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Budhiraja, Amarjit; Dupuis, Paul; Maroulas, Vasileios

Large deviations for infinite dimensional stochastic dynamical systems. (English) Zbl 1155.60024

Ann. Probab. 36, No. 4, 1390-1420 (2008).
This work contains a well presented result of the large deviation principle for stochastic partial differential equations. The approach avoids the use of approximations and is based on a variational representation for functionals of Brownian motion. The results are applicable to a variety of infinite dimensional SPDEs driven by an infinite dimensional cylindrical Wiener process.
Proofs of large deviations properties are reduced to basic qualitative properties (existence, uniqueness and tightness) of certain perturbations of the original process.
Reviewer: Dirk Blömker (Augsburg)

Cited in 7 Reviews
Cited in 176 Documents

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60F10 Large deviations
37L55 Infinite-dimensional random dynamical systems; stochastic equations

Keywords:

large deviations; Brownian sheet; Freidlin-Wentzell; LDP; large deviation principle; stochastic partial differential equations; stochastic evolution equations; stochastic reaction-diffusion equation; small noise asymptotics; infinite dimensional Brownian motion
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References:

[1] Azencott, R. (1980). Grandes deviations et applications. École d ’ Été de Probabilités de Saint-Flour VII. Lecture Notes in Math. 774 1-176. Springer, Berlin. · Zbl 0435.60028
[2] Cardon-Webber, C. (1999). Large deviations for Burger’s type SPDE. Stochastic Process. Appl. 84 53-70. · Zbl 0996.60073 · doi:10.1016/S0304-4149(99)00047-2
[3] Budhiraja, A. and Dupuis, P. (2000). A variational representation for positive functional of infinite dimensional Brownian motions. Probab. Math. Statist. 20 39-61. · Zbl 0994.60028
[4] Cerrai, S. and Rockner, M. (2004). Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Ann. Probab. 32 1100-1139. · Zbl 1054.60065 · doi:10.1214/aop/1079021473
[5] Chenal, F. and Millet, A. (1997). Uniform large deviations for parabolic SPDEs and applications. Stochastic Process. Appl. 72 161-187. · Zbl 0942.60056 · doi:10.1016/S0304-4149(97)00091-4
[6] Chow, P. L. (1992). Large deviation problem for some parabolic Itô equations. Comm. Pure Appl. Math. 45 97-120. · Zbl 0739.60055 · doi:10.1002/cpa.3160450105
[7] Da Prato, G. and Zabcyk, J. (1992). Stochastic Equations in Infinite Dimensions . Cambridge Univ. Press. · Zbl 0761.60052 · doi:10.1017/CBO9780511666223
[8] Dupuis, P. and Ellis, R. (1997). A Weak Convergence Approach to the Theory of Large Deviations . Wiley, New York. · Zbl 0904.60001
[9] Faris, W. G. and Jonas-Lasino, G. (1982). Large fluctuations for a nonlinear heat equation with noise. J. Phys. A 15 3025-3055. · Zbl 0496.60060 · doi:10.1088/0305-4470/15/10/011
[10] Feng, J. (2006). Large deviations for diffusion and Hamilton-Jacobi equation in Hilbert space. Ann. Probab. 34 321-385. · Zbl 1091.60002
[11] Feng, J. and Kurtz, T. G. (2006). Large Deviations of Stochastic Processes . Amer. Math. Soc., Providence, RI. · Zbl 1113.60002
[12] Freidlin, M. (1988). Random perturbations of reaction-diffusion equations: The quasi deterministic approximation. Trans. Amer. Math. Soc. 305 665-697. JSTOR: · Zbl 0673.35049 · doi:10.2307/2000884
[13] Freidlin, M. I. and Wentzell, A. D. (1984). Random Perturbations of Dynamical Systems . Springer, New York. · Zbl 0522.60055
[14] Garsia, A. (1972). Continuity properties of Gaussian processes with multidimensional time parameter. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 2 369-374. Univ. California Press, Berkeley. · Zbl 0272.60034
[15] Ibragimov, I. (1983). On smoothness conditions for trajectories of random functions. Theory Probab. Appl. 28 240-262. · Zbl 0533.60046 · doi:10.1137/1128023
[16] Imauikin, V. and Komech, A. I. (1988). Large deviations of solutions of nonlinear stochastic equations. Trudy Sem. Petrovsk. 13 177-196. · Zbl 0675.60022
[17] Kallenberg, O. (2002). Foundations of Modern Probability , 2nd ed. Springer, New York. · Zbl 0996.60001
[18] Kallianpur, G. and Xiong, J. (1995). Stochastic Differential Equations in Infinite-Dimensional Spaces . IMS, Hayward, CA. · Zbl 0845.60008
[19] Kallianpur, G. and Xiong, J. (1996). Large deviations for a class of stochastic partial differential equations. Ann. Probab. 24 320-345. · Zbl 0854.60026 · doi:10.1214/aop/1042644719
[20] Kotelenez, P. (1992). Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equations. Stochastics Stochastics Rep. 41 177-199. · Zbl 0766.60078
[21] Kushner, H. J. and Dupuis, P. (1992). Numerical Methods for Stochastic Control Problems in Continuous Time . Springer, New York. · Zbl 0754.65068
[22] Peszat, S. (1994). Large deviation estimates for stochastic evolution equations. Probab. Theory Related Fields 98 113-136. · Zbl 0792.60057 · doi:10.1007/BF01311351
[23] Ren, J. and Zhang, X. (2005). Schilder theorem for the Brownian motion on the diffeomorphism group of the circle. J. Funct. Anal. 224 107-133. · Zbl 1080.60024 · doi:10.1016/j.jfa.2004.08.006
[24] Ren, J. and Zhang, X. (2005). Freidlin-Wentzell’s large deviations for homeomorphism flows of non-Lipschitz SDEs. Bull. Sci. Math. 129 643-655. · Zbl 1086.60036 · doi:10.1016/j.bulsci.2004.12.005
[25] Sowers, R. B. (1992). Large deviations for a reaction-diffusion equation with non-Gaussian perturbations. Ann. Probab. 20 504-537. · Zbl 0749.60059 · doi:10.1214/aop/1176989939
[26] Sritharan, S. S. and Sundar, P. (2006). Large deviations for the two dimensional Navier-Stokes equations with multiplicative noise. Stochastic Process. Appl. 116 1636-1659. · Zbl 1117.60064 · doi:10.1016/j.spa.2006.04.001
[27] Wang, W. and Duan, J. Reductions and deviations for stochastic differential equations under fast dynamical boundary conditions. · Zbl 1166.60038 · doi:10.1080/07362990802679166
[28] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. École d’Été de Probabilités de Saint Flour XIV-1984. Lecture Notes in Math. 1180 265-439. Springer, Berlin. · Zbl 0608.60060
[29] Xiong, J. (1996). Large deviations for diffusion processes in duals of nuclear spaces. Appl. Math. Optim. 34 1-27. · Zbl 0865.60018 · doi:10.1007/BF01182470
[30] Zabczyk, J. (1988). On large deviations for stochastic evolution equations. Stochastic Systems and Optimization ( Warsaw , 1988 ). Lecture Notes in Control and Inform. Sci. 136 240-253. Springer, Berlin. · Zbl 0696.60056 · doi:10.1007/BFb0002685
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