## Electronic Communications in Probability

### Discrete maximal regularity of an implicit Euler–Maruyama scheme with non-uniform time discretisation for a class of stochastic partial differential equations

Yoshihito Kazashi

#### Abstract

An implicit Euler–Maruyama method with non-uniform step-size applied to a class of stochastic partial differential equations is studied. A spectral method is used for the spatial discretization and the truncation of the Wiener process. A discrete analogue of maximal $L^2$-regularity of the scheme and the discretised stochastic convolution is established, which has the same form as their continuous counterpart.

#### Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 29, 14 pp.

Dates
Accepted: 4 April 2018
First available in Project Euclid: 28 April 2018

https://projecteuclid.org/euclid.ecp/1524881137

Digital Object Identifier
doi:10.1214/18-ECP130

Mathematical Reviews number (MathSciNet)
MR3798240

Zentralblatt MATH identifier
1390.60236

#### Citation

Kazashi, Yoshihito. Discrete maximal regularity of an implicit Euler–Maruyama scheme with non-uniform time discretisation for a class of stochastic partial differential equations. Electron. Commun. Probab. 23 (2018), paper no. 29, 14 pp. doi:10.1214/18-ECP130. https://projecteuclid.org/euclid.ecp/1524881137

#### References

• [1] R. P. Agarwal, C. Cuevas, and C. Lizama, Regularity of Difference Equations on Banach Spaces, Springer International Publishing, Cham, 2014.
• [2] H. Amann, Linear and Quasilinear Parabolic Problems. Volume I: Abstract Linear Theory, Monographs in Mathematics, vol. 89, Birkhäuser, Boston, 1995.
• [3] V. V. Anh, P. Broadbridge, A. Olenko, and Y. G. Wang, On approximation for fractional stochastic partial differential equations on the sphere, Stoch. Environ. Res. Risk Assess. (2018). doi:10.1007/s00477-018-1517-1
• [4] A. Ashyralyev and P. E. Sobolevskii, Well-Posedness of Parabolic Difference Equations, Birkhäuser, Basel, 1994.
• [5] P. Auscher, J. van Neerven, and P. Portal, Conical stochastic maximal $L^p$-regularity for $1\le p<\infty$, Math. Ann. (2014).
• [6] P. Baldi, D. Marinucci, and V. S. Varadarajan, On the characterization of isotropic Gaussian fields on homogeneous spaces of compact groups, Electron. Commun. Probab. 12 (2007), 291–302.
• [7] S. Blunck, Maximal regularity of discrete and continuous time evolution equations, Stud. Math. 146 (2001), 157–176.
• [8] G. Da Prato, Regularity results of a convolution stochastic integral and applications to parabolic stochastic equations in a Hilbert space, Conf. del Semin. di Mat. dell’Università di Bari (1982), no. 182, 17.
• [9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2nd ed., vol. 152, Cambridge University Press, 2014.
• [10] L. Gawarecki and V. Mandrekar, Stochastic Differential Equations in Infinite Dimensions, Probability and Its Applications, Springer Berlin Heidelberg, Berlin, Heidelberg, 2011.
• [11] A. Jentzen and P. E. Kloeden, Taylor Approximations for Stochastic Partial Differential Equations, SIAM, Philadelphia, 2011.
• [12] T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
• [13] T. Kemmochi, Discrete maximal regularity for abstract Cauchy problems, Stud. Math. 234 (2016), 241–263.
• [14] T. Kemmochi and N. Saito, Discrete maximal regularity and the finite element method for parabolic equations, Numer. Math. 138 (2018), 905–937.
• [15] B. Kovács, B. Li, and C. Lubich, A-stable time discretizations preserve maximal parabolic regularity, SIAM J. Numer. Anal. 54 (2016), 3600–3624.
• [16] R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Lecture Notes in Mathematics, vol. 2093, Springer International Publishing, 2014.
• [17] R. Kruse and S. Larsson, Optimal regularity for semilinear stochastic partial differential equations with multiplicative noise, Electron. J. Probab. 17 (2012).
• [18] P. C. Kunstmann and L. Weis, Maximal $L_{p}$-regularity for parabolic equations, Fourier multiplier theorems and $H^{\infty }$-functional calculus, Functional Analytic Methods for Evolution Equations, Lecture Notes in Math., vol. 1855, Springer, Berlin, 2004, 65–311.
• [19] A. Lang and C. Schwab, Isotropic Gaussian random fields on the sphere: Regularity, fast simulation and stochastic partial differential equations, Ann. Appl. Probab. 25 (2015), 3047–3094.
• [20] D. Leykekhman and B. Vexler, Discrete maximal parabolic regularity for Galerkin finite element methods, Numer. Math. 135 (2017), 923–952.
• [21] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Modern Birkhäuser Classics, Birkhäuser/Springer, Basel, 1995.
• [22] D. Marinucci and G. Peccati, Random Fields on the Sphere, London Mathematical Society Lecture Note Series, vol. 389, Cambridge University Press, Cambridge, 2011.
• [23] T. Müller-Gronbach and K. Ritter, An implicit Euler scheme with non-uniform time discretization for heat equations with multiplicative noise, BIT Numer. Math. 47 (2007), 393–418.
• [24] T. Müller-Gronbach and K. Ritter, Lower bounds and nonuniform time discretization for approximation of stochastic heat equations, Found. Comput. Math. 7 (2007), 135–181.
• [25] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, vol. 143, Springer-Verlag, New York, 2013.
• [26] J. van Neerven, M. Veraar, and L. Weis, Maximal $L^p$-regularity for stochastic evolution equations, SIAM J. Math. Anal. 44 (2012), 1372–1414.
• [27] J. van Neerven, M. Veraar, and L. Weis, Stochastic maximal $L^p$-regularity, Ann. Probab. 40 (2012), 788–812.
• [28] J. van Neerven, M. Veraar, and L. Weis, On the $R$-boundedness of stochastic convolution operators, Positivity. 19 (2015), 355–384.
• [29] K. Yosida, Functional Analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995.