Approximation of quasipotentials and exit problems for multidimensional RDE's with noise
Authors:
Sandra Cerrai and Mark Freidlin
Journal:
Trans. Amer. Math. Soc. 363 (2011), 38533892
MSC (2010):
Primary 60H15, 60F10, 35K57, 49J45
Published electronically:
February 10, 2011
MathSciNet review:
2775830
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We deal with a class of reactiondiffusion equations, in space dimension , perturbed by a Gaussian noise which is white in time and colored in space. We assume that the noise has a small correlation radius , so that it converges to the white noise , as . By using arguments of convergence, we prove that, under suitable assumptions, the quasipotential converges to the quasipotential , corresponding to spacetime white noise, in spite of the fact that the equation perturbed by spacetime 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.
Sandra
Cerrai, Second order PDE’s in finite and infinite
dimension, Lecture Notes in Mathematics, vol. 1762,
SpringerVerlag, Berlin, 2001. A probabilistic approach. MR 1840644
(2002j:35327)
 2.
Sandra
Cerrai, Stochastic reactiondiffusion systems with multiplicative
noise and nonLipschitz reaction term, Probab. Theory Related Fields
125 (2003), no. 2, 271–304. MR 1961346
(2004a:60117), http://dx.doi.org/10.1007/s0044000202306
 3.
Sandra
Cerrai, Asymptotic behavior of systems of stochastic partial
differential equations with multiplicative noise, Stochastic partial
differential equations and applications—VII, Lect. Notes Pure Appl.
Math., vol. 245, Chapman & Hall/CRC, Boca Raton, FL, 2006,
pp. 61–75. MR 2227220
(2008e:60187), http://dx.doi.org/10.1201/9781420028720.ch7
 4.
Sandra
Cerrai and Michael
Röckner, Large deviations for stochastic reactiondiffusion
systems with multiplicative noise and nonLipschitz reaction term,
Ann. Probab. 32 (2004), no. 1B, 1100–1139. MR 2044675
(2005b:60063), http://dx.doi.org/10.1214/aop/1079021473
 5.
Sandra
Cerrai and Michael
Röckner, Large deviations for invariant measures of stochastic
reactiondiffusion systems with multiplicative noise and nonLipschitz
reaction term, Ann. Inst. H. Poincaré Probab. Statist.
41 (2005), no. 1, 69–105 (English, with English
and French summaries). MR 2110447
(2005k:60084), http://dx.doi.org/10.1016/j.anihpb.2004.03.001
 6.
Giuseppe
Da Prato and Jerzy
Zabczyk, Stochastic equations in infinite dimensions,
Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge
University Press, Cambridge, 1992. MR 1207136
(95g:60073)
 7.
G.
Da Prato, A.
J. Pritchard, and J.
Zabczyk, On minimum energy problems, SIAM J. Control Optim.
29 (1991), no. 1, 209–221. MR 1088228
(92b:49049), http://dx.doi.org/10.1137/0329012
 8.
Gianni
Dal Maso, An introduction to Γconvergence, Progress in
Nonlinear Differential Equations and their Applications, 8, Birkhäuser
Boston, Inc., Boston, MA, 1993. MR 1201152
(94a:49001)
 9.
Amir
Dembo and Ofer
Zeitouni, Large deviations techniques and applications, 2nd
ed., Applications of Mathematics (New York), vol. 38, SpringerVerlag,
New York, 1998. MR 1619036
(99d:60030)
 10.
M.
I. Freidlin and A.
D. Wentzell, Random perturbations of dynamical systems, 2nd
ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles
of Mathematical Sciences], vol. 260, SpringerVerlag, New York, 1998.
Translated from the 1979 Russian original by Joseph Szücs. MR 1652127
(99h:60128)
 11.
Eric
Gautier, Exit from a basin of attraction for stochastic weakly
damped nonlinear Schrödinger equations, Ann. Probab.
36 (2008), no. 3, 896–930. MR 2408578
(2009d:60206), http://dx.doi.org/10.1214/07AOP344
 12.
Thomas
Runst and Winfried
Sickel, Sobolev spaces of fractional order, Nemytskij operators,
and nonlinear partial differential equations, de Gruyter Series in
Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter &
Co., Berlin, 1996. MR 1419319
(98a:47071)
 13.
Richard
Sowers, Large deviations for the invariant measure of a
reactiondiffusion equation with nonGaussian perturbations, Probab.
