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

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

doi:10.3934/dcds.2026132

Article Contents

2026, Volume 57: 403-442. Doi: 10.3934/dcds.2026132

This volume Previous Article Dynamical spectrum of power-free integers in quadratic number fields and beyond Next Article Global well-posedness of a nonlinear Fokker-Planck type model of grain growth

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

1.
Department of Mathematics, Xi'an Polytechnic University, Shaanxi 710048, China
2.
Universite de Pau et des Pays de l'Adour, E2S UPPA, CNRS, LMAP, Pau, France
3.
School of Science, Xi'an Technological University, Shaanxi 710021, China

^*Corresponding author: Charles-Edouard Bréhier
^*Corresponding author: Charles-Edouard Bréhier

Received: December 4, 2025

Revised: May 19, 2026

Published online: July 13, 2026

This work was supported by the Natural Science Foundation of Shaanxi Province (Grant No.2025JC-YBMS-277), Shaanxi Fundamental Science Research Project for Mathematics and Physics (Grant No.23JSY044).

Abstract / Introduction Full Text(HTML) Figure(3) Related Papers Cited by

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 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 an infinite-dimensional fractional Ornstein–Uhlenbeck process depending on the value of the Hurst parameter $ H $. Numerical experiments illustrate the main result of this manuscript.

Keywords:

Stochastic partial differential equations,
fractional Brownian motion,
splitting schemes,
strong error estimates,
non-Lipschitz nonlinearities.

Mathematics Subject Classification: Primary: 65C30, 60H35; Secondary: 60H15, 60G22.

Citation:

Full Text(HTML)

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\} $.

Download: Full-size image PowerPoint slide

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\} $.

Download: Full-size image PowerPoint slide

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\} $.

