The Kardar-Parisi-Zhang equation (KPZ equation) is solved via Cole-Hopf transformation $h=logu$, where u is the solution of the multiplicative stochastic heat equation(SHE). In [CD20, CSZ20, G20], they consider the solution of two dimensional KPZ equation via the solution $u_{\mathit{\epsilon }}$ of SHE with the flat initial condition and with noise which is mollified in space on scale in ε and its strength is weakened as ${\mathit{\beta }}_{\mathit{\epsilon }}=\hat{\mathit{\beta }}\sqrt{\frac{2\mathrm{\pi }}{-log\mathit{\epsilon }}}$, and they prove that when $\hat{\mathit{\beta }}\in (0,1)$, $\frac{1}{{\mathit{\beta }}_{\mathit{\epsilon }}}(log{u}_{\mathit{\epsilon }}-\mathbb{E}[log{u}_{\mathit{\epsilon }}])$ converges in distribution as a random field to a solution of Edwards-Wilkinson equation.

In this paper, we consider a stochastic heat equation $u_{\mathit{\epsilon }}$ with a general initial condition $u_0$ and its transformation $F(u_{\mathit{\epsilon }})$ for F in a class of functions $\mathfrak{F}$, which contains $F(x)=x^p$ ($0<p\le 1$) and $F(x)=logx$. Then, we prove that $\frac{1}{{\mathit{\beta }}_{\mathit{\epsilon }}}(F(u_{\mathit{\epsilon }}(t,\cdot ))-\mathbb{E}[F(u_{\mathit{\epsilon }}(t,\cdot ))])$ converges in distribution as a random field to a centered Gaussian field jointly in finitely many $F\in \mathfrak{F}$, t, and $u_0$. In particular, we show the fluctuations of solutions of stochastic heat equations and KPZ equations jointly converge to solutions of SPDEs which depend on $u_0$.

Our main tools are Itô’s formula, the martingale central limit theorem, and the homogenization argument as in [CNN22]. To this end, we also prove a local limit theorem for the partition function of intermediate disorder $2d$ directed polymers.

Let $M_{\mathit{\gamma }}$ be a subcritical Gaussian multiplicative chaos measure associated with a general log-correlated Gaussian field defined on a bounded domain $D\subset {\mathbb{R}}^d$, $d\ge 1$. We find an explicit formula for its singularity spectrum by showing that $M_{\mathit{\gamma }}$ satisfies almost surely the multifractal formalism, i.e., we prove that its singularity spectrum is almost surely equal to the Legendre–Fenchel transform of its $L^q$-spectrum. Then applying this result, we compute the lower singularity spectrum of the multifractal random walk and of the Liouville Brownian motion.

Let ${(X_i)}_{i\ge 1}$ and ${(Y_i)}_{i\ge 1}$ be two independent sequences of independent identically distributed (iid) random variables taking their values in a common finite alphabet and having the same law. Let $L{C}_n$ be the length of the longest common subsequences of the two random words $X_1\cdots X_n$ and $Y_1\cdots Y_n$. Under a lower bound assumption on the order of its variance, $L{C}_n$ is shown to satisfy a central limit theorem. This is in contrast to the limiting distribution of the length of the longest common subsequences in two independent uniform random permutations of $\{1,\dots ,n\}$, which is shown to be the Tracy-Widom distribution.

We study the invariant measures of infinite systems of stochastic differential equations (SDEs) indexed by the vertices of a regular tree. These invariant measures correspond to Gibbs measures associated with certain continuous specifications, and we focus specifically on those measures which are homogeneous Markov random fields. We characterize the joint law at any two adjacent vertices in terms of a new two-dimensional SDE system, called the “local equation”, which exhibits an unusual dependence on a conditional law. Exploiting an alternative characterization in terms of an eigenfunction-type fixed point problem, we derive existence and uniqueness results for invariant measures of the local equation and infinite SDE system. This machinery is put to use in two examples. First, we give a detailed analysis of the surprisingly subtle case of linear coefficients, which yields a new way to derive the famous Kesten-McKay law for the spectral measure of the regular tree. Second, we construct solutions of tree-indexed SDE systems with nearest-neighbor repulsion effects, similar to Dyson’s Brownian motion.

We prove that, for every $0\le t\le 1$, the limiting distribution of the scale-normalized number of key comparisons used by the celebrated algorithm QuickQuant to find the tth quantile in a randomly ordered list has a Lipschitz continuous density function $f_t$ that is bounded above by 10. Furthermore, this density $f_t(x)$ is positive for every $x>min\{t,1-t\}$ and, uniformly in t, enjoys superexponential decay in the right tail. We also prove that the survival function $1-F_t(x)={\int }_x^{\mathrm{\infty }}{f}_t(y)\mathrm{d}y$ and the density function $f_t(x)$ both have the right tail asymptotics $exp[-xlnx-xlnlnx+O(x)]$. We use the right-tail asymptotics to bound large deviations for the scale-normalized number of key comparisons used by QuickQuant.

We consider the equilibrium perturbations for two stochastic systems: the d-dimensional generalized exclusion process and the one-dimensional chain of anharmonic oscillators. We add a perturbation of order $N^{-\mathit{\alpha }}$ to the equilibrium profile and speed up the process by $N^{1+\mathrm{\kappa }}$ for parameters $0<\mathrm{\kappa }\le \mathit{\alpha }$. Under some additional constraints on κ and α, we show the perturbed quantities evolve according to the Burgers equation in the exclusion process, and to two decoupled Burgers equations in the anharmonic chain, both in the smooth regime.

We extend the new approach introduced in [24] and [25] for dealing with stochastic Volterra equations using the ideas of Rough Path theory and prove global existence and uniqueness results. The main idea of this approach is simple: Instead of the iterated integrals of a path comprising the data necessary to solve any equation driven by that path, now iterated integral convolutions with the Volterra kernel comprise said data. This leads to the corresponding abstract objects called Volterra-type Rough Paths, as well as the notion of the convolution product, an extension of the natural tensor product used in Rough Path Theory.

We prove that the reverse characteristic polynomial $det(I_n-z{A}_n)$ of a random $n\times n$ matrix $A_n$ with iid $\mathrm{Bernoulli}(d∕n)$ entries converges in distribution towards the random infinite product

$$\prod _{\ell =1}^{\mathrm{\infty }}{(1-z^{\ell })}^{Y_{\ell }}$$

where $Y_{\ell }$ are independent $\mathrm{Poisson}(d^{\ell }∕\ell )$ random variables. We show that this random function is a Poisson analog of more classical Gaussian objects such as the Gaussian holomorphic chaos. As a byproduct, we obtain new simple proofs of previous results on the asymptotic behaviour of extremal eigenvalues of sparse Erdős-Rényi digraphs: for every $d>1$, the greatest eigenvalue of $A_n$ is close to d and the second greatest is smaller than $\sqrt{d}$, a Ramanujan-like property for irregular digraphs. For $d<1$, the only non-zero eigenvalues of $A_n$ converge to a Poisson multipoint process on the unit circle.

Our results also extend to the semi-sparse regime where d is allowed to grow to ∞ with n, slower than $n^{o(1)}$. We show that the reverse characteristic polynomial converges towards a more classical object written in terms of the exponential of a log-correlated real Gaussian field. In the semi-sparse regime, the empirical spectral distribution of $A_n∕\sqrt{d_n}$ converges to the circle distribution; as a consequence of our results, the second eigenvalue sticks to the edge of the circle.

