Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-08T15:30:03.510Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Approximation of Stationary Distributions for Stochastic Partial Differential Equations

Part of: Stochastic analysis Parabolic equations and systems Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  30 January 2018

Jianhai Bao  and
Chenggui Yuan
Jianhai Bao*
Affiliation:
Central South University
Chenggui Yuan*
Affiliation:
Swansea University
*
∗ Postal address: Department of Mathematics, Central South University, Changsha, 410075, P. R. China. Email address: [email protected]
∗∗ Postal address: Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, UK. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we discuss an exponential integrator scheme, based on spatial discretization and time discretization, for a class of stochastic partial differential equations. We show that the scheme has a unique stationary distribution whenever the step size is sufficiently small, and that the weak limit of the stationary distribution of the scheme as the step size tends to 0 is in fact the stationary distribution of the corresponding stochastic partial differential equations.

Keywords

Stochastic partial differential equationmild solutionstationary distributionexponential integrator schemenumerical approximation

MSC classification

Primary: 60H15: Stochastic partial differential equations
Secondary: 65C30: Stochastic differential and integral equations 35K90: Abstract parabolic equations
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 858 - 873
Copyright
© Applied Probability Trust 

References

Bao, J., Hou, Z. and Yuan, C. (2010). Stability in distribution of mild solutions to stochastic partial differential equations. Proc. Amer. Math. Soc. 138, 2169–2180.Google Scholar
Bréhier, C. É. (2014). Approximation of the invariant measure with a Euler scheme for stochastic PDEs driven by space-time white noise. Potential Anal. 40, 1–40.Google Scholar
Caraballo, T. and Kloeden, P E. (2006). The pathwise numerical approximation of stationary solutions of semilinear stochastic evolution equations. Appl. Math. Optim. 54, 401–415.CrossRefGoogle Scholar
Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Cambridge University Press.CrossRefGoogle Scholar
Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite-Dimensional Systems. Cambridge University Press.Google Scholar
Da Prato, G., Jentzen, A. and Röckner, M. (2011). A mild Itô formula for SPDEs. Preprint. Available at http://uk.arxiv.org/abs/1009.3526v3.Google Scholar
Debussche, A. (2011). Weak approximation of stochastic partial differential equations: the nonlinear case. Math. Comp. 80, 89–117.Google Scholar
Grecksch, W. and Kloeden, P. E. (1996). Time-discretised Galerkin approximations of parabolic stochastic PDEs. Bull. Austral. Math. Soc. 54, 79–85.CrossRefGoogle Scholar
Gyöngy, I. (1998). Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. I. Potential Anal. 9, 1–25.Google Scholar
Gyöngy, I. (1999). Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. II. Potential Anal. 11, 1–37.CrossRefGoogle Scholar
Hardy, G. H., Littlewood, J. E. and Pólya, G. (1952). Inequalities, 2nd edn. Cambridge University Press.Google Scholar
Hausenblas, E. (2002). Numerical analysis of semilinear stochastic evolution equations in Banach spaces. J. Comput. Appl. Math. 147, 485–516.Google Scholar
Hausenblas, E. (2003). Approximation for semilinear stochastic evolution equations. Potential Anal. 18, 141–186.Google Scholar
Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.Google Scholar
Jentzen, A. (2011). Higher order pathwise numerical approximations of SPDEs with additive noise. SIAM J. Numer. Anal. 49, 642–667.CrossRefGoogle Scholar
Jentzen, A. and Kloeden, P. E. (2011). Taylor Approximations for Stochastic Partial Differential Equations. Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
Kato, T. (1966). Perturbation Theory for Linear Operators. Springer, New York.Google Scholar
Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Berlin.Google Scholar
Kloeden, P. E., Lord, G. J., Neuenkirch, A. and Shardlow, T. (2011). The exponential integrator scheme for stochastic partial differential equations: pathwise error bounds. J. Comput. Appl. Math. 235, 1245–1260.Google Scholar
Mao, X. and Yuan, C. (2006). Stochastic Differential Equations with Markovian Switching. Imperial College Press, London.CrossRefGoogle Scholar
Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations (Appl. Math. Sci. 44) Springer, New York.Google Scholar
Schurz, H. (1997). Stability, Stationarity, and Boundedness of Some Implicit Numerical Methods for Stochastic Differential Equations and Applications. Logos, Berlin.Google Scholar
Shardlow, T. (1999). Numerical methods for stochastic parabolic PDEs. Numer. Funct. Anal. Optim. 20, 121–145.Google Scholar
Yevik, A. and Zhao, H. (2011). Numerical approximations to the stationary solutions of stochastic differential equations. SIAM J. Numer. Anal. 49, 1397–1416.Google Scholar
Yoo, H. (2000). Semi-discretization of stochastic partial differential equations on {\rm R}^1 by a finite-difference method. Math. Comput. 69, 653–666.Google Scholar
Yuan, C. and Mao, X. (2004). Stability in distribution of numerical solutions for stochastic differential equations. Stoch. Anal. Appl. 22, 1133–1150.Google Scholar