Perturbations of Parabolic Equations and Diffusion Processes with Degeneration: Boundary Problems, Metastability, and Homogenization

TL;DR

This paper investigates the behavior of diffusion processes with boundary degeneracy and small perturbations, analyzing their impact on parabolic equations, metastability phenomena, and homogenization in boundary value problems.

Contribution

It introduces a detailed analysis of degenerate diffusion processes with perturbations, extending understanding of boundary effects, metastability, and homogenization in parabolic PDEs.

Findings

Asymptotic behavior depends on the time scale as perturbation size tends to zero.

Boundary degeneracy influences stabilization of solutions to parabolic equations.

Homogenization results for operators with degeneracy are established.

Abstract

We study diffusion processes that are stopped or reflected on the boundary of a domain. The generator of the process is assumed to contain two parts: the main part that degenerates on the boundary in a direction orthogonal to the boundary and a small non-degenerate perturbation. The behavior of such processes determines the stabilization of solutions to the corresponding parabolic equations with a small parameter. Metastability effects arise in this case: the asymptotics of solutions, as the size of the perturbation tends to zero, depends on the time scale. Initial-boundary value problems with both the Dirichlet and the Neumann boundary conditions are considered. We also consider periodic homogenization for operators with degeneration.