A Probabilistic Representation of the Solution to a 1D Evolution Equation in a Medium with Negative Index

In this work we investigate a 1D evolution equation involving a divergence form operator where the diffusion coefficient inside the divergence changes 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 associated second 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 coefficient is 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 PDE and we report the associated numerical test results.