We give a counterexample to a conjecture of Hammersley and Welsh (1965) about the convexity of the time constant in first-passage percolation, as a functional on the space of distribution functions. The present counterexample only works for first-passage percolation on…

This work was motivated by the recent work by H. Dette, J. Pitman and W.J. Studden on a new duality relation for random walks. In this note we consider the diffusion process limit of their result, and use the alternative…

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

The asymmetric $k$-step exclusion processes are the simplest interacting particle systems whose hydrodynamic equation may exhibit both increasing and decreasing entropic shocks under Euler scaling. We prove that, under Riemann initial condition with right density zero and adequate left density,…

We obtain new results concerning poisoning/nonpoisoning in a catalytic model which has previously been introduced and studied. We show that poisoning can occur even when the arrival rate of one gas is smaller than the sum of the arrival rates…

We prove that the density fluctuations for a zero-range process evolving on the $d$-dimensional supercritical percolation cluster, with $d\geq{3}$, are given by a generalized Ornstein-Uhlenbeck process in the space of distributions $\mathscr{S}'(\mathbb{R}^d)$.

Using the recently discovered strong negative dependence properties of the symmetric exclusion process, we derive general conditions for when the normalized current of particles between regions converges to the Gaussian distribution. The main novelty is that the results do not…

The law for the time $\tau_{a}$ spent by a standard Brownian excursion above a given level $a > 0$ is found using Ito excursion theory. This is achieved by conditioning the excursion to have exactly one mark of an independent…

We derive a unified stochastic picture for the duality of a resampling-selection model with a branching-coalescing particle process (cf. MR2123250) and for the self-duality of Feller's branching diffusion with logistic growth (cf. MR2308333). The two dual processes are approximated by…

It has been shown that certain types of random walks in random potentials and Brownian motion in Poissonian potentials undergo a phase transition from sub-ballistic to ballistic behavior when the strength of the underlying drift is increased. The ballistic behavior…