On the Peaks of a Stochastic Heat Equation on a Sphere with a Large Radius

TL;DR

This paper investigates the behavior of the peaks of solutions to a stochastic heat equation on spheres with increasing radius, revealing how the maximum amplitude scales with the sphere's size under certain noise and initial conditions.

Contribution

It provides asymptotic bounds for the maximum of the solution on large spheres, extending understanding of stochastic PDEs on curved geometries with log-scale noise correlations.

Findings

Maximum amplitude scales as a power of log R with explicit exponents.

Upper and lower bounds for the solution's peaks are established with high probability.

Results depend on the noise covariance structure and initial conditions.

Abstract

For every $R>0$, consider the stochastic heat equation $\partial_{t} u_{R}(t\,,x)=\tfrac12 \Delta_{S_{R}^{2}}u_{R}(t\,,x)+\sigma(u_{R}(t\,,x)) \xi_{R}(t\,,x)$ on $S_{R}^{2}$, where $\xi_{R}=\dot{W_{R}}$ are centered Gaussian noises with the covariance structure given by $E [\dot{W_{R}}(t,x)\dot{W_{R}}(s,y)]=h_{R}(x,y)\delta_{0}(t-s)$, where $h_{R}$ is symmetric and semi-positive definite and there exist some fixed constants $-2< C_{h_{up}}< 2$ and $\frac{1}{2}C_{h_{up}}-1 <C_{h_{lo}}\le C_{h_{up}}$ such that for all $R>0$ and $x\,,y \in S_{R}^{2}$, $(\log R)^{C_{h_{lo}}/2}=h_{lo}(R)\leq h_{R}(x,y) \leq h_{up}(R)=(\log R)^{C_{h_{up}}/2}$, $\Delta_{S_{R}^{2}}$ denotes the Laplace-Beltrami operator defined on $S_{R}^{2}$ and $\sigma:R \mapsto R$ is Lipschitz continuous, positive and uniformly bounded away from $0$ and $\infty$. Under the assumption that $u_{R,0}(x)=u_{R}(0\,,x)$ is a nonrandom continuous function on $x \in S_{R}^{2}$ and the initial condition that there exists a finite positive $U$ such that $\sup_{R>0}\sup_{x \in S_{R}^{2}}\vert u_{R,0}(x)\vert \le U$, we prove that for every finite positive $t$, there exist finite positive constants $C_{low}(t)$ and $C_{up}(t)$ which only depend on $t$ such that as $R \to \infty$, $\sup_{x \in S_{R}^{2}}\vert u_{R}(t\,,x)\vert$ is asymptotically bounded below by $C_{low}(t)(\log R)^{1/4+C_{h_{lo}}/4-C_{h_{up}}/8}$ and asymptotically bounded above by $C_{up}(t)(\log R)^{1/2+C_{h_{up}}/4}$ with high probability.