论文标题
在具有负数为负指数的培养基中，解决方案对1D进化方程的概率表示
A probabilistic representation of the solution to a 1D evolution equation in a medium with negative index
论文作者
论文摘要
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.我们表明，可以得出与方程相关的半群的光谱表示，这导致了基本解决方案的首次表达。 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相关的数值测试结果。
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.