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	S. Allen and J. Cahn, A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metal. Mater., 27 (1979), 1085-1095. doi: 10.1016/0001-6160(79)90196-2.
[2]	S. Becker, B. Gess, A. Jentzen and P. E. Kloeden, Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen–Cahn equations, Stochastics Partial Diff. Equat.: Anal. Comput., 11 (2023), 211-268.
[3]	S. Becker and A. Jentzen, Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg-Landau equations, Stochastic Processes Appl., 129 (2019), 28-69. doi: 10.1016/j.spa.2018.02.008.
[4]	F. Biagini, Y. Z. Hu, B. Øksendal and T. S. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Probab. Appl. (N. Y.), Springer-Verlag London, Ltd., London, 2008.
[5]	C. E. Bréhier, J. B. Cui and J. L. Hong, Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation, IMA J. Numer. Anal., 39 (2019), 2096-2134. doi: 10.1093/imanum/dry052.
[6]	C. E. Bréhier and L. Goudenège, Analysis of some splitting schemes for the stochastic Allen–Cahn equation, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4169-4190.
[7]	C. E. Bréhier and L. Goudenège, Weak convergence rates of splitting schemes for the stochastic Allen–Cahn equation, BIT Numer. Math., 60 (2020), 543-582. doi: 10.1007/s10543-019-00788-x.
[8]	J. B. Cui and J. L. Hong, Strong and weak convergence rates of a spatial approximation for stochastic partial differential equation with one-sided Lipschitz coefficient, SIAM J. Numer. Anal., 57 (2019), 1815-1841. doi: 10.1137/18M1215554.
[9]	X. L. Ding and D. H. Wang, Stability theory of stochastic evolution equations with multiplicative fractional Brownian motions in Hilbert spaces, Chaos, Solitons and Fractals, 187 (2024), Paper No. 115435, 10 pp. doi: 10.1016/j.chaos.2024.115435.
[10]	X. L. Ding and D. H. Wang, Regularity analysis for SEEs with multiplicative fBms and strong convergence for a fully discrete scheme, IMA J. Numer. Anal., 44 (2024), 1435-1463. doi: 10.1093/imanum/drad019.
[11]	T. E. Duncan, Y. Z. Hu and B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion I. theory, SIAM J. Control Optim., 38 (2000), 582-612. doi: 10.1137/S036301299834171X.
[12]	T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Semilinear stochastic equations in a Hilbert space with a fractional Brownian motion, SIAM J. Math. Anal., 40 (2009), 2286-2315. doi: 10.1137/08071764X.
[13]	X. L. Feng, P. J. Li and X. Wang, An inverse random source problem for the time fractional diffusion equation driven by a fractional Brownian motion, Inverse Problem, 36 (2020), 045008, 30 pp. doi: 10.1088/1361-6420/ab6503.
[14]	D. J. Higham, X. R. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), 1041-1063. doi: 10.1137/S0036142901389530.
[15]	J. L. Hong and C. Y. Huang, Super-convergence analysis on exponential integrator for stochastic heat equation driven by additive fractional Brownian motion, arXiv: 2007.02223v1[math. NA], 2020.
[16]	Y. Z. Hu, Integral transformations and anticipative calculus for fractional Brownian motions, Methods of the American Mathematical Society, 175 (2005), viii+127 pp.
[17]	R. Kohn, F. Otto, M. G. Reznikoff and E. Vanden-Eijnden, Action minimization and sharp-interface limits for the stochastic Allen–Cahn equation, Comm. Pure Appl. Math., 60 (2007), 393-438. doi: 10.1002/cpa.20144.
[18]	M. Kovács, S. Larsson and F. Lindgren, On the backward Euler approximation of the stochastic Allen–Cahn equation, J. Appl. Probab., 52 (2015), 323-338.
[19]	M. Kovács, S. Larsson and F. Lindgren., On the discretisation in time of the stochastic Allen–Cahn equation, Math. Nachrichten, 291 (2018), 966-995. doi: 10.1002/mana.201600283.
[20]	G. S. Ladde and L. Wu, Development of nonlinear stochastic models by using stock price data and basic statistics, Neutral Parallel Sci. Comput., 18 (2010), 269-282.
[21]	Z. H. Liu, $L^p$-convergence rate of backward Euler schemes for monotone SDEs, BIT Numer. Math., 62 (2022), 1573-1590. doi: 10.1007/s10543-022-00923-1.
[22]	Z. Liu and Z. Qiao, Wong-Zakai approximations of stochastic Allen–Cahn equation, Int. J. Numer. Anal. Model., 16 (2019), 681-694.
[23]	A. Lobashev, D. Guskov and K. Polovnikov, Generative inpainting of incomplete Euclidean distance matrices of trajectories generated by a fractional Brownian motion, Sci. Reports, 15 (2025), Article number: 19145. doi: 10.1038/s41598-025-97893-5.
[24]	B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion, J. Funct. Anal., 202 (2003), 277-305. doi: 10.1016/S0022-1236(02)00065-4.
[25]	Y. Mishura and G. Shevchenko, The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional, Brownian motion. Stochastics, 80 (2008), 489-511. doi: 10.1080/17442500802024892.
[26]	A. Neuenkirch, Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion, Stochastic Process Appl., 118 (2008), 2294-2333. doi: 10.1016/j.spa.2008.01.002.
[27]	D. Nualart, The Malliavin Calculus and Related Topics, Second edition, Probab. Appl. (N. Y.), Springer-Verlag, Berlin, 2006.
[28]	G. D. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Second Edition, Encyclopedia Math. Appl., 152, Cambridge University Press, Cambridge, 2014.
[29]	J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.1090/S0002-9947-1984-0743749-9.
[30]	R. S. Qi and X. J. Wang, Optimal error estimates of Galerkin finite element methods for stochastic Allen–Cahn equation with additive noise, J. Sci. Comput., 80 (2019), 1171-1194. doi: 10.1007/s10915-019-00973-8.
[31]	X. J. Wang, R. S. Qi and F. Z. Jiang, Sharp mean-square regularity results for SPDEs with fractional noise and optimal convergence rates for the numerical approximations, BIT Numer. Math., 57 (2017), 557-585. doi: 10.1007/s10543-016-0639-4.
[32]	K. Yosida, Functional Analysis, , 6th edition, Grundlehren der Mathematischen Wissenschaften, 123, Springer-Verlag, Berlin-New York, 1980.
[33]	S. Zamani, Reaction-diffusion equations with polynomial drifts driven by fractional Brownian motions, Stochastic Anal. Appl., 28 (2010), 1020-1039. doi: 10.1080/07362994.2010.515483.
[34]	H. Zhou, Y. Z. Hu and Y. H. Liu, Backward Euler method for stochastic differential equations with non-Lipschitz coefficients driven by fractional Brownian motion, BIT Numer. Math., 63 (2023), Paper No. 40, 37 pp. doi: 10.1007/s10543-023-00981-z.