model, the disease free equilibrium is an absorbing state, which is reached soon or later by the process. However, for a large population size, i.e. when the stochastic model is close to its deterministic limit, the time needed for the stochastic perturbations to stop the epidemic may be enormous. In this paper, we discuss how the Central Limit Theorem, Moderate and Large Deviations allow us to give estimates of the extinction time of the epidemic.

In this work we develop stochastic control methods for the study of large deviation principles (LDP) for certain interacting particle systems. Although such methods have been well studied for analyzing large deviation properties of small noise stochastic dynamical systems [7] and of weakly interacting particle systems [6], this is the first work to implement the approach for Brownian particle systems with a local interaction. As an application of these methods we give a new proof of the large deviation principle from the hydrodynamic limit for the Ginzburg-Landau model studied in [10]. Along the way, we establish regularity properties of the densities of certain controlled Markov processes and certain results relating large deviation principles and Laplace principles in non-Polish topological spaces that are of independent interest. The proof of the LDP is based on characterizing subsequential hydrodynamic limits of controlled diffusions with nearest neighbor interaction that arise from a variational representation of certain Laplace functionals. This proof also yields a new representation for the rate function which is very natural from a control theoretic point of view. Proof techniques are very similar to those used for the law of large number analysis, namely in the proof of convergence to the hydrodynamic limit (cf. [15]). Specifically, the key step in the proof is establishing suitable bounds on relative entropies and Dirichlet forms associated with certain controlled laws. This general approach has the promise to be applicable to other interacting Brownian systems as well.

We introduce a new Wright-Fisher type model for seed banks incorporating “simultaneous switching”, which is motivated by recent work on microbial dormancy ([21], [28]). We show that the simultaneous switching mechanism leads to a new jump-diffusion limit for the scaled frequency processes, extending the classical Wright-Fisher and seed bank diffusion limits. We further establish a new dual coalescent structure with multiple activation and deactivation events of lineages. While this seems reminiscent of multiple merger events in general exchangeable coalescents, it actually leads to an entirely new class of coalescent processes with unique qualitative and quantitative behaviour. To illustrate this, we provide a novel kind of condition for coming down from infinity for these coalescents, applying a recent approach of Griffiths [12].

The free energy for multiple systems of spherical spin glasses with constrained overlaps was first studied in [10]. In [24] the authors proved an upper bound of the constrained free energy using Guerra’s interpolation. In this paper, we prove this upper bound is sharp. Our approach combines the ideas of the Aizenman–Sims–Starr scheme in [4] and the synchronization mechanism used in the vector spin models in [22] and [23]. We derive a vector version of the Aizenman–Sims–Starr scheme for spherical spin glass and use the synchronization property of arrays obeying the overlap-matrix form of the Ghirlanda–Guerra identities to prove the matching lower bound.

In last passage percolation models, the energy of a path is maximized over all directed paths with given endpoints in a random environment, and the maximizing paths are called geodesics. The geodesics and their energy can be scaled so that transformed geodesics cross unit distance and have fluctuations and scaled energy of unit order. Here we consider Poissonian last passage percolation, a model lying in the KPZ universality class, and refer to scaled geodesics as polymers and their scaled energies as weights. Polymers may be viewed as random functions of the vertical coordinate and, when they are, we show that they have modulus of continuity whose order is at most $t^{2/3}\big (\log t^{-1}\big )^{1/3}$. The power of one-third in the logarithm may be expected to be sharp and in a related problem we show that it is: among polymers in the unit box whose endpoints have vertical separation $t$ (and a horizontal separation of the same order), the maximum transversal fluctuation has order $t^{2/3}\big (\log t^{-1}\big )^{1/3}$. Regarding the orthogonal direction, in which growth occurs, we show that, when one endpoint of the polymer is fixed at $(0,0)$ and the other is varied vertically over $(0,z)$, $z\in [1,2]$, the resulting random weight profile has sharp modulus of continuity of order $t^{1/3}\big (\log t^{-1}\big )^{2/3}$. In this way, we identify exponent pairs of $(2/3,1/3)$ and $(1/3,2/3)$ in power law and polylogarithmic correction, respectively for polymer fluctuation, and polymer weight under vertical endpoint perturbation. The two exponent pairs describe [9, 10, 8] the fluctuation of the boundary separating two phases in subcritical planar random cluster models.

We consider a one-dimensional random walk among biased i.i.d. conductances, in the case where the random walk is transient but sub-ballistic: this occurs when the conductances have a heavy-tail at $+\infty $ or at $0$. We prove that the scaling limit of the process is the inverse of an $\alpha $-stable subordinator, which indicates an aging phenomenon, expressed in terms of the generalized arcsine law. In analogy with the case of an i.i.d. random environment studied in details in [ESZ09a, ESTZ13], some “traps” are responsible for the slowdown of the random walk. However, the phenomenology is somehow different (and richer) here. In particular, three types of traps may occur, depending on the fine properties of the tails of the conductances: (i) a very large conductance (a well in the potential); (ii) a very small conductance (a wall in the potential); (iii) the combination of a large conductance followed shortly after by a small conductance (a well-and-wall in the potential).

In this paper we obtain Berry–Esseén bounds on partial sums of functionals of heavy-tailed moving averages, including the linear fractional stable noise, stable fractional ARIMA processes and stable Ornstein–Uhlenbeck processes. Our rates are obtained for the Wasserstein and Kolmogorov distances, and depend strongly on the interplay between the memory of the process, which is controlled by a parameter $\alpha $, and its tail-index, which is controlled by a parameter $\beta $. In fact, we obtain the classical $1/\sqrt {n}$ rate of convergence when the tails are not too heavy and the memory is not too strong, more precisely, when $\alpha \beta >3$ or $\alpha \beta >4$ in the case of Wasserstein and Kolmogorov distance, respectively.

Our quantitative bounds rely on a new second-order Poincaré inequality on the Poisson space, which we derive through a combination of Stein’s method and Malliavin calculus. This inequality improves and generalizes a result by Last, Peccati, Schulte [Probab. Theory Relat. Fields 165 (2016)].

We consider the thick points of random walk, i.e. points where the local time is a fraction of the maximum. In two dimensions, we answer a question of [19] and compute the number of thick points of planar random walk, assuming that the increments are symmetric and have a finite moment of order two. The proof provides a streamlined argument based on the connection to the Gaussian free field and works in a very general setting including isoradial graphs. In higher dimensions, we study the scaling limit of the set of thick points. In particular, we show that the rescaled number of thick points converges to a nondegenerate random variable and that the centred maximum of the local times converges to a randomly shifted Gumbel distribution.

We consider one-dependent random walks on ${\mathbb{Z} }^{d}$ in random hypergeometric environment for $d\ge 3$. These are memory-one walks in a large class of environments parameterized by positive weights on directed edges and on pairs of directed edges which includes the class of Dirichlet environments as a special case. We show that the walk is a.s. transient for any choice of the parameters, and moreover that the return time has some finite positive moment. We then give a characterization for the existence of an invariant measure for the process from the point of view of the walker which is absolutely continuous with respect to the initial distribution on the environment in terms of a function $\kappa $ of the initial weights. These results generalize [Sab11] and [Sab13] on random walks in Dirichlet environment. It turns out that $\kappa $ coincides with the corresponding parameter in the Dirichlet case, and so in particular the existence of such invariant measures is independent of the weights on pairs of directed edges, and determined solely by the weights on directed edges.

We consider two related linear PDE’s perturbed by a fractional Brownian motion. We allow the drift to be discontinuous, in which case the corresponding deterministic equation is ill-posed. However, the noise will be shown to have a regularizing effect on the equations in the sense that we can prove existence of solutions for almost all paths of the fractional Brownian motion.

We prove the existence and uniqueness of entropy solutions for nonlinear diffusion equations with nonlinear conservative gradient noise. As particular applications our results include stochastic porous media equations, as well as the one-dimensional stochastic mean curvature flow in graph form.

This work is devoted to the study of semimartingales on the dual of a general nuclear space. We start by establishing conditions for a cylindrical semimartingale in the strong dual $\Phi '$ of a nuclear space $\Phi $ to have a $\Phi '$-valued semimartingale version whose paths are right-continuous with left limits. Results of similar nature but for more specific classes of cylindrical semimartingales and examples are also provided. Later, we will show that under some general conditions every semimartingale taking values in the dual of a nuclear space has a canonical representation. The concept of predictable characteristics is introduced and is used to establish necessary and sufficient conditions for a $\Phi '$-valued semimartingale to be a $\Phi '$-valued Lévy process.

We provide necessary and sufficient conditions for convergence of exponential integrals of Markov additive processes. By contrast with the classical Lévy case studied by Erickson and Maller we have to distinguish between almost sure convergence and convergence in probability. Our proofs rely on recent results on perpetuities in a Markovian environment by Alsmeyer and Buckmann.

We introduce a stochastic version of Gubinelli’s sewing lemma ([18]), providing a sufficient condition for the convergence in moments of some random Riemann sums. Compared with the deterministic sewing lemma, adaptiveness is required and the regularity restriction is improved by a half. The limiting process exhibits a Doob-Meyer-type decomposition. Relations with Itô calculus are established. To illustrate further potential applications, we use the stochastic sewing lemma in studying stochastic differential equations driven by Brownian motions or fractional Brownian motions with irregular drifts.

We introduce a class of kinetic and anisotropic random motions $(x_{t}^{\sigma },v_{t}^{\sigma })_{t \geq 0}$ on the unit tangent bundle $T^{1} \mathcal{M} $ of a general Riemannian manifold $(\mathcal{M} ,g)$, where $\sigma $ is a positive parameter quantifying the amount of noise affecting the dynamics. As the latter goes to infinity, we then show that the time rescaled process $(x_{\sigma ^{2} t}^{\sigma })_{t \geq 0}$ converges in law to an explicit anisotropic Brownian motion on $\mathcal{M} $. Our approach is essentially based on the strong mixing properties of the underlying velocity process and on rough paths techniques, allowing us to reduce the general case to its Euclidean analogue. Using these methods, we are able to recover a range of classical results.

We establish a sharp estimate on the negative moments of the smallest eigenvalue of the Malliavin matrix $\gamma _{Z}$ of $Z := (u(s, y), u(t, x) - u(s, y))$, where $u$ is the solution to a system of $d$ non-linear stochastic heat equations in spatial dimension $k \geq 1$. We also obtain the optimal exponents for the $L^{p}$-modulus of continuity of the increments of the solution and of its Malliavin derivatives. These lead to optimal lower bounds on hitting probabilities of the process $\{u(t, x): (t, x) \in [0, \infty [ \times \mathbb{R} ^{k}\}$ in the non-Gaussian case in terms of Newtonian capacity, and improve a result in Dalang, Khoshnevisan and Nualart [Stoch PDE: Anal Comp 1 (2013) 94–151].

We use analytical methods to construct the two-parameter Feller semigroup associated with a Markov process on a line with a moving membrane such that at the points on both sides of the membrane it coincides with the ordinary diffusion processes given there and its behavior after reaching the membrane is described by a kind of nonlocal Feller-Wentzell conjugation condition.

We consider Diffusion Limited Aggregation (DLA) in a two-dimensional wedge. We prove that if the angle of the wedge is smaller than $\pi /4$, there is some $a>2$ such that almost surely, for all $R$ large enough, after time $R^{a}$ all new particles attached to the DLA will be at distance larger than $R$ from the origin. Furthermore, we provide estimates on the size of $R$ under which this holds. This means that DLA stabilizes in growing balls, thus allowing a definition of the infinite DLA in a wedge via a finite time process.

We give a recursive construction of the stationary distribution of multi-type asymmetric simple exclusion processes on a finite ring or on the infinite line $\mathbb{Z} $. The construction can be interpreted in terms of “multi-line diagrams” or systems of queues in tandem. Let $q$ be the asymmetry parameter of the system. The queueing construction generalises the one previously known for the totally asymmetric ($q=0$) case, by introducing queues in which each potential service is unused with probability $q^{k}$ when the queue-length is $k$. The analysis is based on the matrix product representation of Prolhac, Evans and Mallick. Consequences of the construction include: a simple method for sampling exactly from the stationary distribution for the system on a ring; results on common denominators of the stationary probabilities, expressed as rational functions of $q$ with non-negative integer coefficients; and probabilistic descriptions of “convoy formation” phenomena in large systems.

Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general càdlàg semimartingales taking values in Lie groups are defined and investigated. The considered set of SDEs, first introduced by S. Cohen, includes affine and Marcus type SDEs as well as smooth SDEs driven by Lévy processes and iterated random maps. A natural extension to this general setting of reduction and reconstruction theory for symmetric SDEs is provided. Our theorems imply as special cases non trivial invariance results concerning a class of affine iterated random maps as well as (weak) symmetries for numerical schemes (of Euler and Milstein type) for Brownian motion driven SDEs.

