# Exponential convergence of solutions for random Hamilton-Jacobi equations

## TL;DR

This paper proves that solutions to certain randomly kicked Hamilton-Jacobi equations on the torus converge exponentially fast to a stationary solution, advancing understanding in stochastic PDEs.

## Contribution

It establishes exponential convergence for solutions of multi-dimensional random Hamilton-Jacobi equations, completing a significant research program.

## Key findings

- Solutions converge exponentially fast to stationary solutions almost surely.
- The results extend previous work to multi-dimensional settings.
- Completes the theoretical framework for stochastic Hamilton-Jacobi equations.

## Abstract

We show that for a family of randomly kicked Hamiton-Jacobi equations on the torus, almost surely, the solution of an initial value problem converges exponentially fast to the unique stationary solution. Combined with the results in \cite{IK03} and \cite{KZ12}, this completes the program started in \cite{EKMS00} for the multi-dimensional setting.