On the peaks of a stochastic heat equation on a sphere with a large radius

Abstract

For every $R>0$, consider the stochastic heat equation \[ \partial _{t} u_{R}(t\,,x)=\tfrac {1}{2} \Delta _{S_{R}^{2}}u_{R}(t\,,x)+\sigma (u_{R}(t\,,x)) \xi _{R}(t\,,x) \] on $S_{R}^{2}$, where $\xi _{R}=\dot {W_{R}}$ are centered Gaussian noises with the covariance structure given by $\mathrm {E}[\dot {W_{R}}(t,x)\dot {W_{R}}(s,y)]=h_{R}(x,y)\delta _{0}(t-s)$, where $h_{R}$ is symmetric and semi-positive definite and there exist some fixed constants $-2< C_{h_{up}}< 2$ and $\tfrac {1}{2} C_{h_{up}}-1<C_{h_{down}} \leqslant C_{h_{up}}$ such that for all $R>0$ and $x\,,y \in S_{R}^{2}$, $(\log R)^{C_{h_{down}}/2}=h_{down}(R)\leqslant h_{R}(x,y) \leqslant h_{up}(R)=(\log R)^{C_{h_{up}}/2}$, $\Delta _{S_{R}^{2}}$ denotes the Laplace-Beltrami operator defined on $S_{R}^{2}$ and $\sigma :\mathbb {R}\mapsto \mathbb {R}$ 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 $\sup _{R>0}\sup _{x \in S_{R}^{2}}\vert u_{R,0}(x)\vert \leqslant U$, we prove that for every finite positive $t$, there exist finite positive constants $C_{down}(t)$ 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 $C_{down}(t)(\log R)^{1/4+C_{h_{down}}/4-C_{h_{up}}/8}$ and asymptotically bounded above by $C_{up}(t)(\log R)^{1/2+C_{h_{up}}/4}$ with high probability.