Electronic Journal of Probability

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

Weicong Su

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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.

Article information

Electron. J. Probab., Volume 25 (2020), paper no. 5, 38 pp.

Received: 10 January 2019
Accepted: 10 January 2020
First available in Project Euclid: 24 January 2020

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Primary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60G15: Gaussian processes

stochastic partial differential equations spheres

Creative Commons Attribution 4.0 International License.


Su, Weicong. On the peaks of a stochastic heat equation on a sphere with a large radius. Electron. J. Probab. 25 (2020), paper no. 5, 38 pp. doi:10.1214/20-EJP415. https://projecteuclid.org/euclid.ejp/1579835021

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