We present an alternative particle picture for super-stable motion. It is based on a non-local branching mechanism in discrete time and only trivial space motion.

The trajectory of the ball in a soccer game is modelled by the Brownian motion on a cylinder, subject to elastic reflections at the boundary points (as proposed in [KPY]). The score is then the number of windings of the…

We prove the law of large numbers for $U$-statistics whose underlying sequence of random variables satisfies an absolute regularity condition ($\beta$-mixing condition) under suboptimal conditions.

We show that if a sequence of domains $D_k$ increases to a domain $D$ then the reflected Brownian motions in $D_k$'s converge to the reflected Brownian motion in $D$, under mild technical assumptions.

In this short note we show that the paths two independent loop-erased random walks in four dimensions intersect infinitely often. We actually prove the stronger result that the cut-points of the two walks intersect infinitely often.

For a function $\ell$ satisfying suitable integrability (but not continuity) requirements, we construct a process $(B^\ell_u, 0 \leq u \leq 1)$ interpretable as Brownian excursion conditioned to have local time $\ell(\cdot)$ at time $1$. The construction is achieved by first…

We sharpen a bound in a theorem of Russell Lyons for percolation on a tree and associated random walk.

Fractional Brownian motion belongs to a class of long memory Gaussian processes that can be represented as linear functionals of an infinite dimensional Markov process. This leads naturally to: An efficient algorithm to approximate the process. An ergodic theorem which…

The lattice gas model in equilibrium is considered. We give a lower bound of the density-density time correlation for large time, which involves the bulk diffusion matrix in a physically natural way.

The law of a random tree constructed within a Brownian excursion is calculated conditional on knowing the occupation measure of the excursion. In previous work David Aldous has used random walk approximations to obtain this result. Here it is deduced…