This paper is concerned with the existence and asymptotic stability in the \,$p$-th moment ofmild solutions of nonlinear impulsive stochastic delay neutral partial functional integro-differential equations. We suppose that the linear part possesses a resolvent operator in the sense given by Grimmer and the nonlinear terms are assumed to be Lipschitz continuous. A fixed point approach is used to achieve the required result. An example is provided to illustrate the theory developed in this work.

In a recent paper, Shah and Zaman proposed the rumor center as an effective rumor source estimator for rumor spreading on random graphs. They proved for a very general random tree model that the detection probability remains positive as the number of nodes to which the rumor has spread tends to infinity. Moreover, they derived explicit asymptotic formulas for the detection probability of random $d$-regular trees and random geometric trees. In this paper, we derive asymptotic formulas for the detection probability of grown simple families of random increasing trees. These families of random trees contain important random tree models as special cases, e.g., binary search trees, recursive trees and plane-oriented recursive trees. Our results show that the detection probability varies from $0$ to $1$ across these families. Moreover, a brief discussion of the rumor center for unordered trees is given as well.

The vacant set of random interlacements at level $u>0$, introduced in [Sznitman 2009], is a percolation model on $\mathbb{Z}^d$, $d \geq 3$ which arises as the set of sites avoided by a Poissonian cloud of doubly infinite trajectories, where $u$ is a parameter controlling the density of the cloud. It was proved in [Sznitman 2009] and [Sidoravicius, Sznitman 2010] that for any $d \geq 3$ there exists a positive and finite threshold $u_*$ such that if $u<u_*$ then the vacant set percolates and if $u>u_*$ then the vacant set does not percolate. We give an elementary proof of these facts. Our method also gives simple upper and lower bounds on the value of $u_*$ for any $d \geq 3$.

Transport-entropy inequalities are considered in terms of Rényi informational divergence.

Let $\rho$ and $\mu$ be two probability measures on $\mathbb{R}$ which are not the Dirac mass at $0$. We denote by $H(\mu|\rho)$ the relative entropy of $\mu$ with respect to $\rho$. We prove that, if $\rho$ is symmetric and $\mu$ has a finite first moment, then \[ H(\mu|\rho)\geq \frac{\displaystyle{\left(\int_{\mathbb{R}}z\,d\mu(z)\right)^2}}{\displaystyle{2\int_{\mathbb{R}}z^2\,d\mu(z)}}\,,\] with equality if and only if $\mu=\rho$. We give an applicaion to the Curie-Weiss model of self-organized criticality.

The classical result by Itô on the existence of strong solutions of stochastic differential equations (SDEs) with Lipschitz coefficients can be extended to the case where the drift is only measurable and bounded. These generalizations are based on techniques presented by Zvonkin and Veretennikov, which rely on the uniform ellipticity of the diffusion coefficient.In this paper we study the case of degenerate ellipticity and give sufficient conditions for the existence of a solution. The conditions on the diffusion coefficient are more general than previous results and we gain fundamental insight into the geometric properties of the discontinuity of the drift on the one hand and the diffusion vector field on the other hand. Besides presenting existence results for the degenerate elliptic situation, we give an example illustrating the difficulties in obtaining more general results than those given.The particular types of SDEs considered arise naturally in the framework of combined optimal control and filtering problems.

Kingman's coalescent is a random tree that arises from classical population genetic models such as the Moran model. Concerning the structure of the tree-top there are two well-known laws of large numbers: (i) The (shortest) distance, denoted by $T_n$, from the tree-top to the level when there are $n$ lines in the tree satisfies $nT_n \xrightarrow{n\to\infty} 2$ almost surely; (ii) At time $T_n$, the population is naturally partitioned in exactly $n$ families where individuals belong to the same family if they have a common ancestor at time $T_n$ in the past. If $F_{i,n}$ denotes the relative size of the $i$th family, then $n(F_{1,n}^2 + \cdots + F_{n,n}^2) \xrightarrow{n\to \infty}2$ almost surely. For both laws of large numbers we prove corresponding large deviations results. For (i), the rate of the large deviations is $n$ and we can give the rate function explicitly. For (ii), the rate is $n$ for downwards deviations and $\sqrt n$ for upwards deviations. In both cases we give exact rate functions.

The signature of a $d$-dimensional Brownian motion is a sequence of iterated Stratonovich integrals along the Brownian paths, an object taking values in the tensor algebra over $\mathbb{R}^{d}$. In this article, we derive the exact rate of convergence for the expected signatures of piecewise linear approximations to Brownian motion. The computation is based on the identification of the set of words whose coefficients are of the leading order, and the convergence is concentrated on this subset of words. Moreover, under the choice of $l^{1}$ tensor norm, we give the explicit value of the leading term constant.

In this note we give a direct proof of the Gaussian integrability of distance function as $\mu e^{\delta d^2(x,x_0)} < \infty$ for some $\delta>0$ provided the Lyapunov condition holds for symmetric diffusion operators, which answers a question by Cattiaux, Guillin, and Wu. The similar argument still works for diffusions processes with unbounded diffusion coefficients and for jump processes such as birth-death chains. An analogous discussion is also made under the Gozlan's condition.

