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.

翻译：本文研究了一维含散度型算子的演化方程，其中散度内的扩散系数变号，类似于超材料模型中的情形。我们聚焦于该演化方程基本解的构造，由于关联的二阶算子非椭圆，该构造过程与标准抛物型偏微分方程不同。研究表明，可推导出方程关联半群的谱表示，从而获得基本解的首个表达式。我们还推导出基于伪斜布朗运动的概率表示。该构造推广了扩散系数分段常数但保持正数时由吸收型斜布朗运动导出的形式。我们证明伪斜布朗运动可通过重缩放的伪非对称随机游走逼近，进而导出求解该偏微分方程的若干数值格式，并给出相关数值测试结果。