Neumann Boundary Problem for Parabolic Partial Differential Equations with Divergence Terms

TL;DR

This paper establishes existence and uniqueness for a Neumann boundary problem involving a parabolic PDE with a singular divergence term, using probabilistic methods via backward stochastic differential equations.

Contribution

It introduces a novel probabilistic approach to handle singular divergence terms in parabolic PDEs with Neumann boundary conditions.

Findings

Proves existence and uniqueness of solutions for the PDE.

Develops a penalization method to approximate solutions via BSDEs.

Links PDE solutions to limits of BSDE sequences.

Abstract

We prove an existence and uniqueness result for Neumann boundary problem of a parabolic partial differential equation (PDE for short) with a singular nonlinear divergence term which can only be understood in a weak sense. A probabilistic approach is applied by studying the backward stochastic differential equations (BSDEs for short) corresponding to the PDEs, the solution of which turns out to be a limit of a sequence of BSDEs constructed by penalization method.