Fundamental solutions of singular SPDEs

Abstract

This paper deals with some models of mathematical physics, where random fluctuations are modeled by white noise or other singular Gaussian generalized processes. White noise, as the distributional derivative od Brownian motion, which is the most important case of a Lévy process, is defined in the framework of Hida distribution spaces. The Fourier transformation in the framework of singular generalized stochastic processes is introduced and its applications to solving stochastic differential equations involving Wick products and singularities such as the Dirac delta distribution are presented. Explicit solutions are obtained in form of a chaos expansion in the Kondratiev white noise space, while the coefficients of the expansion are tempered distributions. Stochastic differential equations of the form P( ω, D) ◊ u( x, ω) = A( x, ω) are considered, where A is a singular generalized stochastic process and P( ω, D) is a partial differential operator with random coefficients. We introduce the Wick-convolution operator ★ which enables us to express the solution as u = s★ A ◊ I ◊(−1), where s denotes the fundamental solution and I is the unit random variable. In particular, the stochastic Helmholtz equation is solved, which in physical interpretation describes waves propagating with a random speed from randomly appearing point sources.