Evolution equations driven by a fractional Brownian motion

Ergodicity and Parameter Estimates for Infinite-Dimensional Fractional Ornstein-Uhlenbeck Process

“…Grecksch and Anh [8] consider a semi-linear stochastic parabolic equation with additive noise term and prove existence and uniqueness of the solution. Semilinear evolution equations with a covariance type fractional Brownian motion with a non-additive noise term are studied by Maslowski and Nualart in [14] and Nualart and Vuillermot in [16], where existence, uniqueness and pathwise regularity are established. Hu, Oksendal and Zhang [11] treat the elliptic equation (the Poisson problem) and the heat equation with multiparameter fractional Gaussian noise (cf.…”

Cylindrical fractional Brownian motion in Banach spaces

“…In the special case of Hilbert spaces, quite some literature on stochastic evolution equations with fBm noise can be found -see amongst others Grecksch and Anh [13], Duncan and coauthors in a series of papers [9,10,11,12], Tindel et al [28], Maslowski and Nualart [17], Gubinelli et al [14]. When restricted to the Hilbert space case, our approach is to some extent similar to the one in [9,11,12].…”

“…For comparison, we apply our methods to an example often considered in the literature and typically formulated in a Hilbert space setting. There are several works in the literature devoted to a similar or related problem, such as Brzeźniak et al [7] for the case of a Banach space, and Grecksch and Anh [13], Duncan and coauthors in a series of papers [9,10,11,12], Tindel et al [28], Maslowski and Nualart [17], Gubinelli et al [14] in the Hilbert space. Among these the papers [7] by Brzeźniak et al and [28] by Tindel et al are the most related to our work, and thus, it might be worth to comment on them in more detail in the sequel: in [7] the authors consider an abstract Cauchy problems in Banach spaces driven by a cylindrical Liouville fBm.…”

Stochastic fractional perturbed control systems with fractional Brownian motion and Sobolev stochastic non local conditions

“…In particular, many types of stochastic differential equations driven by fBm in infinite dimension received much attention, for example, Maslowski and Nualart [8] studied nonlinear stochastic evolution equations in a Hilbert space driven by cylindrical fractional Brownian motion with Hurst parameter H > 1 2 and nuclear covariance operator using techniques of fractional calculus with semigroup estimates. Boufoussi and Hajji [9] proved the existence and uniqueness of mild solutions of a neutral stochastic differential equations with nite delay, driven by a fractional Brownian motion in a Hilbert space and established some conditions ensuring the exponential decay to zero in mean square for the mild solution.…”