Stochastic and partial differential equations on non-smooth time-dependent domains

TL;DR

This paper establishes existence, uniqueness, and comparison principles for stochastic differential equations and fully nonlinear PDEs on non-smooth, time-dependent domains with oblique boundary conditions, extending prior work to dynamic settings.

Contribution

It generalizes existing results to non-smooth, time-dependent domains, providing new proofs for existence and uniqueness of solutions in this complex setting.

Findings

Proved existence and uniqueness of strong solutions to stochastic differential equations with oblique reflection.

Established a comparison principle for fully nonlinear second-order PDEs with oblique boundary conditions.

Extended previous results to more general, time-dependent domain settings.

Abstract

In this article, we consider non-smooth time-dependent domains and single-valued, smoothly varying directions of reflection at the boundary. In this setting, we first prove existence and uniqueness of strong solutions to stochastic differential equations with oblique reflection. Secondly, we prove, using the theory of viscosity solutions, a comparison principle for fully nonlinear second-order parabolic partial differential equations with oblique derivative boundary conditions. As a consequence, we obtain uniqueness, and, by barrier construction and Perron's method, we also conclude existence of viscosity solutions. Our results generalize two articles by Dupuis and Ishii to time-dependent domains.