APPROXIMATION OF QUASI-POTENTIALS AND EXIT PROBLEMS FOR MULTIDIMENSIONAL RDE'S WITH NOISE

We deal with a class of reaction-diffusion equations, in space dimension d > 1, perturbed by a Gaussian noise ∂w δ /∂t which is white in time and colored in space. We assume that the noise has a small correlation radius δ, so that it converges to the white noise ∂W/∂t, as δ ↓ 0. By using arguments of Γ-convergence, we prove that, under suitable assumptions, the quasi-potential V δ converges to the quasi-potential V, corresponding to space-time white noise, in spite of the fact that the equation perturbed by space-time white noise has no solution. We apply these results to the asymptotic estimate of the mean of the exit time of the solution of the stochastic problem from a basin of attraction of an asymptotically stable point for the unperturbed problem.

[1]  T. Kurtz,et al.  Stochastic equations in infinite dimensions , 2006 .

[2]  Giuseppe Da Prato,et al.  Stochastic Partial Differential Equations and Applications - VII , 2005 .

[3]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[4]  H. Triebel.  Interpolation Theory, Function Spaces, Differential Operators , 1978 .

[5]  M. Röckner,et al.  Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipshitz reaction term , 2004 .

[6]  S. Cerrai.  Second Order Pde's in Finite and Infinite Dimension: A Probabilistic Approach , 2001 .

[7]  S. Cerrai.  Asymptotic Behavior of Systems of Stochastic Partial Differential Equations with Multiplicative Noise , 2005 .

[8]  Richard B. Sowers,et al.  Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations , 1992 .

[9]  S. Varadhan,et al.  A probabilistic approach to , 1974 .

[10]  S. Cerrai.  Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term , 2003 .

[11]  G. D. Maso,et al.  An Introduction to-convergence , 1993 .

[12]  Michael Röckner,et al.  Large deviations for invariant measures of stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term , 2005 .

[13]  Winfried Sickel,et al.  Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations , 1996, de Gruyter series in nonlinear analysis and applications.

[14]  E. Gautier.  Exit from a basin of attraction for stochastic weakly damped nonlinear Schr , 2006, math/0602350.

[15]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[16]  J. Zabczyk,et al.  On minimum energy problems , 1991 .