Some properties of superprocesses under a stochastic flow
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 2, pp. 477-490.

Nous montrons que, sous un flot stochastique en dimension un, un superprocess a une densité par rapport à la mesure de Lebesgue. Nous déduisons une équation différentielle stochastique satisfaite par la densité. Nous montrons ensuite la régularité de la solution en utilisant la theorie de Krylov pour les EDPS linéaires dans Lp.

For a superprocess under a stochastic flow in one dimension, we prove that it has a density with respect to the Lebesgue measure. A stochastic partial differential equation is derived for the density. The regularity of the solution is then proved by using Krylov's Lp-theory for linear SPDE.

DOI : 10.1214/08-AIHP171
Classification : 60G57, 60H15, 60J80
Mots-clés : superprocess, random environment, snake representation, stochastic partial differential equation 
@article{AIHPB_2009__45_2_477_0,
     author = {Lee, Kijung and Mueller, Carl and Xiong, Jie},
     title = {Some properties of superprocesses under a stochastic flow},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {477--490},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {2},
     year = {2009},
     doi = {10.1214/08-AIHP171},
     mrnumber = {2521410},
     zbl = {1171.60011},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/08-AIHP171/}
}
TY  - JOUR
AU  - Lee, Kijung
AU  - Mueller, Carl
AU  - Xiong, Jie
TI  - Some properties of superprocesses under a stochastic flow
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2009
SP  - 477
EP  - 490
VL  - 45
IS  - 2
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/08-AIHP171/
DO  - 10.1214/08-AIHP171
LA  - en
ID  - AIHPB_2009__45_2_477_0
ER  - 
%0 Journal Article
%A Lee, Kijung
%A Mueller, Carl
%A Xiong, Jie
%T Some properties of superprocesses under a stochastic flow
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2009
%P 477-490
%V 45
%N 2
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/08-AIHP171/
%R 10.1214/08-AIHP171
%G en
%F AIHPB_2009__45_2_477_0
Lee, Kijung; Mueller, Carl; Xiong, Jie. Some properties of superprocesses under a stochastic flow. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 2, pp. 477-490. doi : 10.1214/08-AIHP171. http://www.numdam.org/articles/10.1214/08-AIHP171/

[1] D. A. Dawson, Z. Li and H. Wang. Superprocesses with dependent spatial motion and general branching densities. Electron. J. Probab. 6 (2001) 1-33. | MR | Zbl

[2] D. A. Dawson, J. Vaillancourt and H. Wang. Stochastic partial differential equations for a class of interacting measure-valued diffusions. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) 167-180. | Numdam | MR | Zbl

[3] A. Friedman. Stochastic Differential Equations and Applications, Vol. 1. Academic Press, New York, 1975. | MR | Zbl

[4] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North Holland/Kodansha, Amsterdam, 1989. | MR | Zbl

[5] G. Kallianpur and J. Xiong. Stochastic differential equations on infinite dimensional spaces. IMS Lecture Notes-Monograph Series, Vol. 26, 1995. | MR | Zbl

[6] N. Konno and T. Shiga. Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Related Fields 79 (1988) 201-225. | MR | Zbl

[7] N. V. Krylov. An analytic approach to SPDEs, Stochastic partial differential equations: six perspectives, Math. Surveys Monogr. 64 185-242. Amer. Math. Soc., Providence, RI, 1999. | MR | Zbl

[8] G. Skoulakis and R. J. Adler. Superprocesses over a stochastic flow. Ann. Appl. Probab. 11 (2001) 488-543. | MR | Zbl

[9] J. B. Walsh. An Introduction to Stochastic Partial Differential Equations. In École d'été de probabilités de Saint-Flour, XIV-1984 256-439. Lecture Notes in Math. 1180. Springer-Verlag, Berlin, 1986. | MR | Zbl

[10] H. Wang. State classification for a class of measure-valued branching diffusions in a Brownian medium. Probab. Theory Related Fields 109 (1997) 39-55. | MR | Zbl

[11] H. Wang. A class of measure-valued branching diffusions in a random medium. Stochastic Anal. Appl. 16 (1998) 753-786. | MR | Zbl

[12] J. Xiong. A stochastic log-Laplace equation. Ann. Probab. 32 (2004) 2362-2388. | MR | Zbl

[13] J. Xiong. Long-term behavior for superprocesses over a stochastic flow. Electron. Comm. Probab. 9 (2004) 36-52. | EuDML | MR | Zbl

[14] J. Xiong and X. Zhou. Superprocess over a stochastic flow with superprocess catalyst. Internat. J. Pure Appl. Mathematics 17 (2004) 353-382. | MR | Zbl

Cité par Sources :