Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The strong Feller property for singular stochastic PDEs

M. Hairer and J. Mattingly

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Abstract

We show that the Markov semigroups generated by a large class of singular stochastic PDEs satisfy the strong Feller property. These include for example the KPZ equation and the dynamical $\Phi^{4}_{3}$ model. As a corollary, we prove that the Brownian bridge measure is the unique invariant measure for the KPZ equation with periodic boundary conditions.

Résumé

Nous montrons que les semi-groupes de Markov engendrés par une classe large d’EDPs stochastiques singulières satisfont la propriété forte de Feller. Cette classe inclut par exemple l’équation KPZ et le modèle $\Phi^{4}_{3}$. Nous montrons comme corollaire que la distribution du pont Brownien est l’unique mesure invariante pour l’équation KPZ avec conditions frontières périodiques.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 3 (2018), 1314-1340.

Dates
Received: 11 October 2016
Revised: 5 April 2017
Accepted: 23 April 2017
First available in Project Euclid: 11 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1531296021

Digital Object Identifier
doi:10.1214/17-AIHP840

Mathematical Reviews number (MathSciNet)
MR3825883

Zentralblatt MATH identifier
06976077

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 37L55: Infinite-dimensional random dynamical systems; stochastic equations [See also 35R60, 60H10, 60H15] 81S20: Stochastic quantization

Keywords
Strong Feller Random dynamical systems Rough stochastic PDEs Ergodicity Stochastic quantisation Girsanov

Citation

Hairer, M.; Mattingly, J. The strong Feller property for singular stochastic PDEs. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 3, 1314--1340. doi:10.1214/17-AIHP840. https://projecteuclid.org/euclid.aihp/1531296021


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