We prove an invariance principle for linearly edge reinforced random walks on γ-stable critical Galton-Watson trees, where $\mathit{\gamma }\in (1,2]$ and where the edge joining x to its parent has rescaled initial weight $d{(O,x)}^{\mathit{\alpha }}$ for some $\mathit{\alpha }\le 1$. This corresponds to the recurrent regime of initial weights. We then establish fine asymptotics for the limit process. In the transient regime, we also give an upper bound on the random walk displacement in the discrete setting, showing that the edge reinforced random walk never has positive speed, even when the initial edge weights are strongly biased away from the root.

We study the behavior of a tracer particle driven by a one-dimensional fluctuating potential, defined initially as a Brownian motion, and evolving in time according to the heat equation. We obtain two main results. First, in the short time limit, we show that the fluctuations of the particle become Gaussian and sub-diffusive, with dynamical exponent $3∕4$. Second, in the long time limit, we show that the particle is trapped by the local minima of the potential and evolves diffusively i.e. with exponent $1∕2$.

In 2002, Johansson conjectured that the maximum of the Airy${}_2$ process minus the parabola $x^2$ is almost surely achieved at a unique location [Joh03, Conjecture 1.5]. This result was proved a decade later by Corwin and Hammond [CH14, Theorem 4.3]; Moreno Flores, Quastel and Remenik [FQR13]; and Pimentel [Pim14]. Up to scaling, the Airy${}_2$ process minus the parabola $x^2$ arises as the fixed time spatial marginal of the KPZ fixed point when started from narrow wedge initial data. We extend this maximizer uniqueness result to the fixed time spatial marginal of the KPZ fixed point when begun from any element of a very broad class of initial data. None of these results rules out the possibility that at random times, the KPZ fixed point spatial marginal violates maximizer uniqueness. To understand this possibility, we study the probability that the KPZ fixed point has, at a given time, two or more locations where its value is close to the maximum, obtaining quantitative upper and lower bounds in terms of the degree of closeness for a very broad class of initial data. We also compute a quantity akin to the joint density of the locations of two maximizers and the maximum value. As a consequence, the set of times of maximizer non-uniqueness almost surely has Hausdorff dimension at most two-thirds. Our analysis relies on the exact formula for the distribution function of the KPZ fixed point obtained by Matetski, Quastel and Remenik in [MQR21], the variational formula for the KPZ fixed point involving the Airy sheet constructed by Dauvergne, Ortmann and Virág in [DOV], and the Brownian Gibbs property for the Airy${}_2$ process minus the parabola $x^2$ demonstrated by Corwin and Hammond in [CH14].

Several long-time limit theorems of one-dimensional Lévy processes weighted and normalized by functions of the local time are studied. The long-time limits are taken via certain families of random times, called clocks: exponential clock, hitting time clock, two-point hitting time clock and inverse local time clock. The limit measure can be characterized via a certain martingale expressed by an invariant function for the process killed upon hitting zero. The limit processes may differ according to the choice of the clocks when the original Lévy process is recurrent and of finite variance.

We study the statistically invariant structures of the nonlinear generalized Langevin equation (GLE) with a power-law memory kernel. For a broad class of memory kernels, including those in the subdiffusive regime, we construct solutions of the GLE using a Gibbsian framework, which does not rely on existing Markovian approximations. Moreover, we provide conditions on the decay of the memory to ensure uniqueness of statistically steady states, generalizing previous known results for the GLE under particular kernels as a sum of exponentials.

We establish several equivalent characterisations of the anchored isoperimetric dimension of supercritical clusters in Bernoulli bond percolation on transitive graphs. We deduce from these characterisations together with a theorem of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin (Duke Math. J. 2020) that if G is a transient transitive graph then the infinite clusters of Bernoulli percolation on G are transient for p sufficiently close to 1. It remains open to extend this result down to the critical probability. Along the way we establish two new cluster repulsion inequalities that are of independent interest.

Let $({\mathrm{\xi }}_1,{\mathrm{\eta }}_1)$, $({\mathrm{\xi }}_2,{\mathrm{\eta }}_2),\dots$ be independent identically distributed ${\mathbb{R}}^2$-valued random vectors. Assuming that ${\mathrm{\xi }}_1$ has zero mean and finite variance and imposing three distinct groups of assumptions on the distribution of ${\mathrm{\eta }}_1$ we prove three functional limit theorems for the logarithm of convergent discounted perpetuities ${\sum }_{k\ge 0}{\mathrm{e}}^{{\mathrm{\xi }}_1+\dots +{\mathrm{\xi }}_k-ak}{\mathrm{\eta }}_{k+1}$ as $a\to 0+$. Also, we prove a law of the iterated logarithm which corresponds to one of the aforementioned functional limit theorems. The present paper continues a line of research initiated in the paper Iksanov, Nikitin and Samoillenko (2022), which focused on limit theorems for a different type of convergent discounted perpetuities.

In this work, we characterize cluster-invariant point processes for critical branching spatial processes on ${\mathbb{R}}^d$ for all large enough d when the motion law is α-stable or has a finite discrete range. More precisely, when the motion is α-stable with $\mathit{\alpha }\le 2$ and the offspring law μ of the branching process has an heavy tail such that $\mathrm{\mu }(k)\sim k^{-2-\mathit{\beta }}$, then we need the dimension d to be strictly larger than the critical dimension $\mathit{\alpha }∕\mathit{\beta }$. In particular, when the motion is Brownian and the offspring law μ has a second moment, this critical dimension is 2. Contrary to the previous work of Bramson, Cox and Greven in [4] whose proof used PDE techniques, our proof uses probabilistic tools only.

The paper addresses probabilistic aspects of the KPZ equation and stochastic Burgers equation by providing a solution theory that builds on the energy solution theory [15, 19, 20, 23]. The perspective we adopt is to study the stochastic Burgers equation by writing its solution as a probabilistic solution [22] plus a term that can be studied with deterministic PDE considerations. One motivation is universality of KPZ and stochastic Burgers equations for a certain class of stochastic PDE growth models, first studied in [28]. For this, we prove universality for SPDEs with general nonlinearities, thereby extending [28, 29], and for many non-stationary initial data, thereby extending [21].

Our perspective lets us also prove explicit rates of convergence to white noise invariant measure of stochastic Burgers for non-stationary initial data, in particular extending the spectral gap result of [23] beyond stationary initial data, though for non-stationary data our convergence will be measured in Wasserstein distance and relative entropy, not via the spectral gap as in [23]. Actually, we extend the spectral gap in [23] to a log-Sobolev inequality. Our methods can also analyze fractional stochastic Burgers equations [23]; we discuss this briefly.

Lastly, we note our perspective on the KPZ and stochastic Burgers equations gives a first intrinsic notion of solutions for general continuous initial data, in contrast to Hölder regular data needed for regularity structures [26], paracontrolled distributions [14], and Hölder-regular Brownian bridge data for energy solutions [19, 20].

