Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise
Tóm tắt
The Freidlin-Wentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. As examples, the main results are applied to derive the large deviation principle for different types of SPDE such as stochastic reaction-diffusion equations, stochastic porous media equations and fast diffusion equations, and the stochastic p-Laplace equation in Hilbert space. The weak convergence approach is employed in the proof to establish the Laplace principle, which is equivalent to the large deviation principle in our framework.
Tài liệu tham khảo
Aronson, D.G.: The porous medium equation. In: Lecture Notes in Mathematics, vol. 1224, pp. 1–46. Springer, Berlin (1986)
Azencott, R.G.: Grandes deviations et applications. In: Ecole d’Eté de Probabilités de Saint-Flour VII. Lecture Notes in Mathematics, vol. 774. Springer, Berlin (1980)
Bensoussan, A.: Filtrage Optimale des Systemes Linéaires. Dunod, Paris (1971)
Bensoussan, A., Temam, R.: Equations aux derives partielles stochastiques non linéaires. Isr. J. Math. 11, 95–129 (1972)
Budhiraja, A., Dupuis, P.: A variational representation for positive functionals of infinite dimensional Brownian motion. Probab. Math. Stat. 20, 39–61 (2000)
Budhiraja, A., Dupuis, P., Maroulas, V.: Large deviations for infinite dimensional stochastic dynamical systems. Ann. Probab. 36, 1390–1420 (2008)
Bryc, W.: Large deviations by the asymptotic value method. In: Pinsky, M. (ed.) Diffusion Processes and Related Problems in Analysis, vol. 1, pp. 447–472. Birkhäuser, Boston (1990)
Cerrai, S., Röckner, M.: Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Ann. Probab. 32, 1100–1139 (2004)
Chow, P.L.: Large deviation problem for some parabolic Itô equations. Commun. Pure Appl. Math. 45, 97–120 (1992)
Da Prato, G., Röckner, M.: Weak solutions to stochastic porous media equations. J. Evol. Equ. 4, 249–271 (2004)
Da Prato, G., Zabczyk, J.: Stochastic Equations, Infinite Dimensions. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1992)
Da Prato, G., Röckner, M., Rozovskii, Wang, F.-Y.: Strong solutions to stochastic generalized porous media equations: existence, uniqueness and ergodicity. Commun. Partial Differ. Equ. 31(2), 277–291 (2006)
Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Springer, New York (2000)
Donsker, M.D., Varadhan, S.R.S.: Asymptotic evalution of certain Markov process expectations for large time, I. Commun. Pure Appl. Math. 28, 1–47 (1975)
Donsker, M.D., Varadhan, S.R.S.: Asymptotic evalution of certain Markov process expectations for large time, II. Commun. Pure Appl. Math. 28, 279–301 (1975)
Donsker, M.D., Varadhan, S.R.S.: Asymptotic evalution of certain Markov process expectations for large time, III. Commun. Pure Appl. Math. 29, 389–461 (1977)
Duan, J., Millet, A.: Large deviations for the Boussinesq equations under random influences. Stoch. Process. Appl. (to appear)
Dupuis, P., Ellis, R.: A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York (1997)
Feng, J., Kurtz, T.G.: Large Deviations of Stochastic Processes. Mathematical Surveys and Monographs, vol. 131. American Mathematical Society, Providence (2006)
Freidlin, M.I.: Random perturbations of reaction-diffusion equations: the quasi-deterministic approximations. Trans. Am. Math. Soc. 305, 665–697 (1988)
Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260. Springer, New York (1984). Translated from the Russian by Joseph Szu”cs
Gyöngy, I., Millet, A.: On discretization schemes for stochastic evolution equations. Potential Anal. 23, 99–134 (2005)
Krylov, N.V., Rozovskii, B.L.: Stochastic evolution equations. Translated from Itogi Naukii Tekhniki, Seriya Sovremennye Problemy Matematiki 14, 71–146 (1979), Plenum Publishing Corp. (1981)
Liu, W.: Harnack inequality and applications for stochastic evolution equations with monotone drifts. SFB-Preprint 09-023
Liu, W., Wang, F.-Y.: Harnack inequality and strong Feller property for stochastic fast diffusion equations. J. Math. Anal. Appl. 342, 651–662 (2008)
Pardoux, E.: Equations aux dérivées partielles stochastiques non linéaires monotones. Thesis, Université Paris XI (1975)
Peszat, S.: Large deviation principle for stochastic evolution equations. Probab. Theory Relat. Fields 98, 113–136 (1994)
Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1905. Springer, Berlin (2007)
Pukhalskii, A.A.: On the theory of large deviations. Theory Probab. Appl. 38, 490–497 (1993)
Ren, J., Zhang, X.: Freidlin-Wentzell large deviations for homeomorphism flows of non-Lipschitz SDE. Bull. Sci. 129, 643–655 (2005)
Ren, J., Zhang, X.: Schilder theorem for the Brownian motion on the diffeomorphism group of the circle. J. Funct. Anal. 224, 107–133 (2005)
Ren, J., Röckner, M., Wang, F.-Y.: Stochastic generalized porous media and fast diffusion equations. J. Differ. Equ. 238, 118–152 (2007)
Röckner, M., Wang, F.-Y., Wu, L.: Large deviations for stochastic generalized porous media equations. Stoch. Process. Appl. 116, 1677–1689 (2006)
Röckner, M., Schmuland, B., Zhang, X.: Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions. Condens. Matter Phys. 54, 247–259 (2008)
Sritharan, S.S., Sundar, P.: Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise. Stoch. Process. Appl. 116, 1636–1659 (2006)
Stroock, D.W.: An Introduction to the Theory of Large Deviations. Springer, New York (1984)
Varadhan, S.R.S.: Asymptotic probabilities and differential equations. Commun. Pure Appl. Math. 19, 261–286 (1966)
Varadhan, S.R.S.: Large Deviations and Applications. CBMS, vol. 46. SIAM, Philadelphia (1984)
Walsh, J.B.: An introduction to stochastic partial differential equations. In: Hennequin, P.L. (ed.) Ecole d’Ete de Probabilite de Saint-Flour XIV. Lecture Notes in Mathematics, vol. 1180, pp. 265–439. Springer, Berlin (1984)
Wang, F.-Y.: Harnack inequality and applications for stochastic generalized porous media equations. Ann. Probab. 35, 1333–1350 (2007)
Wu, L.: On large deviations for moving average processes. In: Lai, T.L., Yang, H.L., Yung, S.P. (eds.) Probability, Finance and Insurance. The Proceeding of a Workshop at the University of Hong-Kong, 15–17 July 2002, pp. 15–49. World Scientific, Singapore (2004)
Zeidler, E.: Nonlinear Functional Analysis and its Applications, II/B, Nonlinear Monotone Operators. Springer, New York (1990)
Zhang, X.: On Stochastic evolution equations with non-Lipschitz coefficients. Preprint