Nonlinear Dirichlet problem of non-local branching processes

TL;DR

This paper introduces a probabilistic approach to solving nonlinear Dirichlet problems with discontinuous boundary data using nonlocal branching processes, enabling controlled convergence at boundaries.

Contribution

It provides a novel probabilistic representation for solutions to nonlinear Dirichlet problems with discontinuous boundary conditions, leveraging nonlocal branching processes.

Findings

Probabilistic representation of solutions using nonlocal branching processes

Method handles discontinuous boundary data effectively

Controlled convergence replaces pointwise convergence at boundaries

Abstract

We present a method of solving a nonlinear Dirichlet problem with discontinuous boundary data and we give a probabilistic representation of the solution using the nonlocal branching process associated with the nonlinear term of the operator. Instead of the pointwise convergence of the solution to the given boundary data we use the controlled convergence which allows to have discontinuities at the boundary.