Let $W_t$ be Brownian motion in the plane started at the origin and let θ be the first exit time of the unit disk $D_1$. Let

$${\mathrm{\mu }}_{\mathit{\theta }}(x,\mathit{ϵ})=\frac{1}{\mathrm{\pi }{\mathit{ϵ}}^2}{\int }_0^{\mathit{\theta }}{1}_{\{B(x,\mathit{ϵ})\}}(W_t)dt,$$

and set ${\mathrm{\mu }}_{\mathit{\theta }}^{\ast }(\mathit{ϵ})={sup}_{x\in D_1}{\mathrm{\mu }}_{\mathit{\theta }}(x,\mathit{ϵ})$. We show that

$$\sqrt{{\mathrm{\mu }}_{\mathit{\theta }}^{\ast }(\mathit{ϵ})}-\sqrt{2∕\mathrm{\pi }}\left(log{\mathit{ϵ}}^{-1}-\frac{1}{2}loglog{\mathit{ϵ}}^{-1}\right)$$

is tight.

We use the abstract method of (local) martingale problems in order to give criteria for convergence of stochastic processes. Extending previous notions, the formulation we use is neither restricted to Markov processes (or semimartingales), nor to continuous or càdlàg paths. We illustrate our findings both, by finding generalizations of known results, and proving new results. For the latter, we work on processes with fixed times of discontinuity.

Let $\{L_t^z\}$ be the jointly continuous local times of a one-dimensional Brownian motion and let $L_t^{\ast }={sup}_{z\in \mathbb{R}}{L}_t^z$. Let $V_t$ be any point z such that $L_t^z=L_t^{\ast }$, a most visited site of Brownian motion. We prove that if $\mathit{\gamma }>1$, then

$$\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\frac{|V_t|}{\sqrt{t}∕{(logt)}^{\mathit{\gamma }}}=\mathrm{\infty },\text{a.s.},$$

with an analogous result for simple random walk. This proves a conjecture of Lifshits and Shi.

If ${(X^N)}_{N=1,2,\mathrm{...}}$ is a sequence of Markov processes which solve the martingale problems for some operators ${(G^N)}_{N=1,2,\mathrm{...}}$, it is a classical task to derive a limit result as $N\to \mathrm{\infty }$, in particular a weak process limit with limiting operator G. For slow-fast systems $X^N=(V^N,Z^N)$ where $V^N$ is slow and $Z^N$ is fast, $G^N$ consists of two (or more) terms, and we are interested in weak convergence of $V^N$ to some Markov process V. In this case, for some $f\in \mathcal{D}(G)$, the domain of G, depending only on v, the limit $Gf$ can sometimes be derived by using some $g_N\to 0$ (depending on v and z), and study convergence of $G^N(f+g_N)\to Gf$. We develop this method further in order to obtain functional Laws of Large Numbers (LLNs) and Central Limit Theorems (CLTs). We then apply our general result to various examples from Chemical Reaction Network theory. We show that we can rederive most limits previously obtained, but also provide new results in the case when the fast-subsystem is first order. In particular, we allow that fast species to be consumed faster than they are produced, and we derive a CLT for Hill dynamics with coefficient 2.

For Markov processes with absorption, we provide general criteria ensuring the existence and the exponential non-uniform convergence in weighted total variation norm to a quasi-stationary distribution. We also characterize a subset of its domain of attraction by an integrability condition, prove the existence of a right eigenvector for the semigroup of the process and the existence and exponential ergodicity of the Q-process. These results are applied to one-dimensional and multi-dimensional diffusion processes, to pure jump continuous time processes, to reducible processes with several communication classes, to perturbed dynamical systems and discrete time processes evolving in discrete state spaces.

A serious typographical error in [1] is corrected.

We prove several stability estimates, comparing solutions driven by different $(b^i,{\mathit{\sigma }}^i)$, both for Itô and Stratonovich SDEs, possibly depending on negative Sobolev norms of the difference $b^1-b^2$. We then discuss several applications of these results to McKean–Vlasov SDEs, criteria for strong compactness of solutions and Wong–Zakai type theorems.

In this paper, we introduce a framework for studying a subshift of finite type (SFT) with noise, allowing some amount of forbidden patterns to appear. Using the Besicovitch distance, which permits a global comparison of configurations, we then study the closeness of measures on noisy configurations to the non-noisy case as the amount of noise goes to 0. Our first main result is the full classification of the (in)stability in the one-dimensional case. Our second main result is a stability property under Bernoulli noise for higher-dimensional periodic SFTs, which we finally extend to an aperiodic example through a variant of the Robinson tiling.

A stochastic sewing lemma which is applicable for processes taking values in Banach spaces is introduced. Applications to additive functionals of fractional Brownian motion of distributional type are discussed.

We derive estimates for the largest and smallest singular values of sparse rectangular $N\times n$ random matrices, assuming ${lim}_{N,n\to \mathrm{\infty }}\frac{n}{N}=y\in (0,1)$. We consider a model with sparsity parameter $p_N$ such that $N{p}_N\sim {log}^{\mathit{\alpha }}N$ for some $\mathit{\alpha }>1$, and assume that the moments of the matrix elements satisfy the condition $\mathbb{E}|X_{jk}{|}^{4+\mathit{\delta }}\le C<\mathrm{\infty }$. We assume also that the entries of matrices we consider are truncated at the level ${(N{p}_N)}^{\frac{1}{2}-\mathit{\varkappa }}$ with $\mathit{\varkappa }:=\frac{\mathit{\delta }}{2(4+\mathit{\delta })}$.

In this paper, the scaling limit of random connected cubic planar graphs (respectively multigraphs) is shown to be the Brownian sphere.

The proof consists in essentially two main steps. First, thanks to the known decomposition of cubic planar graphs into their 3-connected components, the metric structure of a random cubic planar graph is shown to be well approximated by its unique 3-connected component of linear size, with modified distances.

Then, Whitney’s theorem ensures that a 3-connected cubic planar graph is the dual of a simple triangulation, for which it is known that the scaling limit is the Brownian sphere. Curien and Le Gall have recently developed a framework to study the modification of distances in general triangulations and in their dual. By extending this framework to simple triangulations, it is shown that 3-connected cubic planar graphs with modified distances converge jointly with their dual triangulation to the Brownian sphere.

We consider a Feller branching diffusion process X with drift c having 0 as a slowly reflecting (sticky) boundary point with a stickiness parameter $1∕\mathrm{\mu }\in (0,\mathrm{\infty })$. We show that (i) the process X can be characterised as a unique weak solution to the SDE system

$$d{X}_t=(b{X}_t+c)I(X_t>0)dt+\sqrt{2a{X}_t}d{B}_t$$

$$I(X_t=0)dt=\frac{1}{\mathrm{\mu }}d{\ell }_t^0(X)$$

where $b\in \mathbb{R}$ and $0<c<a$ are given and fixed, B is a standard Brownian motion, and ${\ell }^0(X)$ is a diffusion local time process of X at 0, and (ii) the transition density function of X can be expressed in the closed form by means of a convolution integral involving a new special function and a modified Bessel function of the second kind. The new special function embodies the stickiness of X entirely and reduces to the Mittag-Leffler function when $b=0$. We determine a (sticky) boundary condition at zero that characterises the transition density function of X as a unique solution to the Kolmogorov forward/backward equation of X. Letting $\mathrm{\mu }\downarrow 0$ (absorption) and $\mathrm{\mu }\uparrow \mathrm{\infty }$ (instantaneous reflection) the closed-form expression for the transition density function of X reduces to the ones found by Feller [6] and Molchanov [14] respectively. The results derived for sticky Feller diffusions translate over to yield closed-form expressions for the transition density functions of (a) sticky Cox-Ingersoll-Ross processes and (b) sticky reflecting Vasicek processes that can be used to model slowly reflecting interest rates.

