Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise

In this paper, some explicit solutions are given for stochastic differential equations in a Hilbert space with a multiplicative fractional Gaussian noise. This noise is the formal derivative of a fractional Brownian motion with the Hurst parameter in the interval ( 1 / 2 , 1 ) . These solutions can be weak, strong or mild depending on the specific assumptions. The problem of stochastic stability of these equations is considered and for various notions of stability, sufficient conditions are given for stability. The noise may stabilize or destabilize the corresponding deterministic solutions. Various examples of stochastic partial differential equations are given that satisfy the assumptions for explicit solutions or stability.