Abstract
We investigate a class of abstract functional stochastic evolution equations driven by a fractional Brownian motion in a real separable Hilbert space. Global existence results concerning mild solutions are formulated under various growth and compactness conditions. Continuous dependence estimates and convergence results are also established. Analysis of three stochastic partial differential equations, including a second-order stochastic evolution equation arising in the modeling of wave phenomena and a nonlinear diffusion equation, is provided to illustrate the applicability of the general theory.
Highlights
The purpose of this paper is to study the global existence and convergence properties of mild solutions to a class of abstract functional stochastic evolution equations of the general form dx (t) = (Ax (t) + F (x) (t)) dt + g (t) dβH (t), 0 ≤ t ≤ T, (1)x (0) = x0, in a real separable Hilbert space U
For every t ≥ 0, βH(t) = ∑∞ j=1 βjH(t)ej is a Uvalued fBm, where the convergence is understood to be in the mean-square sense
We develop existence results for (1) in which the Lipschitz condition on F is replaced by the combination of continuity and a sublinear growth condition
Summary
The purpose of this paper is to study the global existence and convergence properties of mild solutions to a class of abstract functional stochastic evolution equations of the general form dx (t) = (Ax (t) + F (x) (t)) dt + g (t) dβH (t) ,. We collect some preliminary information about certain function spaces, linear semigroups, probability measures, the definition of fBm, and the stochastic integral driven by a fBmin Section 2. The paper concludes with a discussion of three different stochastic partial differential equations in Section 6 as an illustration of the abstract theory
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