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 $\partial_{t} u_{R}(t\,,x)=\tfrac12 \Delta_{S_{R}^{2}}u_{R}(t\,,x)+\sigma(u_{R}(t\,,x)) \xi_{R}(t\,,x)$ 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.