Dissipation in Parabolic SPDEs II: Oscillation and decay of the solution

TL;DR

This paper studies the long-term behavior of solutions to a stochastic heat equation with specific boundary conditions, showing decay rates and oscillation properties of the solution's logarithm under certain conditions.

Contribution

It proves that the oscillation of the logarithm of the solution decays sublinearly over time and establishes exponential decay when the noise coefficient is linear.

Findings

Logarithmic oscillation decays sublinearly over time

All limit points of scaled logarithms coincide almost surely

Solutions decay exponentially when  is linear

Abstract

We consider a stochastic heat equation of the type, $\partial_t u = \partial^2_x u + \sigma(u)\dot{W}$ on $(0\,,\infty)\times[-1\,,1]$ with periodic boundary conditions and on-degenerate positive initial data, where $\sigma:\mathbb{R} \to\mathbb{R}$ is a non-random Lipschitz continuous function and $\dot{W}$ denotes space-time white noise. If additionally $\sigma(0)=0$ then the solution is known to be strictly positive; see Mueller '91. In that case, we prove that the oscillation of the logarithm of the solution decays sublinearly as time tends to infinity. Among other things, it follows that, with probability one, all limit points of $t^{-1}\, \sup_{x\in[-1,1]}\, \log u(t\,,x)$ and $t^{-1}\, \inf_{x\in[-1,1]}\, \log u(t\,,x)$ must coincide. As a consequence of this fact, we prove that, when $\sigma$ is linear, there is a.s. only one such limit point and hence the entire path decays almost surely at an exponential rate.