Elena Kosygina

Limit laws of transient excited random walks on integers

We consider excited random walks (ERWs) on Z with a bounded number of i.i.d. cookies per site without the non-negativity assumption on the drifts induced by the cookies. Kosygina and Zerner [15] have shown that when the total expected drift per site, delta, is larger than 1 then ERW is transient to the right and, moreover, for delta > 4 under the averaged measure it obeys the Central Limit Theorem. We show that when delta is an element of (2,4] the limiting behavior of an appropriately centered and scaled excited random walk under the averaged measure is described by a strictly stable law with parameter delta/2. Our method also extends the results obtained by Basdevant and Singh [2] for delta is an element of (1,2] under the non-negativity assumption to the setting which allows both positive and negative cookies.

Crossing velocities for an annealed random walk in a random potential

We consider a random walk in an i.i.d. non-negative potential on the d-dimensional integer lattice. The walk starts at the origin and is conditioned to hit a remote location y on the lattice. We prove that the expected time under the annealed path measure needed by the random walk to reach y grows only linearly in the distance from y to the origin. In dimension 1 we show the existence of the asymptotic positive speed

Homogenization of viscous and non-viscous HJ equations: a remark and an application

AbstractIt was pointed out by P.-L. Lions, G. Papanicolaou, and S.R.S. Varadhan in their seminal paper (1987) that, for first order Hamilton–Jacobi (HJ) equations, homogenization starting with affine initial data implies homogenization for general uniformly continuous initial data. The argument makes use of some properties of the HJ semi-group, in particular, the finite speed of propagation. This property is lost for viscous HJ equations. In this paper we prove the above mentioned implication in both viscous and non-viscous cases. Our proof relies on a variant of Evans’s perturbed test function method. As an application, we show homogenization in the stationary ergodic setting for viscous and non-viscous HJ equations in one space dimension with non-convex Hamiltonians of specific form. The results are new in the viscous case.