Isotropic Gaussian Random Fields On The Ellipsoid:Regularity And Stochastic Partial Differential Heat Conduction Equation

Isotropic Gaussian random fields on the sphere are characterized by Karhunen-Loeve expansions with respect to the spherical harmonic functions and the angular power spectrum.The smoothness of the covariance is connected to the decay of the angular power spectrum.Based on these properties of the sphere,we can obtain sample Holder continuity and sample differentiability of the random fields on the ellipsoid towards the homeomorphis-m between sphere and ellipsoid.Rates of convergence of their finitely truncated Karhunen-Loeve expansions in terms of the covariance spectrum are established.Besides,we will give the general result on manifolds mainly Riemann manifold towards similar way.Finally,the stochastic heat equation on the ellipsoid driven by additive,isotropic Wiener noise is consid-ered and we give the strong convergence rate.