Bilinear equations in Hilbert space driven by paths of low regularity

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>-th variation along a sequence of partitions in the sense given by Cont and Perkowski [<i>Trans. Amer. Math. Soc. Ser. B</i>, <b>6</b> (2019) 161–186] (<inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula> being an even positive integer). Typical functions that satisfy this condition are trajectories of the fractional Brownian motion with Hurst parameter <inline-formula><tex-math id="M3">\begin{document}$H=1 / p$\end{document}</tex-math></inline-formula>. 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.