Stochastic Homogenization for Nonlinear First- and Second- PDE and Applications

Panagiotis Sougandis
The University of Texas at Austin
Department of Mathematics

Abstract: The theory of homogenization of first- and second-order fully nonlinear partial differential equations in random media has attracted lately a lot of attention. Averaging problems in random environments arise naturally in a variety of applications like front propagation, phase transitions, combustion, percolation and large deviation of diffusion processes.

Viscosity solutions have been employed successfully to study periodic/almost periodic homogenization. The key step is the analysis of an auxiliary macroscopic problem, known in this context as the cell problem, which defines the effective equation (nonlinearity). The fundamental difference between the periodic/almost periodic and random settings is that the latter lacks the necessary compactness to employ pde techniques. The macroscopic problem does not have, in general, a solution. There is therefore a need to develop an alternative methodology to identify the effective equation.

In these lectures I will present in detail the main difficulties and the new results, explain the new methodology and discuss the applications.