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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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.


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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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