The Neumann problem for fully nonlinear SPDE

TL;DR

This paper extends the theory of pathwise viscosity solutions to fully nonlinear stochastic PDEs with Neumann boundary conditions, establishing existence, uniqueness, and stability results, and applies these to analyze long-term behavior of perturbed mean-curvature flows.

Contribution

It introduces a generalized framework for solving fully nonlinear SPDEs with Neumann boundary conditions, including a comparison theorem and applications to geometric flows.

Findings

Comparison theorem ensures solution uniqueness.

Solutions depend continuously on the noise.

Application to long-term behavior of stochastic mean-curvature flow.

Abstract

We generalize the notion of pathwise viscosity solutions, put forward by Lions and Souganidis to study fully nonlinear stochastic partial differential equations, to equations set on a sub-domain with Neumann boundary conditions. Under a convexity assumption on the domain, we obtain a comparison theorem which yields existence and uniqueness of solutions as well as continuity with respect to the driving noise. As an application, we study the long time behaviour of a stochastically perturbed mean-curvature flow in a cylinder-like domain with right angle contact boundary condition.