Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-15T03:49:29.517Z Has data issue: false hasContentIssue false
  • English
  • Français

On the discretization in time of parabolic stochastic partial differential equations

Published online by Cambridge University Press:  15 April 2002

Jacques Printems
Jacques Printems*
Affiliation:
Centre de Mathématiques de l'Université de Paris 12, EA 2343, Université de Paris 12, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France. (printems@univ-paris12.fr)
Get access

Abstract

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.

Keywords

Stochastic partial differential equationssemi-discretized scheme for stochastic partial differential equationsEuler scheme.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 6 , November 2001, pp. 1055 - 1078
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bensoussan, A. and Temam, R., Équations stochastiques du type Navier-Stoke. J. Funct. Anal. 13 (1973) 195-222. CrossRef
Bramble, J.H., Schatz, A.H., Thomée, V. and Wahlbin, L.B., Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations. SIAM J. Numer. Anal. 14 (1977) 218-241. CrossRef
C. Cardon-Weber, Autour d'équations aux dérivées partielles stochastiques à dérives non-Lipschitziennes. Thèse, Université Paris VI, Paris (2000).
Crouzeix, M. and Thomée, V., On the discretization in time of semilinear parabolic equations with nonsmooth initial data. Math. Comput. 49 (1987) 359-377. CrossRef
G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation. Nonlinear Anal., Theory Methods. Appl. 26 (1996) 241-263.
Da Prato, G., Debussche, A. and Temam, R., Stochastic Burgers' equation. Nonlinear Differ. Equ. Appl. 1 (1994) 389-402. CrossRef
Da Prato, G. and Gatarek, D., Stochastic Burgers equation with correlated noise. Stochastics Stochastics Rep. 52 (1995) 29-41. CrossRef
G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, in Encyclopedia of Mathematics and its Application. Cambridge University Press, Cambridge (1992).
Flandoli, F. and Gatarek, D., Martingale and stationary solutions for stochastic Navier-Stokes equations. Probab. Theory Relat. Fields 102 (1995) 367-391. CrossRef
Gyöngy, I., Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. I. Potential Anal. 9 (1998) 1-25. CrossRef
Gyöngy, I., Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. II. Potential Anal. 11 (1999) 1-37. CrossRef
Gyöngy, I. and Nualart, D., Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise. Potential Anal. 7 (1997) 725-757. CrossRef
Gyöngy, I., Existence and uniqueness results for semilinear stochastic partial differential equations. Stoch. Process Appl. 73 (1998) 271-299. CrossRef
Johnson, C., Larsson, S., Thomée, V. and Wahlbin, L.B., Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data. Math. Comput. 49 (1987) 331-357. CrossRef
P.E. Kloeden and E. Platten, Numerical solution of stochastic differential equations, in Applications of Mathematics 23, Springer-Verlag, Berlin, Heidelberg, New York (1992).
Krylov, N. and Rozovski, B.L., Stochastic Evolution equations. J. Sov. Math. 16 (1981) 1233-1277. CrossRef
Le Roux, M.-N., Semidiscretization in Time for Parabolic Problems. Math. Comput. 33 (1979) 919-931. CrossRef
Milstein, G.N., Approximate integration of stochastic differential equations. Theor. Prob. Appl. 19 (1974) 557-562.
E. Pardoux, Équations aux dérivées partielles stochastiques non linéaires monotones. Étude de solutions fortes de type Ito. Thèse, Université Paris XI, Orsay (1975).
B.L. Rozozski, Stochastic evolution equations. Linear theory and application to nonlinear filtering. Kluwer, Dordrecht, The Netherlands (1990).
Shardlow, T., Numerical methods for stochastic parabolic PDEs. Numer. Funct. Anal. Optimization 20 (1999) 121-145. CrossRef
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.
Talay, D., 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. CrossRef
M. Viot, Solutions faibles aux équations aux dérivées partielles stochastiques non linéaires. Thèse, Université Pierre et Marie Curie, Paris (1976).
J. B. Walsh, An introduction to stochastic partial differential equations, in Lectures Notes in Mathematics 1180 (1986) 265-437.