Stochastic evolution equations driven by Liouville fractional Brownian motion

Brzeźniak, Zdzisław ; van Neerven, Jan ; Salopek, Donna

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 1, pp. 1-27.

Résumé

Let $H$ be a Hilbert space and $E$ a Banach space. We set up a theory of stochastic integration of $\mathcal{L}(H,E)$-valued functions with respect to $H$-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter $0\beta 1$. For $0\beta \frac{1}{2}$ we show that a function $\mathrm{\Phi }:(0,T)\to \mathcal{L}(H,E)$ is stochastically integrable with respect to an $H$-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an $H$-cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations $$ {\rm d}U(t) = AU(t) {\rm d}t + B {\rm d}W_H^\beta (t) $$ driven by an $H$-cylindrical Liouville fractional Brownian motion, and prove existence, uniqueness and space-time regularity of mild solutions under various assumptions on the Banach space $E$, the operators $A:\mathcal{D}\mathcal{(}\mathcal{A})\to \mathcal{E}$ and $B:H\to E$, and the Hurst parameter $\beta$. As an application it is shown that second-order parabolic SPDEs on bounded domains in ${\mathbb{R}}^d$, driven by space-time noise which is white in space and Liouville fractional in time, admit a mild solution if $\frac{1}{4}d\beta 1$.

Détail

DOI : 10.1007/s10587-012-0011-z

Classification : 35R60, 47D06, 60G18, 60H05
Mots-clés : (Liouville) fractional Brownian motion; fractional integration; stochastic evolution equations