We prove that the SLE${}_{\mathrm{\kappa }}$ loop measure arises naturally from the conformal welding of two γ-Liouville quantum gravity (LQG) disks for ${\mathit{\gamma }}^2=\mathrm{\kappa }\in (0,4)$. The proof relies on our companion work on conformal welding of LQG disks and uses as an essential tool the concept of uniform embedding of LQG surfaces. Combining our result with work of Gwynne and Miller, we get that random quadrangulations decorated by a self-avoiding polygon converge in the scaling limit to the LQG sphere decorated by the SLE${}_{8∕3}$ loop. Our result is also a key input to recent work of the first and third coauthors on the integrability of the conformal loop ensemble.

We consider the spatially inhomogeneous Moran model with seed-banks introduced in [20]. Populations comprising active and dormant individuals are spatially structured in colonies labeled by ${\mathbb{Z}}^d,d\ge 1$. The population sizes are sampled from a translation-invariant, ergodic, uniformly elliptic field that constitutes a static random environment. Individuals carry one of two types: ♡ and ♠. Dormant individual resides in what is called a seed-bank. Active individuals exchange type from the seed-bank of their own colony, and resample type by choosing a parent uniformly at random from the distinct active populations according to a symmetric migration kernel. In [20] by exploiting a dual process given by an interacting coalescing particle system, we showed that the spatial system exhibits a dichotomy between clustering (mono-type equilibrium) and coexistence (multi-type equilibrium). In this paper, we identify the domain of attraction for each mono-type equilibrium in the clustering regime for an arbitrary fixed environment. Furthermore, we show that in dimensions $d\le 2$, when the migration kernel is recurrent, for almost all realization of the environment, the system with an initially consistent type-distribution converges weakly to a mono-type equilibrium in which the probability of fixation to the all type-♡ configuration does not depend on the environment. An explicit formula for the fixation probability is given in terms of an annealed average of the type-♡ densities in the active and the dormant population, biased by the ratio of the two population sizes at the target colony.

Primary techniques employed in the proofs include stochastic duality and the environment process viewed from particle, introduced in [24] for random walk in random environment on a strip. A spectral analysis of Markov operator yields quenched weak convergence of the environment process associated with the single-particle dual process to a reversible ergodic distribution, which we transfer to the spatial system of populations by using duality.

We consider a stochastic individual-based population model with competition, trait-structure affecting reproduction and survival, and changing environment. The changes of traits are described by jump processes, and the dynamics can be approximated in large population by a non-linear PDE with a non-local mutation operator. Using the fact that this PDE admits a non-trivial stationary solution, we can approximate the non-linear stochastic population process by a linear birth-death process where the interactions are frozen, as long as the population remains close to this equilibrium. This allows us to derive, when the population is large, the equation satisfied by the ancestral lineage of an individual uniformly sampled at a fixed time T, which is the path constituted of the traits of the ancestors of this individual in past times $t\le T$. This process is a time inhomogeneous Markov process, but we show that the time reversal of this process possesses a very simple structure (e.g. time-homogeneous and independent of T). This extends recent results where the authors studied a similar model with a Laplacian operator but where the methods essentially relied on the Gaussian nature of the mutations.

We establish the mean-field bounds $\mathit{\gamma }\ge 1$, $\mathit{\delta }\ge 2$ and $△\ge 2$ on the critical exponents of the Poisson-Boolean continuum percolation model under a moment condition on the radii; these were previously known only in the special case of fixed radii (in the case of γ), or not at all (in the case of δ and △). We deduce these as consequences of the mean-field bound $\mathit{\beta }\le 1$, recently established under the same moment condition [8], using a relative entropy method introduced by the authors in previous work [7].

We study the size of the largest component of two models of random graphs with prescribed degree sequence, the configuration model (CM) and the uniform model (UM), in the (barely) subcritical regime. For the CM, we give upper bounds that are asymptotically tight for certain degree sequences. These bounds hold under mild conditions on the sequence and improve previous results of Hatami and Molloy on the barely subcritical regime. For the UM, we give weaker upper bounds that are tight up to logarithmic terms but require no assumptions on the degree sequence. In particular, the latter result applies to degree sequences with infinite variance in the subcritical regime.

We study the level-set of the zero-average Gaussian Free Field on a uniform random d-regular graph above an arbitrary level $h\in (-\mathrm{\infty },h_{\star })$, where $h_{\star }$ is the level-set percolation threshold of the GFF on the d-regular tree ${\mathbb{T}}_d$. We prove that w.h.p as the number n of vertices of the graph diverges, the GFF has a unique giant connected component ${\mathcal{C}}_1^{(n)}$ of size $\mathrm{\eta }(h)n+o(n)$, where $\mathrm{\eta }(h)$ is the probability that the root percolates in the corresponding GFF level-set on ${\mathbb{T}}_d$. This gives a positive answer to the conjecture of [4] for most regular graphs. We also prove that the second largest component has size $\mathrm{\Theta }(logn)$.

Moreover, we show that ${\mathcal{C}}_1^{(n)}$ shares the following similarities with the giant component of the supercritical Erdős-Rényi random graph. First, the diameter and the typical distance between vertices are $\mathrm{\Theta }(logn)$. Second, the 2-core and the kernel encompass a given positive proportion of the vertices. Third, the local structure is a branching process conditioned to survive, namely the level-set percolation cluster of the root in ${\mathbb{T}}_d$ (in the Erdős-Rényi case, it is known to be a Galton-Watson tree with a Poisson distribution for the offspring).

Two-sided sharp Green function estimates are obtained for second order uniformly elliptic operators in non-divergence form with Dini continuous coefficients in bounded $C^{1,1}$ domains, which are shown to be comparable to that of the Dirichlet Laplace operator in the domain. The first and second order derivative estimates of the Green functions are also derived. Moreover, boundary Harnack inequality with an explicit boundary decay rate and interior Schauder’s estimates for these differential operators are established, which may be of independent interest.

We consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions $d+1$ with $d\ge 3$ being the spatial dimension. For this random walk we prove an annealed local central limit theorem and a quenched local limit theorem. The latter shows that the quenched transition probabilities of the random walk converge to the annealed transition probabilities reweighted by a function of the medium centred at the target site. This function is the density of the unique measure which is invariant for the point of view of the particle, is absolutely continuous with respect to the annealed measure and satisfies certain concentration properties.

Consider Bernoulli bond percolation on a locally finite, connected graph G and let $p_{\mathrm{cut}}$ be the threshold corresponding to a “first-moment method” lower bound. Kahn (Electron. Comm. Probab. Volume 8, 184-187. (2003)) constructed a counter-example to Lyons’ conjecture of $p_{\mathrm{cut}}=p_c$ and proposed a modification. Here we give a positive answer to Kahn’s modified question. The key observation is that in Kahn’s modification, the new expectation quantity also appears in the differential inequality of one-arm events. This links the question to a lemma of Duminil-Copin and Tassion (Comm. Math. Phys. Volume 343, 725-745. (2016)). We also study some applications for Bernoulli percolation on periodic trees.

