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.

中文翻译：

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低规则性路径驱动的希尔伯特空间中的双线性方程

在本文中，考虑并解决了由低规则性路径驱动的希尔伯特空间中的一些双线性演化方程。驱动路径是标量值并且是连续的，并且假设它们沿着由Cont和Perkowski [ Trans。阿米尔。数学。Soc。老师 B，6（2019）161–186]（$ p $是偶数正整数）。满足该条件的典型函数是分数阶布朗运动的轨迹，其Hurst参数为$ H = 1 / p $。如果存在对某个非自治初始值问题的解决方案，则表明存在对双线性问题的强大解决方案。随后，给出了解决该初始值问题的解的充分条件。将抽象结果应用于抛物线形和双曲线型的带乘分数分数阶的几个随机偏微分方程，并在路径意义上对其进行了明确求解。