Asymptotic properties of stochastic partial differential equations in the sublinear regime

TL;DR

This paper studies the asymptotic behavior of solutions to stochastic heat equations with sublinear diffusion, providing new bounds and insights into intermittency effects under weak diffusion conditions.

Contribution

It establishes non-trivial moment bounds and spatial asymptotics for solutions with sublinear diffusion, bridging gaps between previous linear and bounded diffusion cases.

Findings

Derived moment upper bounds for solutions.

Characterized almost sure spatial asymptotics.

Applicable to stochastic wave and fractional diffusion equations.

Abstract

In this paper, we investigate stochastic heat equation with sublinear diffusion coefficients. By assuming certain concavity of the diffusion coefficient, we establish non-trivial moment upper bounds and almost sure spatial asymptotic properties for the solutions. These results shed light on the smoothing intermittency effect under weak diffusion (i.e., sublinear growth) previously observed by Zeldovich et al. [Zel+87]. The sample-path spatial asymptotics obtained in this paper partially bridge a gap in earlier works of Conus et al. [CJK13; Con+13], which focused on two extreme scenarios: a linear diffusion coefficient and a bounded diffusion coefficient. Our approach is highly robust and applicable to a variety of stochastic partial differential equations, including the one-dimensional stochastic wave equation and the stochastic fractional diffusion equations.