Functional inequalities and an application for parabolic stochastic partial differential equations containing rotation

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4 Citations (Scopus)

Abstract

The main purpose of this paper is to establish a gradient estimate and a parabolic Harnack inequality for the non-symmetric transition semigroup with respect to the Gibbs measure on a path space. This semigroup is related to a diffusion process which is represented by the solution of a certain parabolic stochastic partial differential equation (SPDE, in abbreviation) containing rotation. We also discuss the relationship between the Gibbs measure and stationary measures of our dynamics. For the proof of our functional inequalities, we formulate a suitable domain of the infinitesimal generator for the semigroup. As an application of our results, we study a certain lower estimate on the transition probability for our dynamics.

Original languageEnglish
Pages (from-to)687-725
Number of pages39
JournalBulletin des Sciences Mathematiques
Volume128
Issue number8
DOIs
Publication statusPublished - 2004 Sep 1
Externally publishedYes

Fingerprint

Functional Inequalities
Gibbs Measure
Stochastic Partial Differential Equations
Parabolic Partial Differential Equations
Semigroup
Transition Semigroup
Abbreviation
Stationary Measure
Path Space
Gradient Estimate
Harnack Inequality
Infinitesimal Generator
Transition Probability
Diffusion Process
Estimate
Relationships

Keywords

  • Gibbs measure
  • Gradient estimate
  • Parabolic Harnack inequality
  • Rotation
  • SPDE
  • Transition probability

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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AB - The main purpose of this paper is to establish a gradient estimate and a parabolic Harnack inequality for the non-symmetric transition semigroup with respect to the Gibbs measure on a path space. This semigroup is related to a diffusion process which is represented by the solution of a certain parabolic stochastic partial differential equation (SPDE, in abbreviation) containing rotation. We also discuss the relationship between the Gibbs measure and stationary measures of our dynamics. For the proof of our functional inequalities, we formulate a suitable domain of the infinitesimal generator for the semigroup. As an application of our results, we study a certain lower estimate on the transition probability for our dynamics.

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