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 ${\partial }_t{u}_R(t,x)=\frac{1}{2}{\Delta }_{S_R^2}{u}_R(t,x)+\sigma (u_R(t,x)){\xi }_R(t,x)$ on $S_R^2$, where ${\xi }_R=\dot{W_R}$ are centered Gaussian noises with the covariance structure given by $E[\dot{W_R}(t,x)\dot{W_R}(s,y)]=h_R(x,y){\delta }_0(t-s)$, 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 $x,y\in S_R^2$, $(\log R{)}^{C_{h_{lo}}/2}=h_{lo}(R)\le h_R(x,y)\le h_{up}(R)=(\log R{)}^{C_{h_{up}}/2}$, ${\Delta }_{S_R^2}$ 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 ${\sup }_{R>0}{\sup }_{x\in S_R^2}|u_{R,0}(x)|\le U$, 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 $R\to \infty$, ${\sup }_{x\in S_R^2}|u_R(t,x)|$ 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.