In the setting of stochastic dynamical systems that eventually go extinct, the quasi-stationary distributions are useful to understand the long-term behavior of a system before evanescence. For a broad class of applicable continuous-time Markov processes on countably infinite state spaces, known as reaction networks, we introduce the inferred notion of absorbing and endorsed sets, and obtain sufficient conditions for the existence and uniqueness of a quasi-stationary distribution within each such endorsed set. In particular, we obtain sufficient conditions for the existence of a globally attracting quasi-stationary distribution in the space of probability measures on the set of endorsed states. Furthermore, under these conditions, the convergence from any initial distribution to the quasi-stationary distribution is exponential in the total variation norm.

We establish large deviation principles (LDPs) for empirical measures associated with a sequence of Gibbs distributions on $n$-particle configurations, each of which is defined in terms of an inverse temperature $\beta _{n}$ and an energy functional consisting of a (possibly singular) interaction potential and a (possibly weakly) confining potential. Under fairly general assumptions on the potentials, we use a common framework to establish LDPs both with speeds $\beta _{n}/n \rightarrow \infty $, in which case the rate function is expressed in terms of a functional involving the potentials, and with speed $\beta _{n} =n$, when the rate function contains an additional entropic term. Such LDPs are motivated by questions arising in random matrix theory, sampling, simulated annealing and asymptotic convex geometry. Our approach, which uses the weak convergence method developed by Dupuis and Ellis, establishes LDPs with respect to stronger Wasserstein-type topologies. Our results address several interesting examples not covered by previous works, including the case of a weakly confining potential, which allows for rate functions with minimizers that do not have compact support, thus resolving several open questions raised in a work of Chafaï et al.

We study limits of the largest connected components (viewed as metric spaces) obtained by critical percolation on uniformly chosen graphs and configuration models with heavy-tailed degrees. For rank-one inhomogeneous random graphs, such results were derived by Bhamidi, van der Hofstad, Sen (2018) [15]. We develop general principles under which the identical scaling limits as in [15] can be obtained. Of independent interest, we derive refined asymptotics for various susceptibility functions and the maximal diameter in the barely subcritical regime.

This paper consists of two parts. In the first part, we focus on the average of a functional over shifted Gaussian homogeneous noise and as the averaging domain covers the whole space, we establish a Breuer-Major type Gaussian fluctuation based on various assumptions on the covariance kernel and/or the spectral measure. Our methodology for the first part begins with the application of Malliavin calculus around Nualart-Peccati’s Fourth Moment Theorem, and in addition we apply the Fourier techniques as well as a soft approximation argument based on Bessel functions of first kind.

The same methodology leads us to investigate a closely related problem in the second part. We study the spatial average of a linear stochastic heat equation driven by space-time Gaussian colored noise. The temporal covariance kernel $\gamma _{0}$ is assumed to be locally integrable in this paper. If the spatial covariance kernel is nonnegative and integrable on the whole space, then the spatial average admits the Gaussian fluctuation; with some extra mild integrability condition on $\gamma _{0}$, we are able to provide a functional central limit theorem. These results complement recent studies on the spatial average for SPDEs. Our analysis also allows us to consider the case where the spatial covariance kernel is not integrable: For example, in the case of the Riesz kernel, the first chaotic component of the spatial average is dominant so that the Gaussian fluctuation also holds true.

We examine an interacting particle system on trees commonly referred to as the frog model. For its initial state, it begins with a single active particle at the root and i.i.d. $\mathrm{Poiss} (\lambda )$ many inactive particles at each non-root vertex. Active particles perform discrete time simple random walk and in the process activate any inactive particles they encounter. We show that for every non-amenable tree with bounded degree there exists a phase transition from transience to recurrence (with a non-trivial intermediate phase sometimes sandwiched in between) as $\lambda $ varies.

We study a class of stochastic partial integral-differential equations with an asymmetrical non-local operator $\frac{1} {2}\Delta +a^{\alpha }\Delta ^{\frac{\alpha } {2}}+b\cdot \nabla $ and a distribution expressed as divergence of a measurable field. For $0<\alpha <2$, the existence and uniqueness of solution is proved by analytical method, and a probabilistic interpretation, similar to the Feynman-Kac formula, is presented for $ 0<\alpha <1$. The method of backward doubly stochastic differential equations is also extended in this work.

We consider a Lévy process $X=(X(t))_{t\ge 0}$ in a generalised Feller class at 0, and study the exit position, $\left \vert X(T(r))\right \vert $, as $X$ leaves, and the position, $\left \vert X(T( r) -)\right \vert $, just prior to its leaving, at time $T(r)$, a two-sided region with boundaries at $\pm r$, $r>0$. Conditions are known for $X$ to be in the Feller class $FC_{0}$ at zero, by which we mean that each sequence $t_{k}\downarrow 0$ contains a subsequence through which $X(t_{k})$, after norming by a nonstochastic function, converges to an a.s. finite nondegenerate random variable. We use these conditions on $X$ to characterise similar properties for the normed positions $\left \vert X(T( r))\right \vert /r$ and $\left \vert X(T( r) -)\right \vert /r$, and also for the normed jump $\left \vert \Delta X(T(r))/r\right \vert = \left \vert X(T(r))-X(T(r)-)\right \vert /r$ (“the jump causing ruin"), as convergence takes place through sequences $r_{k}\downarrow 0$. We go on to give conditions for the continuity of distributions of the limiting random variables obtained in this way.

We consider the two-dimensional simple random walk conditioned on never hitting the origin, which is, formally speaking, the Doob’s $h$-transform of the simple random walk with respect to the potential kernel. We then study the behavior of the future minimum distance of the walk to the origin, and also prove that two independent copies of the conditioned walk, although both transient, will nevertheless meet infinitely many times a.s.

The dead leaves model (DLM) provides a random tessellation of $d$-space, representing the visible portions of fallen leaves on the ground when $d=2$. For $d=1$, we establish formulae for the intensity, two-point correlations, and asymptotic covariances for the point process of cell boundaries, along with a functional CLT. For $d=2$ we establish analogous results for the random surface measure of cell boundaries, and also determine the intensity of cells in a more general setting than in earlier work of Cowan and Tsang. We introduce a general notion of dead leaves random measures and give formulae for means, asymptotic variances and functional CLTs for these measures; this has applications to various other quantities associated with the DLM.

We study the question of when a $\{0,1\}$-valued threshold process associated to a mean zero Gaussian or a symmetric stable vector corresponds to a divide and color (DC) process. This means that the process corresponding to fixing a threshold level $h$ and letting a 1 correspond to the variable being larger than $h$ arises from a random partition of the index set followed by coloring all elements in each partition element 1 or 0 with probabilities $p$ and $1-p$, independently for different partition elements.

While it turns out that all discrete Gaussian free fields yield a DC process when the threshold is zero, for general $n$-dimensional mean zero, variance one Gaussian vectors with nonnegative covariances, this is true in general when $n=3$ but false for $n=4$.

The behavior is quite different depending on whether the threshold level $h$ is zero or not and we show that there is no general monotonicity in $h$ in either direction. We also show that all constant variance discrete Gaussian free fields with a finite number of variables yield DC processes for large thresholds.

In the stable case, for the simplest nontrivial symmetric stable vector with three variables, we obtain a phase transition in the stability exponent $\alpha $ at the surprising value of $1/2$; if the index of stability is larger than $1/2$, then the process yields a DC process for large $h$ while if the index of stability is smaller than $1/2$, then this is not the case.

We study occupancy counts for the critical nearest-neighbor branching random walk on the $d$-dimensional lattice, conditioned on non-extinction. For $d\geq 3$, Lalley and Zheng [4] showed that the properly scaled joint distribution of the number of sites occupied by $j$ generation-$n$ particles, $j=1,2,\ldots $, converges in distribution as $n$ goes to infinity, to a deterministic multiple of a single exponential random variable. The limiting exponential variable can be understood as the classical Yaglom limit of the total population size of generation $n$. Here we study the second order fluctuations around this limit, first, by providing a rate of convergence in the Wasserstein metric that holds for all $d\geq 3$, and second, by showing that for $d\geq 7$, the weak limit of the scaled joint differences between the number of occupancy-$j$ sites and appropriate multiples of the total population size converge in the Wasserstein metric to a multivariate symmetric Laplace distribution. We also provide a rate of convergence for this latter result.

Consider the Markov process taking values in the partitions of $\mathbb{N} $ such that each pair of blocks merges at rate one, and each integer is eroded, i.e., becomes a singleton block, at rate $d$. This is a special case of exchangeable fragmentation-coalescence process, called Kingman’s coalescent with erosion. We provide a new construction of the stationary distribution of this process as a sample from a standard flow of bridges. This allows us to give a representation of the asymptotic frequencies of this stationary distribution in terms of a sequence of independent diffusions. Moreover, we introduce a new process valued in the partitions of $\mathbb{Z} $ called Kingman’s coalescent with immigration, where pairs of blocks coalesce at rate one, and new blocks of size one immigrate according to a Poisson process of intensity $d$. By coupling Kingman’s coalescents with erosion and with immigration, we are able to show that the size of a block chosen uniformly at random from the stationary distribution of the restriction of Kingman’s coalescent with erosion to $\{1, \dots , n\}$ converges as $n\to \infty $ to the total progeny of a critical binary branching process.

We show that a space-time rescaling of the spatial Lamba-Fleming-Viot process of Barton and Etheridge converges to super-Brownian motion. This can be viewed as an extension of a result of Chetwynd-Diggle and Etheridge [5]. In that work the scaled impact factors (which govern the event based dynamics) vanish in the limit, here we drop that requirement. The analysis is particularly interesting in the biologically relevant two-dimensional case.

We consider a family of stochastic models of evolving two-dimensional Young diagrams, given in terms of certain energies, with Gibbs invariant measures. ‘Static’ scaling limits of the shape functions, under these Gibbs measures, have been shown in the literature. The purpose of this article is to study corresponding, but less understood, ‘dynamical’ limits. We show that the hydrodynamic scaling limits of the diagram shape functions may be described by different types of parabolic PDEs, depending on the energy structure.

In this article, we study the behavior of consecutive values of random completely multiplicative functions $(X_{n})_{n \geq 1}$ whose values are i.i.d. at primes. We prove that for $X_{2}$ uniform on the unit circle, or uniform on the set of roots of unity of a given order, and for fixed $k \geq 1$, $X_{n+1}, \dots , X_{n+k}$ are independent if $n$ is large enough. Moreover, with the same assumption, we prove the almost sure convergence of the empirical measure $N^{-1} \sum _{n=1}^{N} \delta _{(X_{n+1}, \dots , X_{n+k})}$ when $N$ goes to infinity, with an estimate of the rate of convergence. At the end of the paper, we also show that for any probability distribution on the unit circle followed by $X_{2}$, the empirical measure converges almost surely when $k=1$.

We consider a random Gaussian ensemble of Laplace eigenfunctions on the 3D torus, and investigate the 1-dimensional Hausdorff measure (‘length’) of nodal intersections against a smooth 2-dimensional toral sub-manifold (‘surface’). A prior result of ours prescribed the expected length, universally proportional to the area of the reference surface, times the wavenumber, independent of the geometry.

In this paper, for surfaces contained in a plane, we give an upper bound for the nodal intersection length variance, depending on the arithmetic properties of the plane. The bound is established via estimates on the number of lattice points in specific regions of the sphere.

Interacting particle systems can often be constructed from a graphical representation, by applying local maps at the times of associated Poisson processes. This leads to a natural coupling of systems started in different initial states. We consider interacting particle systems on the complete graph in the mean-field limit, i.e., as the number of vertices tends to infinity. We are not only interested in the mean-field limit of a single process, but mainly in how several coupled processes behave in the limit. This turns out to be closely related to recursive tree processes as studied by Aldous and Bandyopadyay in discrete time. We here develop an analogue theory for recursive tree processes in continuous time. We illustrate the abstract theory on an example of a particle system with cooperative branching. This yields an interesting new example of a recursive tree process that is not endogenous.

The convergence to the stationary regime is studied for Stochastic Differential Equations driven by an additive Gaussian noise and evolving in a semi-contractive environment, i.e. when the drift is only contractive out of a compact set but does not have repulsive regions. In this setting, we develop a synchronous coupling strategy to obtain sub-exponential bounds on the rate of convergence to equilibrium in Wasserstein distance. Then by a coalescent coupling close to terminal time, we derive a similar bound in total variation distance.

We prove that for a random walk on the real line whose increments have zero mean and are either integer-valued or spread out (i.e. the distributions of steps of the walk are eventually non-singular), the Markov chain of overshoots above a fixed level converges in total variation to its stationary distribution. We find the explicit form of this distribution heuristically and then prove its invariance using a time-reversal argument. If, in addition, the increments of the walk are in the domain of attraction of a non-one-sided $\alpha $-stable law with index $\alpha \in (1,2)$ (resp. have finite variance), we establish geometric (resp. uniform) ergodicity for the Markov chain of overshoots. All the convergence results above are also valid for the Markov chain obtained by sampling the walk at the entrance times into an interval.

