Evolutionary equations driven by fractional Brownian motion

Abstract

We examine the stochastic parabolic integral equation $$\begin{aligned} u+A(k_1*u) = \sum _{k=1}^{\infty }\mathop \int \limits ^t_0 k_2(t-s)g^k(s,\omega ,x)\, \delta \beta ^k_s \end{aligned}$$driven by the family \(\{\beta ^k_s\}_{k=1}^{\infty }\) of i.i.d. fractional Brownian motions, with Hurst index \(H\in (\frac{1}{2},1)\). The solution \(u\) is a function of \(t,\omega , x\); with \(t>0, \omega \) in a probability space, and \(x\in \Delta \), a \(\sigma \)-finite measure space with positive measure \(\Lambda \). 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 \(t^{\alpha -1}, t^{\gamma -1}\), with \(\alpha \in (0,2), \gamma \in (\frac{1}{2},2)\), respectively. The operator \(A\) is a nonnegative linear operator of \(\mathrm{dom }(A)\subset L_p(\Delta )\) into \(L_p(\Delta )\), for some \(p\in [2,\infty ).\) 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.