Evolution equations driven by a fractional Brownian motion. (English) Zbl 1027.60060

The authors study stochastic evolution equations driven by a fractional Brownian motion (fBm) of the form \[ dX_{t}=(A X_{t}+F(X_{t})) dt + G(X_{t}) dB_{t}^{H}, \quad X_{0}=x_{0}, t\in [0,T], \] in a Hilbert space \(V\), where \(A\) is the infinitesimal generator of an analytic semigroup on \(V\), \(B^{H}\) is a \(V\)-valued fBm with Hurst parameter \(H>1/2\) and with nuclear covariance operator. Existence and uniqueness of mild solution are established under some regularity and growth conditions on the coefficients \(F\) and \(G\) and for some values of \(H\). Moreover, an existence result is proved under less restrictive assumptions on the coefficients and the space dimension \(d\). The proofs of these results are based on the approach developed by D. Nualart and A. Răşcanu [Collect. Math. 53, 55-81 (2002; Zbl 1018.60057)] and they combine techniques of fractional calculus and semigroup estimates.
As application, the authors deal with stochastic parabolic equations driven by a fractional white noise with nuclear covariance: \[ \frac{\partial u}{\partial t}= Lu+f(u)+\Phi(u)\frac{\partial B^{H}}{\partial t}, \] with some boundary conditions, where \(L\) is a uniformly elliptic operator, the drift \(f\) is supposed to be continuous with at most linear growth and \(\Phi\) is Lipschitz continuous. Finally, it is worth quoting from the authors: “It may be surprising that if the coefficient \(\Phi\) is Lipschitz and bounded, we find the restriction \(H>\frac{d}{4}\) for having a function space valued solution, which is the same as in T. E. Duncan, B. Maslowski and B. Pasik-Duncan [Stoch. Dyn. 2, 225-250 (2002; Zbl 1040.60054)] for the case of an additive noise with identity covariance”.


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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