Stochastic Heat Equation with Colored Noise on Torus

The subject of this doctoral dissertation is the stochastic heat equation on compact manifolds. In general the solution for the stochastic heat equation in dimension greater than 1 requires regularity structures to analyze. White noise fails Dalang’s condition in higher dimensions, and the solution to the stochastic heat equation driven by this noise will be a distribution. In joint work with my advisor we developed a two parameter (α,ρ) family of driving noise, colored in space and white in time, which is adapted to manifold geometry. The noise correlation is defined on the set of Laplace-Beltrami operator eigenvectors, which is a complete orthonormal basis for L2(M). The Wiener calculus then allows one to define the solution using the Skorohod integral, which can also be interpreted as a Walsh integral. The parameter α controls how smooth the noise is; α=0 corresponds to white noise. The parameter ρ controls the lower bound on the noise correlation function; it can be tuned to make the correlation function strictly positive. We focus in particular on the parabolic Anderson model on the flat torus in d-dimensions because the eigenvectors of the Laplace-Beltrami operator correspond to the familiar Fourier series expansion. Using the Itô-Burkholder isometry and the Picard iteration, we express the moments of the solution as a sum of a series of terms Ln. Ln+1 can be analyzed as a (slightly modified) convolution of Ln with the density of pinned Brownian motion. This correspondence allows us to generalize our techniques from the torus to a compact manifold, where the pinned Brownian motion can also be defined. We bound the terms of this sum, establishing an upper bound on the second moment that is exponential in time. This establishes existence and uniqueness results for the solution. Lower bounds on solution moments are also of interest to investigate the phenomenon of intermittency. When the solution moments have exponential lower bounds, the magnitude of the solution exhibits a multi-fractal structure where high peaks intermittent in time appear at different scales. We obtain exponential lower bounds on the solution when ρ>0 by applying the stochastic Feynman-Kac formula and using the ergodic property of Brownian motion on compact manifolds.