We consider the three-dimensional magnetohydrodynamics system forced by noise that is white in both time and space. Its complexity due to four non-linear terms makes its analysis very intricate. Nevertheless, taking advantage of its structure and adapting the theory of paracontrolled distributions from [30], we prove its local well-posedness. A first challenge is to find an appropriate paracontrolled ansatz which must consist of both the velocity and the magnetic fields. Second challenge is that for some non-linear terms, renormalizations cannot be achieved individually; we overcome this obstacle by strategically coupling certain terms together rather than separately. Our proof is also inspired by the work of [70].

Motivated by structural, reduced-form and hybrid models of the third party and counterparty credit risk, we study a generalized backward stochastic differential equations (BSDE) up to a random time horizon ϑ, which is not a stopping time with respect to a reference filtration. In contrast to the existing literature in the area of credit risk modeling, we do not impose specific assumptions on the random time ϑ and we study the existence of solutions to BSDE and reflected BSDE with a random time horizon through the method of reduction. For this purpose, we also examine BSDE and reflected BSDE with a làdlàg driver where the driver is allowed to have a finite number of jumps overlapping with jumps of the martingale part. Theoretical results are illustrated by particular instances of a random time and explicit BSDEs in either the Brownian or Brownian-Poisson filtration.

We show the equivalence of three properties for an infinitely divisible distribution: the subexponentiality of the density, the subexponentiality of the density of its Lévy measure and the tail equivalence between the density and its Lévy measure density, under monotonic-type assumptions on the Lévy measure density. The key assumption is that tail of the Lévy measure density is asymptotic to a non-increasing function or is almost decreasing. Our conditions are natural and cover a rather wide class of infinitely divisible distributions. Several significant properties for analyzing the subexponentiality of densities have been derived such as closure properties of [ convolution, convolution roots and asymptotic equivalence ] and the factorization property. Moreover, we illustrate that the results are applicable for developing the statistical inference of subexponential infinitely divisible distributions which are absolutely continuous.

We consider growing random networks ${\{{\mathcal{G}}_n\}}_{n\ge 1}$ where, at each time, a new vertex attaches itself to a collection of existing vertices via a fixed number $m\ge 1$ of edges, with probability proportional to a function f (called attachment function) of their degree. It was shown in [BB21] that such network models exhibit two distinct regimes: (i) the persistent regime, corresponding to ${\sum }_{i=1}^{\mathrm{\infty }}f{(i)}^{-2}<\mathrm{\infty }$, where the top K maximal degree vertices fixate over time for any given K, and (ii) the non-persistent regime, with ${\sum }_{i=1}^{\mathrm{\infty }}f{(i)}^{-2}=\mathrm{\infty }$, where the identities of these vertices keep changing infinitely often over time. In this article, we develop root finding algorithms using the empirical degree structure and local network information based on a snapshot of such a network at some large time. In the persistent regime, the algorithm is purely based on degree centrality, that is, for a given error tolerance $\mathit{\epsilon }\in (0,1)$, there exists $K_{\mathit{\epsilon }}\in \mathbb{N}$ such that for any $n\ge 1$, the confidence set for the root in ${\mathcal{G}}_n$, which contains the root with probability at least $1-\mathit{\epsilon }$, consists of the top $K_{\mathit{\epsilon }}$ maximal degree vertices. In particular, the size of the confidence set is stable in the network size. Upper and lower bounds on $K_{\mathit{\epsilon }}$ are explicitly characterized in terms of the error tolerance ε and the attachment function f. In the non-persistent regime, for an appropriate choice of $r_n\to \mathrm{\infty }$ at a rate much smaller than the diameter of the network, the neighborhood of radius $r_n$ around the maximal degree vertex is shown to contain the root with high probability, and a size estimate for this set is obtained. It is shown that, when $f(k)=k^{\mathit{\alpha }},k\ge 1$, for any $\mathit{\alpha }\in (0,1∕2]$, this size grows at a smaller rate than any positive power of the network size.

Consider a non-autonomous continuous-time linear system in which the time-dependent matrix determining the dynamics is piecewise constant and takes finitely many values $A_1,\dots ,A_N$. This paper studies the equality cases between the maximal Lyapunov exponent associated with the set of matrices $\{A_1,\dots ,A_N\}$, on the one hand, and the corresponding ones for piecewise deterministic Markov processes with modes $A_1,\dots ,A_N$, on the other hand. A fundamental step in this study consists in establishing a result of independent interest, namely, that any sequence of Markov processes associated with the matrices $A_1,\dots ,A_N$ converges, up to extracting a subsequence, to a Markov process associated with a suitable convex combination of those matrices.

We consider the following family of random trigonometric polynomials of the form $S_n(\mathit{\theta }):=\frac{1}{\sqrt{n}}{\sum }_{k=1}^n{a}_{k}cos(k\mathit{\theta })+b_{k}sin(k\mathit{\theta })$, where the variables ${\{a_k,b_k\}}_{k\ge 1}$ are i.i.d. and symmetric, defined on a common probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$. In the seminal paper [SZ54], Salem and Zygmund proved that for Rademacher coefficients, $\mathbb{P}-$almost surely and for all $t\in \mathbb{R}$

$$\underset{n\to +\mathrm{\infty }}{lim}\frac{1}{2\mathrm{\pi }}{\int }_0^{2\mathrm{\pi }}{e}^{it{S}_n(\mathit{\theta })}d\mathit{\theta }=e^{-t^2∕2}.$$

To the best of our knowledge, the natural question of the fluctuations in the above almost sure limit has not been tackled so far and is precisely the object of this article. Namely, for general i.i.d. symmetric random coefficients having a finite sixth-moment and for a large class of continuous test functions ϕ we prove that

$$\sqrt{n}\left(\frac{1}{2\mathrm{\pi }}{\int }_0^{2\mathrm{\pi }}\mathit{\varphi }(S_n(\mathit{\theta }))d\mathit{\theta }-{\int }_{\mathbb{R}}\mathit{\varphi }(t)\frac{e^{-\frac{t^2}{2}}dt}{\sqrt{2\mathrm{\pi }}}\right)\underset{n\to \mathrm{\infty }}{\overset{\text{Law}}{\to }}\mathcal{N}\left(0,{\mathrm{\Sigma }}_{\mathit{\varphi }}^2\right),$$

where the limit variance is given by ${\mathrm{\Sigma }}_{\mathit{\varphi }}^2:={\mathit{\sigma }}_{\mathit{\varphi }}^2+\frac{c_2{(\mathit{\varphi })}^2}{2}\left(\mathbb{E}[a_1^4]-3\right)$. Here, the constant ${\mathit{\sigma }}_{\mathit{\varphi }}^2$ is explicit and corresponds to the limit variance in the case of Gaussian coefficients and $c_2(\mathit{\varphi })$ is the coefficient of order 2 in the decomposition of ϕ in the Hermite polynomial basis. It thus turns out that the fluctuations are Gaussian and not universal since they involve the kurtosis of the random coefficients.

The method we develop here is robust and allows to use Wiener chaos expansions for proving CLTs for functionals of general, non-necessarily Gaussian, random fields. It combines contributions from the Malliavin–Stein method, noise stability results, and algebraic considerations around Newton sums and the Newton–Girard formula.

This addendum clarifies Definition 5.1 in section 5.3 of the previously published paper Electron. J. Probab. 27, 1–27 (2022).

