On a Stochastic Parabolic Integral Equation

In this article we analyze the stochastic parabolic integral equation $$ u\left( {t,x,\omega } \right) = c_\alpha t^{ - 1 + \alpha } *\Delta u + \sum\limits_{k = 1}^\infty {\smallint _0^t g^k \left( {s,x,\omega } \right)} dw_s^k , $$ where t ≥ 0, x ∈ ℝ d , α ∈ (1/2, 1) and ω ∈ Ω. We take w k t ⌝ k = 1, 2, . . . to be a family of independent $$ \mathcal{F}_t $$ -adapted Wiener processes defined on a probability space $$ \left( {\Omega ,\mathcal{F},P} \right) $$ . Here $$ \mathcal{F}_t \subset \mathcal{F}{\text{and }}\mathcal{F}_t $$ is an increasing filtration. By applying and modifying the method of Krylov we obtain existence and regularity results in L p -spaces, p ≥ 2.