Probability density for a hyperbolic SPDE with time dependent coefficients
ESAIM: Probability and Statistics, Tome 11 (2007), pp. 365-380.

We prove the existence and smoothness of density for the solution of a hyperbolic SPDE with free term coefficients depending on time, under hypoelliptic non degeneracy conditions. The result extends those proved in Cattiaux and Mesnager, PTRF 123 (2002) 453-483 to an infinite dimensional setting.

DOI : 10.1051/ps:2007024
Classification : 60H07, 60H15, 60G60
Mots-clés : Malliavin calculus, stochastic partial differential equations, two-parameter processes 
@article{PS_2007__11__365_0,
     author = {Sanz-Sol\'e, Marta and Torrecilla-Tarantino, Iv\'an},
     title = {Probability density for a hyperbolic {SPDE} with time dependent coefficients},
     journal = {ESAIM: Probability and Statistics},
     pages = {365--380},
     publisher = {EDP-Sciences},
     volume = {11},
     year = {2007},
     doi = {10.1051/ps:2007024},
     mrnumber = {2339298},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps:2007024/}
}
TY  - JOUR
AU  - Sanz-Solé, Marta
AU  - Torrecilla-Tarantino, Iván
TI  - Probability density for a hyperbolic SPDE with time dependent coefficients
JO  - ESAIM: Probability and Statistics
PY  - 2007
SP  - 365
EP  - 380
VL  - 11
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps:2007024/
DO  - 10.1051/ps:2007024
LA  - en
ID  - PS_2007__11__365_0
ER  - 
%0 Journal Article
%A Sanz-Solé, Marta
%A Torrecilla-Tarantino, Iván
%T Probability density for a hyperbolic SPDE with time dependent coefficients
%J ESAIM: Probability and Statistics
%D 2007
%P 365-380
%V 11
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps:2007024/
%R 10.1051/ps:2007024
%G en
%F PS_2007__11__365_0
Sanz-Solé, Marta; Torrecilla-Tarantino, Iván. Probability density for a hyperbolic SPDE with time dependent coefficients. ESAIM: Probability and Statistics, Tome 11 (2007), pp. 365-380. doi : 10.1051/ps:2007024. http://archive.numdam.org/articles/10.1051/ps:2007024/

[1] R. Cairoli and J.B. Walsh, Stochastic integrals in the plane. Acta Mathematica 134 (1975) 111-183. | Zbl

[2] P. Cattiaux and L. Mesnager, Hypoelliptic non-homogeneous diffusions. PTRF 123 (2002) 453-483. | Zbl

[3] M. Chen and X. Zhou, Applications of Malliavin calculus to stochastic differential equations with time-dependent coefficients. Acta Appli. Math. Sinica 7 (1991) 193-216. | Zbl

[4] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag (1988). | MR | Zbl

[5] P. Malliavin, Stochastic calculus of variations and hypoelliptic operators, in Proc. Inter. Symp. on Stoch. Diff. Equations, Kyoto 1976, Tokyo and Wiley, New York (1978) 195-263. | Zbl

[6] J.R. Norris, Simplified Malliavin calculus, in Séminaire de Probabilités XX. LNM 1204 (1986) 101-130. | Numdam | Zbl

[7] D. Nualart, The Malliavin Calculus and Related Topics. Probability and its Applications. Springer-Verlag, 2nd Edition (2006). | MR | Zbl

[8] D. Nualart and M. Sanz, Malliavin calculus for two-parameter Wiener functionals. Z. für Wahrscheinlichkeitstheorie verw. Gebiete 70 (1985) 573-590. | Zbl

[9] P.E. Protter, Stochastic Integration and Differential Equations. Applications of Mathematics. Stochastic Modelling and Applied Probability. Springer, 2nd Edition 21 (2004). | MR | Zbl

[10] D.W. Stroock, Some applications of stochastic calculus to partial differential equations, in École d'Été de Probabilités de Saint Flour. LNM 976 (1983) 267-382. | Zbl

[11] S. Taniguchi, Applications of Malliavin's calculus to time-dependent systems of heat equations. Osaka J. Math. 22 (1985) 307-320. | Zbl

Cité par Sources :