Bilinear equations in Hilbert space driven by paths of low regularity

Petr Čoupek; María J. Garrido-Atienza

doi:10.3934/dcdsb.2020230

Article Contents

2021, Volume 26, Issue 1: 121-154. Doi: 10.3934/dcdsb.2020230

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Bilinear equations in Hilbert space driven by paths of low regularity

Petr Čoupek^1, and
María J. Garrido-Atienza^2, ,

1.
Charles University, Faculty of Mathematics and Physics, Sokolovská 83, Prague 8,186 75, Czech Republic
2.
Universidad de Sevilla, Dpto. Ecuaciones Diferenciales y Análisis numérico, Avda. Reina Mercedes s/n, 41012-Sevilla, Spain

^* Corresponding author: María J. Garrido-Atienza
^* Corresponding author: María J. Garrido-Atienza

Received: December 31, 2019

Revised: May 31, 2020

Early access: July 2020

Published: January 2021

The first author is supported by the Czech Science Foundation, project GAČR 19-07140S. The second author is supported by Ministerio de Ciencia, Innovación y Universidades, Grant No. PGC2018-096540-I00

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Abstract

In the article, some bilinear evolution equations in Hilbert space driven by paths of low regularity are considered and solved explicitly. The driving paths are scalar-valued and continuous, and they are assumed to have a finite $ p $-th variation along a sequence of partitions in the sense given by Cont and Perkowski [Trans. Amer. Math. Soc. Ser. B, 6 (2019) 161–186] ($ p $ being an even positive integer). Typical functions that satisfy this condition are trajectories of the fractional Brownian motion with Hurst parameter $H=1 / p$. A strong solution to the bilinear problem is shown to exist if there is a solution to a certain non–autonomous initial value problem. Subsequently, sufficient conditions for the existence of the solution to this initial value problem are given. The abstract results are applied to several stochastic partial differential equations with multiplicative fractional noise, both of the parabolic and hyperbolic type, that are solved explicitly in a pathwise sense.

Keywords:

Itô-Föllmer calculus,
p-th variation along a sequence of partitions,
multiplicative noise,
fractional Brownian motion,
rough paths.

Mathematics Subject Classification: Primary: 60H15, 60G22; Secondary: 34F05, 47D06.

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