Let $\big(M_k, Q_k\big)_{k\in\mathbb{N}}$ be independent copies of an $\mathbb{R}^2$-valued random vector. It is known that if $Y_n:=Q_1+M_1Q_2+\ldots+M_1\cdot\ldots\cdot M_{n-1}Q_n$ converges a.s. to a random variable $Y$, then the law of $Y$ satisfies the stochastic fixed-point equation $Y \overset{d}{=} Q_1+M_1Y$, where $(Q_1, M_1)$ is independent of $Y$. In the present paper we consider the situation when $|Y_n|$ diverges to $\infty$ in probability because $|Q_1|$ takes large values with high probability, whereas the multiplicative random walk with steps $M_k$'s tends to zero a.s. Under a regular variation assumption we show that $\log |Y_n|$, properly scaled and normalized, converge weakly in the Skorokhod space equipped with the $J_1$-topology to an extremal process. A similar result also holds for the corresponding Markov chains. Proofs rely upon a deterministic result which establishes the $J_1$-convergence of certain sums to a maximal function and subsequent use of the Skorokhod representation theorem.

Let $Z^{H}$ be the Rosenblatt process with the representation$$Z^H_t=\int_0^t\int_0^tL^H(t,s,r)dB_{s}dB_{r},$$where $B$ is a standard Brownian motion, $\frac12<H<1$ and $L^H$ is a given kernel. By reviewing the kernel $L^H$ we construct its approximation of multiple Wiener integrals of the form $$\int_{0}^{t}\int_{0}^{t}\left\{k_{1}(sr)^{-\frac{1}{2}H} +k_{2}(s\vee r)^{\frac{1}{2}H}(s\wedge r)^{-\frac{1}{2}H}|s-r|^{H-1}\right\}dB_{s}dB_{r}, \;\;k_1,k_2\geq 0.$$We find an optimal approximation of $Z^{H}$ via calculating accurately the values of $k_1,k_2$.

We construct a finitely generated group $G$ without the Liouville property such that the return probability of a random walk satisfies $p_{2n}(e,e) \gtrsim e^{-n^{1/2+ o(1)}}$. This shows that the constant $1/2$ in a recent theorem by Saloff-Coste and Zheng, saying that return probability exponent less than $1/2$ implies the Liouville property, cannot be improved. Our construction is based on permutational wreath products over tree-like Schreier graphs and the analysis of large deviations of inverted orbits on such graphs.

We establish existence and uniqueness of the solution to the cavity equation for the random assignment problem in pseudo-dimension $d>1$, as conjectured by Aldous and Bandyopadhyay (Annals of Applied Probability, 2005) and Wästlund (Annals of Mathematics, 2012). This fills the last remaining gap in the proof of the original Mézard-Parisi prediction for this problem (Journal de Physique Lettres, 1985).

For a class of stochastic partial differential equations studied by Conus and Dalang, we prove the existence of density of the probability law of the solution at a given point $(t,x)$, and that the density belongs to some Besov space. The proof relies on a method developed by Debussche and Romito.The result can be applied to the solution of the stochastic wave equation with multiplicative noise, Lipschitz coefficients and any spatial dimension $d\ge 1$, and also to the heat equation.This provides an extension of earlier results.

In recent papers it has been demonstrated that sampling a Gibbs distribution from an appropriate time-irreversible Langevin process is,from several points of view, advantageous when compared to sampling from a time-reversible one. Adding an appropriate irreversible drift to the overdamped Langevin equation results in a larger large deviations rate function for the empirical measure of the process, a smaller variance for the long time average of observables of the process, as well as a larger spectral gap. In this work, we concentrate on irreversible Langevin samplers with a drift of increasing intensity. The asymptotic variance is monotonically decreasing with respect to the growth of the drift and we characterize its limiting behavior. For a Gibbs measure whose potential has one or more critical points, adding a large irreversible drift results in a decomposition of the process in a slow and fast component with fast motion along the level sets of the potential and slow motion in the orthogonal direction. This result helps understanding the variance reduction, which can be explained at the process level by the induced fast motion of the process along the level sets of the potential. Correspondingly the limit of the asymptotic variance is the asymptotic variance of the limiting slow motion which is a diffusion process on a graph.

We review the construction of flows associated to Tanaka’s SDE. Using the skew Brownian motion, we give an easy proof of the classification of these flows by means of probability measures on [0; 1]. Our arguments also simplify some already available proofs.

This paper concerns the number $Z_n$ of sites visited up to time $n$ by a random walk $S_n$ having zero mean and moving on the two dimensional square lattice ${\bf Z}^2$. Asymptotic evaluation of the conditional expectation of $Z_n$ for large $n$ given that $S_n=x$ is carried out under some exponential moment condition. It gives an explicit form of the leading term valid uniformly in $(x, n)$, $|x|< cn$ for each $c>0$.

Processes sharing the same bridges are said to belong to the same reciprocal class. In this article we analyze reciprocal classes of Markov counting processes by identifying their reciprocal invariant and we characterize them as the set of counting processes satisfying some duality formula.

In this note, we construct an example of a sequence of n-fold product chains which does display cutoff neither for the total-variation distance nor for the separation distance. In addition we show that this type of product chains necessarily displays pre-cutoff.

A famous result of Bayer and Diaconis is that the Gilbert-Shannon-Reeds (GSR) model for the riffle shuffle of n cards mixes in $\frac{3}{2}\log_2(n)$ steps and that for $52$ cards about $7$ shuffles suffices to mix the deck. In this paper,we study variants of the GSR shuffle that have been proposed to model more realistically how people actually shuffle a deck of cards. The clumpy riffle shuffle and dealer riffle shuffle differ from the GSR model in that when a card is dropped from one hand, the conditional probability that the next card is dropped from the same hand is higher/lower than for the GSR model. Until now, no nontrivial rigorous results have been known for the clumpy shuffle or dealer shuffle. In this paper we show that the mixing time is $O(\log^4(n))$.

