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We prove that the local time process of a planar simple random walk, when time is scaled logarithmically, converges to a non-degenerate pure jump process. The convergence takes place in the Skorokhod space with respect to the $M1$ topology and fails to hold in the $J1$ topology.
Fix $p>1$, not necessarily integer, with $p(d-2)<d$. We study the $p$-fold self-intersection local time of a simple random walk on the lattice $Z^d$ up to time $t$. This is the $p$-norm of the vector of the walkers local times, $ell_t$. We derive pre
Given a sequence of lattice approximations $D_Nsubsetmathbb Z^2$ of a bounded continuum domain $Dsubsetmathbb R^2$ with the vertices outside $D_N$ fused together into one boundary vertex $varrho$, we consider discrete-time simple random walks in $D_N
We consider a one dimensional random walk in random environment that is uniformly biased to one direction. In addition to the transition probability, the jump rate of the random walk is assumed to be spatially inhomogeneous and random. We study the p
The aim of this paper is the study of the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite dimensional generating function G and a maximum principle which, we
Lecture Notes. Minicourse given at the workshop Activated Random Walks, DLA, and related topics at IMeRA-Marseille, March 2015.