  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.
