We consider the stirring process in the interval $\Lambda_N:=[-N,N]$ of $\mathbb Z$ with  births and deaths taking place in the intervals $I_+:=(N-K,N]$, and respectively $I_-:=[-N,-N+K)$, $1 \le K <N$. We prove bounds on the truncated moments uniform in $N$ which…

It is proved that the solutions to the slow diffusion stochastic porous media equation $dX-{\Delta}( |X|^{m-1}X )dt=\sigma(X)dW_t,$ $ 1< m\le 5,$ in $\mathcal{O}\subset\mathbb{R}^d,\ d=1,2,3,$ have the property of finite speed of propagation of disturbances for $\mathbb{P}\text{-a.s.}$ ${\omega}\in{\Omega}$ on a sufficiently…

We give rates of convergence in the strong invariance principle for stationary sequences satisfying some projective criteria. The conditions are expressed in terms of conditional expectations of partial sums of the initial sequence. Our results apply to a large variety…

This article is devoted to the construction of a solution for the "skew inhomogeneous Brownian motion" equation, which first appear in a seminal paper by Sophie Weinryb (1983). We investigate some laws related to the constructed process. In particular, using…

We prove a shape theorem for the internal (graph) distance on the interlacement set $\mathcal{I}^u$ of the random interlacement model on $\mathbb Z^d$, $d\ge 3$. We provide large deviation estimates for the internal distance of distant points in this set, and use these estimates…

We consider semilinear obstacle problem with measure data associated with uniformly elliptic divergence form operator. We prove existence and uniqueness of entropy solution of the problem and provide stochastic representation of the solution in terms of some generalized reflected backward…

The location of the unique supremum of a stationary process on an interval does not need to be uniformly distributed over that interval. We describe all possible distributions of the supremum location for a broad class of such stationary processes. We show that, in…

We study the cover time $\tau_{\mathrm{cov}}$ by (continuous-time) random walk on the 2D box of side length $n$ with wired boundary or on the 2D torus,and show that in both cases with probability approaching $1$ as $n$ increases,$\sqrt{\tau_{\mathrm{cov}}}=\sqrt{2n^2 [\sqrt{2/\pi} \log…

We give a necessary and sucient condition for transport entropy inequalities in dimension one. As an application, we construct a new example of a probability distribution verifying Talagrand's T2 inequality and not the logarithmic Sobolev inequality.

We study the asymptotic behaviour of a class of self-attracting motions on $\mathbb{R}^d$. We prove the decrease of the free energy related to the system and mix it together with stochastic approximation methods. We finally obtain the (limit-quotient) ergodicity of…