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, ω); x ∈ ℝ d , ω ∈ Ω with general d-dimensional Hurst parameter H = (H 1,…, H d ) ∈ (0, 1) d . As an illustration we solve the stochastic Poisson problem ΔU(x) = −W H (x); x ∈ D, U = 0 on ∂D, where the potential W H (x) is d-parameter fractional white noise given by W H (x) = (∂ d B H (x))/(∂x 1…∂x d ), and D ⊂ ℝ d is a given bounded smooth domain. We also solve the linear stochastic heat equation . 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.

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