## Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term.(English)Zbl 1054.60065

Following a pioneering paper by Freidlin and subsequent work by Sowers as well as Peszat, large deviations upper and lower bounds are established for some stochastic reaction-diffusion systems considered in their small noise limit. The main innovations of the present paper come from the fact that the reaction coefficients are no more assumed to be globally Lipschitz, but only locally Lipschitz and of polynomial growth, whereas no global boundedness assumption is made on the diffusion coefficients, which are rather assumed to be globally Lipschitz. Under such generous hypotheses, several new difficulties have to be overcome in the establishment of a large deviations principle; in particular, the authors are forced to leave the context of Hilbert spaces by working on a space of continuous functions instead of an $$L^2$$ space, and providing some exponential estimates in a sup-norm rather than in an $$L^2$$ norm. In return, these large deviations results are valid for stochastic reaction-diffusion systems having polynomial reaction terms and an unbounded multiplicative noise term, such as e.g. systems of stochastic Ginzburg-Landau equations with multiplicative noise.

### MSC:

 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60F10 Large deviations
Full Text:

### References:

 [1] Azencott, R. (1980). Grandes deviations et applications. Ecole d’Eté de Probabilités de Saint-Flour VII. Lecture Notes in Math. 774 1–176. Springer, New York. · Zbl 0435.60028 [2] Cardon-Weber, C. (1999). Large deviations for Burger’s type SPDE. Stochastic Process. Appl. 84 53–70. · Zbl 0996.60073 [3] Cerrai, S. (2001). Second Order PDE’s in Finite and Infinite Dimension. A Probabilistic Approach. Lecture Notes in Math. 1762 . Springer, New York. · Zbl 0983.60004 [4] Cerrai, S. (2003). Stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term. Probab. Theory Related Fields 125 271–304. · Zbl 1027.60064 [5] Chenal, F. and Millet, A. (1997). Uniform large deviations for parabolic SPDE’s and applications. Stochastic Process. Appl. 72 161–187. · Zbl 0942.60056 [6] Chow, P. L. and Menaldi, J. L. (1990). Exponential estimates in exit probability for some diffusion processes in Hilbert spaces. Stochastics Stochastics Rep. 29 377–393. · Zbl 0699.60047 [7] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions . Cambridge Univ. Press. · Zbl 0761.60052 [8] Davies, E. B. (1989). Heat Kernels and Spectral Theory . Cambridge Univ. Press. · Zbl 0699.35006 [9] Faris, W. G. and Jona-Lasinio, G. (1982). Large fluctuations for a non linear heat equation with noise. J. Phys. A 15 3025–3055. · Zbl 0496.60060 [10] Freidlin, M. I. (1988). Random perturbations of reaction–diffusion equations : The quasi deterministic approximation. Trans. Amer. Math. Soc. 305 665–697. JSTOR: · Zbl 0673.35049 [11] Freidlin, M. I. and Wentzell, A. D. (1984). Random Perturbations of Dynamical Systems . Springer, New York. · Zbl 0522.60055 [12] Kallianpur, G. and Xiong, J. (1996). Large deviations for a class of stochastic partial differential equations. Ann. Probab. 24 320–345. · Zbl 0854.60026 [13] Leandre, R. (1993). A simple proof for a large deviation theorem. In Barcelona Seminar on Stochastic Analysis (D. Nualart and M. Sanz Solé, eds.) 72–76. Birkhäuser, Basel. · Zbl 0789.60023 [14] Lunardi, A. (1995). Analytic Semigroups and Optimal Regularity in Parabolic Problems . Birkhäuser, Basel. · Zbl 0816.35001 [15] Peszat, S. (1992). Exponential tail estimates for infinite-dimensional stochastic convolutions. Bull. Polish Acad. Sci. Math. 40 323–333. · Zbl 0818.60050 [16] Peszat, S. (1994). Large deviation estimates for stochastic evolution equations. Probab. Theory Related Fields 98 113–136. · Zbl 0792.60057 [17] Priouret, P. (1982). Remarques sur les petites perturbations de systemes dynamiques. Séminaire de Probabilités XVI. Lecture Notes in Math. 920 184–200. Springer, New York. · Zbl 0484.60021 [18] Sowers, R. (1992). Large deviations for a reaction–diffusion equation with non-Gaussian perturbation. Ann. Probab. 20 504–537. JSTOR: · Zbl 0749.60059 [19] Walsh, J. B. (1992). An introduction to stochastic partial differential equations. Ecole d’Eté de Probabilité de Saint-Flour XIV. Lecture Notes in Math. 1180 265–439. Springer, New York. · Zbl 0608.60060 [20] Zabczyk, J. (1988). On large deviations for stochastic evolution equations. Stochastic Systems and Optimization. Lecture Notes in Control and Inform. Sci. Springer, Berlin. · Zbl 0696.60056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.