Quasilinear SPDEs in divergence-form

TL;DR

This paper develops a solution theory in Hölder spaces for quasilinear stochastic PDEs with additive noise, using deterministic PDE bounds to establish regularity and probabilistic bounds on solutions.

Contribution

It introduces a novel approach combining deterministic PDE lemmas with stochastic analysis to handle irregular noise in divergence-form quasilinear SPDEs.

Findings

Established a priori Hölder bounds for divergence-form PDEs with irregular right-hand sides.

Derived stretched exponential bounds for the Hölder semi-norms of solutions to stochastic equations.

Provided a new framework for analyzing regularity of quasilinear SPDEs driven by white-in-time noise.

Abstract

We develop a solution theory in H\"older spaces for a quasilinear stochastic PDE driven by an additive noise. The key ingredients are two deterministic PDE Lemmas which establish a priori H\"older bounds for an equation with irregular right hand side written in divergence form. We apply these deterministic bounds to the case of a noise term which is white in time and trace class in space to obtain stretched exponential bounds for the H\"older semi-norms of the solution for the stochastic equation.