On the spatio-temporal increments of nonlinear parabolic SPDEs and the open KPZ equation

TL;DR

This paper analyzes the fine regularity and oscillation properties of solutions to nonlinear parabolic SPDEs and the open KPZ equation, providing precise moduli of continuity and probabilistic estimates.

Contribution

It introduces new local non-determinism results and detailed linearization error estimates, extending regularity results to boundary conditions and the open KPZ equation.

Findings

Identified exact local and uniform spatio-temporal moduli of continuity.

Established small-ball probability estimates and Chung-type laws of the iterated logarithm.

Extended results to the open KPZ equation with inhomogeneous Neumann boundary conditions.

Abstract

We study spatio-temporal increments of the solutions to nonlinear parabolic SPDEs on a bounded interval with Dirichlet, Neumann, or Robin boundary conditions. We identify the exact local and uniform spatio-temporal moduli of continuity for the sample functions of the solutions. These moduli of continuity results imply the existence of random points in space-time at which spatio-temporal oscillations are exceptionally large. We also establish small-ball probability estimates and Chung-type laws of the iterated logarithm for spatio-temporal increments. Our method yields extension of some of these results to the open KPZ equation on the unit interval with inhomogeneous Neumann boundary conditions. Our key ingredients include new strong local non-determinism results for linear stochastic heat equation under various types of boundary conditions, and detailed estimates for the errors in linearization of spatio-temporal increments of the solution to the nonlinear equation.