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

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.

💡 Research Summary

This research paper presents a significant advancement in the numerical analysis of semilinear stochastic evolution equations (SEEs) driven by cylindrical fractional noise. The primary challenge addressed in this study is the presence of non-globally Lipschitz nonlinearities, which often lead to the instability or divergence of standard numerical integrators, such as the Euler-Maruyama method. To tackle this, the authors introduce a sophisticated time-splitting scheme designed to handle the complexities of both the nonlinearity and the fractional noise structure.

The core methodology involves a strategic decomposition of the nonlinear term into two distinct components: a one-sided, non-globally Lipschitz continuous function and a globally Lipschitz continuous function. The innovation lies in the treatment of these two parts. For the more volatile, non-globally Lipschitz component, the authors utilize the exact flow of an associated differential equation. This approach effectively bypasses the error accumulation and instability typically associated with approximating highly nonlinear terms. For the second, more well-behaved component, an explicit Euler approximation is employed, ensuring computational efficiency without sacrificing the overall stability of the scheme.

A major theoretical contribution of this paper is the establishment of mean-square, strong error estimates for the proposed splitting scheme. The authors rigorously prove that the order of convergence is $H - 1/4$, where $H$ represents the Hurst index ($1/4 < H < 1$). This result is particularly noteworthy as it directly links the convergence rate to the memory properties and smoothness of the fractional noise. To achieve this proof, the researchers developed new regularity results for both real-valued and infinite-dimensional fractional Ornstein-Uhlenbeck processes. These findings are crucial for understanding how the long-range dependence inherent in fractional noise affects the precision of numerical approximations in infinite-dimensional settings.

The paper concludes with numerical experiments that validate the theoretical findings. The simulations demonstrate that the proposed splitting scheme maintains the predicted convergence order and performs robustly even under the challenging conditions of non-Lipschitz nonlinearities. By bridging the gap between complex stochastic theory and practical numerical implementation, this work provides a powerful tool for scientists and engineers modeling systems with long-range dependence and non-linear dynamics, such as those found in fluid mechanics, mathematical finance, and biological modeling.