We are interested in nodes with fixed outdegrees in large conditioned Galton-Watson trees. We first study the scaling limits of processes coding the evolution of the number of such nodes in different explorations of the tree (lexicographical order and contour order) starting from the root. We give necessary and sufficient conditions for the limiting processes to be centered, thus measuring the linearity defect of the evolution of the number of nodes with fixed outdegrees. This extends results by Labarbe & Marckert in the case of the contour-ordered counting process of leaves in uniform plane trees. Then, we extend results obtained by Janson concerning the asymptotic normality of the number of nodes with fixed outdegrees.

We study level-set percolation of the Gaussian free field on the infinite $d$-regular tree for fixed $d\geq 3$. Denoting by $h_{\star }$ the critical value, we obtain the following results: for $h>h_{\star }$ we derive estimates on conditional exponential moments of the size of a fixed connected component of the level set above level $h$; for $h<h_{\star }$ we prove that the number of vertices connected over distance $k$ above level $h$ to a fixed vertex grows exponentially in $k$ with positive probability. Furthermore, we show that the percolation probability is a continuous function of the level $h$, at least away from the critical value $h_{\star }$. Along the way we also obtain matching upper and lower bounds on the eigenfunctions involved in the spectral characterisation of the critical value $h_{\star }$ and link the probability of a non-vanishing limit of the martingale used therein to the percolation probability. A number of the results derived here are applied in the accompanying paper [1].

We consider the upper and lower tail probabilities for the centered (by time$/24$) and scaled (according to KPZ time$^{1/3}$ scaling) one-point distribution of the Cole-Hopf solution of the KPZ equation when started with initial data drawn from a very general class. For the lower tail, we prove an upper bound which demonstrates a crossover from super-exponential decay with exponent $3$ in the shallow tail to an exponent $5/2$ in the deep tail. For the upper tail, we prove super-exponential decay bounds with exponent $3/2$ at all depths in the tail.

We establish a novel bijective encoding that represents permutations as forests of decorated (or enriched) trees. This allows us to prove local convergence of uniform random permutations from substitution-closed classes satisfying a criticality constraint. It also enables us to reprove and strengthen permuton limits for these classes in a new way, that uses a semi-local version of Aldous’ skeleton decomposition for size-constrained Galton–Watson trees.

We introduce a new model of preferential attachment with fitness, and establish a time reversed duality between the model and a system of branching-coalescing particles. Using this duality, we give a clear and concise explanation for the condensation phenomenon, in which unusually fit vertices may obtain abnormally high degree: it arises from a growth-extinction dichotomy within the branching part of the dual.

We show further that the condensation is extensive. As the graph grows, unusually fit vertices become, each only for a limited time, neighbouring to a non-vanishing proportion of the current graph.

Existence of stationary point vortices solution to the damped and stochastically driven Euler’s equation on the two dimensional torus is proved, by taking limits of solutions with finitely many vortices. A central limit scaling is used to show in a similar manner the existence of stationary solutions with white noise marginals.

The irreducible decomposition of successive restriction and induction of irreducible representations of a symmetric group gives rise to a Markov chain on Young diagrams keeping the Plancherel measure invariant. Starting from this Res-Ind chain, we introduce a not necessarily Markovian continuous time random walk on Young diagrams by considering a general pausing time distribution between jumps according to the transition probability of the Res-Ind chain. We show that, under appropriate assumptions for the pausing time distribution, a diffusive scaling limit brings us concentration at a certain limit shape depending on macroscopic time which leads to a similar consequence to the exponentially distributed case studied in our earlier work. The time evolution of the limit shape is well described by using free probability theory. On the other hand, we illustrate an anomalous phenomenon observed with a pausing time obeying a one-sided stable distribution, heavy-tailed without the mean, in which a nontrivial behavior appears under a non-diffusive regime of the scaling limit.

We study two different versions of the simple exclusion process on augmented Galton–Watson trees, the constant speed model and the variable speed model. In both cases, the simple exclusion process starts from an equilibrium distribution with non-vanishing particle density. Moreover, we assume to have initially a particle in the root, the tagged particle. We show for both models that the tagged particle has a positive linear speed and we give explicit formulas for the speeds.

We obtain factorial moment identities for the Charlier, Meixner and Krawtchouk orthogonal polynomial ensembles. Building on earlier results by Ledoux [Elect. J. Probab. 10, (2005)], we find hypergeometric representations for the factorial moments when the reference measure is Poisson (Charlier ensemble) and geometric (a particular case of the Meixner ensemble). In these cases, if the number of particles is suitably randomised, the factorial moments have a polynomial property, and satisfy three-term recurrence relations and differential equations. In particular, the normalised factorial moments of the randomised ensembles are precisely related to the moments of the corresponding equilibrium measures. We also briefly outline how these results can be interpreted as Cauchy-type identities for certain Schur measures.

This paper introduces the Attracting Random Walks model, which describes the dynamics of a system of particles on a graph with $n$ vertices. At each step, a single particle moves to an adjacent vertex (or stays at the current one) with probability proportional to the exponent of the number of other particles at a vertex. From an applied standpoint, the model captures the rich get richer phenomenon. We show that the Markov chain exhibits a phase transition in mixing time, as the parameter governing the attraction is varied. Namely, mixing time is $O(n\log n)$ when the temperature is sufficiently high and $\exp (\Omega (n))$ when temperature is sufficiently low. When $\mathcal {G}$ is the complete graph, the model is a projection of the Potts model, whose mixing properties and the critical temperature have been known previously. However, for any other graph our model is non-reversible and does not seem to admit a simple Gibbsian description of a stationary distribution. Notably, we demonstrate existence of the dynamic phase transition without decomposing the stationary distribution into phases.

We provide a simple abstract formalism of integration by parts under which we obtain some regularization lemmas. These lemmas apply to any sequence of random variables $(F_{n})$ which are smooth and non-degenerated in some sense and enable one to upgrade the distance of convergence from smooth Wasserstein distances to total variation in a quantitative way. This is a well studied topic and one can consult for instance [3, 11, 14, 20] and the references therein for an overview of this issue. Each of the aforementioned references share the fact that some non-degeneracy is required along the whole sequence. We provide here the first result removing this costly assumption as we require only non-degeneracy at the limit. The price to pay is to control the smooth Wasserstein distance between the Malliavin matrix of the sequence and its limit, which is particularly easy in the context of Gaussian limit as the Malliavin matrix is deterministic. We then recover, in a slightly weaker form, the main findings of [19]. Another application concerns the approximation of the semi-group of a diffusion process by the Euler scheme in a quantitative way and under the Hörmander condition.

This is a continuation of our earlier work [Stochastic Processes and their Applications, 129(1), pp.102–128, 2019] on the random walk in random scenery and in random layered conductance. We complete the picture of upper deviation of the random walk in random scenery, and also prove a bound on lower deviation probability. Based on these results, we determine asymptotics of the return probability, a certain moderate deviation probability, and the Green function of the random walk in random layered conductance.

We study a model, introduced initially by Gates and Westcott [11] to describe crystal growth evolution, which belongs to the Anisotropic KPZ universality class [19]. It can be thought of as a $(2+1)$-dimensional generalisation of the well known ($1+1$)-dimensional Polynuclear Growth Model (PNG). We show the full hydrodynamic limit of this process i.e the convergence of the random interface height profile after ballistic space-time scaling to the viscosity solution of a Hamilton-Jacobi PDE: $\partial _{t}u = v(\nabla u)$ with $v$ an explicit non-convex speed function. The convergence holds in the strong almost sure sense.

We study the supercritical contact process on Galton-Watson trees and periodic trees. We prove that if the contact process survives weakly then it dominates a supercritical Crump-Mode-Jagers branching process. Hence the number of infected sites grows exponentially fast. As a consequence we conclude that the contact process dies out at the critical value $\lambda _{1}$ for weak survival, and the survival probability $p(\lambda )$ is continuous with respect to the infection rate $\lambda $. Applying this fact, we show the contact process on a general periodic tree experiences two phase transitions in the sense that $\lambda _{1}<\lambda _{2}$, which confirms a conjecture of Stacey’s [12]. We also prove that if the contact process survives strongly at $\lambda $ then it survives strongly at a $\lambda '<\lambda $, which implies that the process does not survive strongly at the critical value $\lambda _{2}$ for strong survival.

The Box-Ball System (BBS) is a one-dimensional cellular automaton in the configuration space $\{0,1\}^{\mathbb{Z} }$ introduced by Takahashi and Satsuma [8], who identified conserved quantities called solitons. Ferrari, Nguyen, Rolla and Wang [4] map a configuration to a family of soliton components, indexed by the soliton sizes $k\ge 1$. Building over this decomposition, we give an explicit construction of a large family of invariant measures for the BBS that are also shift invariant, including Ising-like Markov and Bernoulli product measures. The construction is based on the concatenation of iid excursions of the associated walk trajectory. Each excursion has the property that the law of its $k$ component given the larger components is product of a finite number of geometric distributions with a parameter depending on $k$. As a consequence, the law of each component of the resulting ball configuration is product of identically distributed geometric random variables, and the components are independent. This last property implies invariance for BBS, as shown by [4].

We study weak convergence of a sequence of point processes to a scale-invariant simple point process. For a deterministic sequence $(z_{n})_{n\in \mathbb {N}}$ of positive real numbers increasing to infinity as $n \to \infty $ and a sequence $(X_{k})_{k\in \mathbb {N}}$ of independent non-negative integer-valued random variables, we consider the sequence of point processes \[ \nu _{n}=\sum _{k=1}^{\infty }X_{k} \delta _{z_{k}/z_{n}}, \quad n \in \mathbb {N}, \] and prove that, under some general conditions, it converges vaguely in distribution to a scale-invariant Poisson process $\eta _{c}$ on $(0,\infty )$ with the intensity measure having the density $ct^{-1}$, $t\in (0,\infty )$. An important motivating example from probabilistic number theory relies on choosing $X_{k} \sim {\mathrm {Geom}}(1-1/p_{k})$ and $z_{k}=\log p_{k}$, $k \in \mathbb {N}$, where $(p_{k})_{k \in \mathbb {N}}$ is an enumeration of the primes in increasing order. We derive a general result on convergence of the integrals $\int _{0}^{1} t \nu _{n}(dt)$ to the integral $\int _{0}^{1} t \eta _{c}(dt)$, the latter having a generalized Dickman distribution, thus providing a new way of proving Dickman convergence results.

We extend our results to the multivariate setting and provide sufficient conditions for vague convergence in distribution for a broad class of sequences of point processes obtained by mapping the points from $(0,\infty )$ to $\mathbb {R}^{d}$ via multiplication by i.i.d. random vectors. In addition, we introduce a new class of multivariate Dickman distributions which naturally extends the univariate setting.

We study the contact process running in the one-dimensional lattice undergoing dynamical percolation, where edges open at rate $vp$ and close at rate $v(1-p)$. Our goal is to explore how the speed of the environment, $v$, affects the behavior of the process. Among our main results we find that: 1. For small enough $v$ the process dies out, while for large $v$ the process behaves like a contact process on $\mathbb {Z}$ with rate $\lambda p$, where $\lambda $ is the birth rate of each particle, so in particular it survives if $\lambda $ is large. 2. For fixed $v$ and small enough $p$ the network becomes immune, in the sense that the process dies out for any infection rate $\lambda $, while if $p$ is sufficiently close to $1$ then for all $v>0$ survival is possible for large enough $\lambda $. 3. Even though the first two points suggest that larger values of $v$ favor survival, this is not necessarily the case for small $v$: when the number of initially infected sites is large enough, the infection survives for a larger expected time in a static environment than in the case of $v$ positive but small. Some of these results hold also in the setting of general (infinite) vertex-transitive regular graphs.

We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed edge weights. We find classes with different behaviour depending on a sequence of parameters $(s_{n})_{n\geq 1}$ that quantifies the extreme-value behavior of small weights. We consider both $n$-independent as well as $n$-dependent edge weights and illustrate our results in many examples.

In particular, we investigate the case where $s_{n}\rightarrow \infty $, and focus on the exploration process that grows the smallest-weight tree from a vertex. We establish that the smallest-weight tree process locally converges to the invasion percolation cluster on the Poisson-weighted infinite tree, and we identify the scaling limit of the weight of the smallest-weight path between two uniform vertices. In addition, we show that over a long time interval, the growth of the smallest-weight tree maintains the same volume-height scaling exponent – volume proportional to the square of the height – found in critical Galton–Watson branching trees and critical Erdős-Rényi random graphs.

The strong rate of convergence of the Euler-Maruyama scheme for nondegenerate SDEs with irregular drift coefficients is considered. In the case of $\alpha $-Hölder drift in the recent literature the rate $\alpha /2$ was proved in many related situations. By exploiting the regularising effect of the noise more efficiently, we show that the rate is in fact arbitrarily close to $1/2$ for all $\alpha >0$. The result extends to Dini continuous coefficients, while in $d=1$ also to all bounded measurable coefficients.

