abstract:
The aim of this talk is to study the long-time dynamics of mild solutions to retarded stochastic evolution systems driven by a Hilbert-valued Brownian motion. We do not assume that the noise is given in additive form or that it is a non-trivial multiplicative noise, but instead we assume that the diffusion coefficient is smooth in some sense.
In that way, and by using an integration by parts formula, the stochastic integral can be expressed in terms of non-stochastic integrals and the noisy path. As we will show, this latter term causes that in a first moment only a local mild solution can be obtained. Nevertheless, by using appropriate stopping times, we will be able to derive the existence and uniqueness of a global mild solution. Furthermore, we shall show that the global mild solution generates a random dynamical system that, under an appropriate smallness condition for the time lag, have associated a random attractor. |