Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems
By H. Gao, María J. Garrido-Atienza, B. Schmalfuss. Discrete & Continuous Dynamical Systems - A, 2013
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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 [17], 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.
Subfields: Economics and Econometrics, Finance
View Paper: https://www.semanticscholar.org/paper/cc5d29040eb93b9b36ba869605c181176bf836c9
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- Semilinear Caputo time-fractional pseudo-parabolic equations (2021, background): “(See [16], Lemma 8, page 9) Let ω1 > −1, ω2 > −1 such that ω1 + ω2 ≥ −1, h > 0 and t ∈ [0, T ].”