Spatial fluctuation for stochastic heat equation with Hölder coefficients

TL;DR

This work analyzes the spatial fluctuations of nonnegative solutions to the stochastic heat equation with Hölder diffusion $u^\\gamma \\xi$, focusing on the regimes $\\gamma=\\tfrac12$ and $\\gamma\\in(\\tfrac34,1)$. It develops an associativity framework by approximating the Hölder coefficient with Lipschitz surrogates and passing to a weak limit, enabling CLT-type results and spatial ergodicity without relying on Malliavin calculus. The authors establish a central limit theorem for the spatial integral $S_{N,t}$ and a precise large deviation limit in the $\\gamma=\\tfrac12$ case, along with a lower bound on spatial growth under a tail assumption. Collectively, the results advance the understanding of spatial fluctuations and growth in nonlinear SPDEs with rough coefficients and introduce a robust associativity-based approach for non-Lipschitz diffusion.

Abstract

In this paper, we establish associativity, spatial ergodicity and a central limit theorem for certain nonnegative solutions to the stochastic heat equation $\partial_t u=\frac12\partial_x^2 u+ u^γξ$ with $γ\in (0, 1)$. When $γ=\frac12$, we derive a limit for the moment generating function of the spatial integral and provide a lower bound on the spatial growth of the solution.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• proof : Proof of Theorem \ref{['th:association']}
• Remark 2.1
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['th:CLT']}: spatial ergodicity
• Proposition 3.2
• proof
• proof : Proof of Theorem \ref{['eq:SHE']}: CLT