Analysis of splitting schemes for stochastic evolution equations with non-Lipschitz nonlinearities driven by fractional noise

TL;DR

This paper introduces a new splitting scheme for stochastic evolution equations with non-Lipschitz nonlinearities driven by fractional noise, providing strong error estimates and demonstrating convergence depending on the Hurst parameter.

Contribution

It develops a novel time-splitting method combining exact flow and Euler approximation for fractional noise-driven equations with non-Lipschitz nonlinearities, along with new regularity results.

Findings

Proves mean-square strong error estimates with convergence order H-1/4.

Establishes new regularity results for fractional Ornstein-Uhlenbeck processes.

Numerical experiments confirm theoretical convergence rates.

Abstract

We propose a novel time-splitting scheme for a class of semilinear stochastic evolution equations driven by cylindrical fractional noise. The nonlinearity is decomposed as the sum of a one-sided, non-globally, Lipschitz continuous function, and of a globally Lipschitz continuous function. The proposed scheme is based on a splitting strategy, where the first nonlinearity is treated using the exact flow of an associated differential equation, and the second one is treated by an explicit Euler approximation. We prove mean-square, strong error estimates for the proposed scheme and show that the order of convergence is $H-1/4$, where $H\in(1/4,1)$ is the Hurst index. For the proof, we establish new regularity results for real-valued and infinite dimensional fractional Ornstein-Uhlenbeck process depending on the value of the Hurst parameter $H$. Numerical experiments illustrate the main result of this manuscript.