In the paper we prove that, for $\mathrm{\kappa }\in (0,8)$, the n-point boundary Green’s function of exponent $\frac{8}{\mathrm{\kappa }}-1$ for chordal SLE${}_{\mathrm{\kappa }}$ exists. We also prove that the convergence is uniform over compact sets and the Green’s function is continuous. We also give up-to-constant bounds for the Green’s function.

We study the following model for a diploid population of constant size N: Every individual carries a random number of (genetic) elements. Upon a reproduction event each of the two parents passes each element independently with probability $\frac{1}{2}$ on to the offspring. We study the process $X^N=(X^N(1),X^N(2),\mathrm{...})$, where $X_t^N(k)$ is the frequency of individuals at time t that carry k elements, and prove convergence (in some weak sense) of $X^N$ jointly with its empirical first moment $Z^N$ to the “slow-fast” system $(Z,X)$, where $X_t=\text{Poi}(Z_t)$ and Z evolves according to a critical Feller branching process. We discuss heuristics explaining this finding and some extensions and limitations.

We prove a center manifold theorem for rough partial differential equations (rough PDEs). The class of rough PDEs we consider contains as a key subclass reaction-diffusion equations driven by nonlinear multiplicative noise, where the stochastic forcing is given by a γ-Hölder rough path, for $\mathit{\gamma }\in (1∕3,1∕2]$. Our proof technique relies upon the theory of rough paths and analytic semigroups in combination with a discretized Lyapunov-Perron-type method in a suitable scale of interpolation spaces. The resulting center manifold is a random manifold in the sense of the theory of random dynamical systems (RDS). We also illustrate our main theorem for reaction-diffusion equations as well as for the Swift-Hohenberg equation.

Growth-fragmentation processes model the evolution of positive masses which undergo binary divisions. The aim of this paper is twofold. First, we extend the theory of growth-fragmentation processes to allow signed mass. Among other things, we introduce genealogical martingales and establish a spinal decomposition for the associated cell system, following [BBCK18]. Then, we study a particular family of such self-similar signed growth-fragmentation processes which arise when cutting half-planar excursions at horizontal levels. When restricting this process to the positive masses in a special case, we recover part of the family introduced by Bertoin, Budd, Curien and Kortchemski in [BBCK18].

Excluding some special cases, computing the critical inverse-temperature ${\mathit{\beta }}_c$ of a mixed p-spin spin glass model is a difficult task. The only known method to calculate its value for a general model requires the full power of the Parisi formula. On the other hand, an easy application of the second moment method to the partition function yields an explicit lower bound ${\mathit{\beta }}_m\le {\mathit{\beta }}_c$ to the critical inverse-temperature. Interestingly, in the important case of the Sherrington-Kirkpatrick model ${\mathit{\beta }}_m={\mathit{\beta }}_c$. In this work we consider the multi-species spherical mixed p-spin models without external field, and characterize by a simple condition the models for which the second moment method works in the whole replica symmetric phase, namely, models such that ${\mathit{\beta }}_m={\mathit{\beta }}_c$. In particular, for those models we obtain the value of ${\mathit{\beta }}_c$.

In this paper, we obtain stability results for backward stochastic differential equations with jumps (BSDEs) in a very general framework. More specifically, we consider a convergent sequence of standard data, each associated to their own filtration, and we prove that the associated sequence of (unique) solutions is also convergent. The current result extends earlier contributions in the literature of stability of BSDEs and unifies several frameworks for numerical approximations of BSDEs and their implementations.

Two-pointed quantum disks with a weight parameter $W>0$ are a family of finite-area random surfaces that arise naturally in Liouville quantum gravity. In this paper we show that conformally welding two quantum disks according to their boundary lengths gives another quantum disk decorated with an independent chordal ${SLE}_{\mathrm{\kappa }}({\mathit{\rho }}_{-};{\mathit{\rho }}_{+})$ curve. This is the finite-volume counterpart of the classical result of Sheffield (2010) and Duplantier-Miller-Sheffield (2014) on the welding of infinite-area two-pointed quantum surfaces called quantum wedges, which is fundamental to the mating-of-trees theory. Our results can be used to give unified proofs of the mating-of-trees theorems for the quantum disk and the quantum sphere, in addition to a mating-of-trees description of the weight $W=\frac{{\mathit{\gamma }}^2}{2}$ quantum disk. Moreover, it serves as a key ingredient in our companion work, which proves an exact formula for ${SLE}_{\mathrm{\kappa }}({\mathit{\rho }}_{-};{\mathit{\rho }}_{+})$ using conformal welding of random surfaces and a conformal welding result giving the so-called SLE loop.

A Lévy process is said to creep through a curve if, at its first passage time across this curve, the process reaches it with positive probability. We first study this property for bivariate subordinators. Given the graph $\{(t,f(t)):t\ge 0\}$ of any continuous, non increasing function f such that $f(0)>0$, we give an expression of the probability that a bivariate subordinator $(Y,Z)$ issued from 0 creeps through this graph in terms of its renewal function and the drifts of the components Y and Z. We apply this result to the creeping probability of any real Lévy process through the graph of any continuous, non increasing function at a time where the process also reaches its past supremum. This probability involves the density of the renewal function of the bivariate upward ladder process as well as its drift coefficients. We also investigate the case of Lévy processes conditioned to stay positive creeping at their last passage time below the graph of a function. Then we provide some examples and we give an application to the probability of creeping through fixed levels by stable Ornstein-Uhlenbeck processes. We also raise a couple of open questions along the text.

We establish the existence and uniqueness of pathwise strong solutions to the stochastic 3D primitive equations with only horizontal viscosity and diffusivity driven by transport noise on a cylindrical domain $M=(-h,0)\times G$, $G\subset {\mathbb{R}}^2$ bounded and smooth, with the physical Dirichlet boundary conditions on the lateral part of the boundary. Compared to the deterministic case where the uniqueness of z-weak solutions holds in $L^2$, more regular initial data are necessary to establish uniqueness in the anisotropic space $H_z^1{L}_{xy}^2$ so that the existence of local pathwise solutions can be deduced from the Gyöngy-Krylov theorem. Global existence is established using the logarithmic Sobolev embedding, the stochastic Gronwall lemma and an iterated stopping time argument.

We consider the Generalized Gibbs ensembles of the Ablowitz-Ladik lattice and of the Schur flow. We derive large deviations principles for the distribution of the empirical measures for these ensembles. As a consequence, we deduce their almost sure convergence. Moreover, we are able to characterize their limits in terms of the equilibrium measure of the Circular, and the Jacobi beta ensemble, respectively.

Three-velocity ballistic annihilation is an interacting system in which stationary, left-, and right-moving particles are placed at random throughout the real line and mutually annihilate upon colliding. We introduce a coalescing variant in which collisions may generate new particles. For a symmetric three-parameter family of such systems, we compute the survival probability of stationary particles at a given initial density. This allows us to describe a phase-transition for stationary particle survival.

In this article, we establish new central limit theorems for Ruppert-Polyak averaged stochastic gradient descent schemes. Compared to previous work we do not assume that convergence occurs to an isolated attractor but instead allow convergence to a stable manifold. On the stable manifold the target function is constant and the oscillations of the iterates in the tangential direction may be significantly larger than the ones in the normal direction. We still recover a central limit theorem for the averaged scheme in the normal direction with the same rates as in the case of isolated attractors. In the setting where the magnitude of the random perturbation is of constant order, our research covers step-sizes ${\mathit{\gamma }}_n=C_{\mathit{\gamma }}{n}^{-\mathit{\gamma }}$with $C_{\mathit{\gamma }}>0$ and $\mathit{\gamma }\in (\frac{3}{4},1)$. In particular, we show that the beneficial effect of averaging prevails in more general situations.

