Delay rough evolution equations

TL;DR

This paper establishes the existence, stability, and convergence of solutions for delay rough partial differential equations, with applications to delay stochastic PDEs driven by fractional Brownian motion.

Contribution

It introduces new results on the existence, stability, and zero-delay convergence of solutions for delay rough PDEs, extending to stochastic cases with fractional Brownian motion.

Findings

Proved existence and stability of DRPDE solutions.

Showed convergence of DRPDE solutions to RPDE solutions as delay approaches zero.

Applied results to delay stochastic PDEs with fractional Brownian motion.

Abstract

In this paper, we accomplish the existence and stability of the solution of a class of delay rough partial differential equations (DRPDEs). Moreover, we prove that the solution of DRPDEs can converge to that of RPDEs in sense of some distance as the delay tends to zero. As applications, we employ the main results to the study of a class of delay stochastic partial differential equations driven by fractional Brownian motion with Hurst parameter $\alpha\in(\frac{1}{3},\frac{1}{2}]$.