Homogenization of random walk in asymmetric random environment

Abstract:

In this paper, the author investigates the scaling limit of a partial difference equation on the d dimensional integer lattice $\Z^d$, corresponding to a translation invariant random walk perturbed by a random vector field. In the case when the translation invariant walk scales to a Cauchy process he proves convergence to an effective equation on $\R^d$. The effective equation corresponds to a Cauchy process perturbed by a constant vector field. In the case when the translation invariant walk scales to Brownian motion he shows that the scaling limit, if it exists, depends on dimension. For $d=1,2$ he provides evidence that the scaling limit cannot be diffusion.

University of Michigan, Department of Mathematics, Ann Arbor, MI 48109-1109
conlon@math.lsa.umich.edu
http://www.math.lsa.umich.edu/~conlon/