The random interlacement point process (introduced in [47], generalized in [50]) is a Poisson point process on the space of labeled doubly infinite nearest neighbour trajectories modulo time-shift on a transient graph G. We show that the random interlacement point process on any transient transitive graph G is a factor of i.i.d., i.e., it can be constructed from a family of i.i.d. random variables indexed by vertices of the graph via an equivariant measurable map. Our proof uses a variant of the soft local time method (introduced in [37]) to construct the interlacement point process as the almost sure limit of a sequence of finite-length variants of the model with increasing length. We also discuss a more direct method of proving that the interlacement point process is a factor of i.i.d. which works if and only if G is non-unimodular.

This paper provides a general framework for Stein’s density method for multivariate continuous distributions. The approach associates to any probability density function a canonical operator and Stein class, as well as an infinite collection of operators and classes which we call standardizations. These in turn spawn an entire family of Stein identities and characterizations for any continuous distribution on ${\mathbb{R}}^d$, among which we highlight those based on the score function and the Stein kernel. A feature of these operators is that they do not depend on normalizing constants. A new definition of Stein kernel is introduced and examined; integral formulas are obtained through a connection with mass transport, as well as ready-to-use explicit formulas for elliptical distributions. The flexibility of the kernels is used to compare in Stein discrepancy (and therefore 2-Wasserstein distance) between two normal distributions, Student and normal distributions, as well as two normal-gamma distributions. Upper and lower bounds on the 1-Wasserstein distance between continuous distributions are provided, and computed for a variety of examples: comparison between different normal distributions (improving on existing bounds in some regimes), posterior distributions with different priors in a Bayesian setting (including logistic regression), centred Azzalini–Dalla Valle distributions. Finally the notion of weak Stein equation and weak Stein factors is introduced. Bounds for solutions of the weak Stein equation are obtained for Lipschitz test functions if the distribution admits a Poincaré constant. We use these bounds to compare different copulas on the unit square in 1-Wasserstein distance.

The energy and geometry of maximizing paths in integrable last passage percolation models are governed by the characteristic KPZ scaling exponents of one-third and two-thirds. When represented in scaled coordinates that respect these exponents, this random field of paths may be viewed as a complex energy landscape. We investigate the structure of valleys and connecting pathways in this landscape. The routed weight profile $\mathbb{R}\to \mathbb{R}$associates to $x\in \mathbb{R}$ the maximum scaled energy obtainable by a path whose scaled journey from $(0,0)$ to $(0,1)$ passes through the point $(x,1∕2)$. Developing tools of Brownian Gibbs analysis from [Ham22] and [CHH19], we prove an assertion of strong similarity of this profile for Brownian last passage percolation to Brownian motion of rate two on the unit-order scale. A sharp estimate on the rarity that two macroscopically different routes in the energy landscape offer energies close to the global maximum results. We prove robust assertions concerning modulus of continuity for the energy and geometry of scaled maximizing paths, that develop the results and approach of [HS20], delivering estimates valid on all scales above the microscopic. The geometry of excursions of near ground states about the maximizing path is investigated: indeed, we estimate the energetic shortfall of scaled paths forced to closely mimic the geometry of the maximizing route while remaining disjoint from it. We also provide bounds on the approximate gradient of the maximizing path, viewed as a function, ruling out sharp steep movement down to the microscopic scale. Our results find application in a companion study [GH20a] of the stability, and fragility, of last passage percolation under a dynamical perturbation.

We introduce and study interval partition diffusions with Poisson–Dirichlet$(\mathit{\alpha },\mathit{\theta })$ stationary distribution for parameters $\mathit{\alpha }\in (0,1)$ and $\mathit{\theta }\ge 0$. This extends previous work on the cases $(\mathit{\alpha },0)$ and $(\mathit{\alpha },\mathit{\alpha })$ and builds on our recent work on measure-valued diffusions. Our methods for dealing with general $\mathit{\theta }\ge 0$ allow us to strengthen previous work on the special cases to include initial interval partitions with dust. In contrast to the measure-valued setting, we can show that this extended process is a Feller process improving on the Hunt property established in that setting. These processes can be viewed as diffusions on the boundary of a branching graph of integer compositions. Indeed, by studying their infinitesimal generator on suitable quasi-symmetric functions, we relate them to diffusions obtained as scaling limits of composition-valued up-down chains.

For massless vertex-transitive transient graphs, the percolation phase transition for the level sets of the Gaussian free field on the associated continuous cable system is particularly well understood, and in particular the associated critical parameter ${\tilde{h}}_{\ast }$ is always equal to zero. On general transient graphs, two weak conditions on the graph $\mathcal{G}$ are given in [12], each of which implies one of the two inequalities ${\tilde{h}}_{\ast }\le 0$ and ${\tilde{h}}_{\ast }\ge 0$. In this article, we give two counterexamples to show that none of these two conditions are necessary, prove that the strict inequality ${\tilde{h}}_{\ast }<0$ is typical on massive graphs with bounded weights, and provide an example of a graph on which ${\tilde{h}}_{\ast }=\mathrm{\infty }$. On the way, we obtain another characterization of random interlacements on massive graphs, as well as an isomorphism between the Gaussian free field and the Doob $\mathbf{h}$-transform of random interlacements, and between the two-dimensional pinned free field and random interlacements.

Let $n\ge 1$, $K>0$, and let $X=(X_1,X_2,\dots ,X_n)$ be a random vector in ${\mathbb{R}}^n$ with independent K–subgaussian components. We show that for every 1–Lipschitz convex function f in ${\mathbb{R}}^n$ (the Lipschitzness with respect to the Euclidean metric), $\forall t>0$,

$$max\left(\mathbb{P}\right\{f(X)-\mathrm{Med}f(X)\ge t\},\mathbb{P}\{f(X)-\mathrm{Med}f(X)\le -t\left\}\right)\le exp(-\frac{c{t}^2}{K^{2}log(2+\frac{K^{2}n}{t^2})}),$$

where $c>0$ is a universal constant. The estimates are optimal in the sense that for every $n\ge \tilde{C}$ and $t>0$ there exist a product probability distribution X in ${\mathbb{R}}^n$ with K–subgaussian components, and a 1–Lipschitz convex function f, with

$$\mathbb{P}\left\{\right|f(X)-\mathrm{Med}f(X)|\ge t\}\ge \tilde{c}exp(-\frac{\tilde{C}{t}^2}{K^{2}log(2+\frac{K^{2}n}{t^2})}).$$

The obtained deviation estimates for subgaussian variables are in sharp contrast with the case of variables with bounded $\Vert X_i{\Vert }_{{\mathit{\psi }}_p}$–quasi-norms for $p\in (0,2)$.

We characterize Gaussian estimates for transition probability of a discrete time Markov chain in terms of geometric properties of the underlying state space. In particular, we show that the following are equivalent: (1) Two sided Gaussian bounds on heat kernel (2) A scale invariant Parabolic Harnack inequality (3) Volume doubling property and a scale invariant Poincaré inequality. The underlying state space is a metric measure space, a setting that includes both manifolds and graphs as special cases. An important feature of our work is that our techniques are robust to small perturbations of the underlying space and the Markov kernel. In particular, we show the stability of the above properties under quasi-isometries. We discuss various applications and examples.

