Strong convergence of parabolic rate 1 of discretisations of stochastic Allen-Cahn-type equations

Gerencsér, Máté; Singh, Harprit

doi:10.1090/tran/9029

Strong convergence of parabolic rate $1$ of discretisations of stochastic Allen-Cahn-type equations
HTML articles powered by AMS MathViewer

by Máté Gerencsér and Harprit Singh;

Trans. Amer. Math. Soc. 377 (2024), 1851-1881

DOI: https://doi.org/10.1090/tran/9029

Published electronically: December 12, 2023

HTML | PDF

Abstract:

Consider the approximation of stochastic Allen-Cahn-type equations (i.e. $1+1$-dimensional space-time white noise-driven stochastic PDEs with polynomial nonlinearities $F$ such that $F(\pm \infty )=\mp \infty$) by a fully discrete space-time explicit finite difference scheme. The consensus in literature, supported by rigorous lower bounds, is that strong convergence rate $1/2$ with respect to the parabolic grid meshsize is expected to be optimal. We show that one can reach almost sure convergence rate arbitrarily close to (and no better than) $1$ when measuring the error in appropriate negative Besov norms, by temporarily ‘pretending’ that the SPDE is singular.

References

Aureli Alabert and István Gyöngy, On numerical approximation of stochastic Burgers’ equation, From stochastic calculus to mathematical finance, Springer, Berlin, 2006, pp. 1–15. MR 2233532, DOI 10.1007/978-3-540-30788-4_{1}
Sebastian Becker, Benjamin Gess, Arnulf Jentzen, and Peter E. Kloeden, Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations, Stoch. Partial Differ. Equ. Anal. Comput. 11 (2023), no. 1, 211–268. MR 4563700, DOI 10.1007/s40072-021-00226-6
Charles-Edouard Bréhier and Ludovic Goudenège, Analysis of some splitting schemes for the stochastic Allen-Cahn equation, Discrete Contin. Dyn. Syst. Ser. B 24 (2019), no. 8, 4169–4190. MR 3986273, DOI 10.3934/dcdsb.2019077
Oleg Butkovsky, Konstantinos Dareiotis, and Máté Gerencsér, Optimal rate of convergence for approximations of SPDEs with non-regular drift, SIAM Journal on Numerical Analysis 61 (2023), no. 2, DOI 10.1137/21M1454213
Giuseppe Da Prato and Arnaud Debussche, Strong solutions to the stochastic quantization equations, Ann. Probab. 31 (2003), no. 4, 1900–1916. MR 2016604, DOI 10.1214/aop/1068646370
A. M. Davie and J. G. Gaines, Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations, Math. Comp. 70 (2001), no. 233, 121–134. MR 1803132, DOI 10.1090/S0025-5718-00-01224-2
William G. Faris and Giovanni Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise, J. Phys. A 15 (1982), no. 10, 3025–3055. MR 684578, DOI 10.1088/0305-4470/15/10/011
Máté Gerencsér and István Gyöngy, Finite difference schemes for stochastic partial differential equations in Sobolev spaces, Appl. Math. Optim. 72 (2015), no. 1, 77–100. MR 3369398, DOI 10.1007/s00245-014-9272-2
Massimiliano Gubinelli, Peter Imkeller, and Nicolas Perkowski, Paracontrolled distributions and singular PDEs, Forum Math. Pi 3 (2015), e6, 75. MR 3406823, DOI 10.1017/fmp.2015.2
István Gyöngy, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. II, Potential Anal. 11 (1999), no. 1, 1–37. MR 1699161, DOI 10.1023/A:1008699504438
M. Hairer, A theory of regularity structures, Invent. Math. 198 (2014), no. 2, 269–504. MR 3274562, DOI 10.1007/s00222-014-0505-4
M. Hairer, An introduction to stochastic PDEs (lecture notes), May 2023, https://www.hairer.org/notes/SPDEs_Course.pdf, Accessed: 23-05-22.
M. Hairer and K. Matetski, Optimal rate of convergence for stochastic Burgers-type equations, Stoch. Partial Differ. Equ. Anal. Comput. 4 (2016), no. 2, 402–437. MR 3498987, DOI 10.1007/s40072-015-0067-5
M. Hairer and K. Matetski, Discretisations of rough stochastic PDEs, Ann. Probab. 46 (2018), no. 3.
Khoa Lê, A stochastic sewing lemma and applications, Electron. J. Probab. 25 (2020), Paper No. 38, 55. MR 4089788, DOI 10.1214/20-ejp442
Zhihui Liu and Zhonghua Qiao, Strong approximation of monotone stochastic partial differential equations driven by white noise, IMA J. Numer. Anal. 40 (2020), no. 2, 1074–1093. MR 4092279, DOI 10.1093/imanum/dry088
Ting Ma and Rong Chan Zhu, Convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations on 2D torus, Acta Math. Sin. (Engl. Ser.) 37 (2021), no. 3, 471–490. MR 4229518, DOI 10.1007/s10114-020-9367-4
Thomas Müller-Gronbach, Optimal pointwise approximation of SDEs based on Brownian motion at discrete points, Ann. Appl. Probab. 14 (2004), no. 4, 1605–1642. MR 2099646, DOI 10.1214/105051604000000954
Xiaojie Wang, An efficient explicit full-discrete scheme for strong approximation of stochastic Allen-Cahn equation, Stochastic Process. Appl. 130 (2020), no. 10, 6271–6299. MR 4140034, DOI 10.1016/j.spa.2020.05.011
Rongchan Zhu and Xiangchan Zhu, Lattice approximation to the dynamical $\Phi _3^4$ model, Ann. Probab. 46 (2018), no. 1, 397–455. MR 3758734, DOI 10.1214/17-AOP1188
A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 236587