A stochastic model for solar flares: on the peaks of a stochastic heat equation on a sphere with a large radius

The reason for the existence of solar flares, which are sudden flashes of increased brightness on the sun, is of interest in astronomical research but not fully understood to date. We may think of solar flares as spots where the temperature is exceptionally high. Then we can model heat flow on a large sphere by a heat equation in a random environment. The solution to that equation describes the temperatures on the sphere. Our goal is to estimate the highest temperature on the sphere. To formulate our probabilistic heat-flow model, consider for every R > 0 the stochastic heat equation @tuR(t ; x) = 1 2
S2R uR(t ; x) + (uR(t ; x))R(t ; x) on S2R , where R = _WR is a centered Gaussian noise with the covariance structure given by E[ _WR(t; x) _WR(s; y)] = hR(x; y)0(t 􀀀 s), where hR is symmetric and semi-positive definite and there exist some fixed constants 􀀀2 < Chup < 2 and 1 2Chup 􀀀1 < Chlo 6 Chup such that for all R > 0 and x ; y 2 S2R , (logR)Chlo =2 = hlo(R) 6 hR(x; y) 6 hup(R) = (logR)Chup=2,
S2R denotes the Laplace-Beltrami operator defined on S2R and  : R 7! R is Lipschitz continuous, positive, and uniformly bounded away from 0 and 1. Under the assumption that uR;0(x) = uR(0 ; x) is a nonrandom continuous function on x 2 S2R and that there exists a finite positive U such that supR>0 supx2S2R juR;0(x)j 6 U, we prove that for every finite positive t, there exist finite positive constants Clow(t) and Cup(t), which only depend on t such that as R ! 1, supx2S2R juR(t ; x)j is asymptotically bounded below by Clow(t)(logR)1=4+Chlo =4􀀀Chup=8 and asymptotically bounded above by Cup(t)(logR)1=2+Chup=4 with high probability. Since uR(t ; x) is nonnegative by a comparison principle, the result in this dissertation establishes a relationship between the highest temperature on the surface of a sphere and its radius. The highest temperature on a large sphere is referred to as the peak of the stochastic heat equation on the sphere. The peaks of the stochastic heat equation on spheres of different radii can serve as a simplified iii iv model for solar flares on stars of different sizes.