Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$

María J. Garrido–Atienza; Kening Lu; Björn  Schmalfuss

doi:10.3934/dcdsb.2015.20.2553

Article Contents

2015, Volume 20, Issue 8: 2553-2581. Doi: 10.3934/dcdsb.2015.20.2553

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Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$

1.
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
2.
346 TMCB Brigham Young University, Provo, UT 84602
3.
Institut für Stochastik, Friedrich Schiller Universität Jena, Ernst Abbe Platz 2, 77043, Jena

Received: October 31, 2014

Revised: February 28, 2015

Published: September 2015

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Abstract

In this article we are concerned with the study of the existence and uniqueness of pathwise mild solutions to evolutions equations driven by a Hölder continuous function with Hölder exponent in $(1/3,1/2)$. Our stochastic integral is a generalization of the well-known Young integral. To be more precise, the integral is defined by using a fractional integration by parts formula and it involves a tensor for which we need to formulate a new equation. From this it turns out that we have to solve a system consisting of a path and an area equations. In this paper we prove the existence of a unique local solution of the system of equations. The results can be applied to stochastic evolution equations with a non-linear diffusion coefficient driven by a fractional Brownian motion with Hurst parameter in $(1/3,1/2]$, which in particular includes white noise.

Keywords:

Stochastic PDEs,
Hilbert-space valued fractional Brownian motion,
pathwise solutions.

Mathematics Subject Classification: Primary: 60H15; Secondary: 60H05, 60G22, 26A33, 26A42.

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