局部和非局部混合发展方程
Mixing local and nonlocal evolution equations
摘要：
本文研究了随机过程及其相关的发展方程的均匀化问题，其中我们混合了局部部分（由在边界上具有反射的布朗运动给出）和非局部部分（由具有光滑核的跳跃过程给出）.我们考虑一个序列的分区（固定）空间域分为两个部分（本地和非本地）的混合方式，他们都有积极的密度在每一点的限制。在适当的假设下的序列的分区，我们证明了收敛的相关密度（即解决方案的发展方程与耦合的本地和非本地部分在两个不同的区域的域）的唯一解决方案的极限演化系统中，本地部分消失和非本地部分生存，但分为两个不同的组件。我们还得到了与分区相关联的过程的分布依分布收敛，并证明了极限过程有一个密度对与密度的极限重合。
英文摘要：
In this paper we study the homogenization of a stochastic process and its associated evolution equations in which we mix a local part (given by a Brownian motion with a reflection on the boundary) and a nonlocal part (given by a jump process with a smooth kernel). We consider a sequence of partitions of the (fixed) spacial domain into two parts (local and nonlocal) that are mixed in such a way that they both have positive density at every point in the limit. Under adequate hypotheses on the sequence of partitions, we prove convergence of the associated densities (that are solutions to an evolution equation with coupled local and nonlocal parts in two different regions of the domain) to the unique solution to a limit evolution system in which the local part disappears and the nonlocal part survives but divided into two different components. We also obtain convergence in distributions of the processes associated to the partitions and prove that the limit process has a density pair that coincides with the limit of the densities.
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