Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems

Abstract

This article is devoted to the existence and uniqueness of pathwise solutions to stochastic evolution equations, driven by a Hölder continuous function with Hölder 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 BH with Hurst parameter H > 1/2. In contrast to the article by Maslowski and Nualart, we present here an existence and uniqueness result in the space of H¨older 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 H¨older 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 BH 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.