Let $\{G_n: n\geq 1\}$ be a sequence of simple graphs. Suppose $G_n$ has $m_n$ edges and each vertex of $G_n$ is colored independently and uniformly at random with $c_n$ colors. Recently, Bhattacharya, Diaconis and Mukherjee (2013) proved universal limit theorems for the number of monochromatic edges in $G_n$. Their proof was by the method of moments, and therefore was not able to produce rates of convergence. By a non-trivial application of Stein's method, we prove that there exists a universal error bound for their central limit theorem. The error bound depends only on $m_n$ and $c_n$, regardless of the graph structure.

We obtain the leading orders of the maximum and the minimum of local times for the simple random walk on the two-dimensional torus at time proportional to the cover time. We also estimate the number of points with large (or small) values of the local times. These are analogues of estimates on the two-dimensional Gaussian free fields by Bolthausen, Deuschel, and Giacomin [Ann. Probab., 29 (2001)] and Daviaud [Ann. Probab., 34 (2006)], but we have different exponents from the case of the Gaussian free field.

We prove an optimal diffusive decay of the environment viewed by the particle in random walk among random independent conductances, with, as a main assumption, finite second moment of the conductance. Our proof, using the analytic approach of Gloria, Neukamm and Otto, is very short and elementary.

In this article we derive formulas for the probability $\mathbb{P}(\sup_{t\leq T} X(t)>u)$, $T>0$ and $\mathbb{P}(\sup_{t<\infty} X(t)>u)$ where $X$ is a spectrally positive Lévy process with infinite variation. The formulas are generalizations of the well-known Takács formulas for stochastic processes with non-negative and interchangeable increments. Moreover, we find the joint distribution of $\inf_{t\leq T} Y(t)$ and $Y(T)$ where $Y$ is a spectrally negative Lévy process.

We consider the Bouchaud trap model on the integers in the case that the trap distribution has a slowly varying tail at infinity. We prove that the model eventually localises on exactly two sites with overwhelming probability. This is a stronger form of localisation than has previously been established in the literature for the Bouchaud trap model on the integers in the case of regularly varying traps. Underlying this result is the fact that the sum of a sequence of i.i.d. random variables with a slowly varying tail is asymptotically dominated by the maximal term.<!–EndFragment–>

In this note, we study one-dimensional reflected backward stochastic differential equations (RBSDEs) driven by Countable Brownian Motions with one continuous barrier and continuous generators. Via a comparison theorem, we provide the existence of a minimal and a maximal solution to this kind of equations.

We prove uniform estimates for the expected value of averages of order statistics of matrices interms of their largest entries. As an application, we obtain similar probabilistic estimates for $\ell_p$ norms via real interpolation.

We consider an $\epsilon K$ transversal perturbing vector field in a foliated Brownian motion defined in a foliated tubular neighbourhood of an embedded compact submanifold in $\mathbb{R}^3$. We study the effective behaviour of the system under this $\epsilon$ perturbation. If the perturbing vector field $K$ is proportional to the Gaussian curvature at the corresponding leaf, we have that the transversal component, after rescaling the time by $t/\epsilon$, approaches a linear increasing behaviour proportional to the Euler characteristic of $M$, as $\epsilon$ goes to zero. An estimate of the rate of convergence is presented.

In his 2011 work, Maas has shown that the law of anytime-reversible continuous-time Markov chain with finite state space evolves like a gradient flow of the relative entropy with respect to its stationary distribution. In this work we show the converse to the above by showing that if the relative law of a Markov chain with finite state space evolves like a gradient flow of the relative entropy functional, it must be time-reversible. When we allow general functionals in place of the relative entropy, we show that the law of a Markov chain evolves as gradient flow if and only if the generator of the Markov chain is real diagonalisable. Finally, we discuss what aspects of the functional are uniquely determined by the Markov chain.

In this paper we study the problems of invariant and ergodic expectations under $G$-expectation framework. In particular, the stochastic differential equations driven by $G$-Brownian motion have the unique invariant and ergodic expectations. Moreover, the invariant and ergodic expectations of $G$-SDEs are also sublinear expectations. However, the invariant expectations may not coincide with the ergodic expectations, which is different from the classical case.

In a recent paper, we constructed a stationary $1-$dependent $4-$coloring of the integers that is invariant under permutations of the colors. This was the first stationary $k-$dependent $q-$coloring for any $k$ and $q$. When the analogous construction is carried out for $q>4$ colors, the resulting process is not $k-$dependent for any $k$. We construct here a process that is symmetric in the colors and $1-$dependent for every $q\geq 4$. The construction uses a recursion involving Chebyshev polynomials evaluated at $\sqrt{q}/2$.

The purpose of this note is to provide a coupling of weak limits in distribution of sequence of (normalized) multiplicative cascade measures under strong disorder in terms of the extremes of an associated branching random walk, assuming i.i.d positive, non-lattice bond weights and a second moment condition. The solution is expressed as an almost sure coupling of random probability measures in the disorder parameter $\beta > \beta_c$ through the introduction of a tree-indexed random field of derivative martingales, and the Brunet-Derrida-Madaule decorated Poisson point process. A number of corollaries are provided to illustrate the utility of this construction.

We study the Incipient Infinite Cluster (IIC) of high-dimensional bond percolation on $\mathbb{Z}^d$. We prove that the mass dimension of IIC almost surely equals $4$ and the volume growth exponent of IIC almost surely equals $2$.

