On diffusions in media with pockets of large diffusivity

TL;DR

This paper studies the asymptotic behavior of diffusion processes in media with large diffusivity pockets, revealing a limiting process that simplifies the complex media into a diffusion on a reduced space with boundary conditions.

Contribution

It introduces a new limiting diffusion process for media with pockets of large diffusivity and derives associated boundary value problems for PDEs.

Findings

Limiting process as pockets' diffusivity tends to infinity.

New boundary conditions for PDEs in media with large diffusivity pockets.

Explicit characterization of the process on the reduced space.

Abstract

We consider diffusion processes in media with pockets of large diffusivity. The asymptotic behavior of such processes is described when the diffusion coefficients in the pockets tend to infinity. The limiting process is identified as a diffusion on the space where each of the pockets is treated as a single point, and certain conditions on the behavior of the process on the boundary of the pockets are imposed. Calculation of various probabilities and expectations related to the limiting process leads to new initial-boundary (and boundary) problems for the corresponding parabolic (and elliptic) PDEs.