Long-time behaviour of stochastic Hamilton-Jacobi equations

TL;DR

This paper investigates the long-term behavior of stochastic Hamilton-Jacobi equations, revealing how noise influences solution decay and convergence, with implications for nonlinear stochastic PDEs.

Contribution

It introduces new results on the regularization by noise phenomenon and long-time convergence for stochastic Hamilton-Jacobi equations, including the stochastic mean curvature flow.

Findings

Noise accelerates decay of solutions in mean curvature flow

Solutions converge over time in inhomogeneous stochastic Hamilton-Jacobi equations

New theoretical insights into nonlinear stochastic PDEs

Abstract

The long-time behavior of stochastic Hamilton-Jacobi equations is analyzed, including the stochastic mean curvature flow as a special case. In a variety of settings, new and sharpened results are obtained. Among them are (i) a regularization by noise phenomenon for the mean curvature flow with homogeneous noise which establishes that the inclusion of noise speeds up the decay of solutions, and (ii) the long-time convergence of solutions to spatially inhomogeneous stochastic Hamilton-Jacobi equations. A number of motivating examples about nonlinear stochastic partial differential equations are presented in the appendix.