This note studies the martingale property of a nonnegative, continuous local martingale $Z$, given as a nonanticipative functional of a solution to a stochastic differential equation. The condition states that $Z$ is a (uniformly integrable) martingale if and only if an integral test of a related functional holds.

We derive Edgeworth-type expansions for Skorohod and Itô integrals with respect to Brownian motion, based on cumulant operators defined by the Malliavin calculus. As a consequence we obtain Stein approximation bounds for stochastic integrals, which apply to SDE solutions and to multiple stochastic integrals.

We use the Malliavin calculus to prove a new abstract concentration inequality result for zero mean, Malliavin differentiable random variables which admit densities. We demonstrate the applicability of the result by deriving two new concrete concentration inequalities, one relating to an integral functional of a fractional Brownian motion process, and the other relating to the centered maximum of a finite sum of Normal random variables. These concentration inequalities are, to the best of our knowledge, largely unattainable via existing methods other than those which are the subject of this paper.

This work is concerned with the analysis of a stochastic approximation algorithm for the simulation of quasi-stationary distributions on finite state spaces. This is a generalization of a methodintroduced by Aldous, Flannery and Palacios. It is shown that the asymptotic behavior of the empirical occupation measure of this process is precisely related to the asymptotic behavior of somedeterministic dynamical system induced by a vector field on the unit simplex. This approach provides new proof of convergence as well as precise asymptotic rates for this type of algorithm. Inthe last part, our convergence results are compared with those of a particle system algorithm (adiscrete-time version of the Fleming-Viot algorithm).

Let $M$ be a manifold equipped (locally) with a pair of complementary foliations. In Catuogno, da Silva and Ruffino (Stoch. Dyn. 2013), it is shown that, up to a stopping time $\tau$, a stochastic flow of local diffeomorphisms $\varphi_t$ in $M$ can be decomposed in diffeomorphisms that preserves this foliations. In this article we present techniques which allows us to extend the time of this decomposition. For this extension, we use two techniques: In the first one, assuming that the vector fields of the system commute with each other, we apply Marcus equation to jump nondecomposable diffeomorphisms. The second approach deals with the general case: we introduce a "stop and go" technique that allows us to construct a process that follows the original flow in the "good zones" for the decomposition, and remains paused in "bad zones". Among other applications, our results open the possibility of studying the asymptotic behaviour of each component.

Motivated by evaluating the limiting distribution of randomly biased random walks on trees, we compute the exact value of a negative moment of the maximal drawdown of the standard Brownian meander.

The purpose of this note is to prove that the flatness of an invariant manifold for a semilinear stochastic partial differential equation driven by Lévy processes is at least equal to the number of driving sources with small jumps. We illustrate our findings by means of an example.

We give a “direct” coupling proof of strict monotonicity of the speed for 1-dimensional multi-excited random walks with positive speed. This reproves (and extends) a recent result of Peterson without using branching processes.

This note highlights a special class of mean field games in which the coefficients satisfy a convolution-type structural condition. A mean field game of this type with common noise is related to a certain mean field game without common noise by a simple transformation, which permits a tractable construction of a solution of the problem with common noise from a solution of the problem without. This yields new existence results for common noise problems, notably allowing controlled volatility, as well as new uniqueness results for problems without common noise.

We propose a simple construction of the solution to the continuum parabolic Anderson model on $\mathbf{R}^2$ which does not rely on any elaborate arguments and makes extensive use of the linearity of the equation. A logarithmic renormalisation is required to counterbalance the divergent product appearing in the equation. Furthermore, we use time-dependent weights in our spaces of distributions in order to construct the solution on the unbounded space $\mathbf{R}^2$.

We obtain non-asymptotic lower bounds on the least singular value of ${\mathbf X}_{pn}^\top/\sqrt{n}$, where ${\mathbf X}_{pn}$ is a $p\times n$ random matrix whose columns are independent copies of an isotropic random vector $X_p$ in $ {\mathbb R}^p$. We assume that there exist $M>0$ and $\alpha\in\leqslant M/t^{2+\alpha}$ for all $t>0$ and any unit vector $v\in{\mathbb R}^p$. These bounds depend on $y=p/n,$ $\alpha$, $M$ and are asymptotically optimal up to a constant factor.

The Green's function for the chordal Schramm-Loewner evolution $SLE_\kappa$for $0 < \kappa < 8$, gives the normalizedprobability of getting near points. We give up-to-constant bounds for the two-point Green's function.

This erratum corrects the statement of the main result in <a href="http://dx.doi.org/10.1214/ECP.v17-2133">ECP volume 17 paper 40</a>, which establishes that the mulit-time particle densitites for certain intercating particle systems on the real line are extended Pfaffian point processes.

We study the extremes of a class of Gaussian fields with in-built hierarchical structure. The number of scales in the underlying trees depends on a parameter $\alpha \in \left[0,1\right]$: choosing $\alpha=0$ yields the random energy model by Derrida (REM), whereas $\alpha=1$ corresponds to the branching random walk (BRW). When the parameter $\alpha$ increases, the level of the maximum of the field decreases smoothly from the REM- to the BRW-value. However, as long as $\alpha<1$ strictly, the limiting extremal process is always Poissonian.

Consider a random geometric graph defined on $n$ vertices uniformly distributedin the $d$-dimensional unit torus. Two vertices are connected if their distance is less than a "visibility radius" $r_n$. We consider Bluetooth networks that are locally sparsified random geometric graphs. Each vertex selects $c$ of its neighbors in the random geometric graph at random and connects only to the selected points. We show that if the visibility radius is at least of the order of $n^{-(1-\delta)/d}$for some $\delta > 0$, then a constant value of $c$ is sufficient forthe graph to be connected, with high probability. It suffices to take$c \ge \sqrt{(1+\epsilon)/\delta} + K$ for any positive $\epsilon$ where $K$is a constant depending on $d$ only. On the other hand, with $c\le \sqrt{(1-\epsilon)/\delta}$,the graph is disconnected, with high probability.

