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LARGE DEVIATIONS FOR INFINITE-DIMENSIONAL STOCHASTIC SYSTEMS WITH JUMPS

Part of: Limit theorems Infinite-dimensional dissipative dynamical systems

Published online by Cambridge University Press:  21 December 2010

Vasileios Maroulas
Vasileios Maroulas*
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A (email: [email protected])
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Abstract

Uniform large deviation principles for positive functionals of all equivalent types of infinite-dimensional Brownian motions acting together with a Poisson random measure are established. The core of our approach is a variational representation formula, which for an infinite sequence of independent and identically distributed real Brownian motions and a Poisson random measure was shown in [A. Budhiraja, P. Dupuis and V. Maroulas, Variational representations for continuous time processes. Ann. Inst. H. Poincaré (to appear)].

MSC classification

Secondary: 60F10: Large deviations 37L55: Infinite-dimensional random dynamical systems; stochastic equations
Type
Research Article
Information
Mathematika , Volume 57 , Issue 1 , January 2011 , pp. 175 - 192
Copyright
Copyright © University College London 2011

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References

[1]Bessaih, H. and Millet, A., Large deviation principle and inviscid shell models. Electron. J. Probab. 14 (2009), 2551–2579.CrossRefGoogle Scholar
[2]Budhiraja, A. and Dupuis, P., A variational representation for positive functional of infinite dimensional Brownian motions. Probab. Math. Statist. 20 (2000), 39–61.Google Scholar
[3]Budhiraja, A., Dupuis, P. and Maroulas, V., Large deviations for infinite dimensional stochastic dynamical systems. Ann. Probab. 36(4) (2008), 1390–1420.CrossRefGoogle Scholar
[4]Budhiraja, A., Dupuis, P. and Maroulas, V., Large deviations for stochastic flows of diffeomorphisms. Bernoulli 36(1) (2010), 234–257.Google Scholar
[5]Budhiraja, A., Dupuis, P. and Maroulas, V., Variational representations for continuous time processes. Ann. Inst. H. Poincaré (to appear).Google Scholar
[6]Chueshov, I. and Millet, A, Stochastic 2D hydrodynamical type systems: well posedness and large deviations. Appl. Math. Optim. 61(3) (2010), 379–420.CrossRefGoogle Scholar
[7]Dembo, A. and Zeitouni, O., Large Deviations Techniques and Applications, Academic Press (San Diego, CA, 1989).Google Scholar
[8]Deuschel, J.-D. and Stroock, D., Large Deviations, Academic Press (San Diego, CA, 1989).Google Scholar
[9]Du, A., Duan, J. and Gao, H., Small probability events for two-layer geophysical flows under uncertainty. Preprint.Google Scholar
[10]Duan, J. and Millet, A., Large deviations for the Boussinesq equations under random influences. Stochastic Process. Appl. 119(6) (2009), 2052–2081.CrossRefGoogle Scholar
[11]Dupuis, P. and Ellis, R., A Weak Convergence Approach to the Theory of Large Deviations, Wiley (New York, 1997).CrossRefGoogle Scholar
[12]Freidlin, M. I. and Wentzell, A. D., Random Perturbations of Dynamical Systems, Springer (New York, 1984).CrossRefGoogle Scholar
[13]Liu, W., Large deviations for stochastic evolution equations with small multiplicative noise. Appl. Math. Optim. 61(1) (2010), 27–56.CrossRefGoogle Scholar
[14]Manna, U., Sritharan, S. S. and Sundar, P., Large deviations for the stochastic shell model of turbulence. Nonlinear Differential Equations Appl. 16(4) (2009), 493–521.CrossRefGoogle Scholar
[15]Ren, J. and Zhang, X., Freidlin–Wentzell’s large deviations for homeomorphism flows of non-Lipschitz SDEs. Bull. Sci. Math. 129 (2005), 643–655.CrossRefGoogle Scholar
[16]Ren, J. and Zhang, X., Schilder theorem for the Brownian motion on the diffeomorphism group of the circle. J. Funct. Anal. 224(1) (2005), 107–133.CrossRefGoogle Scholar
[17]Rockner, M., Zhang, T. and Zhang, X., Large deviations for stochastic tamed 3D Navier–Stokes equations. Appl. Math. Optim. 61(2) (2010), 267–285.CrossRefGoogle Scholar
[18]Royden, H. L., Real Analysis, Prentice Hall (Englewood Cliffs, NJ, 1988).Google Scholar
[19]Sritharan, S. S. and Sundar, P., Large deviations for the two dimensional Navier–Stokes equations with multiplicative noise. Stochastic Process. Appl. 116 (2006), 1636–1659.CrossRefGoogle Scholar
[20]Varadhan, S. R. S., Large Deviations and Applications, SIAM (Philadelphia, 1984).CrossRefGoogle Scholar
[21]Wang, W. and Duan, J., Reductions and deviations for stochastic partial differential equations under fast dynamical boundary conditions. Stochastic Anal. Appl. 27(3) (2009), 431–459.CrossRefGoogle Scholar
[22]Yang, D. and Hou, Z., Large deviations for the stochastic derivative Ginzburg–Landau equation with multiplicative noise. Phys. D 237(1) (2008), 82–91.CrossRefGoogle Scholar
[23]Zhang, X., Euler schemes and large deviations for stochastic Volterra equations with singular kernels. J. Differential Equations 244(9) (2008), 2226–2250.CrossRefGoogle Scholar
[24]Zhang, X., Clark–Ocone formula and variational representation for Poisson functionals. Ann. Probab. 37(2) (2009), 506–529.CrossRefGoogle Scholar
[25]Zhang, X., Stochastic Volterra equations in Banach spaces and stochastic partial differential equations. J. Funct. Anal. 258(4) (2010), 1361–1425.CrossRefGoogle Scholar