Homogenization of Random Media: Random Walks, Diffusions and Stochastic Interface Models

Abstract

This thesis concerns homogenization results, in particular scaling limits and heat kernel estimates, for random processes moving in random environments and for stochastic interface models. The first chapter will survey recent research and introduce three models of interest: the random conductance model, the Ginzburg-Landau ∇φ model, and the symmetric diffusion process in a random medium. In the second chapter we present some novel research on the random conductance model; a random walk on an infinite lattice, usually taken to be Ζ^d with nearest neighbour edges, whose law is determined by random weights on the edges. In the setting of degenerate, ergodic weights and general speed measure, we present a quenched local limit theorem for this model. This states that for almost every instance of the random environment, the heat kernel, once suitably rescaled, converges to that of Brownian motion with a deterministic, non-degenerate covariance matrix. The quenched local limit theorem is proven under ergodicity and moment conditions on the environment. Under stronger, non-optimal moment conditions, we also prove annealed local limit theorems for the static RCM with general speed measure and for the dynamic RCM. The dynamic model allows for the random weights, or conductances, to vary with time. Our focus turns to the Ginzburg-Landau gradient model in the subsequent chapter. This is a model for a stochastic interface separating two distinct thermodynamic phases, using an infinite system of coupled stochastic differential equations (SDE). Our main assumption is that the potential in the SDE system is strictly convex with second derivative uniformly bounded below. The aforementioned annealed local limit theorem for the dynamic RCM is applied via a coupling relation to prove a scaling limit result for the space-time covariances in the Ginzburg-Landau model. We also show that the associated Gibbs distribution scales to a Gaussian free field. In the final chapter, we study a symmetric diffusion process in divergence form in a stationary and ergodic random environment. This is a continuum analogue of the random conductance model and similar analytical techniques are applicable here. The coefficients are assumed to be degenerate and unbounded but satisfy a moment condition. We derive upper off-diagonal estimates on the heat kernel of this process for general speed measure. Lower off-diagonal estimates are also proven for a natural choice of speed measure under an additional decorrelation assumption on the environment. Finally, using these estimates, a scaling limit for the Green’s function is derived.