A rotor-router walk on a graph is a deterministic process, in which each vertex is endowed with a rotor that points to one of the neighbors. A particle located at some vertex first rotates the rotor in a prescribed order, and then it is routed to the neighbor the rotor is now pointing at. In the current work we make a step toward in understanding the behavior of rotor router walks on random trees. More precisely, we consider random i.i.d. initial configurations of rotors on Galton-Watson trees T, i.e. on a family tree arising from a Galton-Watson process, and give a classification in recurrence and transience for rotor-router walks on these trees.

Equip each point $x$ of a homogeneous Poisson point process $\mathcal{P}$ on $\mathbb{R}$ with $D_x$ edge stubs, where the $D_x$ are i.i.d. positive integer-valued random variables with distribution given by $\mu$. Following the stable multi-matching scheme introduced by Deijfen, Häggström and Holroyd [1], we pair off edge stubs in a series of rounds to form the edge set of a graph $G$ on the vertex set $\mathcal{P}$. In this note, we answer questions of Deijfen, Holroyd and Peres [2] and Deijfen, Häggström and Holroyd [1] on percolation (the existence of an infinite connected component) in $G$. We prove that percolation may occur a.s. even if $\mu$ has support over odd integers. Furthermore, we show that for any $\varepsilon \gt 0$, there exists a distribution $\mu$ such that $\mu(\{1\})\gt 1 -\varepsilon$, but percolation still occurs a.s.

We consider a renewal process with regularly varying stationary and weakly dependent steps, and prove that the steps made before a given time $t$, satisfy an interesting invariance principle. Namely, together with the age of the renewal process at time $t$, they converge after scaling to the Pitman–Yor distribution. We further discuss how our results extend the classical Dynkin–Lamperti theorem.

We give a positive answer to a conjecture on the uniqueness of harmonic functions in the quarter plane stated by K. Raschel. More precisely we prove the existence and uniqueness of a positive discrete harmonic function for a random walk satisfying finite range, centering and ellipticity conditions, killed at the boundary of an orthant in Zd. Our methodsallow on the other hand to generalize from the quarter plane to orthants in higher dimensions and to treat the spatially inhomogeneous walks.

Let $(\mathcal{X},\mathcal{E})$, $(\mathcal{Y},\mathcal{F})$ and $(\mathcal{Z},\mathcal{G})$ be measurable spaces. Suppose we are given two probability measures $\gamma$ and $\tau$, with $\gamma$ defined on $(\mathcal{X}\times\mathcal{Y},\mathcal{E}\otimes\mathcal{F}$ and $\tau$ on $(\mathcal{X}\times\mathcal{Z},\mathcal{E}\otimes\mathcal{G})$. Conditions for the existence of random variables $X,Y,Z$, defined on the same probability space $(\Omega,\mathcal{A},P)$ and satisfying

$$(X,Y)\sim\gamma\,\text{ and }\,(X,Z)\sim\tau,$$

are given. The probability $P$ may be finitely additive or $\sigma$-additive. As an application, a version of Skorohod representation theorem is proved. Such a version does not require separability of the limit probability law, and answers (in a finitely additive setting) a question raised in preceding works.

The classical skew-product decomposition of planar Brownian motionrepresents the process in polar coordinates as an autonomously Markovian radial part and an angular part that is an independent Brownian motion on the unit circle time-changed according to the radial part. Theorem 4 of L09 gives a broad generalization of this fact to a setting where there is a diffusion on a manifold $X$ with a distribution that is equivariant under the smooth action ofa Lie group $K$. Under appropriate conditions, there is a decomposition into an autonomously Markovian "radial" part that lives on the space of orbits of $K$ and an "angular" part that is an independent Brownian motion on the homogeneous space $K/M$, where $M$ is the isotropy subgroup of a point of $x$, that is time-changed with a time-change that is adapted to the filtration of the radial part. We present two apparent counterexamples to Theorem 4 of L09. In the first counterexample the angular part is not a time-change of any Brownian motionon $K/M$, whereas in the second counterexample the angular part is the time-change of a Brownian motion on $K/M$ but this Brownian motion is not independent of the radial part. In both of these examples $K/M$ has dimension $1$. The statement and proof of Theorem 4 in L09 remain valid when $K/M$ has dimension greater than $1$. Our examples raise the question of what conditions lead to the usual sort of skew-product decomposition when $K/M$ has dimension $1$ and what conditions lead to there being no decomposition at all or one in which the angular part is a time-changed Brownian motion but this Brownian motion is not independent of the radial part.

We study the weighted version of the interchange process where a permutation receives weight $\theta^{\#\mathrm{cycles}}$. For $\theta=2$ this is Tóth's representation of the quantum Heisenberg ferromagnet on the complete graph. We prove, for $\theta>1$, that large cycles appear at "low temperature".

We present a technique for approximating generic normalization constants subject to constraints. The method is then applied to derive the exact asymptotics for the conditional normalization constant of constrained exponential random graphs.

