The fundamental solution of a 1D evolution equation with a sign changing diffusion coefficient

In this work we investigate a 1D evolution equation involving a divergence form operator where the diffusion coefficient inside the divergence is sign changing. Equivalently the evolution equation of interest can be interpreted as behaving locally like a heat equation, and involving a transmission condition at some interface that prescribes in particular a change of sign of the first order space derivatives across the interface. We especially focus on the construction of fundamental solutions for the evolution equation. As the second order operator involved in the evolution equation is not elliptic, this cannot be performed by standard tools for parabolic PDEs. However we manage in a first time to provide a spectral representation of the semigroup associated to the equation, which leads to a first expression of the fundamental solution. In a second time, examining the case when the diffusion coefficient is piecewise constant but remains positive, we do probabilistic computations involving the killed Skew Brownian Motion (SBM), that provide a certain explicit expression of the fundamental solution for the positive case. It turns out that this expression also provides a fundamental solution for the case when the coefficient is sign changing, and can be interpreted as defining a pseudo SBM. This pseudo SBM can be approached by a rescaled pseudo asymmetric random walk. We infer from these different results various approximation schemes that we test numerically.