PATHWISE SOLUTIONS OF SPDES DRIVEN BY HOLDER-CONTINUOUS INTEGRATORS WITH EXPONENT LARGER THAN 1/2 AND RANDOM DYNAMICAL SYSTEMS

This article is devoted to the existence and uniqueness of pathwise solutions to stochastic evolution equations, driven by a Holder continuous function with Holder exponent in (1/2, 1), and with nontrivial multiplicative noise. As a particular situation, we shall consider the case where the equation is driven by a fractional Brownian motion B-H with Hurst parameter H > 1/2. In contrast to the article by Maslowski and Nualart [71], we present here an existence and uniqueness result in the space of Holder continuous functions with values in a Hilbert space V. If the initial condition is in the latter space this forces us to consider solutions in a different space, which is a generalization of the Holder continuous functions. That space of functions is appropriate to introduce a non-autonomous dynamical system generated by the corresponding solution to the equation. In fact, when choosing B-H as the driving process, we shall prove that the dynamical system will turn out to be a random dynamical system, defined over the ergodic metric dynamical system generated by the infinite dimensional fractional Brownian motion.