Let $u^{\beta}(x,y)$ be the $\beta$-potential density of a transient Lévy process $\overline Y$ and $X_{\alpha}=\{X_{\alpha,x}, x\in R \}$ be the $\alpha$-permanental process determined by $u^{\beta}(x,y)$. Let $\overline L=\{\overline L_{t}^{x}, (t,x),\in R_{+}\times R \}$ be the local time process of $\overline Y$ and let $G=\{G_{x}, x\in R\}$ be the stationary mean zero Gaussian process with covariance $u^{\beta}(x,y)+ u^{\beta}(y,x)$. Then the processes $X_{\alpha}$, $\overline L$ and $G$ are either all continuous almost surely or all unbounded almost surely. Therefore, the well known necessary and sufficient condition for the continuity of $\overline L$ and $G$ is also a necessary and sufficient condition for the continuity of $X_{\alpha}$.

Let $X$ denote a spectrally positive stable process of index $\alpha \in $.

We present an elementary treatment of the Optional Decomposition Theorem for continuous semimartingales and general filtrations. This treatment does not assume the existence of equivalent local martingale measure(s), only that of strictly positive local martingale deflator(s).

We consider Brownian diffusions on the real line, interacting through rank-dependent drifts. It is known that in the mean-field limit, such particle systems behave like independent copies of a so-called nonlinear diffusion process. We prove a similar asymptotic behaviour at the level of stationary distributions. Our proof is based on explicit expressions for the Laplace transforms of the stationary distributions of both the particle system and the nonlinear diffusion process, and yields convergence of the marginal distributions in Wasserstein distances of all orders. We highlight the consequences of this result on the study of rank-based models of equity markets, such as the Atlas model.

In this note, we provide a new characterization of Aldous' Brownian continuum random tree as the unique fixed point of a certain natural operation on continuum trees (which gives rise to a recursive distributional equation). We also show that this fixed point is attractive.

We calculate the probability $p_c$ that the maximum of a reflected Brownian motion $U$ is achieved on a complete excursion, i.e. $p_c:=P\big(\overline{U}(t)=U^*(t)\big)$ where $\overline{U}(t)$ (respectively $U^*(t)$) is the maximum of the process $U$ over the time interval $[0,t]$ (respectively $\big[0,g(t)\big]$ where $g(t)$ is the last zero of $U$ before $t$).

We prove that in any recurrent reversible random rooted graph, two independent simple random walks started at the same vertex collide infinitely often almost surely. This applies to the Uniform Infinite Planar Triangulation and Quadrangulation and to the Incipient Infinite Cluster in $\mathbb{Z}^2$.

We prove that an irreducible aperiodic Markov chain is geometrically ergodic if and only if any separately bounded functional of the stationary chain satisfies an appropriate subgaussian deviation inequality from its mean.

By choosing some special (random) initial data, we prove that with probability $1$, the stochastic shadow Gierer-Meinhardt system blows up pointwisely in finite time.We also give a (random) upper bound for the blowup time and some estimates about this bound. By increasing the amplitude of the initial data, we can get the blowup in any short time with positive probability.

We prove the central limit theorem (CLT) for a sequence of independent zero-mean random variables $\xi_j$, perturbed by predictable multiplicative factors $\lambda_j$ with values in intervals $[\underline\lambda_j,\overline\lambda_j]$. It is assumed that the sequences $\underline\lambda_j$, $\overline\lambda_j$ are bounded and satisfy some stabilization condition. Under the classical Lindeberg condition we show that the CLT limit, corresponding to a "worst'' sequence $\lambda_j$, is described by the solution $v$ of one-dimensional $G$-heat equation. The main part of the proof follows Peng's approach to the CLT under sublinear expectations, and utilizes Hölder regularity properties of $v$. Under the lack of such properties, we use the technique of half-relaxed limits from the theory of viscosity solutions.

A transient stochastic process is considered strongly transient if conditioned on returning to the starting location, the expected time it takes to return the the starting location is finite. We characterize strong transience for a one-dimensional random walk in a random environment. We show that under the quenched measure transience is equivalent to strong transience, while under the averaged measure strong transience is equivalent to ballisticity (transience with non zero limiting speed).

An explicit formula for the mean spectral measure of a random Jacobi matrix is derived. The matrix may be regarded as the limit of Gaussian beta ensemble (G$\beta$E) matrices as the matrix size $N$ tends to infinity with the constraint that $N \beta $ is a constant.

Given a homogenous Poisson point process in the plane, we prove that it is possible to partition the plane into bounded connected cells of equal volume, in a translation-invariant way, with each point of the process contained in exactly one cell. Moreover, the diameter $D$ of the cell containing the origin satisfies the essentially optimal tail bound $\mathbb{P}(D>r)<c/r$. We give two variants of the construction. The first has the curious property that any two cells are at positive distance from each other. In the second, any bounded region of the plane intersects only finitely many cells almost surely.

In this paper two examples of two independent centered Gaussian processes are given such that at least one of them is not a semimartingale but their sum is a martingale.

We study the following backward stochastic differential equation on finite time horizon driven by an integer-valued random measure $\mu$ on $\mathbb R_+\times E$, where $E$ is a Lusin space, with compensator $\nu(dt,dx)=dA_t\,\phi_t(dx)$:\[Y_t = \xi + \int_{(t,T]} f(s,Y_{s-},Z_s(\cdot))\, d A_s - \int_{(t,T]} \int_E Z_s(x) \, (\mu-\nu)(ds,dx),\qquad 0\leq t\leq T.\]The generator $f$ satisfies, as usual, a uniform Lipschitz condition with respect to its last two arguments. In the literature, the existence and uniqueness for the above equation in the present general setting has only been established when $A$ is continuous or deterministic. The general case, i.e. $A$ is a right-continuous nondecreasing predictable process, is addressed in this paper.

