论文标题
三个不同平滑核的非局部进化问题的均质化
Homogenization for nonlocal evolution problems with three different smooth kernels
论文作者
论文摘要
在本文中，我们考虑了与跳跃过程相关的进化问题的均质化，该跳动过程涉及三个不同的平滑核，这些光滑核控制着域的不同部分的跳跃。我们假设空间域分为两个子域的序列$ a_n \ cup b_n $，我们有三个不同的平滑核，一个将跳跃从$ a_n $到$ a_n $，第二个控制跳跃的$ a_n $，将跳跃从$ b_n $和$ b_n $和第三个互动的跳跃和$ a_n $ a_n $ a_n $ a_n和$ a_n之间的互动。 $χ_{a_n}（x）\ to x（x）$在$ l^\ infty $中弱（然后$χ_{b_n}（x）（x）\至1-x（x）\ in $ l^\ infty $中的$ l^\ infty $中的$ n \ as $ n \作为$ n \ to \ infty $，并在\ infty $中给出了$ limient $ limient $ limitive $ u _ $ u _ l^limite $ l^limitive l^2 2 2 2三个内核和极限函数$ x $出现。当初始条件在某个时刻是三角洲时，$δ{\ bar {x}} $（这对应于$ \ bar {x} $开始的过程，我们表明，$ \ bar {x} \ in a_jj} $ n_j} $ n $ n $ \ n $足够大。 我们还根据随机过程来提供对该演化方程的概率解释，该过程描述了根据三种不同核的粒子运动的运动，该粒子在$ω$中跳跃，并表明基础过程在分布中将分布收敛到与限制方程相关的极限过程。 我们将分析集中在Neumann类型边界条件上，并在最后简要描述如何处理Dirichlet边界条件。
In this paper we consider the homogenization of the evolution problem associated with a jump process that involves three different smooth kernels that govern the jumps to/from different parts of the domain. We assume that the spacial domain is divided into a sequence of two subdomains $A_n \cup B_n$ and we have three different smooth kernels, one that controls the jumps from $A_n$ to $A_n$, a second one that controls the jumps from $B_n$ to $B_n$ and the third one that governs the interactions between $A_n$ and $B_n$.Assuming that $χ_{A_n} (x) \to X(x)$ weakly in $L^\infty$ (and then $χ_{B_n} (x) \to 1-X(x)$ weakly in $L^\infty$) as $n \to \infty$ and that the initial condition is given by a density $u_0$ in $L^2$ we show that there is an homogenized limit system in which the three kernels and the limit function $X$ appear. When the initial condition is a delta at one point, $δ_{\bar{x}}$ (this corresponds to the process that starts at $\bar{x}$) we show that there is convergence along subsequences such that $\bar{x} \in A_{n_j}$ or $\bar{x} \in B_{n_j}$ for every $n_j$ large enough. We also provide a probabilistic interpretation of this evolution equation in terms of a stochastic process that describes the movement of a particle that jumps in $Ω$ according to the three different kernels and show that the underlying process converges in distribution to a limit process associated with the limit equation. We focus our analysis in Neumann type boundary conditions and briefly describe at the end how to deal with Dirichlet boundary conditions.
