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.