Malliavin calculus for infinite-dimensional systems with additive noise

Yuri Bakhtin, Jonathan C. Mattingly

Research output: Contribution to journalArticle

Abstract

We consider an infinite-dimensional dynamical system with polynomial nonlinearity and additive noise given by a finite number of Wiener processes. By studying how randomness is spread by the dynamics, we develop in this setting a partial counterpart of Hörmander's classical theory of Hypoelliptic operators. We study the distributions of finite-dimensional projections of the solutions and give conditions that provide existence and smoothness of densities of these distributions with respect to the Lebesgue measure. We also apply our results to concrete SPDEs such as a Stochastic Reaction Diffusion Equation and the Stochastic 2D Navier-Stokes System.

Original languageEnglish (US)
Pages (from-to)307-353
Number of pages47
JournalJournal of Functional Analysis
Volume249
Issue number2
DOIs
StatePublished - Aug 15 2007

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Malliavin Calculus
Infinite-dimensional Systems
Additive Noise
Stochastic Reaction-diffusion Equations
Hypoelliptic Operators
Infinite Dimensional Dynamical System
2-D Systems
Navier-Stokes System
Wiener Process
Lebesgue Measure
Randomness
Smoothness
Projection
Nonlinearity
Partial
Polynomial

Keywords

  • Degenerate stochastic partial differential equations
  • Malliavin calculus
  • Smooth densities
  • SPDEs
  • Stochastic evolution equations

ASJC Scopus subject areas

  • Analysis

Cite this

Malliavin calculus for infinite-dimensional systems with additive noise. / Bakhtin, Yuri; Mattingly, Jonathan C.

In: Journal of Functional Analysis, Vol. 249, No. 2, 15.08.2007, p. 307-353.

Research output: Contribution to journalArticle

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