We give a probabilistic representation for the solution to a nonlinear evolution equation induced by a measure-valued branching process. We first construct the involved branching processes on the set of all finite configurations of a given set, with a killing rate induced by a continuous additive functional, and with a non-local branching procedure given by a sequence of Markovian kernels. The main application is to prove stochastic aspects for a nonlinear evolution equation related to the Neumann problem and the surface measure on the boundary, which corresponds to the reflecting Brownian motion as base movement, taking the killing rate given by the local time on the boundary. We use specific potential theoretical tools.

我们给出了由量值分支过程引起的非线性演化方程解的概率表示。我们首先在给定集合的所有有限构型的集合上构造涉及的分支过程，其杀灭率由连续的加和函数诱导，并具有由马尔可夫核序列提供的非局部分支过程。主要应用是证明与诺伊曼（Neumann）问题和边界上的表面测度有关的非线性演化方程的随机方面，该方程对应于反射布朗运动作为基本运动，并采用边界上局部时间给定的杀死率。我们使用特定的潜在理论工具。