We investigate the evolution of the random interfaces in a two dimensional Potts model at zero temperature under Glauber dynamics for some particular initial conditions. We prove that under space-time diffusive scaling the shape of the interfaces converges in probability…

We provide several inequalities on the asymptotic behavior of the harmonic measure of the first exit position of a $d$-dimensional symmetric stable process from a unbounded convex domain. Our results on the harmonic measure will determine the asymptotic behavior of…

A reflected Brownian motion in the Gelfand-Tsetlin cone is used to construct Dyson's process of non-colliding Brownian motions. The key step of the construction is to consider two interlaced families of Brownian paths with paths belonging to the second family…

Chain records is a new type of multidimensional record. We discuss how often the chain records occur when the background sampling is from the unit cube with uniform distribution (or, more generally, from an arbitrary continuous product distribution in d…

A general version of the Yamada-Watanabe and Engelbert results relating existence and uniqueness of strong and weak solutions for stochastic equations is given. The results apply to a wide variety of stochastic equations including classical stochastic differential equations, stochastic partial…

Large deviations and isoperimetric inequalities are considered for probability distributions, satisfying convexity conditions of the Brunn-Minkowski-type

We study the isoperimetric problem for product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate half-spaces are approximate solutions of the isoperimetric problem

Let $\{X_1,\cdots ,X_N\}$ be a set of $N$ independent random variables, and let $S_n$ be a sum of $n$ random variables chosen without replacement from the set $\{X_1, \cdots , X_N\}$ with equal probabilities. In this paper we give a…

We introduce a class of rooted infinite self-similar graphs containing the well known Fibonacci graph and graphs associated with Pisot numbers. We consider directed random walks on these graphs and study their entropy and their limit measures. We prove that…

We examine the total number of collisions $C_n$ in the $\Lambda$-coalescent process which starts with $n$ particles. A linear growth and a stable limit law for $C_n$ are shown under the assumption of a power-like behaviour of the measure $\Lambda$…