Theory Related Fields 92 (1992), no. 3,
393–421. MR 1165518
(93h:60043), http://dx.doi.org/10.1007/BF01300562
 14.
Hans
Triebel, Interpolation theory, function spaces, differential
operators, NorthHolland Mathematical Library, vol. 18,
NorthHolland Publishing Co., AmsterdamNew York, 1978. MR 503903
(80i:46032b)
 1.
 S. Cerrai, SECOND ORDER PDE'S IN FINITE AND INFINITE DIMENSION. A PROBABILISTIC APPROACH, Lecture Notes in Mathematics Series 1762, SpringerVerlag (2001). MR 1840644 (2002j:35327)
 2.
 S. Cerrai, Stochastic reactiondiffusion systems with multiplicative noise and nonLipschitz reaction term, Probability Theory and Related Fields 125 (2003), pp. 271304. MR 1961346 (2004a:60117)
 3.
 S. Cerrai, Asymptotic behavior of systems of SPDE's with multiplicative noise, Stochastic Partial Differential Equations and Applications VII, Lecture Notes in Pure and Applied Mathematics 245 (2005), Chapman and Hall/CRC Press, pp. 6175. MR 2227220 (2008e:60187)
 4.
 S. Cerrai, M. Röckner, Large deviations for stochastic reactiondiffusion systems with multiplicative noise and nonLipschitz reaction term, Annals of Probability 32 (2004), pp. 11001139. MR 2044675 (2005b:60063)
 5.
 S. Cerrai, M. Röckner, Large deviations for invariant measures of stochastic reactiondiffusion systems with multiplicative noise and nonLipschitz reaction term, Annales de l'Institut Henri Poincaré (Probabilités et Statistiques) 41 (2005), pp. 69105. MR 2110447 (2005k:60084)
 6.
 G. Da Prato, J. Zabczyk, STOCHASTIC EQUATIONS IN INFINITE DIMENSIONS, Cambridge University Press, Cambridge (1992). MR 1207136 (95g:60073)
 7.
 G. Da Prato, A. J. Pritchard, J. Zabczyk, On minimum energy problems, SIAM Journal on Control and Optimization 29 (1991), pp. 209221. MR 1088228 (92b:49049)
 8.
 G. Del Maso, AN INTRODUCTION TO CONVERGENCE, Birkäuser Boston (1993). MR 1201152 (94a:49001)
 9.
 A. Dembo, O. Zeitouni, LARGE DEVIATIONS TECHNIQUES AND APPLICATIONS, Second Edition, SpringerVerlag (1998). MR 1619036 (99d:60030)
 10.
 M.I. Freidlin, A.D. Wentzell, RANDOM PERTURBATIONS OF DYNAMICAL SYSTEMS, Second edition, SpringerVerlag, New York (1998). MR 1652127 (99h:60128)
 11.
 E. Gautier, Exit from a basin of attraction for stochastic weakly damped nonlinear Schrödinger equations, Annals of Probability 36 (2008), pp. 896930. MR 2408578 (2009d:60206)
 12.
 T. Runst, W. Sickel, SOBOLEV SPACES OF FRACTIONAL ORDER, NEMYTSKIJ OPERATORS AND NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, Walter de Gruyter, Berlin, New York (1996). MR 1419319 (98a:47071)
 13.
 R. Sowers, Large deviations for the invariant measure of a reactiondiffusion equation with nonGaussian perturbations, Probability Theory and Related Fields 92 (1992), pp. 393421. MR 1165518 (93h:60043)
 14.
 H. Triebel, INTERPOLATION THEORY, FUNCTION SPACES, DIFFERENTIAL OPERATORS, NorthHolland, Amsterdam (1978). MR 503903 (80i:46032b)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2010):
60H15,
60F10,
35K57,
49J45
Retrieve articles in all journals
with MSC (2010):
60H15,
60F10,
35K57,
49J45
Additional Information
Sandra Cerrai
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742
Mark Freidlin
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742
DOI:
http://dx.doi.org/10.1090/S000299472011053523
PII:
S 00029947(2011)053523
Received by editor(s):
June 5, 2009
Received by editor(s) in revised form:
March 19, 2010
Published electronically:
February 10, 2011
Additional Notes:
The second author was partially supported by an NSF grant
Article copyright:
© Copyright 2011
American Mathematical Society
