On a Stochastic Parabolic Integral Equation

Abstract

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.

To the memory of Günter Lumer