Cylindrical Fractional Brownian Motion in Banach Spaces

TL;DR

This paper introduces cylindrical fractional Brownian motions in Banach spaces, develops a stochastic integration theory for them, and applies it to solve stochastic Cauchy problems driven by such processes.

Contribution

It provides the first framework for stochastic integration with cylindrical fractional Brownian motions in Banach spaces and applies it to stochastic differential equations.

Findings

Established a series representation for cylindrical fractional Brownian motion.

Developed a stochastic integral for deterministic operator-valued integrands.

Applied the theory to solve stochastic Cauchy problems in Banach spaces.

Abstract

In this article we introduce cylindrical fractional Brownian motions in Banach spaces and develop the related stochastic integration theory. Here a cylindrical fractional Brownian motion is understood in the classical framework of cylindrical random variables and cylindrical measures. The developed stochastic integral for deterministic operator valued integrands is based on a series representation of the cylindrical fractional Brownian motion, which is analogous to the Karhunen-Lo\`eve expansion for genuine stochastic processes. In the last part we apply our results to study the abstract stochastic Cauchy problem in a Banach space driven by cylindrical fractional Brownian motion.