Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 8 (2003) > Paper 12 open journal systems 


Large Deviation Principle for a Stochastic Heat Equation With Spatially Correlated Noise

David Marquez-carreras, Universitat de Barcelona
Monica Sarra, Universitat de Barcelona


Abstract
In this paper we prove a large deviation principle (ldp) for a perturbed stochastic heat equation defined on $[0,T]times [0,1]^d$. This equation is driven by a Gaussian noise, white in time and correlated in space. Firstly, we show the H"older continuity for the solution of the stochastic heat equation. Secondly, we check that our Gaussian process satisfies a ldp and some requirements on the skeleton of the solution. Finally, we prove the called Freidlin-Wentzell inequality. In order to obtain all these results we need precise estimates of the fundamental solution of this equation.


Full text: PDF

Pages: 1-39

Published on: July 18, 2003
Bibliography
  1. Azencott R., Grandes d'eviations et applications, Ecole d''Et'e de Probabilit'es de Saint Flour VIII-1978, Lectures Notes in Math. 774 (1980) 1-176, Springer-Verlag, Berlin. MR:81m:58085
  2. Chenal F. Principes de grandes d'eviations pour des 'equations aux d'eriv'ees partielles stochastiques et applications, Th`ese de doctorat de l'Universit'e Paris 6, Paris.
  3. Chenal F. and Millet A. Uniform large deviations for parabolic SPDE's and applications, Stochastic Processes and their Applications 72 (1997) 161-186. MR:98m:60038
  4. Dalang R.C. Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e's, Electronic Journal of Probability Vol.4 (1999) 1-29. http://www.math.washington.edu/~ejpecp/EjpVol4/paper6. MR:2000b:60132
  5. Dalang R.C. and Frangos N.E. The stochastic wave equation in two spatial dimensions, Annals of Probabability 26-1 (1998) 187-212. MR:99c:60127
  6. Dembo A. and Zeitouni O. Large deviations techniques and applications, (1983) Jones and Barlett Publishers, Boston.
  7. Deuschel J.D. and Stroock D.W. Large deviations, Pure and Applied Mathematics, vol 137, (1989) Academic Press, Boston. MR:90h:60026
  8. Eidelman S.D. and Ivasisen S.D. Investigation of the Green matrix for a homogeneous parabolic boundary value problem, Trans. Moscow Math. Soc. 23 (1970) 179-242. MR:51#3697
  9. Franzova N. Long time existence for the heat equation with a spatially correlated noise term, Stochastic analysis and Applications 17-2 (1999) 169-190. MR:99m:35278
  10. Freidlin M.I. and Wentzell A.D. Random perturbation of dynamical systems, (1984) Springer-Verlag, New York. MR:85a:60064
  11. Friedman, A. (1964) Partial differential equations of parabolic type, (1964) Prentice-Hall, New York. MR: 31#6062
  12. Karczewska A. and Zabczyk J. Stochastic PDE's with function-valued solutions, to appear in Clément Ph., den Hollander F., van Neerven J., and de Pagter B. (Eds.), "Infinite-Dimensional Stochastic Analysis", Proceedings of the Colloquium of the Royal Netherlands Academy of Arts and Sciences, (1999) Amsterdam. MR:2002h:60132
  13. Márquez-Carreras D., Mellouk M. and Sarrá M. On stochastic partial differential equations with spatially correlated noise: smoothness of the law, Stochastic Processes and their Applications 93 (2001) 269-284. MR: 2002e:60089
  14. Márquez-Carreras D. and Sanz-Solé M.} Smal perturbations in a hyperbolic stochastic partial differential equation, Stochastic Processes and their Applications 68 (1997) 133-154. MR:98d:60124
  15. Millet A. and Morien P.L. On a stochastic wave equation in two space dimension regularity of the solution and its density, Prépublications de Mathématiques de l'Université de Paris 10 98/5.
  16. Millet A. and Sanz-Solé M. A stochastic wave equation in two space dimension: Smoothness of the law, Annals of Probability 27-2 (1999) 803-844. MR:2001e:60130
  17. Peszat S. and Zabczyk J. Stochastic evolution equations with a spatially homogeneous Wiener process, Stochastic Processes and their Applications 72 (1997) 187-204. MR:99k:60166
  18. Peszat S. and Zabczyk J. Nonlinear stochastic wave and heat equations, Preprint of Institut of Mathematics. Polish Academy of Sciences 584 (1998).
  19. Priouret P. Remarques sur les petits perturbations de syst&ecute;mes dynamiques, Séminaire de Probabilités XVI, Lecture Notes in Math. 920 (1982) 184-200, Springer, Berlin. MR:84g:58096
  20. Sanz-Solé M. and Sarrá M. Path properties of a class of Gaussian processes with applications to spde's, Canadian Mathematical Society, Conference Proceedings 28 (2000) 303-316. MR:2001m:60148
  21. Sanz-Solé M. and Sarrá M. H"older continuity for the stochastic heat equation with spatially correlated noise, Progress in Probability 52 (2002) 259-268. MR:1 958 822
  22. Sowers R.B. Large deviations for a reaction-diffusion equation with non-Gaussian perturbations, Annals of Probability 20 (1992) 504-537. MR:93e:60125
  23. Walsh J.B. An introduction to stochastic partial differential equations, École d'Été de Prob. de St-Flour XIV-1984, Lect. Notes in Math. 1180, (1986) Springer-Verlag. MR:88a:60114
  24. Watanabe S. Stochastic differential equation and Malliavin Calculus, Tata Institue, (1984) Springer-Verlag. MR:86b:60113
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489