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In this work we investigate a 1D evolution equation involving a divergenceform operator where the diffusion coefficient inside the divergence is changingsign, as in models for metamaterials.We focus on the construction of afundamental solution for the evolution equation,which does not proceed as inthe case of standard parabolic PDE's, since the associatedsecond order operatoris not elliptic. We show that a spectral representation of the semigroupassociated to the equation can be derived, which leads to a first expression ofthe fundamental solution. We also derive a probabilistic representation interms of a pseudo Skew Brownian Motion (SBM).This construction generalizes thatderived from the killed SBM when the diffusion coefficientis piecewise constantbut remains positive.We show that the pseudo SBM can be approached by arescaled pseudo asymmetric random walk,which allows us to derive severalnumerical schemes for the resolution of the PDEand we report the associatednumerical test results.

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