We identify the local scaling limit of multiple boundary-to-boundary branches in a uniform spanning tree (UST) as a local multiple $\mathrm {SLE}(2)$, i.e., an $\mathrm {SLE}(2)$ process weighted by a suitable partition function. By recent results, this also characterizes the “global” scaling limit of the full collection of full curves. The identification is based on a martingale observable in the UST with $N$ branches, obtained by weighting the well-known martingale in the UST with one branch by the discrete partition functions of the models. The obtained weighting transforms of the discrete martingales and the limiting SLE processes, respectively, only rely on a discrete domain Markov property and (essentially) the convergence of partition functions. We illustrate their generalizability by sketching an analogous convergence proof for a boundary-visiting UST branch and a boundary-visiting $\mathrm {SLE}(2)$.

We consider a chain of $n$ coupled oscillators placed on a one-dimensional lattice with periodic boundary conditions. The interaction between particles is determined by a weakly anharmonic potential $V_{n} = r^{2}/2 + \sigma _{n}U(r)$, where $U$ has bounded second derivative and $\sigma _{n}$ vanishes as $n \to \infty $. The dynamics is perturbed by noises acting only on the positions, such that the total momentum and length are the only conserved quantities. With relative entropy technique, we prove for dynamics out of equilibrium that, if $\sigma _{n}$ decays sufficiently fast, the fluctuation field of the conserved quantities converges in law to a linear $p$-system in the hyperbolic space-time scaling limit. The transition speed is spatially homogeneous due to the vanishing anharmonicity. We also present a quantitative bound for the speed of convergence to the corresponding hydrodynamic limit.

We establish estimates for the coalescence time of semi-infinite directed geodesics in the planar corner growth model with i.i.d. exponential weights. There are four estimates: upper and lower bounds on the probabilities of both fast and slow coalescence on the correct spatial scale with exponent $3/2$. Our proofs utilize a geodesic duality introduced by Pimentel and properties of the increment-stationary last-passage percolation process. For fast coalescence our bounds are new and they have matching optimal exponential order of magnitude. For slow coalescence we reproduce bounds proved earlier with integrable probability inputs, except that our upper bound misses the optimal order by a logarithmic factor.

In the present paper, we study central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a point of view of discrete geometric analysis developed by Kotani and Sunada. We establish a semigroup CLT for a non-symmetric random walk on a nilpotent covering graph. Realizing the nilpotent covering graph into a nilpotent Lie group through a discrete harmonic map, we give a geometric characterization of the limit semigroup on the nilpotent Lie group. More precisely, we show that the limit semigroup is generated by the sub-Laplacian with a non-trivial drift on the nilpotent Lie group equipped with the Albanese metric. The drift term arises from the non-symmetry of the random walk and it vanishes when the random walk is symmetric. Furthermore, by imposing the “centered condition”, we establish a functional CLT (i.e., Donsker-type invariance principle) in a Hölder space over the nilpotent Lie group. The functional CLT is extended to the case where the realization is not necessarily harmonic. We also obtain an explicit representation of the limiting diffusion process on the nilpotent Lie group and discuss a relation with rough path theory. Finally, we give an example of random walks on nilpotent covering graphs with explicit computations.

The loop-erased random walk (LERW) in $ {\mathbb {Z}}^{d}, d \geq 2$, is obtained by erasing loops chronologically from simple random walk. In this paper we show the existence of the two-sided LERW which can be considered as the distribution of the LERW as seen by a point in the “middle” of the path.

This article fills a gap in the mathematical analysis of Adaptive Biasing algorithms, which are extensively used in molecular dynamics computations. Given a reaction coordinate, ideally, the biasing force in the overdamped Langevin dynamics would be given by the gradient of the associated free energy function, which is unknown. We consider an adaptive biased version of the overdamped dynamics, where the biasing force depends on the past of the trajectory and is designed to approximate the free energy.

The main result of this article is the consistency and efficiency of this approach. More precisely we prove the almost sure convergence of the biasing force as time goes to infinity, and that the limit is close to the ideal biasing force, as an auxiliary parameter of the algorithm goes to $0$.

The proof is based on interpreting the process as a self-interacting dynamics, and on the study of a non-trivial fixed point problem for the limiting flow obtained using the ODE method.

Plaquette models are short range ferromagnetic spin models that play a key role in the dynamic facilitation approach to the liquid glass transition. In this paper we study the dynamics of the square plaquette model at the smallest of the three critical length scales discovered in [7]. Our main results are estimates of the spectral gap and mixing time for two natural boundary conditions. As a consequence, we observe that these time scales depend heavily on the boundary condition in this scaling regime.

In this article we study a system of $N$ particles, each of them being defined by the couple of a position (in $\mathbb {R}^{d}$) and a so-called orientation which is an element of a compact Riemannian manifold. This orientation can be seen as a generalisation of the velocity in Vicsek-type models such as [20, 16]. We will assume that the orientation of each particle follows a jump process whereas its position evolves deterministically between two jumps. The law of the jump depends on the position of the particle and the orientations of its neighbours. In the limit $N\to +\infty $, we first prove a propagation of chaos result which can be seen as an adaptation of the classical result on McKean-Vlasov systems [53] to Piecewise Deterministic Markov Processes (PDMP). As in [38], we then prove that under a proper rescaling with respect to $N$ of the interaction radius between the agents (moderate interaction), the law of the limiting mean-field system satisfies a BGK equation with localised interactions which has been studied as a model of collective behaviour in [14]. Finally, in the spatially homogeneous case, we give an alternative approach based on martingale arguments.

We derive explicit central moment inequalities for random variables that admit a Stein coupling, such as exchangeable pairs, size–bias couplings or local dependence, among others. The bounds are in terms of moments (not necessarily central) of variables in the Stein coupling, which are typically local in some sense, and therefore easier to bound. In cases where the Stein couplings have the kind of behaviour leading to good normal approximation, the central moments are closely bounded by those of a normal. We show how the bounds can be used to produce concentration inequalities, and compare them to those existing in related settings. Finally, we illustrate the power of the theory by bounding the central moments of sums of neighbourhood statistics in sparse Erdős–Rényi random graphs.

For i.i.d. $d$-dimensional observations $X^{(1)}, X^{(2)}, \ldots $ with independent Exponential$(1)$ coordinates, consider the boundary (relative to the closed positive orthant), or “frontier”, $F_{n}$ of the closed Pareto record-setting (RS) region \[ \mbox {RS}_{n} := \{0 \leq x \in \mathbb {R}^{d}: x \not \prec X^{(i)} \ \text {for all}\ 1 \leq i \leq n\} \] at time $n$, where $0 \leq x$ means that $0 \leq x_{j}$ for $1 \leq j \leq d$ and $x \prec y$ means that $x_{j} < y_{j}$ for $1 \leq j \leq d$. With $x_{+} := \sum _{j = 1}^{d} x_{j}$, let \[ F_{n}^{-} := \min \{x_{+}: x \in F_{n}\} \quad \text {and} \quad F_{n}^{+} := \max \{x_{+}: x \in F_{n}\}, \] and define the width of $F_{n}$ as \[ W_{n} := F_{n}^{+} - F_{n}^{-}. \] We describe typical and almost sure behavior of the processes $F^{+}$, $F^{-}$, and $W$. In particular, we show that $F^{+}_{n} \sim \ln n \sim F^{-}_{n}$ almost surely and that $W_{n}/\ln \ln n$ converges in probability to $d - 1$; and for $d \geq 2$ we show that, almost surely, the set of limit points of the sequence $W_{n}/\ln \ln n$ is the interval $[d - 1, d]$.

We also obtain modifications of our results that are important in connection with efficient simulation of Pareto records. Let $T_{m}$ denote the time that the $m$th record is set. We show that $F^{+}_{T_{m}} \sim (d! m)^{1/d} \sim F^{-}_{T_{m}}$ almost surely and that $W_{T_{m}} / \ln m$ converges in probability to $1 - d^{-1}$; and for $d \geq 2$ we show that, almost surely, the sequence $W_{T_{m}}/\ln m$ has $\liminf $ equal to $1 - d^{-1}$ and $\limsup $ equal to $1$.

Nazarov and Sodin have shown that the number of connected components of the nodal set of a planar Gaussian field in a ball of radius $R$, normalised by area, converges to a constant as $R\to \infty $. This has been generalised to excursion/level sets at arbitrary levels, implying the existence of functionals $c_{ES}(\ell )$ and $c_{LS}(\ell )$ that encode the density of excursion/level set components at the level $\ell $. We prove that these functionals are continuously differentiable for a wide class of fields. This follows from a more general result, which derives differentiability of the functionals from the decay of the probability of ‘four-arm events’ for the field conditioned to have a saddle point at the origin. For some fields, including the important special cases of the Random Plane Wave and the Bargmann-Fock field, we also derive stochastic monotonicity of the conditioned field, which allows us to deduce regions on which $c_{ES}(\ell )$ and $c_{LS}(\ell )$ are monotone.

In this article we derive Talagrand’s $T_{2}$ inequality on the path space w.r.t. the maximum norm for various stochastic processes, including solutions of one-dimensional stochastic differential equations with measurable drifts, backward stochastic differential equations, and the value process of optimal stopping problems.

The proofs do not make use of the Girsanov method, but of pathwise arguments. These are used to show that all our processes of interest are Lipschitz transformations of processes which are known to satisfy desired functional inequalities.

The dynamics of a Markov process are often specified by its infinitesimal generator or, equivalently, its symbol. This paper contains examples of analytic symbols which do not determine the law of the corresponding Markov process uniquely. These examples also show that the law of a polynomial process in the sense of [4, 5, 11] is not necessarily determined by its generator if it has jumps. On the other hand, we show that a combination of smoothness of the symbol and ellipticity warrants uniqueness in law. The proof of this result is based on proving stability of univariate marginals relative to some properly chosen distance.

Using marked Dirichlet processes we characterise the law of the convex minorant of the meander for a certain class of Lévy processes, which includes subordinated stable and symmetric Lévy processes. We apply this characterisation to construct $\varepsilon $-strong simulation ($\varepsilon $SS) algorithms for the convex minorant of stable meanders, the finite dimensional distributions of stable meanders and the convex minorants of weakly stable processes. We prove that the running times of our $\varepsilon $SS algorithms have finite exponential moments. We implement the algorithms in Julia 1.0 (available on GitHub) and present numerical examples supporting our convergence results.

We study the radial part of sub-Riemannian Brownian motion in the context of totally geodesic foliations. Itô’s formula is proved for the radial processes associated to Riemannian distances approximating the Riemannian one. We deduce very general stochastic completeness criteria for the sub-Riemannian Brownian motion. In the context of Sasakian foliations and H-type groups, one can push the analysis further, and taking advantage of the recently proved sub-Laplacian comparison theorems one can compare the radial processes for the sub-Riemannian distance to one-dimensional model diffusions. As a geometric application, we prove Cheng’s type estimates for the Dirichlet eigenvalues of the sub-Riemannian metric balls, a result which seems to be new even in the Heisenberg group.

We show that planar Bargmann-Fock percolation is noise sensitive under the Ornstein-Ulhenbeck process. The proof is based on the randomized algorithm approach introduced by Schramm and Steif ([30]) and gives quantitative polynomial bounds on the noise sensitivity of crossing events for Bargmann-Fock.

A rather counter-intuitive consequence is as follows. Let $F$ be a Bargmann-Fock Gaussian field in $\mathbb {R}^{3}$ and consider two horizontal planes $P_{1},P_{2}$ at small distance $\varepsilon $ from each other. Even though $F$ is a.s. analytic, the above noise sensitivity statement implies that the full restriction of $F$ to $P_{1}$ (i.e. $F_{| P_{1}}$) gives almost no information on the percolation configuration induced by $F_{|P_{2}}$.

As an application of this noise sensitivity analysis, we provide a Schramm-Steif based proof that the near-critical window of level line percolation around $\ell _{c}=0$ is polynomially small. This new approach extends earlier sharp threshold results to a larger family of planar Gaussian fields.

Backward stochastic differential equations extend the martingale representation theorem to the nonlinear setting. This can be seen as path-dependent counterpart of the extension from the heat equation to fully nonlinear parabolic equations in the Markov setting. This paper extends such a nonlinear representation to the context where the random variable of interest is measurable with respect to the information at a finite stopping time. We provide a complete wellposedness theory which covers the semilinear case (backward SDE), the semilinear case with obstacle (reflected backward SDE), and the fully nonlinear case (second order backward SDE).

We study the pairing between zeros and critical points of the polynomial $p_{n}(z) = \prod _{j=1}^{n}(z-X_{j})$, whose roots $X_{1}, \ldots , X_{n}$ are complex-valued random variables. Under a regularity assumption, we show that if the roots are independent and identically distributed, the Wasserstein distance between the empirical distributions of roots and critical points of $p_{n}$ is on the order of $1/n$, up to logarithmic corrections. The proof relies on a careful construction of disjoint random Jordan curves in the complex plane, which allow us to naturally pair roots and nearby critical points. In addition, we establish asymptotic expansions to order $1/n^{2}$ for the locations of the nearest critical points to several fixed roots. This allows us to describe the joint limiting fluctuations of the critical points as $n$ tends to infinity, extending a recent result of Kabluchko and Seidel. Finally, we present a local law that describes the behavior of the critical points when the roots are neither independent nor identically distributed.

