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

Xiao-Li Ding, Charles-Edouard Bréhier, Dehua Wang

TL;DR

This work addresses semilinear SPDEs driven by cylindrical fractional noise with a non-globally Lipschitz drift. It introduces a semi-phase-flow splitting scheme that treats the one-sided nonlinearity F via its exact flow while discretizing the globally Lipschitz part G with an explicit Euler step, and proves mean-square convergence of order $H-1/4$ for Hurst parameter $H\in(1/4,1)$. The analysis hinges on new regularity results for fractional Ornstein–Uhlenbeck processes and separates the regimes $H>1/2$ and $H\in(1/4,1/2)$, with a detailed discrete-time framework and auxiliary processes. Numerical experiments in a 1D parabolic SPDE validate the predicted rate and demonstrate the scheme's practical effectiveness.

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.

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

TL;DR

This work addresses semilinear SPDEs driven by cylindrical fractional noise with a non-globally Lipschitz drift. It introduces a semi-phase-flow splitting scheme that treats the one-sided nonlinearity F via its exact flow while discretizing the globally Lipschitz part G with an explicit Euler step, and proves mean-square convergence of order for Hurst parameter . The analysis hinges on new regularity results for fractional Ornstein–Uhlenbeck processes and separates the regimes and , with a detailed discrete-time framework and auxiliary processes. Numerical experiments in a 1D parabolic SPDE validate the predicted rate and demonstrate the scheme's practical effectiveness.

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 , where 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 . Numerical experiments illustrate the main result of this manuscript.

Paper Structure

This paper contains 22 sections, 21 theorems, 253 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Notation
  4. Linear operator
  5. Nonlinearities
  6. Fractional Brownian motion
  7. Preliminary properties
  8. Fractional Ornstein--Uhlenbeck process
  9. Well-posedness of the SPDE
  10. Numerical scheme
  11. Numerical experiments
  12. Conclusion
  13. Appendix: notation
  14. Appendix: properties of the one-dimensional fractional Ornstein--Uhlenbeck process when $H\in(1/2,1)$
  15. Moment bounds
  16. ...and 7 more sections

Key Result

Proposition 2.2

The semigroup $\bigl(S(t)\bigr)_{t\ge 0}$ satisfies the following properties. $\bullet$ For all $t\geq 0$, $S(t)$ is a bounded linear operator on $\mathbb{H}$, and one has $\|S(t)\|_{{\mathcal{L}}(\mathbb{H})}\leq 1$. $\bullet$ For all $t\geq 0$, $S(t)$ is a bounded linear operator on $E$, and one h Moreover, for all $\rho\in [0,1]$, there exists $C_\rho\in(0,\infty)$ such that one has $\bullet$

Figures (3)

  • Figure 1: Temporal convergence rate of numerical approximations with $f(t,X)=-X^3+X$ and $\varepsilon=1$ for different Hurst parameters $H\in \{0.3,0.5,0.7,0.9\}$.
  • Figure 2: Temporal convergence rate of numerical approximations with $f(t,X)=-X^3+X$ and $\varepsilon=0.01$ for different Hurst parameters $H\in \{0.3,0.5,0.7,0.9\}$.
  • Figure 3: Temporal convergence rate of numerical approximations with $f(t,X)=-X^3+X+\sin(X)+1$ and $\varepsilon=1$ for different Hurst parameters $H\in \{0.3,0.5,0.7,0.9\}$.

Theorems & Definitions (38)

  • Proposition 2.2
  • Lemma 2.4
  • Proposition 3.1
  • proof : Proof of Proposition \ref{['propo:ZH']}
  • Proposition 3.2
  • proof : Proof of Proposition \ref{['propo:ZH_E']}
  • Proposition 3.3
  • proof : Proof of Proposition \ref{['propo:ZHregul']}
  • Proposition 3.4
  • Theorem 4.1
  • ...and 28 more