Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T05:40:41.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Moderate deviation principle for a class of stochastic partial differential equations

Published online by Cambridge University Press:  24 March 2016

Parisa Fatheddin  and
Jie Xiong
Jie Xiong
Affiliation:
Department of Mathematics, FST, University of Macau, PO Box 3001, Macau, China.
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish the moderate deviation principle for the solutions of a class of stochastic partial differential equations with non-Lipschitz continuous coefficients. As an application, we derive the moderate deviation principle for two important population models: super-Brownian motion and the Fleming–Viot process.

Keywords

Moderate deviation principleFleming–Viot processstochastic partial differential equationsuper-Brownian motion
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 1 , March 2016 , pp. 279 - 292
Copyright
Copyright © Applied Probability Trust 2016 

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.)

References

[1]Budhiraja, A., Dupuis, P. and Maroulas, V. (2008). Large deviations for infinite dimensional stochastic dynamical systems. Ann Prob. 36, 1390–1420. CrossRefGoogle Scholar
[2]Dawson, D. A. and Feng, S. (1998). Large deviations for the Fleming–Viot process with neutral mutation and selection. Stoch. Process. Appl. 77, 207–232. (Corrigendum: 88 (2000), 175.) CrossRefGoogle Scholar
[3]Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications, 2nd edn. Springer, Berlin. CrossRefGoogle Scholar
[4]Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. John Wiley, New York. CrossRefGoogle Scholar
[5]Fleischmann, K., Gärtner, J. and Kaj, I. (1996). A Schilder type theorem for super-Brownian motion. Canad. J. Math. 48, 542–568. CrossRefGoogle Scholar
[6]Fatheddin, P. (2015). Central limit theorem for a class of SPDEs. J. Appl. Prob. 52, 786–796. CrossRefGoogle Scholar
[7]Fatheddin, P. and Xiong, J. (2015). Large deviation principle for some measure-valued processes. Stoch. Process. Appl. 125, 970–993. CrossRefGoogle Scholar
[8]Feng, S. and Xiong, J. (2002). Large deviations and quasi-potential of a Fleming–Viot process. Electron. Commun. Prob. 7, 13–25. CrossRefGoogle Scholar
[9]Hong, W.-M. (2002). Moderate deviation for super-Brownian motion with super-Brownian immigration. J. Appl. Prob. 39, 829–838. CrossRefGoogle Scholar
[10]Hong, W.-M. (2008). Moderate deviations for the quenched mean of the super-Brownian motion with random immigration. Sci. China Ser. A 51, 343–350. CrossRefGoogle Scholar
[11]Kurtz, T. G. and Xiong, J. (1999). Particle representations for a class of nonlinear SPDEs. Stoch. Process. Appl. 83, 103–126. CrossRefGoogle Scholar
[12]Mitoma, I. (1985). An ∞-dimensional inhomogeneous Langevin's equation. J. Funct. Anal. 61, 342–359. CrossRefGoogle Scholar
[13]Schied, A. (1997). Moderate deviations and functional LIL for super-Brownian motion. Stoch. Process. Appl. 72, 11–25. CrossRefGoogle Scholar
[14]Xiong, J. (2013). Super-Brownian motion as the unique strong solution to an SPDE. Ann. Prob. 41, 1030–1054. CrossRefGoogle Scholar
[15]Yang, X. and Zhang, M. (2012). Moderate deviation for the single point catalytic super-Brownian motion. Acta Math. Sinica 28, 1799–1808. CrossRefGoogle Scholar
[16]Zhang, M. (2008). Some scaled limit theorems for an immigration super-Brownian motion. Sci. China Ser. A 51, 203–214. CrossRefGoogle Scholar