Evolutionary equations driven by fractional Brownian motion

We examine the stochastic parabolic integral equation u+A(k_1*u) = ∑ _k=1^∞∫ ^t_0 k_2(t-s)g^k(s,ω ,x) δβ ^k_s driven by the family {β ^k_s}_k=1^∞ of i.i.d. fractional Brownian motions, with Hurst index H∈ (1/2,1) . The solution u is a function of t,ω , x ; with t>0, ω in a probability space, and x∈Δ , a σ -finite measure space with positive measure Λ . 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∈ (1/2,2) , respectively. The operator A is a nonnegative linear operator of dom (A)⊂ L_p(Δ ) into L_p(Δ ) , 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.