A probabilistic representation of the solution to a 1D evolution equation in a medium with negative index

TL;DR

This paper develops a probabilistic framework for a 1D evolution equation with sign-changing diffusion coefficients, introducing a pseudo Skew Brownian Motion and numerical schemes for solving such non-elliptic PDEs.

Contribution

It introduces a novel probabilistic representation using pseudo Skew Brownian Motion for PDEs with indefinite diffusion coefficients, extending previous models.

Findings

Derived a spectral representation of the semigroup.

Constructed a probabilistic representation via pseudo SBM.

Developed numerical schemes and tested them successfully.

Abstract

In this work we investigate a 1D evolution equation involving a divergence form operator where the diffusion coefficient inside the divergence is changing sign, as in models for metamaterials.We focus on the construction of a fundamental solution for the evolution equation,which does not proceed as in the case of standard parabolic PDE's, since the associatedsecond order operator is not elliptic. We show that a spectral representation of the semigroup associated to the equation can be derived, which leads to a first expression of the fundamental solution. We also derive a probabilistic representation in terms of a pseudo Skew Brownian Motion (SBM).This construction generalizes that derived from the killed SBM when the diffusion coefficientis piecewise constant but remains positive.We show that the pseudo SBM can be approached by a rescaled pseudo asymmetric random walk,which allows us to derive several numerical schemes for the resolution of the PDEand we report the associated numerical test results.