Probability density for a hyperbolic SPDE with time dependent coefficients

Marta Sanz-Solé; Iván Torrecilla-Tarantino

ESAIM: Probability and Statistics (2007)

  • Volume: 11, page 365-380
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Sanz-Solé, Marta, and Torrecilla-Tarantino, Iván. "Probability density for a hyperbolic SPDE with time dependent coefficients." ESAIM: Probability and Statistics 11 (2007): 365-380. <http://eudml.org/doc/250103>.

@article{Sanz2007,
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. },
author = {Sanz-Solé, Marta, Torrecilla-Tarantino, Iván},
journal = {ESAIM: Probability and Statistics},
keywords = {Malliavin calculus. Stochastic partial differential equations. Two-parameter processes.; Malliavin calculus; stochastic partial differential equations; two-parameter processes},
language = {eng},
month = {8},
pages = {365-380},
publisher = {EDP Sciences},
title = {Probability density for a hyperbolic SPDE with time dependent coefficients},
url = {http://eudml.org/doc/250103},
volume = {11},
year = {2007},
}

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
DA - 2007/8//
PB - EDP Sciences
VL - 11
SP - 365
EP - 380
AB - 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.
LA - eng
KW - Malliavin calculus. Stochastic partial differential equations. Two-parameter processes.; Malliavin calculus; stochastic partial differential equations; two-parameter processes
UR - http://eudml.org/doc/250103
ER -

References

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  1. R. Cairoli and J.B. Walsh, Stochastic integrals in the plane. Acta Mathematica134 (1975) 111–183.  Zbl0334.60026
  2. P. Cattiaux and L. Mesnager, Hypoelliptic non-homogeneous diffusions. PTRF123 (2002) 453–483.  Zbl1009.60058
  3. M. Chen and X. Zhou, Applications of Malliavin calculus to stochastic differential equations with time-dependent coefficients. Acta Appli. Math. Sinica7 (1991) 193–216.  Zbl0751.60048
  4. I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag (1988).  Zbl0638.60065
  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.  
  6. J.R. Norris, Simplified Malliavin calculus, in Séminaire de Probabilités XX.LNM1204 (1986) 101–130.  Zbl0609.60066
  7. D. Nualart, The Malliavin Calculus and Related Topics. Probability and its Applications. Springer-Verlag, 2nd Edition (2006).  Zbl1099.60003
  8. D. Nualart and M. Sanz, Malliavin calculus for two-parameter Wiener functionals. Z. für Wahrscheinlichkeitstheorie verw.Gebiete70 (1985) 573–590.  Zbl0595.60065
  9. P.E. Protter, Stochastic Integration and Differential Equations. Applications of Mathematics. Stochastic Modelling and Applied Probability. Springer, 2nd Edition 21 (2004).  Zbl1041.60005
  10. D.W. Stroock, Some applications of stochastic calculus to partial differential equations, in École d'Été de Probabilités de Saint Flour. LNM976 (1983) 267–382.  
  11. S. Taniguchi, Applications of Malliavin's calculus to time-dependent systems of heat equations. Osaka J. Math.22 (1985) 307–320.  Zbl0583.35055

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