## 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”.

### MSC:

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

### Citations:

Zbl 1018.60057; Zbl 1040.60054
Full Text:

### References:

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