On the discretization in time of parabolic stochastic partial differential equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 6, p. 1055-1078

We first generalize, in an abstract framework, results on the order of convergence of a semi-discretization in time by an implicit Euler scheme of a stochastic parabolic equation. In this part, all the coefficients are globally Lipchitz. The case when the nonlinearity is only locally Lipchitz is then treated. For the sake of simplicity, we restrict our attention to the Burgers equation. We are not able in this case to compute a pathwise order of the approximation, we introduce the weaker notion of order in probability and generalize in that context the results of the globally Lipschitz case.

Classification:  60H15,  60F25,  60F99,  65C20,  60H35
Keywords: stochastic partial differential equations, semi-discretized scheme for stochastic partial differential equations, Euler scheme
@article{M2AN_2001__35_6_1055_0,
author = {Printems, Jacques},
title = {On the discretization in time of parabolic stochastic partial differential equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {35},
number = {6},
year = {2001},
pages = {1055-1078},
zbl = {0991.60051},
mrnumber = {1873517},
language = {en},
url = {http://www.numdam.org/item/M2AN_2001__35_6_1055_0}
}

Printems, Jacques. On the discretization in time of parabolic stochastic partial differential equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 6, pp. 1055-1078. http://www.numdam.org/item/M2AN_2001__35_6_1055_0/

[1] A. Bensoussan and R. Temam, Équations stochastiques du type Navier-Stoke. J. Funct. Anal. 13 (1973) 195-222. | Zbl 0265.60094

[2] J.H. Bramble, A.H. Schatz, V. Thomée and L.B. Wahlbin, Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations. SIAM J. Numer. Anal. 14 (1977) 218-241. | Zbl 0364.65084

[3] C. Cardon-Weber, Autour d'équations aux dérivées partielles stochastiques à dérives non-Lipschitziennes. Thèse, Université Paris VI, Paris (2000).

[4] M. Crouzeix and V. Thomée, On the discretization in time of semilinear parabolic equations with nonsmooth initial data. Math. Comput. 49 (1987) 359-377. | Zbl 0632.65097

[5] G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation. Nonlinear Anal., Theory Methods. Appl. 26 (1996) 241-263. | Zbl 0838.60056

[6] G. Da Prato, A. Debussche and R. Temam, Stochastic Burgers' equation. Nonlinear Differ. Equ. Appl. 1 (1994) 389-402. | Zbl 0824.35112

[7] G. Da Prato and D. Gatarek, Stochastic Burgers equation with correlated noise. Stochastics Stochastics Rep. 52 (1995) 29-41. | Zbl 0853.35138

[8] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, in Encyclopedia of Mathematics and its Application. Cambridge University Press, Cambridge (1992). | MR 1207136 | Zbl 0761.60052

[9] F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations. Probab. Theory Relat. Fields 102 (1995) 367-391. | MR 1339739 | Zbl 0831.60072

[10] I. Gyöngy, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. I. Potential Anal. 9 (1998) 1-25. | Zbl 0915.60069

[11] I. Gyöngy, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. II. Potential Anal. 11 (1999) 1-37. | Zbl 0944.60074

[12] I. Gyöngy and D. Nualart, Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise. Potential Anal. 7 (1997) 725-757. | Zbl 0893.60033

[13] I. Gyöngy, Existence and uniqueness results for semilinear stochastic partial differential equations. Stoch. Process Appl. 73 (1998) 271-299. | Zbl 0942.60058

[14] C. Johnson, S. Larsson, V. Thomée and L.B. Wahlbin, Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data. Math. Comput. 49 (1987) 331-357. | Zbl 0634.65110

[15] P.E. Kloeden and E. Platten, Numerical solution of stochastic differential equations, in Applications of Mathematics 23, Springer-Verlag, Berlin, Heidelberg, New York (1992). | MR 1214374 | Zbl 0752.60043

[16] N. Krylov and B.L. Rozovski, Stochastic Evolution equations. J. Sov. Math. 16 (1981) 1233-1277. | Zbl 0462.60060

[17] M.-N. Le Roux, Semidiscretization in Time for Parabolic Problems. Math. Comput. 33 (1979) 919-931. | Zbl 0417.65049

[18] G.N. Milstein, Approximate integration of stochastic differential equations. Theor. Prob. Appl. 19 (1974) 557-562. G.N. Milstein, Weak approximation of solutions of systems of stochastic differential equations. Theor. Prob. Appl. 30 (1985) 750-766. | Zbl 0314.60039

[19] E. Pardoux, Équations aux dérivées partielles stochastiques non linéaires monotones. Étude de solutions fortes de type Ito. Thèse, Université Paris XI, Orsay (1975).

[20] B.L. Rozozski, Stochastic evolution equations. Linear theory and application to nonlinear filtering. Kluwer, Dordrecht, The Netherlands (1990).

[21] T. Shardlow, Numerical methods for stochastic parabolic PDEs. Numer. Funct. Anal. Optimization 20 (1999) 121-145. | Zbl 0919.65100

[22] D. Talay, Efficient numerical schemes for the approximation of expectation of functionals of the solutions of an stochastic differential equation and applications, in Lecture Notes in Control and Information Science 61, Springer, London, (1984) 294-313. | Zbl 0542.93077

[23] D. Talay, Discrétisation d'une équation différentielle stochastique et calcul approché d'espérance de fonctionnelles de la solution. RAIRO Modél. Math. Anal. Numér. 20 (1986) 141-179. | Numdam | Zbl 0662.65129

[24] M. Viot, Solutions faibles aux équations aux dérivées partielles stochastiques non linéaires. Thèse, Université Pierre et Marie Curie, Paris (1976).

[25] J. B. Walsh, An introduction to stochastic partial differential equations, in Lectures Notes in Mathematics 1180 (1986) 265-437. | Zbl 0608.60060