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

Yong Chen; Hongjun Gao; María J. Garrido–Atienza; Björn Schmalfuss

doi:10.3934/dcds.2014.34.79

Article Contents

2014, Volume 34, Issue 1: 79-98. Doi: 10.3934/dcds.2014.34.79

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

1.
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023
2.
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023
3.
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
4.
Institut für Stochastik, Friedrich Schiller Universität Jena, Ernst Abbe Platz 2, 07737 Jena

Received: November 2012

Revised: March 2013

Published: January 2014

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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 $B^H$ with Hurst parameter $H>1/2$. In contrast to the article by Maslowski and Nualart [17], we present here an existence and uniqueness result in the space of Hölder 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 Hölder 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.

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Mathematics Subject Classification: Primary: 37L55; Secondary: 60H15, 60G22, 37H05, 35R60.

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