In this paper we study the maximal position process of branching Brownian motion in random spatial environment. The random environment is given by a process $\mathrm{\xi }={\left(\mathrm{\xi }(x)\right)}_{x\in \mathbb{R}}$ satisfying certain conditions. We show that the maximum position $M_t$ of particles alive at time t satisfies a quenched strong law of large numbers and an annealed invariance principle.

Suppose that $p,q,r\ge 1$ satisfy the condition $\frac{1}{p}=\frac{1}{q}+\frac{1}{r}$. The paper contains the identification of the best constants $c_{p,q,r}$ and $C_{p,q,r}$ in the estimates

$$\Vert g{\Vert }_{p,\mathrm{\infty }}\le c_{p,q,r}\Vert f{\Vert }_q\Vert v^{\ast }{\Vert }_r,\Vert g{\Vert }_p\le C_{p,q,r}\Vert f{\Vert }_q\Vert v^{\ast }{\Vert }_r$$

where f is an arbitrary Hilbert-space valued martingale, g is its transform by a predictable sequence v, and $v^{\ast }$ is the maximal function of v. This is extended to the more general context of differential subordination for continuous-time processes.

An infinite system of point particles placed in ${ℝ}^d$ is studied. The particles are of two types; they perform random walks in the course of which those of distinct type repel each other. The interaction of this kind induces an effective multi-body attraction of the same type particles, which leads to the multiplicity of states of thermal equilibrium in such systems. The pure states of the system are locally finite counting measures on ${ℝ}^d$. The set of such states ${\mathrm{\Gamma }}^2$ is equipped with the vague topology and the corresponding Borel σ-field. For a special class ${\mathcal{P}}_{\exp }$ of probability measures defined on ${\mathrm{\Gamma }}^2$, we prove the existence of a family $\{P_{t,\mathrm{\mu }}:t\ge 0,\mathrm{\mu }\in {\mathcal{P}}_{\exp }\}$ of probability measures defined on the space of càdlàg paths with values in ${\mathrm{\Gamma }}^2$, which is a unique solution of the restricted martingale problem for the mentioned stochastic dynamics. Thereby, the corresponding Markov process is specified.

We consider a directed polymer model in dimension $1+1$, where the random walk is attracted to stable law and the environment is independent in time variable and correlated in space variable. We obtain the scaling limit in the intermediate disorder regime for partition function, and show that the rescaled point-to-point partition function of directed polymers converges in the space of continuous functions to the solution of a stochastic heat equation driven by time-white spatial-colored noise. The scaling limit of the polymer transition probability is also established in the path space. The proof of the tightness is based on the gradient estimates for symmetric random walks in the domain of normal attraction of α-stable law which are established in this paper.

The attracting inverse-square drift provides a prototypical counterexample to solvability of singular SDEs: if the coefficient of the drift is larger than a certain critical value, then no weak solution exists. We prove a positive result on solvability of singular SDEs where this critical value is attained from below (up to strict inequality) for the entire class of form-bounded drifts. This class contains e.g. the inverse-square drift, the critical Ladyzhenskaya-Prodi-Serrin class. The proof is based on a $L^p$ variant of De Giorgi’s method.

In this paper we use duality techniques to study a coupling of the well-known contact process (CP) and the annihilating branching process. As the latter can be seen as a cancellative version of the contact process, we rebrand it as the cancellative contact process (cCP). Our process of interest will consist of two components, the first being a CP and the second being a cCP. We call this process the double contact process (2CP) and prove that it has (depending on the model parameters) at most one invariant law under which ones are present in both processes. In particular, we can choose the model parameters in such a way that CP and cCP are monotonely coupled. In this case also the above mentioned invariant law will have the property that, under it, ones (modeling “infected individuals”) can only be present in the cCP at sites where there are also ones in the CP. Along the way we extend the dualities for Markov processes discovered in our paper “Commutative monoid duality” to processes on infinite state spaces so that they, in particular, can be used for interacting particle systems.

We consider a Fisher–KPP equation with nonlinear selection driven by a Poisson random measure. We prove that the equation admits a unique wave speed $\mathfrak{s}>0$ given by

$$\frac{{\mathfrak{s}}^2}{2}={\int }_{[0,1]}\frac{log(1+y)}{y}\mathfrak{R}(\mathrm{d}y),$$

where $\mathfrak{R}$ is the intensity of the impacts of the driving noise. Our arguments are based on upper and lower bounds via a quenched duality with a coordinated system of branching Brownian motions.

In this paper, we first study the limits of the additive and derivative martingales of one-dimensional branching Brownian motion in a periodic environment. Then we prove the existence of pulsating traveling wave solutions of the corresponding F-KPP equation in the supercritical and critical cases by representing the solutions probabilistically in terms of the limits of the additive and derivative martingales. We also prove that there is no pulsating traveling wave solution in the subcritical case. Our main tools are the spine decomposition and martingale change of measures.

This paper develops the large deviations theory for the point process associated with the Euclidean volume of k-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial point processes in the unit cube. Two different types of large deviation behaviors of such point processes are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of k-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of ${\mathcal{M}}_0$-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. As applications of our main theorems, we discuss large deviations for the number of Poisson or binomial points of degree at most k in a random geometric graph in the dense regime.

We investigate the parabolic Cauchy problem associated with quantum graphs including Lipschitz or polynomial type nonlinearities and additive Gaussian noise perturbed vertex conditions. The vertex conditions are the standard continuity and Kirchhoff assumptions in each vertex. In the case when only Kirchhoff conditions are perturbed, we can prove existence and uniqueness of a mild solution with continuous paths in the standard state space $\mathcal{H}$ of square integrable functions on the edges. We also show that the solution is Markov and Feller. Furthermore, assuming that the vertex values of the normalized eigenfunctions of the self-adjoint operator governing the problem are uniformly bounded, we show that the mild solution has continuous paths in the fractional domain space associated with the Hamiltonian operator, ${\mathcal{H}}_{\mathit{\alpha }}$ for $\mathit{\alpha }<\frac{1}{4}$. This is the case when the Hamiltonian operator is the standard Laplacian perturbed by a potential. We also show that if noise is present in both type of vertex conditions, then the problem admits a mild solution with continuous paths in the fractional domain space ${\mathcal{H}}_{\mathit{\alpha }}$ with $\mathit{\alpha }<-\frac{1}{4}$ only. These regularity results are the quantum graph analogues obtained by da Prato and Zabczyk [9] in case of a single interval and classical boundary Dirichlet or Neumann noise.

We study the anisotropic version of the Hastings-Levitov model AHL$(\mathrm{\nu })$. Previous results have shown that on bounded time-scales the harmonic measure on the boundary of the cluster converges, in the small-particle limit, to the solution of a deterministic ordinary differential equation. We consider the evolution of the harmonic measure on time-scales which grow logarithmically as the particle size converges to zero and show that, over this time-scale, the leading order behaviour of the harmonic measure becomes random. Specifically, we show that there exists a critical logarithmic time window in which the harmonic measure flow, started from the unstable fixed point, moves stochastically from the unstable point towards a stable fixed point, and we show that the full trajectory can be characterised in terms of a single Gaussian random variable.