Evolutionary equations driven by fractional brownian motion

We examine the stochastic parabolic integral equation driven by the family {β _s k }∞ _k=1 of i.i.d. fractional Brownian motions, with Hurst index H ∈ ( ₂ ¹ , 1). The solution u is a function of t, ω, x; with t > 0, ω in a probability space, and x ∈ ∆, a σ-finite measure space with positive measure ∧. The integrals on the right are stochastic Skorohod integrals; the kernels k ₁ (t ), k ₂ (t ) are powers of t, i.e., multiples of t ^α−1 , t ^γ−1 , with α ∈ (0, 2), γ ∈ ( ₂ ¹ , 2), respectively. The operator A is a nonnegative linear operator of dom( A) ⊂ L _p (∆) into L _p (∆), for some p ∈ [2, ∞). We combine transformation techniques with Malliavin calculus including results by Nualart and Balan to develop an L _p -theory for the equation. Fractional powers of A and of time-derivatives are used to indicate smoothness in space (x), and time (t ), respectively.