Parabolic partial differential equations (PDEs) are widely used in the mathematical modeling of natural phenomena and man-made complex systems. In particular, parabolic PDEs are a fundamental tool to approximately determine fair prices of financial derivatives in the financial engineering industry. The PDEs appearing in financial engineering applications are often nonlinear (e.g., in PDE models which take into account the possibility of a defaulting counterparty) and high-dimensional since the dimension typically corresponds to the number of considered financial assets. A major issue in the scientific literature is that most approximation methods for nonlinear PDEs suffer from the so-called curse of dimensionality in the sense that the computational effort to compute an approximation with a prescribed accuracy grows exponentially in the dimension of the PDE or in the reciprocal of the prescribed approximation accuracy and nearly all approximation methods for nonlinear PDEs in the scientific literature have not been shown not to suffer from the curse of dimensionality. Recently, a new class of approximation schemes for semilinear parabolic PDEs, termed full history recursive multilevel Picard (MLP) algorithms, were introduced and it was proven that MLP algorithms do overcome the curse of dimensionality for semilinear heat equations. In this paper we extend and generalize those findings to a more general class of semilinear PDEs which includes as special cases the important examples of semilinear Black-Scholes equations used in pricing models for financial derivatives with default risks. In particular, we introduce an MLP algorithm for the approximation of solutions of semilinear Black-Scholes equations and prove, under the assumption that the nonlinearity in the PDE is globally Lipschitz continuous, that the computational effort of the proposed method grows at most polynomially in both the dimension and the reciprocal of the prescribed approximation accuracy. We thereby establish, for the first time, that the numerical approximation of solutions of semilinear Black-Scholes equations is a polynomially tractable approximation problem.

We derive the large deviation principle for radial Schramm-Loewner evolution ($\operatorname {SLE}$) on the unit disk with parameter $\kappa \rightarrow \infty $. Restricting to the time interval $[0,1]$, the good rate function is finite only on a certain family of Loewner chains driven by absolutely continuous probability measures $\{\phi _{t}^{2} (\zeta )\, d\zeta \}_{t \in [0,1]}$ on the unit circle and equals $\int _{0}^{1} \int _{S^{1}} |\phi _{t}'|^{2}/2\,d\zeta \,dt$. Our proof relies on the large deviation principle for the long-time average of the Brownian occupation measure by Donsker and Varadhan.

We generalize the coalescing Brownian flow, also known as the Brownian web, considered as a weak flow to allow varying drift and diffusivity in the constituent diffusion processes and call these flows coalescing diffusive flows. We then identify the time-reversal of each coalescing diffusive flow and provide two distinct proofs of this identification. One of which is direct and the other proceeds by generalizing the concept of a localized disturbance flow to allow varying size and shape of disturbances, we show these new flows converge weakly under appropriate conditions to a coalescing diffusive flow and identify their time-reversals.

We consider a subcritical Galton–Watson tree $\mathsf {T}_{n}^{\Omega }$ conditioned on having $n$ vertices with outdegree in a fixed set $\Omega $. The offspring distribution is assumed to have a regularly varying density such that it lies in the domain of attraction of an $\alpha $-stable law for $1<\alpha \le 2$. Our main results consist of a local limit theorem for the maximal degree of $\mathsf {T}_{n}^{\Omega }$, and various limits describing the global shape of $\mathsf {T}_{n}^{\Omega }$. In particular, we describe the joint behaviour of the fringe subtrees dangling from the vertex with maximal degree. We provide applications of our main results to establish limits of graph parameters, such as the height, the non-maximal vertex outdegrees, and fringe subtree statistics.

We prove distributional limit theorems for the length of the largest convex minorant of a one-dimensional random walk with independent identically distributed increments. Depending on the increment law, there are several regimes with different limit distributions for this length. Among other tools, a representation of the convex minorant of a random walk in terms of uniform random permutations is utilized.

In [9] the Hausdorff dimension, $d_{f}$, of $\partial \mathcal {R}$, the topological boundary of the range of super-Brownian motion for dimensions $d=2,3$ was found; $d_{f}=4-2\sqrt {2}$ if $d=2$, and $d_{f}=(9-\sqrt {17})/2$ if $d=3$. We will refine these dimension estimates in a number of ways.

If $L^{x}$ is the total occupation local time of $d$-dimensional super-Brownian motion, $X$, for $d=2$ and $d=3$, we construct a random measure $\mathcal {L}$, called the boundary local time measure, as a rescaling of $L^{x} e^{-\lambda L^{x}} dx$ as $\lambda \to \infty $, thus confirming a conjecture of [19] and further show that the support of $\mathcal {L}$ equals $\partial \mathcal {R}$. This latter result uses a second construction of a boundary local time $\widetilde {\mathcal {L}}$ given in terms of exit measures and we prove that $\widetilde {\mathcal {L}}=c\mathcal {L}$ a.s. for some constant $c>0$. We derive reasonably explicit first and second moment measures for $\mathcal {L}$ in terms of negative dimensional Bessel processes and use them with the energy method to give a more direct proof of the lower bound of the Hausdorff dimension of $\partial \mathcal {R}$ in [9]. The construction requires a refinement of the $L^{2}$ upper bounds in [19] and [9] to exact $L^{2}$ asymptotics. The methods also refine the left tail bounds for $L^{x}$ in [19] to exact asymptotics. We conjecture that the $d_{f}$-dimensional Minkowski content of $\partial \mathcal {R}$ is equal to the total mass of the boundary local time $\mathcal {L}$ up to some constant.

For a smooth vectorial stationary Gaussian random field, $X:\Omega \times \mathbb {R}^{d}\to \mathbb {R}^{d}$, we provided necessary conditions to have a finite second moment for the number of roots of $X(t)-u$. Then, under a more restrictive hypothesis, some sufficient conditions were also given. The results were obtained using a method of proof inspired the one obtained by D. Geman for stationary Gaussian processes. Afterward, the same method is applied to the number of critical points of a scalar random field and to the level set of a vectorial process, $X:\Omega \times \mathbb {R}^{D}\to \mathbb {R}^{d}$, with $D>d$.

This paper concerns the Vertex Reinforced Jump Process (VRJP) and its representations as a Markov process in random environment. In [21], it was shown that the VRJP on finite graphs, under a certain time rescaling, has the distribution of a mixture of Markov jump processes. This representation was extended to infinite graphs in [23], by introducing a random potential $\beta $. In this paper, we show that all possible representations of the VRJP as a mixture of Markov processes can be expressed in a similar form as in [23], using the random field $\beta $ and harmonic functions for an associated operator $H_\beta $. This allows to show that the VRJP on $\mathbb {Z}^{d}$ (with certain initial conditions) has a unique representation, by proving that an associated Martin boundary is trivial. Moreover, on infinite trees, we construct a family of representations, that are all different when the VRJP is transient and the tree is $d$-regular (with $d\geq 3$).

We prove the continuity and strict positivity of the multi-layer extension to the stochastic heat equation introduced in [43] which form a hierarchy of partition functions for the continuum directed random polymer. This shows that the corresponding free energy (logarithm of the partition function) is well defined. This is also a step towards proving the conjecture stated at the end of the above paper that an array of such partition functions has the Markov property.

We prove limit theorems for systems of interacting diffusions on sparse graphs. For example, we deduce a hydrodynamic limit and the propagation of chaos property for the stochastic Kuramoto model with interactions determined by Erdös-Rényi graphs with constant mean degree. The limiting object is related to a potentially infinite system of SDEs defined over a Galton-Watson tree. Our theorems apply more generally, when the sequence of graphs (“decorated" with edge and vertex parameters) converges in the local weak sense. Our main technical result is a locality estimate bounding the influence of far-away diffusions on one another. We also numerically explore the emergence of synchronization phenomena on Galton-Watson random trees, observing rich phase transitions from synchronized to desynchronized activity among nodes at different distances from the root.

We consider a system of stochastic interacting particles in $\mathbb {R}^{d}$ and we describe large deviation asymptotics in a joint mean-field and small-noise limit. Precisely, a large deviation principle (LDP) is established for the empirical measure and the stochastic current, as the number of particles tends to infinity and the noise vanishes, simultaneously. We give a direct proof of the LDP using tilting and subsequently exploiting the link between entropy and large deviations. To this aim we employ consistency of suitable deterministic control problems associated to the stochastic dynamics.

The present paper is devoted to the study of the well-posedness of a type of BSDEs with triangularly quadratic generators. This work is motivated by the recent results obtained by Hu and Tang [14] and Xing and Žitković [28]. By the contraction mapping argument, we first prove that this type of triangularly quadratic BSDEs admits a unique local solution on a small time interval whenever the terminal value is bounded. Under additional assumptions, we build the global solution on the whole time interval by stitching local solutions. Finally, we give solvability results when the generators have path dependence in value process.

For a Lévy process $X$ on a finite time interval consider the probability that it exceeds some fixed threshold $x>0$ while staying below $x$ at the points of a regular grid. We establish exact asymptotic behavior of this probability as the number of grid points tends to infinity. We assume that $X$ has a zooming-in limit, which necessarily is $1/\alpha $-self-similar Lévy process with $\alpha \in (0,2]$, and restrict to $\alpha >1$. Moreover, the moments of the difference of the supremum and the maximum over the grid points are analyzed and their asymptotic behavior is derived. It is also shown that the zooming-in assumption implies certain regularity properties of the ladder process, and the decay rate of the left tail of the supremum distribution is determined.

We consider the PushTASEP (pushing totally asymmetric simple exclusion process, also sometimes called long-range TASEP) with the step initial configuration evolving in an inhomogeneous space. That is, the rate of each particle’s jump depends on the location of this particle. We match the distribution of the height function of this PushTASEP with Schur processes. Using this matching and determinantal structure of Schur processes, we obtain limit shape and fluctuation results which are typical for stochastic particle systems in the Kardar-Parisi-Zhang universality class. PushTASEP is a close relative of the usual TASEP. In inhomogeneous space the former is integrable, while the integrability of the latter is not known.

We consider three classes of linear differential equations on distribution functions, with a fractional order $\alpha \in [0,1]$. The integer case $\alpha =1$ corresponds to the three classical extreme families. In general, we show that there is a unique distribution function solving these equations, whose underlying random variable is expressed in terms of an exponential random variable and an integral transform of an independent $\alpha $-stable subordinator. From the analytical viewpoint, this distribution is in one-to-one correspondence with a Kilbas-Saigo function for the Weibull and Fréchet cases, and with a Le Roy function for the Gumbel case.

It has been known since the pioneering paper of Mark Kac [20], that the asymptotics of Fredholm determinants can be studied using probabilistic methods. We demonstrate the efficacy of Kac’ approach by studying the Fredholm Pfaffian describing the statistics of both non-Hermitian random matrices and annihilating Brownian motions. Namely, we establish the following two results. Firstly, let $\sqrt {N}+\lambda _{max}$ be the largest real eigenvalue of a random $N\times N$ matrix with independent $N(0,1)$ entries (the ‘real Ginibre matrix’). Consider the limiting $N\rightarrow \infty $ distribution $\mathbb {P}[\lambda _{max}<-L]$ of the shifted maximal real eigenvalue $\lambda _{max}$. Then \[ \lim _{L\rightarrow \infty } e^{\frac {1}{2\sqrt {2\pi }}\zeta \left (\frac {3}{2}\right )L} \mathbb {P}\left (\lambda _{max}<-L\right ) =e^{C_{e}}, \] where $\zeta $ is the Riemann zeta-function and \[ C_{e}=\frac {1}{2}\log 2+\frac {1}{4\pi }\sum _{n=1}^{\infty }\frac {1}{n} \left (-\pi +\sum _{m=1}^{n-1}\frac {1}{\sqrt {m(n-m)}}\right ). \] Secondly, let $X_{t}^{(max)}$ be the position of the rightmost particle at time $t$ for a system of annihilating Brownian motions (ABM’s) started from every point of $\mathbb {R}_{-}$. Then \[ \lim _{L\rightarrow \infty } e^{\frac {1}{2\sqrt {2\pi }}\zeta \left (\frac {3}{2}\right )L} \mathbb {P}\left (\frac {X_{t}^{(max)}}{\sqrt {4t}}<-L\right ) =e^{C_{e}}. \] These statements are a sharp counterpart of the results of [22], improved by computing the $O(L^{0})$ term in the asymptotic $L\rightarrow \infty $ expansion of the corresponding Fredholm Pfaffian.

We study long time behavior of integrated trawl processes introduced by Barndorff-Nielsen (2011). The trawl processes form a class of stationary infinitely divisible processes, they are described by an infinitely divisible random measure (Lévy base) and a family of shifts of a fixed set (trawl). We assume that the Lévy base is symmetric and homogeneous and that the trawl set is determined by the trawl function that decays slowly. Depending on the geometry of the trawl set and on the Lévy measure corresponding to the Lévy base we obtain various types of limits in law of the normalized integrated trawl processes for large times. The limit processes are always stable and self-similar with stationary increments. In some cases they have independent increments – they are stable Lévy processes where the index of stability depends on the parameters of the model. We show that stable limits with stability index smaller than $2$ may appear even in cases when the underlying Lévy base has all its moments finite. In other cases, the limit process has dependent increments and it may be considered as a new extension of fractional Brownian motion to the class of stable processes.