We prove that quadratic forms in isotropic random vectors $X$ in $\mathbb{R}^n$, possessing the convex concentration property with constant $K$, satisfy the Hanson-Wright inequality with constant $CK$, where $C$ is an absolute constant, thus eliminating the logarithmic (in the dimension) factors in a recent estimate by Vu and Wang. We also show that the concentration inequality for all Lipschitz functions implies a uniform version of the Hanson-Wright inequality for suprema of quadratic forms (in the spirit of the inequalities by Borell, Arcones-Giné and Ledoux-Talagrand). Previous results of this type relied on stronger isoperimetric properties of $X$ and in some cases provided an upper bound on the deviations rather than a concentration inequality.In the last part of the paper we show that the uniform version of the Hanson-Wright inequality for Gaussian vectors can be used to recover a recent concentration inequality for empirical estimators of the covariance operator of $B$-valued Gaussian variables due to Koltchinskii and Lounici.

Consider a sequence of Poisson point processes of non-trivial loops with certain intensity measures $(\mu^{(n)})_n$, where each $\mu^{(n)}$ is explicitly determined by transition probabilities $p^{(n)}$ of a random walk on a finite state space $V^{(n)}$ together with an additional killing parameter $c^{(n)}=e^{-a\cdot\sharp V^{(n)}+o(\sharp V^{(n)})}$. We are interested in asymptotic behavior of typical loops. Under general assumptions, we study the asymptotics of the length of a loop sampled from the normalized intensity measure $\bar{\mu}^{(n)}$ as $n\rightarrow\infty$. A typical loop is small for $a=0$ and extremely large for $a=\infty$. For $a=}$. We verify our general assumptions for random walk loop soups on discrete tori and truncated regular trees. Finally, we consider random walk loop soups on complete graphs. Here, our general assumptions are violated. In this case, we observe different asymptotic behavior of the length of a typical loop.

We consider both the infinite-volume discrete Gaussian Free Field (DGFF) and the DGFF with zero boundary conditions outside a finite box in dimension larger or equal to 3. We show that the associated extremal process converges to a Poisson point process. The result follows from an application of the Stein-Chen method from Arratia et al. (1989).

We consider a sequence of processes X_n(t)$ defined on half-line $0\leq t<\infty$. We give sufficient conditions for Large Deviation Principle (LDP) to hold in the space of continuous functions with metric $\rho(f,g)=\sup_{t\geq0} |f(t)−g(t)|/(1+t^{1+\kappa})$, $\kappa\geq0$. LDP is established for Random Walks, Diffusions, and CEV model of ruin, all defined on the half-line. LDP in this space is "more precise" than that with the usual metric of uniform convergence on compacts.

We consider a branching Brownian motion evolving in $\mathbb{R}^d$. We prove that the asymptotic behaviour of the maximal displacement is given by a first ballistic order, plus a logarithmic correction that increases with the dimension $d$. The proof is based on simple geometrical evidence. It leads to the interesting following side result: with high probability, for any $d \geq 2$, individuals on the frontier of the process are close parents if and only if they are geographically close.

In this paper we give new exponential inequalities for weighted sums of real-valued independent random variables bounded on the right. Our results are extensions of the results of Bennett (1962) to weighted sums.

We identify the fluctuations of the partition function for a class of random energy models, where the energies are given by the positions of the particles of the complex-valued branching Brownian motion (BBM). Specifically, we provide the weak limit theorems for the partition function in the so-called "glassy phase'' – the regime of parameters, where the behaviour of the partition function is governed by the extrema of BBM. We allow for arbitrary correlations between the real and imaginary parts of the energies. This extends the recent result of Madaule, Rhodes and Vargas (2013), where the uncorrelated case was treated. Inparticular, our result covers the case of the real-valued BBM energy model at complex temperatures.

In this note, we characterize the limiting functions in mod-Gaussian convergence; our approach sheds a new light on the nature of mod-Gaussian convergence as well. Our results in fact more generally apply to mod-* convergence, where * stands for any family of probability distributions whose Fourier transforms do not vanish. We moreover provide new examples, including two new examples of (restricted) mod-Cauchy convergence from arithmetics related to Dedekind sums and the linking number of modular geodesics.

The paper studies first exit times from domains for diffusion processes and their dependence on variations of the boundary. We establish some robustness of the first exit times with respect to the fluctuations of the boundary. More precisely, we present an estimate of the $L_1$-distance between exit times from two regions via expectations of exit times.

We give asymptotics for the left and right tails of the limiting Quicksort distribution. The results agree with, but are less precise than, earlier non-rigorous results by Knessl and Spankowski.

We discuss the properties of three upsilon transforms, which are related to the class of $p$-tempered $\alpha$-stable ($TS^p_\alpha$) distributions. In particular, we characterize their domains and show how they can be represented as compositions of each other. Further, we show that if $-\infty<\beta<\alpha<2$ and $0<q<p<\infty$ then they can be used to transform the Lévy measures of $TS^p_\beta$ distributions into those of $TS^q_\alpha$.

We obtain the Erdős–Rényi type law of large numbers for "nonconventional" sums of the form $S_n=\sum_{m=1}^nF(X_m,X_{2m},...,X_{\ell m})$ where $X_1,X_2,...$ is a sequence of i.i.d. random variables and $F$ is a bounded Borel function. The proof relies on nonconventional large deviations obtained in a previous work.

