Stability of the stochastic heat equation in $L^1([0,1])$

TL;DR

This paper investigates the stability, existence, and uniqueness of solutions to the stochastic heat equation driven by white noise in the space of integrable functions, providing new inequalities and stability results.

Contribution

It introduces an inequality for the $L^1$-norm difference of solutions, establishing strong existence, partial uniqueness, and stability of solutions with initial data in $L^1$.

Findings

Proves strong existence and partial uniqueness of solutions in $L^1$.

Derives stability estimates for solutions with respect to initial conditions.

Analyzes long-term behavior, including invariant distributions and solution confluence.

Abstract

We consider the white-noise driven stochastic heat equation on $[0,\infty)\times[0,1]$ with Lipschitz-continuous drift and diffusion coefficients $b$ and $\sigma$. We derive an inequality for the $L^1([0,1])$-norm of the difference between two solutions. Using some martingale arguments, we show that this inequality provides some {\it a priori} estimates on solutions. This allows us to prove the strong existence and (partial) uniqueness of weak solutions when the initial condition belongs only to $L^1([0,1])$, and the stability of the solution with respect to this initial condition. We also obtain, under some conditions, some results concerning the large time behavior of solutions: uniqueness of the possible invariant distribution and asymptotic confluence of solutions.