Homogenization of a periodic semilinear elliptic degenerate PDE

TL;DR

This paper develops a homogenization technique for a semilinear elliptic PDE with rapidly oscillating coefficients, allowing degeneracy in the second order part, using probabilistic methods linked to BSDEs and diffusion processes.

Contribution

It introduces a novel homogenization approach for degenerate semilinear elliptic PDEs using probabilistic methods and weak convergence of diffusion processes.

Findings

Homogenization achieved for PDEs with degenerate second order terms.

Probabilistic methods effectively handle oscillating coefficients.

Weak convergence of diffusion processes underpins the homogenization process.

Abstract

In this paper a semilinear elliptic PDE with rapidly oscillating coefficients is homogenized. The novetly of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some portion of $\R^d$.\\ Our fully probabilistic method is based on the connection between PDEs and BSDEs with random terminal time and the weak convergence of a class of diffusion processes.