Abstract
We investigate a class of abstract stochastic evolution equations driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space. We establish the existence and uniqueness of a mild solution, a continuous dependence estimate, and various convergence and approximation results. Finally, the analysis of three examples is provided to illustrate the applicability of the general theory.
Highlights
The focus of this investigation is the class of abstract measure-dependent stochastic evolution equations driven by fractional Brownian motion of the general form dx(t) = Ax(t) + f t, x(t), μ(t) dt + g(t)dBH (t), 0 ≤ t ≤ T, x(0) = x0, (1.1)
Stochastic partial functional differential equations naturally arise in the mathematical modeling of phenomena in the natural sciences
Since BH (t) is not a semimartingale unless H = 1/2, the standard stochastic calculus involving the Itointegral cannot be used in the analysis of related stochastic evolution equations
Summary
The purpose of this work is to study the class of abstract stochastic evolution equations obtained by accounting for more general nonlinear perturbations (in the sense of McKean-Vlasov equations, as described in [19]) in the mathematical description of phenomena involving an fBm. In particular, the existence and convergence results we present constitute generalizations of the theory governing standard models arising in the mathematical modeling of nonlinear diffusion processes [1, 16,17,18,19, 22], communication networks [4], Sobolev-type equations arising in the study of consolidation of clay [8], shear in second-order fluids [23], and fluid flow through fissured rocks [24]. As a part of our general discussion, we establish an approximation result concerning the effect of the dependence of the nonlinearity on the probability law of the state process, as well as the noise arising from the stochastic integral, for a special case of (1.1) arising often in applications.
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