论文标题
混合局部和非局部进化方程
Mixing local and nonlocal evolution equations
论文作者
论文摘要
在本文中，我们研究了随机过程的均匀化及其相关的进化方程，在该方程中，我们将局部部分（由布朗运动和边界上的反射给出）和非局部部分（由跳跃过程给出了光滑的内核）。我们将（固定）空间结构域的一系列分区视为两个部分（局部和非局部性），它们的混合方式使它们在极限的每个点都具有正密度。在对分区序列的充分假设下，我们证明了相关密度的收敛性（这是与域的两个不同区域中耦合的局部和非局部部件的进化方程的解决方案）与极限演化系统的唯一解决方案，局部部分消失了，但分为两种不同的组成部分。我们还获得了与分区相关的过程的分布中的收敛性，并证明极限过程具有与密度极限相吻合的密度对。
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.
