We give two algorithms that allow to get arbitrary precision asymptotics for the harmonic potential of a random walk.

In this paper a necesary and sufficient condition is established for the probability that a Lévy process is positive at time $t$ to tend to 1 as $t$ tends to 0. This condition is expressed in terms of the characteristics…

We study the models of competition and spreading of infection for infinite systems of independent random walks. For the competition model, we investigate the question whether one of the spins prevails with probability one. For the infection spreading, we give…

Proofs and two applications of two general results are given concerning the problem of establishing interior smoothness of probabilistic solutions of elliptic degenerate equations.

We prove well-posedness of the martingale problem for an infinite-dimensional degenerate elliptic operator under appropriate Hölder continuity conditions on the coefficients. These martingale problems include large population limits of branching particle systems on a countable state space in which the…

We define and study a family of Markov processes with state space the compact set of all partitions of $N$ that we call exchangeable fragmentation-coalescence processes. They can be viewed as a combination of homogeneous fragmentation as defined by Bertoin…

We encode a certain class of stochastic fragmentation processes, namely self-similar fragmentation processes with a negative index of self-similarity, into a metric family tree which belongs to the family of Continuum Random Trees of Aldous. When the splitting times of…

Following the investigation by U. Haagerup and S. Thorbjornsen, we present a simple differential approach to the limit theorems for empirical spectral distributions of complex random matrices from the Gaussian, Laguerre and Jacobi Unitary Ensembles. In the framework of abstract…

We consider a random field $(X_n)_{n\in\mathbb{Z}^d}$ and investigate when the set $A_h=\{k\in\mathbb{Z}^d; \vert X_k\vert \ge h\}$ has infinite clusters. The main problem is to decide whether the critical level $$h_c=\sup\{h\in R : P(A_h\text{ has an infinite cluster })>0\}$$ is neither…

The Kolmogorov-1934 diffusion is the two-dimensional diffusion generated by real Brownian motion and its time integral. In this paper we construct successful co-adapted couplings for iterated Kolmogorov diffusions defined by adding iterated time integrals as further components to the original…