We consider a natural local dynamic on the set of all rooted planar maps with $n$ edges that is in some sense analogous to “edge flip” Markov chains, which have been considered before on a variety of combinatorial structures (triangulations of the $n$-gon and quadrangulations of the sphere, among others). We provide the first polynomial upper bound for the mixing time of this “edge rotation” chain on planar maps: we show that the spectral gap of the edge rotation chain is bounded below by an appropriate constant times $n^{-11/2}$. In doing so, we provide a partially new proof of the fact that the same bound applies to the spectral gap of edge flips on quadrangulations as defined in [8], which makes it possible to generalise the result of [8] to a variant of the edge flip chain related to edge rotations via Tutte’s bijection.

We consider $n$-point sticky Brownian motions: a family of $n$ diffusions that evolve as independent Brownian motions when they are apart, and interact locally so that the set of coincidence times has positive Lebesgue measure with positive probability. These diffusions can also be seen as $n$ random motions in a random environment whose distribution is given by so-called stochastic flows of kernels. For a specific type of sticky interaction, we prove exact formulas characterizing the stochastic flow and show that in the large deviations regime, the random fluctuations of these stochastic flows are Tracy-Widom GUE distributed. An equivalent formulation of this result states that the extremal particle among $n$ sticky Brownian motions has Tracy-Widom distributed fluctuations in the large $n$ and large time limit. These results are proved by viewing sticky Brownian motions as a (previously known) limit of the exactly solvable beta random walk in random environment.

We consider the spatial $\Lambda $-Fleming-Viot process model for frequencies of genetic types in a population living in $\mathbb {R}^{d}$, with two types of individuals ($0$ and $1$) and natural selection favouring individuals of type $1$. We first prove that the model is well-defined and provide a measure-valued dual process encoding the locations of the “potential ancestors” of a sample taken from such a population, in the same spirit as the dual process for the SLFV without natural selection [7]. We then consider two cases, one in which the dynamics of the process are driven by purely “local” events (that is, reproduction events of bounded radii) and one incorporating large-scale extinction-recolonisation events whose radii have a polynomial tail distribution. In both cases, we consider a sequence of spatial $\Lambda $-Fleming-Viot processes indexed by $n$, and we assume that the fraction of individuals replaced during a reproduction event and the relative frequency of events during which natural selection acts tend to $0$ as $n$ tends to infinity. We choose the decay of these parameters in such a way that when reproduction is only local, the measure-valued process describing the local frequencies of the less favoured type converges in distribution to a (measure-valued) solution to the stochastic Fisher-KPP equation in one dimension, and to a (measure-valued) solution to the deterministic Fisher-KPP equation in more than one dimension. When large-scale extinction-recolonisation events occur, the sequence of processes converges instead to the solution to the analogous equation in which the Laplacian is replaced by a fractional Laplacian (again, noise can be retained in the limit only in one spatial dimension). We also consider the process of “potential ancestors” of a sample of individuals taken from these populations, which we see as (the empirical distribution of) a system of branching and coalescing symmetric jump processes. We show their convergence in distribution towards a system of Brownian or stable motions which branch at some finite rate. In one dimension, in the limit, pairs of particles also coalesce at a rate proportional to their collision local time. In contrast to previous proofs of scaling limits for the spatial $\Lambda $-Fleming-Viot process, here the convergence of the more complex forwards in time processes is used to prove the convergence of the dual process of potential ancestries.

We consider large deviations of empirical measures of diffusion processes. In a first part, we present conditions to obtain a large deviations principle (LDP) for a precise class of unbounded functions. This provides an analogue to the standard Cramér condition in the context of diffusion processes, which turns out to be related to a spectral gap condition for a Witten–Schrödinger operator. Secondly, we study more precisely the properties of the Donsker–Varadhan rate functional associated with the LDP. We revisit and generalize some standard duality results as well as a more original decomposition of the rate functional with respect to the symmetric and antisymmetric parts of the dynamics. Finally, we apply our results to overdamped and underdamped Langevin dynamics, showing the applicability of our framework for degenerate diffusions in unbounded configuration spaces.

We consider the stochastic convection–diffusion equation \[ \partial _{t} \theta (t\,,\bm {x}) =\nu \Delta \theta (t\,,\bm {x}) + V(t\,,x_{1})\partial _{x_{2}} \theta (t\,,\bm {x}), \] for $t>0$ and $\bm {x}=(x_{1}\,,x_{2})\in \mathbb {R}^{2}$, subject to $\theta _{0}$ being a nice initial profile. Here, the velocity field $V$ is assumed to be centered Gaussian with covariance structure \[ \Cov [V(t\,,a)\,,V(s\,,b)]= \delta _{0}(t-s)\rho (a-b)\qquad \text {for all $s,t\geqslant 0$ and $a,b\in \mathbb {R}$}, \] where $\rho $ is a continuous and bounded positive-definite function on $\mathbb {R}$.

We prove a quite general existence/uniqueness/regularity theorem, together with a probabilistic representation of the solution that represents $\theta $ as an expectation functional of an exogenous infinite-dimensional Brownian motion. We use that probabilistic representation in order to study the Itô/Walsh solution, when it exists, and relate it to the Stratonovich solution which is shown to exist for all $\nu >0$.

Our a priori estimates imply the physically-natural fact that, quite generally, the solution dissipates. In fact, very often, \[\label {DR} \mathrm {P}\left \{ \sup _{|x_{1}|\leqslant m}\sup _{\mathstrut x_{2}\in \mathbb {R}} |\theta (t\,,\bm {x})| = O\left ( \frac {1}{\sqrt {t}}\right )\qquad \text {as $t\to \infty $} \right \} =1\qquad \text {for all $m>0$},\tag {0.1} \] and the $O(1/\sqrt {t})$ rate is shown to be unimproveable.

Our probabilistic (Lagrangian) representation is malleable enough to allow us to analyze the Stratonovich solution in two physically-relevant regimes: As $t\to \infty $ and as $\nu \to 0$. Among other things, our analysis leads to a “macroscopic multifractal analysis” of the rate of decay in (0.1) in terms of the reciprocal of the Prandtl (or Schmidt) number, valid in a number of simple though still physically-relevant cases.

Consider a weighted branching process generated by the lengths of intervals obtained by stick-breaking of unit length (a.k.a. the residual allocation model) and associate with each weight a ‘box’. Given the weights ‘balls’ are thrown independently into the boxes of the first generation with probability of hitting a box being equal to its weight. Each ball located in a box of the $j$th generation, independently of the others, hits a daughter box in the $(j+1)$th generation with probability being equal the ratio of the daughter weight and the mother weight. This is what we call nested occupancy scheme in random environment. Restricting attention to a particular generation one obtains the classical Karlin occupancy scheme in random environment.

Assuming that the stick-breaking factor has a uniform distribution on $[0,1]$ and that the number of balls is $n$ we investigate occupancy of intermediate generations, that is, those with indices $\lfloor j_{n} u\rfloor $ for $u>0$, where $j_{n}$ diverges to infinity at a sublogarithmic rate as $n$ becomes large. Denote by $K_{n}(j)$ the number of occupied (ever hit) boxes in the $j$th generation. It is shown that the finite-dimensional distributions of the process $(K_{n}(\lfloor j_{n} u\rfloor ))_{u>0}$, properly normalized and centered, converge weakly to those of an integral functional of a Brownian motion. The case of a more general stick-breaking is also analyzed.

We study scale-invariant Rayleigh Random Flights (“RRF”) in random environments given by planar Scale-Invariant Random Spatial Networks (“SIRSN”) based on speed-marked Poisson line processes. A natural one-parameter family of such RRF (with scale-invariant dynamics) can be viewed as producing “randomly-broken local geodesics” on the SIRSN; we aim to shed some light on a conjecture that a (non-broken) geodesic on such a SIRSN will never come to a complete stop en route. (If true, then all such geodesics can be represented as doubly-infinite sequences of sequentially connected line segments. This would justify a natural procedure for computing geodesics.) The family of these RRF (“SIRSN-RRF”), is introduced via a novel axiomatic theory of abstract scattering representations for Markov chains (itself of independent interest). Palm conditioning (specifically the Mecke-Slivnyak theorem for Palm probabilities of Poisson point processes) and ideas from the ergodic theory of random walks in random environments are used to show that at a critical value of the parameter the speed of the scale-invariant SIRSN-RRF neither diverges to infinity nor tends to zero, thus supporting the conjecture.

By combining the coupling by reflection for Brownian motion with the refined basic coupling for Poisson random measure, we present sufficient conditions for the exponential ergodicity of general continuous-state nonlinear branching processes in both the $L^{1}$-Wasserstein distance and the total variation norm, where the drift term is dissipative only for large distance, and either diffusion noise or jump noise is allowed to be vanished. Sufficient conditions for the corresponding strong ergodicity are also established.

We study the asymptotic behavior of branching diffusion processes in periodic media. For a super-critical branching process, we distinguish two types of behavior for the normalized number of particles in a bounded domain, depending on the distance of the domain from the region where the bulk of the particles is located. At distances that grow linearly in time, we observe intermittency (i.e., the $k$-th moment dominates the $k$-th power of the first moment for some $k$), while, at distances that grow sub-linearly in time, we show that all the moments converge. A key ingredient in our analysis is a sharp estimate of the transition kernel for the branching process, valid up to linear in time distances from the location of the initial particle.

Let $G=(V,E)$ be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on $G$. In recent work, we conjectured that if $G$ is nonamenable then the matrix of critical connection probabilities $T_{p_{c}}(u,v)=\mathbb {P}_{p_{c}}(u\leftrightarrow v)$ is bounded as an operator $T_{p_{c}}:L^{2}(V)\to L^{2}(V)$ and proved that this conjecture holds for several classes of graphs, including all transitive, nonamenable, Gromov hyperbolic graphs. In notation, the conjecture states that $p_{c}<p_{2\to 2}$, where for each $q\in [1,\infty ]$ we define $p_{q\to q}$ to be the supremal value of $p$ for which the operator norm $\|T_{p}\|_{q\to q}$ is finite. We also noted in that work that the conjecture implies two older conjectures, namely that percolation on transitive nonamenable graphs always has a nontrivial nonuniqueness phase, and that critical percolation on the same class of graphs has mean-field critical behaviour.

In this paper we further investigate the consequences of the $L^{2}$ boundedness conjecture. In particular, we prove that the following hold for all transitive graphs: i) The two-point function decays exponentially in the distance for all $p<p_{2\to 2}$; ii) If $p_{c}<p_{2\to 2}$, then the critical exponent governing the extrinsic diameter of a critical cluster is $1$; iii) Below $p_{2\to 2}$, percolation is “ballistic" in the sense that the intrinsic (a.k.a. chemical) distance between two points is exponentially unlikely to be much larger than their extrinsic distance; iv) If $p_{c}<p_{2\to 2}$, then $\|T_{p_{c}}\|_{q\to q} \asymp (q-1)^{-1}$ and $p_{q\to q}-p_{c} \asymp q-1$ as $q\downarrow 1$; v) If $p_{c}<p_{2\to 2}$, then various ‘multiple-arm’ events have probabilities comparable to the upper bound given by the BK inequality. In particular, the probability that the origin is a trifurcation point is of order $(p-p_{c})^{3}$ as $p \downarrow p_{c}$. All of these results are new even in the Gromov hyperbolic case.

Finally, we apply these results together with duality arguments to compute the critical exponents governing the geometry of intrinsic geodesics at the uniqueness threshold of percolation in the hyperbolic plane.

Random union sets $Z$ associated with stationary Poisson processes of $k$-cylinders in $\mathbb {R}^{d}$ are considered. Under general conditions on the typical cylinder base a concentration inequality for the volume of $Z$ restricted to a compact window is derived. Assuming convexity of the typical cylinder base and isotropy of $Z$ a concentration inequality for intrinsic volumes of arbitrary order is established. A number of special cases are discussed, for example the case when the cylinder bases arise from a random rotation of a fixed convex body. Also the situation of expanding windows is studied. Special attention is payed to the case $k=0$, which corresponds to the classical Boolean model.

