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Probability density for a hyperbolic SPDE with time dependent coefficients

Published online by Cambridge University Press:  17 August 2007

Marta Sanz-Solé
Affiliation:
Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain; marta.sanz@ub.edu; itorrecilla@ub.edu
Iván Torrecilla-Tarantino
Affiliation:
Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain; marta.sanz@ub.edu; itorrecilla@ub.edu
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Abstract

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, PTRF123 (2002) 453-483 to an infinite dimensional setting.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2007

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References

Cairoli, R. and Walsh, J.B., Stochastic integrals in the plane. Acta Mathematica 134 (1975) 111183. CrossRef
Cattiaux, P. and Mesnager, L., Hypoelliptic non-homogeneous diffusions. PTRF 123 (2002) 453483.
Chen, M. and Zhou, X., Applications of Malliavin calculus to stochastic differential equations with time-dependent coefficients. Acta Appli. Math. Sinica 7 (1991) 193216. CrossRef
I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag (1988).
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.
J.R. Norris, Simplified Malliavin calculus, in Séminaire de Probabilités XX. LNM 1204 (1986) 101–130.
D. Nualart, The Malliavin Calculus and Related Topics. Probability and its Applications. Springer-Verlag, 2nd Edition (2006).
D. Nualart and M. Sanz, Malliavin calculus for two-parameter Wiener functionals. Z. für Wahrscheinlichkeitstheorie verw. Gebiete 70 (1985) 573–590.
P.E. Protter, Stochastic Integration and Differential Equations. Applications of Mathematics. Stochastic Modelling and Applied Probability. Springer, 2nd Edition 21 (2004).
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.
Taniguchi, S., Applications of Malliavin's calculus to time-dependent systems of heat equations. Osaka J. Math. 22 (1985) 307320.