We compute the tail asymptotics of the product of a beta random variable and a generalized gamma random variable which are independent and have general parameters. A special case of these asymptotics were proved and used in a recent work of Bubeck, Mossel, and Racz in order to determine the tail asymptotics of the maximum degree of the preferential attachment tree. The proof presented here is simpler and highlights why these asymptotics hold.

Let $(Y_n)_{n\ge0}$ be a Mandelbrot's martingale defined as sums of products of random weights indexed by nodes of a Galton-Watson tree, and let $Y$ be its limit. We show a necessary and sufficient condition for the existence of weighted moments of $Y$ of the forms $\mathbb{E}Y^{\alpha}\ell(Y)$, where $\alpha>1$ and $\ell$ is a positive function slowly varying at $\infty$. We also show a sufficient condition in the case of $\alpha=1$. Our results complete those of Alsmeyer and Kuhlbusch (2010) for weighted branching processes by removing their extra conditions on $\ell$.

We consider a self-avoiding walk on the dual $\mathbb{Z}^2$ lattice.This walk can traverse the same square twice but cannot cross the same edge more than once. The weight of each square visited by the walk depends on the way the walk passes through it and the weight of the whole walk is calculated as a product of these weights. We consider a family of critical weights parametrized by angle $\theta\in[\frac{\pi}{3},\frac{2\pi}{3}]$. For $\theta=\frac{\pi}{3}$, this can be mapped to the self-avoiding walk on the honeycomb lattice. The connective constant in this case was proved to be equal to $\sqrt{2+\sqrt{2}}$ by Duminil-Copin and Smirnov. We generalize their result.

We consider both the Bolthausen-Sznitman and the Kingman coalescent restricted to the partitions of <span id="MathJax-Element-1-Frame" class="MathJax"><span id="MathJax-Span-1" class="math"><span id="MathJax-Span-2" class=""><span id="MathJax-Span-3" class="mo">{<span id="MathJax-Span-4" class="mn">1<span id="MathJax-Span-5" class="mo">,<span id="MathJax-Span-6" class="mo">…<span id="MathJax-Span-7" class="mo">,<span id="MathJax-Span-8" class="mi">n<span id="MathJax-Span-9" class="mo">}<span id="MathJax-Span-10" class="mo">. Spectral decompositions of the corresponding generators are derived. As an application we obtain a formula for the Green's functions and a short derivation of the well-known formula for the transition probabilities of the Bolthausen-Sznitman coalescent.

By computations on generating functions, Szekeres proved in 1983 that the law of the diameter of a uniformly distributed rooted labelled tree with $n$ vertices, rescaled by a factor $n^{-1/2}$ , converges to a distribution whose density is explicit. Aldous observed in 1991 that this limiting distribution is the law of the diameter of the Brownian tree. In our article, we provide a computation of this law which is directly based on the normalized Brownian excursion. Moreover, we provide an explicit formula for the joint law of the height and diameter of the Brownian tree, which is a new result.

We study the large deviation rate functional for the empirical distribution of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer that this functional is asymptotically equivalent (in the sense of $\Gamma$-convergence) to the Jordan-Kinderlehrer-Otto functional arising in the Wasserstein gradient flow structure of the Fokker-Planck equation. In higher dimensions, part of this statement (the lower bound) has been recently proved by Duong, Laschos and Renger, but the upper bound remained open, since the proof of Duong et al relies on regularity properties of optimal transport maps that are restricted to one dimension. In this note we present a new proof of the upper bound, thereby generalising the result of Adams et al to arbitrary dimensions.

In this note we show that, in the context of stochastic control systems, the uniform existence of a limit of Cesàro averages implies the existence of uniform limits for averages with respect to a wide class of measures dominated by the Lebesgue measure and satisfying some asymptotic condition. It gives a partial answer to the problem mentioned in Renault (2014) and it provides an alternative method for the approach in Li, Quincampoix, Renault (2015) (in the deterministic control setting). Finally, we mention that the arguments rely essentially on integration-by-parts and is applicable to general deterministic or stochastic control problems.

We prove that the Markov operator corresponding to the two-variable, non-reversible Gibbs sampler has spectrum which is entirely real and non-negative, thus providing a first step towards the spectral analysis of MCMC algorithms in the non-reversible case. We also provide an extension to Metropolis-Hastings components, and connect the spectrum of an algorithm to the spectrum of its marginal chain.

We prove the continuity of the shape governing the asymptotic growth of the supercritical contact process in $\mathbb{Z}^d$, with respect to the infection parameter. The proof is valid in any dimension $d \ge 1$.

<span style="color: #000000;">Recently, by introducing the notion of cumulatively merged partition, M<span style="color: #800000;">énard<span style="color: #000000;"> and Singh provide <span style="color: #000000;">a sufficient condition on graphs ensuring that the critical value of the contact process is positive. In this note, we show that the one-dimensional long range percolation with high exponent satisfies their condition and thus the contact process exhibits a non-trivial phase transition.

We establish an uniform factorial decay estimate for the Taylor approximation of solutions to controlled differential equations in the p-variation metric. As part of the proof, we also obtain a factorial decay estimate for controlled paths which is interesting in its own right.

We explore a method introduced by Chatterjee and Ledoux in a paper on eigenvalues of principle submatrices. The method provides a tool to prove concentration of measure in cases where there is a Markov chain meeting certain conditions, and where the spectral gap of the chain is known. We provide several additional applications of this method. These applications include results on operator compressions using the Kac walk on $SO(n)$ and a Kac walk coupled to a thermostat, and a concentration of measure result for the length of the longest increasing subsequence of a random walk distributed under the invariant measure for the asymmetric exclusion process.