General fractional multiparameter white noise theory and stochastic partial differential equations

Abstract

We present a white noise calculus for d-parameter fractional Brownian motion B-H (x, omega); x is an element of R-d, omega is an element of Omega with general d-dimensional Hurst parameter H = (H-l,..., H-d) is an element of (0, 1)(d). As an illustration we solve the stochastic Poisson problem DeltaU(x) = -W-H(x); x is an element of D, U = 0 on partial derivativeD, where the potential W-H(x) is d-parameter fractional white noise given by W-H (x) = (partial derivative(d) B-H (x)) / (partial derivativex(l)...partial derivativex(d)), and D subset of R-d is a given bounded smooth domain. We also solve the linear stochastic heat equation (partial derivativeU/partial derivativet)(t, x) = 1/2 DeltaU(t, x) + W-H (t, x). For each equation we give sufficient conditions that the solutions U(x) and U(t,x), respectively, are square integrable random variables for all t, x.