In this article, we consider an anisotropic finite-range bond percolation model on $\mathbb {Z}^2$. On each horizontal layer $\{ (x,i)\colon x \in \mathbb {Z}\}$ we have edges $\langle (x,i),(y,i)\rangle $ for $1 \le |x-y|\le N$. There are also vertical edges connecting two nearest neighbor vertices on distinct layers $\langle (x,i),(x,i+1)\rangle $ for $x,i \in \mathbb {Z}$. On this graph we consider the following anisotropic independent percolation model: horizontal edges are open with probability $1/(2N)$, while vertical edges are open with probability $\epsilon $ to be suitably tuned as $N$ grows to infinity. The main result tells that if $\epsilon = \kappa N^{-2/5}$, we see a phase transition in $\kappa $: positive and finite constants $C_1, C_2$ exist so that there is no percolation if $\kappa <C_1$ while percolation occurs for $\kappa >C_2$. The question is motivated by a result on the analogous layered ferromagnetic Ising model at mean field critical temperature [11, J. Stat. Phys. 161, (2015), 91–123] for which the authors showed the existence of multiple Gibbs measures for a fixed value of the vertical interaction and conjectured a change of behavior in $\kappa $ when the vertical interaction suitably vanishes as $\kappa \gamma ^{b}$, where $1/\gamma $ is the range of the horizontal interaction. For the product percolation model we have a value of $b$ that differs from what was conjectured in that paper. The proof relies on the analysis of the scaling limit of the critical branching random walk that dominates the growth process restricted to each horizontal layer and a careful analysis of the true horizontal growth process, which is interesting by itself. This is inspired by works on the long range contact process [17, Probab. Th. Rel. Fields 102, (1995), 519–545]. A renormalization scheme is used for the percolative regime.

We consider the zero-average Gaussian free field on a certain class of finite $d$-regular graphs for fixed $d\geq 3$. This class includes $d$-regular expanders of large girth and typical realisations of random $d$-regular graphs. We show that the level set of the zero-average Gaussian free field above level $h$ exhibits a phase transition at level $h_{\star }$, which agrees with the critical value for level-set percolation of the Gaussian free field on the infinite $d$-regular tree. More precisely, we show that, with probability tending to one as the size of the finite graphs tends to infinity, the level set above level $h$ does not contain any connected component of larger than logarithmic size whenever $h>h_{\star }$, and on the contrary, whenever $h<h_{\star }$, a linear fraction of the vertices is contained in connected components of the level set above level $h$ having a size of at least a small fractional power of the total size of the graph. It remains open whether in the supercritical phase $h<h_{\star }$, as the size of the graphs tends to infinity, one observes the emergence of a (potentially unique) giant connected component of the level set above level $h$. The proofs in this article make use of results from the accompanying paper [2].

Rooted, weighted continuum random trees are used to describe limits of sequences of random discrete trees. Formally, they are random quadruples $(\mathcal {T},d,r,p)$, where $(\mathcal {T},d)$ is a tree-like metric space, $r\in \mathcal {T}$ is a distinguished root, and $p$ is a probability measure on this space. Intuitively, these trees have a combinatorial “underlying branching structure” implied by their topology but otherwise independent of the metric $d$. We explore various ways of making this rigorous, using the weight $p$ to do so without losing the fractal complexity possible in continuum trees. We introduce a notion of mass-structural equivalence and show that two rooted, weighted $\mathbb {R}$-trees are equivalent in this sense if and only if the discrete hierarchies derived by i.i.d. sampling from their weights, in a manner analogous to Kingman’s paintbox, have the same distribution. We introduce a family of trees, called “interval partition trees” that serve as representatives of mass-structure equivalence classes, and which naturally represent the laws of the aforementioned hierarchies.

Applying an inductive technique for Stein and zero bias couplings yields Berry-Esseen theorems for normal approximation for two new examples. The conditions of the main results do not require that the couplings be bounded. Our two applications, one to the Erdős-Rényi random graph with a fixed number of edges, and one to Jack measure on tableaux, demonstrate that the method can handle non-bounded variables with non-trivial global dependence, and can produce bounds in the Kolmogorov metric with the optimal rate.

Consider a spectrally positive Stable$(1\!+\!\alpha )$ process whose jumps we interpret as lifetimes of individuals. We mark the jumps by continuous excursions assigning “sizes” varying during the lifetime. As for Crump–Mode–Jagers processes (with “characteristics”), we consider for each level the collection of individuals alive. We arrange their “sizes” at the crossing height from left to right to form an interval partition. We study the continuity and Markov properties of the interval-partition-valued process indexed by level. From the perspective of the Stable$(1\!+\!\alpha )$ process, this yields new theorems of Ray–Knight-type. From the perspective of branching processes, this yields new, self-similar models with dense sets of birth and death times of (mostly short-lived) individuals. This paper feeds into projects resolving conjectures by Feng and Sun (2010) on the existence of certain measure-valued diffusions with Poisson–Dirichlet stationary laws, and by Aldous (1999) on the existence of a continuum-tree-valued diffusion.

We study the Hamilton-Jacobi equation $f - \lambda Hf = h$, where $H f = e^{-f}Ae^{f}$ and where $A$ is an operator that corresponds to a well-posed martingale problem.

We identify an operator that gives viscosity solutions to the Hamilton-Jacobi equation, and which can therefore be interpreted as the resolvent of $H$. The operator is given in terms of an optimization problem where the running cost is a path-space relative entropy.

Finally, we use the resolvents to give a new proof of the abstract large deviation result of Feng and Kurtz (2006).

We prove that random triangulations of types I, II, and III with a simple boundary under the critical Boltzmann weight converge in the scaling limit to the Brownian disk. The proof uses a bijection due to Poulalhon and Schaeffer between type III triangulations of the $p$-gon and so-called blossoming forests. A variant of this bijection was also used by Addario-Berry and the first author to prove convergence of type III triangulations to the Brownian map, but new ideas are needed to handle the simple boundary. Our result is an ingredient in the program of the second and third authors on the convergence of uniform triangulations under the Cardy embedding.

In this paper, we exhibit a new family of martingale couplings between two one-dimensional probability measures $\mu $ and $\nu $ in the convex order. This family is parametrised by two dimensional probability measures on the unit square with respective marginal densities proportional to the positive and negative parts of the difference between the quantile functions of $\mu $ and $\nu $. It contains the inverse transform martingale coupling which is explicit in terms of the quantile functions of these marginal densities. The integral of $|x-y|$ with respect to each of these couplings is smaller than twice the $\mathcal {W}_{1}$ distance between $\mu $ and $\nu $. When the comonotonous coupling between $\mu $ and $\nu $ is given by a map $T$, the elements of the family minimise $\int _{\mathbb {R}}\vert y-T(x)\vert \,M(dx,dy)$ among all martingale couplings between $\mu $ and $\nu $. When $\mu $ and $\nu $ are in the decreasing (resp. increasing) convex order, the construction is generalised to exhibit super (resp. sub) martingale couplings.

We establish Hölder regularity and gradient estimates for the transition semigroup of the solutions to the following SDE: \[ \mathrm{d} X_{t}=\sigma (t, X_{t-})\mathrm{d} Z_{t}+b (t, X_{t})\mathrm{d} t,\ \ X_{0}=x\in{\mathbb {R}} ^{d}, \] where $( Z_{t})_{t\geqslant 0}$ is a $d$-dimensional cylindrical $\alpha $-stable process with $\alpha \in (0, 2)$, $\sigma (t, x):{\mathbb{R} }_{+}\times{\mathbb {R}} ^{d}\to{\mathbb {R}} ^{d}\otimes{\mathbb {R}} ^{d}$ is bounded measurable, uniformly nondegenerate and Lipschitz continuous in $x$ uniformly in $t$, and $b (t, x):{\mathbb{R} }_{+}\times{\mathbb {R}} ^{d}\to{\mathbb {R}} ^{d}$ is bounded $\beta $-Hölder continuous in $x$ uniformly in $t$ with $\beta \in [0,1]$ satisfying $\alpha +\beta >1$. Moreover, we also show the existence and regularity of the distributional density of $X (t, x)$. Our proof is based on Littlewood-Paley’s theory.

We consider the symmetric simple exclusion process in $\mathbb {Z}^{d}$ with quenched bounded dynamic random conductances and prove its hydrodynamic limit in path space. The main tool is the connection, due to the self-duality of the process, between the invariance principle for single particles starting from all points and the macroscopic behavior of the density field. While the hydrodynamic limit at fixed macroscopic times is obtained via a generalization to the time-inhomogeneous context of the strategy introduced in [41], in order to prove tightness for the sequence of empirical density fields we develop a new criterion based on the notion of uniform conditional stochastic continuity, following [50]. In conclusion, we show that uniform elliptic dynamic conductances provide an example of environments in which the so-called arbitrary starting point invariance principle may be derived from the invariance principle of a single particle starting from the origin. Therefore, our hydrodynamics result applies to the examples of quenched environments considered in, e.g., [1], [3], [6] in combination with the hypothesis of uniform ellipticity.

In this paper, we consider the recurrence of some multi-dimensional diffusion processes in random environments including zero potentials. Previous methods on diffusion processes in random environments are not applicable to the case of such environments. In main theorems, we obtain a sufficient condition to be recurrent for the product of a multi-dimensional diffusion process in semi-selfsimilar random environments and one-dimensional Brownian motion, and also more explicit sufficient conditions in the case of Gaussian random environments and random environments generated by Lévy processes. To prove them, we introduce an index which measures the strength of recurrence of symmetric Markov processes, and give some sufficient conditions for recurrence of direct products of symmetric diffusion processes. The index is given by the Dirichlet forms of the Markov processes.

We establish the stochastic comparison principles, including moment comparison principle as a special case, for solutions to the following nonlinear stochastic heat equation on $\mathbb{R} ^{d}$ \[ \left (\frac{\partial } {\partial t} -\frac{1} {2}\Delta \right ) u(t,x) = \rho (u(t,x)) \:\dot{M} (t,x), \] where $\dot{M} $ is a spatially homogeneous Gaussian noise that is white in time and colored in space, and $\rho $ is a Lipschitz continuous function that vanishes at zero. These results are obtained for rough initial data and under Dalang’s condition, namely, $\int _{\mathbb{R} ^{d}}(1+|\xi |^{2})^{-1}\hat{f} (\text{d} \xi )<\infty $, where $\hat{f} $ is the spectral measure of the noise. We first show that the nonlinear stochastic heat equation can be approximated by systems of interacting diffusions (SDEs) and then, using those approximations, we establish the comparison principles by comparing either the diffusion coefficient $\rho $ or the correlation function of the noise $f$. As corollaries, we obtain Slepian’s inequality for SPDEs and SDEs.

Given $\alpha \in (0, \infty )$ and $r \in (0, \infty )$, let ${\mathcal {D}}_{r, \alpha }$ be the disc of radius $r$ in the hyperbolic plane having curvature $-\alpha ^{2}$. Consider the Poisson point process having uniform intensity density on ${\mathcal {D}}_{R, \alpha }$, with $R = 2 \log (n/ \nu )$, $n \in \mathbb {N}$, and $\nu < n$ a fixed constant. The points are projected onto ${\mathcal {D}}_{R, 1}$, preserving polar coordinates, yielding a Poisson point process ${\mathcal {P}}_{\alpha , n}$ on ${\mathcal {D}}_{R, 1}$. The hyperbolic geometric graph ${\mathcal {G}}_{\alpha , n}$ on ${\mathcal {P}}_{\alpha , n}$ puts an edge between pairs of points of ${\mathcal {P}}_{\alpha , n}$ which are distant at most $R$. This model has been used to express fundamental features of complex networks in terms of an underlying hyperbolic geometry.

For $\alpha \in (1/2, \infty )$ we establish expectation and variance asymptotics as well as asymptotic normality for the number of isolated and extreme points in ${\mathcal {G}}_{\alpha , n}$ as $n \to \infty $. The limit theory and renormalization for the number of isolated points are highly sensitive on the curvature parameter. In particular, for $\alpha \in (1/2, 1)$, the variance is super-linear, for $\alpha = 1$ the variance is linear with a logarithmic correction, whereas for $\alpha \in (1, \infty )$ the variance is linear. The central limit theorem fails for $\alpha \in (1/2, 1)$ but it holds for $\alpha \in (1, \infty )$.

The first main goal of this article is to give a new metrization of the Mukherjee–Varadhan topology, recently introduced as a translation-invariant compactification of the space of probability measures on Euclidean spaces. This new metrization allows us to achieve our second goal which is to extend the recent program of Bates and Chatterjee on localization for the endpoint distribution of discrete directed polymers to polymers based on general random walks in Euclidean spaces. Following their strategy, we study the asymptotic behavior of the endpoint distribution update map and study the set of its distributional fixed points satisfying a variational principle. We show that the distribution concentrated on the zero measure is a unique element in this set if and only if the system is in the high temperature regime. This enables us to prove that the asymptotic clustering (a natural continuous analogue of the asymptotic pure atomicity property) holds in the low temperature regime and that the endpoint distribution is geometrically localized with positive density if and only if the system is in the low temperature regime.

It is shown that a certain functional of a branching process has representations in terms of both a maximisation problem and a minimisation problem. A consequence of these representations is that upper and lower bounds on the functional can be found easily, yielding a non-asymptotic Trotter product formula. As an application, the speed of the right-most particle of a branching Lévy process is calculated.

Under mild conditions on a family of independent random variables $(X_{n})$ we prove that almost sure convergence of a sequence of tetrahedral polynomial chaoses of uniformly bounded degrees in the variables $(X_{n})$ implies the almost sure convergence of their homogeneous parts. This generalizes a recent result due to Poly and Zheng obtained under stronger integrability conditions. In particular for i.i.d. sequences we provide a simple necessary and sufficient condition for this property to hold.

