Potential Analysis

, Volume 9, Issue 1, pp 27–64 | Cite as

Malliavin Calculus for White Noise Driven Parabolic SPDEs

  • Vlad Bally
  • Etienne Pardoux
Article

Abstract

We consider the parabolic SPDE
$$\partial _t X\left( {t,x} \right) = \partial _{_x }^2 X\left( {t,x} \right) + \psi \left( {X\left( {t,x} \right)} \right) + \varphi \left( {X\left( {t,x} \right)} \right)\dot W\left( {t,x} \right),\left( {t,x} \right) \in R_ + \times \left[ {0,1} \right]$$
with the Neuman boundary condition
$${{\partial x}}\left( {t,0} \right) = \frac{{\partial X}} {{\partial x}}\left( {t,1} \right) = 1$$

and some initial condition.

We use the Malliavin calculus in order to prove that, if the coefficients ϕ and ψ are smooth and ϕ > 0, then the law of any vector (X(t,x1),..., X(t,xd)), 0 ≤ x1 ≤ ... ≤ xd ≤ 1, has a smooth, strictly positive density with respect to Lebesgue measure.

Stochastics partial differential equations space time noise Malliavin calculus 

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Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Vlad Bally
    • 1
  • Etienne Pardoux
    • 3
  1. 1.Université du Maine, Equipe de Statistiques et ProcessusLe Mans and
  2. 2.Laboratoire de ProbabilitésUniversité de Paris VI, tour 56ParisFrance
  3. 3.LATP, URA CNRS 225, CMIUniversitéeMarsille Cedex 13France

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