Schauder estimates and classical solutions of the Dirichlet problem for stochastic parabolic equations

TL;DR

This paper establishes Schauder estimates and proves existence and uniqueness of classical solutions for stochastic parabolic equations with Dirichlet boundary conditions in cylindrical domains, under certain regularity assumptions.

Contribution

It provides the first global Schauder estimates for such equations with boundary conditions, extending classical PDE theory to stochastic settings.

Findings

Established global Schauder estimates in stochastic Hölder spaces.

Proved existence and uniqueness of quasi-classical solutions.

Achieved pathwise classical solvability in Hölder classes.

Abstract

We study second-order stochastic parabolic equations in a cylindrical domain with homogeneous Dirichlet boundary conditions. Under a natural compatibility condition on the gradient-type noise, we establish global Schauder estimates in stochastic H\"older spaces for the Dirichlet problem. The coefficients and free terms are assumed to be H\"older continuous in the spatial variables, while only their boundary traces are required to be H\"older in time. As a consequence, we obtain existence and uniqueness of quasi-classical solutions in stochastic H\"older spaces, and further derive pathwise classical solvability in H\"older classes.