We also discuss similar phenomena for sums of multiple Wiener-Itô integrals with respect to Poisson processes, answering a question by Poly and Zheng.

In this paper we study the probability that a $d$ dimensional simple random walk (or the first $L$ steps of it) covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_{1}$ ball. We show that among all such paths, the one that maximizes the covering probability is the monotonic increasing one that stays within distance 1 from the diagonal. As a result, we can obtain an exponential upper bound on the decaying rate of covering probability of any such path when $d\geq 4$. The main tool is a general combinatorial inequality, that is interesting in its own right.

Preferential attachment models form a popular class of growing networks, where incoming vertices are preferably connected to vertices with high degree. We consider a variant of this process, where vertices are equipped with a random initial fitness representing initial inhomogeneities among vertices and the fitness influences the attractiveness of a vertex in an additive way. We consider a heavy-tailed fitness distribution and show that the model exhibits a phase transition depending on the tail exponent of the fitness distribution. In the weak disorder regime, one of the old vertices has maximal degree irrespective of fitness, while for strong disorder the vertex with maximal degree has to satisfy the right balance between fitness and age. Our methods use martingale methods to show concentration of degree evolutions as well as extreme value theory to control the fitness landscape.

We consider a class of space-time coupled evolution equations (CEEs), obtained by a subordination of the heat operator. Our CEEs reformulate and extend known governing equations of non-Markovian processes arising as scaling limits of continuous time random walks, with widespread applications. In particular we allow for initial conditions imposed on the past, general spatial operators on Euclidean domains and a forcing term. We prove existence, uniqueness and stochastic representation for solutions.

In this paper, we prove that the stochastic telegraph equation arises as a scaling limit of the stochastic higher spin six vertex (SHS6V) model with general spin $I/2, J/2$. This extends results of Borodin and Gorin which focused on the $I=J=1$ six vertex case and demonstrates the universality of the stochastic telegraph equation in this context. We also provide a functional extension of the central limit theorem obtained in [BG19, Theorem 6.1]. The main idea is to generalize the four point relation established in [BG19, Theorem 3.1], using fusion.

We prove limit theorems for functionals of a Poisson point process using the Malliavin calculus on the Poisson space. The target distribution is conditionally either a Gaussian vector or a Poisson random variable. The convergence is stable and our conditions are expressed in terms of the Malliavin operators. For conditionally Gaussian limits, we also obtain quantitative bounds, given for the Wasserstein transport distance in the univariate case; and for another probabilistic variational distance in higher dimension. Our work generalizes several limit theorems on the Poisson space, including the seminal works by G. Peccati, J. L. Solé, M. S. Taqqu & F. Utzet. “Stein’s method and normal approximation of Poisson functionals” for Gaussian approximations; and by G. Peccati “The Chen-Stein method for Poisson functionals” for Poisson approximations; as well as the recently established fourth-moment theorem on the Poisson space of C. Döbler & G.Peccati “The fourth moment theorem on the Poisson space”. We give an application to stochastic processes.

We introduce a framework to derive quantitative central limit theorems in the context of non-linear approximation of Gaussian random variables taking values in a separable Hilbert space. In particular, our method provides an alternative to the usual (non-quantitative) finite dimensional distribution convergence and tightness argument for proving functional convergence of stochastic processes. We also derive four moments bounds for Hilbert-valued random variables with possibly infinite chaos expansion, which include, as special cases, all finite-dimensional four moments results for Gaussian approximation in a diffusive context proved earlier by various authors. Our main ingredient is a combination of an infinite-dimensional version of Stein’s method as developed by Shih and the so-called Gamma calculus. As an application, rates of convergence for the functional Breuer-Major theorem are established.

In this paper, we establish the Freidlin-Wentzell type large deviation principle for porous medium-type equations perturbed by small multiplicative noise. The porous medium operator $\Delta (|u|^{m-1}u)$ is allowed. Our proof is based on weak convergence approach.

We consider an interacting particle system, known as the frog model, on infinite Galton–Watson trees allowing offspring 0 and 1. The system starts with one awake particle (frog) at the root of the tree and a random number of sleeping particles at the other vertices. Awake frogs move according to simple random walk on the tree and as soon as they encounter sleeping frogs, those will wake up and move independently according to simple random walk. The frog model is called transient if there are almost surely only finitely many particles returning to the root. In this paper we prove a 0–1-law for transience of the frog model and show the existence of a transient phase for certain classes of Galton–Watson trees.

We consider a recurrent random walk of i.i.d. increments on the one-dimensional integer lattice and obtain a formula relating the hitting distribution of a half-line with the potential function, $a(x)$, of the random walk. Applying it, we derive an asymptotic estimate of $a(x)$ and thereby a criterion for $a(x)$ to be bounded on a half-line. The application is also made to estimate some hitting probabilities as well as to derive asymptotic behaviour for large times of the walk conditioned never to visit the origin.

We obtain estimates for large and moderate deviations for the capacity of the range of a random walk on $\mathbb {Z}^{d}$, in dimension $d\ge 5$, both in the upward and downward directions. The results are analogous to those we obtained for the volume of the range in two companion papers [AS17a, AS19]. Interestingly, the main steps of the strategy we developed for the latter apply in this seemingly different setting, yet the details of the analysis are different.

Under Poincaré-type conditions, upper bounds are explored for the Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law. Based on improved concentration inequalities on high-dimensional Euclidean spheres, the results extend and refine previous results to non-symmetric models.

We study ASEP in a spatially inhomogeneous environment on a torus $ \mathbb {T}^{(N)} = \mathbf {Z}/N\mathbf {Z}$ of $ N $ sites. A given inhomogeneity $ \widetilde { \mathbb {a} } (x)\in (0,\infty ) $, $ x\in \mathbb {T}^{(N)} $, perturbs the overall asymmetric jumping rates $ r<\ell \in (0,1) $ at bonds, so that particles jump from site $x$ to $x+1$ with rate $r \widetilde { \mathbb {a} } (x)$ and from $x+1$ to $x$ with rate $\ell \widetilde { \mathbb {a} } (x)$ (subject to the exclusion rule in both cases). Under the limit $ N\to \infty $, we suitably tune the asymmetry $ (\ell -r) $ to zero like $N^{-\frac {1}{2}}$ and the inhomogeneity $ \widetilde { \mathbb {a} } $ to unity, so that the two compete on equal footing. At the level of the Gärtner (or microscopic Hopf–Cole) transform, we show convergence to a new SPDE — the Stochastic Heat Equation with a mix of spatial and spacetime multiplicative noise. Equivalently, at the level of the height function we show convergence to the Kardar–Parisi–Zhang equation with a mix of spatial and spacetime additive noise.

Our method applies to a general class of $ \widetilde { \mathbb {a} } (x) $, which, in particular, includes i.i.d., fractional-Brownian-motion like, and periodic inhomogeneities. The key technical component of our analysis consists of a host of estimates on the kernel of the semigroup $ \mathcal {Q} (t):=e^{t \mathcal {H} } $ for a Hill-type operator $ \mathcal {H} := \frac {1}{2}\partial _{xx} + \mathcal {A} '(x) $, and its discrete analog, where $ \mathcal {A} $ (and its discrete analog) is a generic Hölder continuous function.

Continuous time Feynman-Kac measures on path spaces are central in applied probability, partial differential equation theory, as well as in quantum physics. This article presents a new duality formula between normalized Feynman-Kac distribution and their mean field particle interpretations. Among others, this formula allows us to design a reversible particle Gibbs-Glauber sampler for continuous time Feynman-Kac integration on path spaces. We also provide new Dyson-Phillips semigroup expansions, as well as novel uniform propagation of chaos estimates for continuous time genealogical tree based particle models with respect to the time horizon and the size of the systems. Our approach is self contained and it is based on a novel stochastic perturbation analysis and backward semigroup techniques. These techniques allow to obtain sharp quantitative estimates of the convergence rate to equilibrium of particle Gibbs-Glauber samplers. To the best of our knowledge these results are the first of this kind for continuous time Feynman-Kac measures.

We characterize the local smoothness and the asymptotic growth rate of the Lévy white noise. We do so by characterizing the weighted Besov spaces in which it is located. We extend known results in two ways. First, we obtain new bounds for the local smoothness via the Blumenthal-Getoor indices of the Lévy white noise. We also deduce the critical local smoothness when the two indices coincide, which is true for symmetric-$\alpha $-stable, compound Poisson, and symmetric-gamma white noises to name a few. Second, we express the critical asymptotic growth rate in terms of the moment properties of the Lévy white noise. Previous analyses only provided lower bounds for both the local smoothness and the asymptotic growth rate. Showing the sharpness of these bounds requires us to determine in which Besov spaces a given Lévy white noise is (almost surely) not. Our methods are based on the wavelet-domain characterization of Besov spaces and precise moment estimates for the wavelet coefficients of the Lévy white noise.

We study existence of probability measure valued jump-diffusions described by martingale problems. We develop a simple device that allows us to embed Wasserstein spaces and other similar spaces of probability measures into locally compact spaces where classical existence theory for martingale problems can be applied. The method allows for general dynamics including drift, diffusion, and possibly infinite-activity jumps. We also develop tools for verifying the required conditions on the generator, including the positive maximum principle and certain continuity and growth conditions. To illustrate the abstract results, we consider large particle systems with mean-field interaction and common noise.

We study the volume of metric balls in Liouville quantum gravity (LQG). For $\gamma \in (0,2)$, it has been known since the early work of Kahane (1985) and Molchan (1996) that the LQG volume of Euclidean balls has finite moments exactly for $p \in (-\infty , 4/\gamma ^{2})$. Here, we prove that the LQG volume of LQG metric balls admits all finite moments. This answers a question of Gwynne and Miller and generalizes a result obtained by Le Gall for the Brownian map, namely, the $\gamma = \sqrt {8/3}$ case. We use this moment bound to show that on a compact set the volume of metric balls of size $r$ is given by $r^{d_{\gamma }+o_{r}(1)}$, where $d_{\gamma }$ is the dimension of the LQG metric space. Using similar techniques, we prove analogous results for the first exit time of Liouville Brownian motion from a metric ball. Gwynne-Miller-Sheffield (2020) prove that the metric measure space structure of $\gamma $-LQG a.s. determines its conformal structure when $\gamma =\sqrt {8/3}$; their argument and our estimate yield the result for all $\gamma \in (0,2)$.

A spectrally positive additive Lévy field is a multidimensional field obtained as the sum $\mathbf {X}_{\mathrm {t}}={\mathrm {X}}^{(1)}_{t_{1}}+{\mathrm {X}}^{(2)}_{t_{2}}+\dots +{\mathrm {X}}^{(d)}_{t_{d}}$, ${\mathrm {t}}=(t_{1},\dots ,t_{d})\in \mathbb {R}_{+}^{d}$, where ${\mathrm {X}}^{(j)}={}^{t} (X^{1,j},\dots ,X^{d,j})$, $j=1,\dots ,d$, are $d$ independent $\mathbb {R}^{d}$-valued Lévy processes issued from $\mathbf {0}={}^{t}(0,0,\dots ,0)$, such that $X^{i,j}$ is non decreasing for $i\neq j$ and $X^{j,j}$ is spectrally positive. It can also be expressed as $\mathbf {X}_{\mathrm {t}}=\mathbb {X}_{\mathrm {t}}\cdot {\mathbf {1}}$, where ${\mathbf {1}}={}^{t}(1,1,\dots ,1)$ and $\mathbb {X}_{\mathrm {t}}=(X^{i,j}_{t_{j}})_{1\leq i,j\leq d}$. The main interest of spaLf’s lies in the Lamperti representation of multitype continuous state branching processes. In this work, we study the law of the first passage times $\mathbf {T}_{\mathrm {r}}$ of such fields at levels $-{\mathrm {r}}$, where ${\mathrm {r}}\in \mathbb {R}_{+}^{d}$. We prove that the field $\{(\mathbf {T}_{\mathrm {r}},\mathbb {X}_{\mathbf {T}_{\mathrm {r}}}),{\mathrm {r}}\in \mathbb {R}_{+}^{d}\}$ has stationary and independent increments and we describe its law in terms of this of the spaLf $\mathbf {X}$. In particular, the Laplace exponent of $(\mathbf {T}_{\mathrm {r}},\mathbb {X}_{\mathbf {T}_{\mathrm {r}}})$ solves a functional equation leaded by the Laplace exponent of $\mathbf {X}$. This equation extends in higher dimension a classical fluctuation identity satisfied by the Laplace exponents of the ladder processes. Then we give an expression of the distribution of $\{(\mathbf {T}_{\mathrm {r}},\mathbb {X}_{\mathbf {T}_{\mathrm {r}}}),{\mathrm {r}}\in \mathbb {R}_{+}^{d}\}$ in terms of the distribution of $\{\mathbb {X}_{\mathrm {t}},{\mathrm {t}}\in \mathbb {R}_{+}^{d}\}$ by the means of a Kemperman-type formula, well-known for spectrally positive Lévy processes.