Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Spectral gap for the stochastic quantization equation on the 2-dimensional torus

Pavlos Tsatsoulis and Hendrik Weber

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We study the long time behavior of the stochastic quantization equation. Extending recent results by Mourrat and Weber (Global well-posedness of the dynamic $\phi^{4}$ in the plane (2015) Preprint) we first establish a strong non-linear dissipative bound that gives control of moments of solutions at all positive times independent of the initial datum. We then establish that solutions give rise to a Markov process whose transition semigroup satisfies the strong Feller property. Following arguments by Chouk and Friz (Support theorem for a singular SPDE: the case of gPAM (2016) Preprint) we also prove a support theorem for the laws of the solutions. Finally all of these results are combined to show that the transition semigroup satisfies the Doeblin criterion which implies exponential convergence to equilibrium.

Along the way we give a simple direct proof of the Markov property of solutions and an independent argument for the existence of an invariant measure using the Krylov–Bogoliubov existence theorem. Our method makes no use of the reversibility of the dynamics or the explicit knowledge of the invariant measure and it is therefore in principle applicable to situations where these are not available, e.g. the vector-valued case.


Nous étudions le comportement sur le long terme de l’équation de quantification stochastique. Dans la continuité de récents résultats par Mourrat et Weber (Global well-posedness of the dynamic $\phi^{4}$ in the plane (2015) Preprint), nous établissons en premier lieu une borne dissipative forte non-linéaire qui contrôle les moments des solutions, pour tout choix de temps, indépendamment des conditions initiales. Nous prouvons ensuite que les solutions génèrent un processus Markovien dont le semigroupe satisfait la propriété de Feller forte. Nous obtenons également un théorème pour le support des lois des solutions grâce à des arguments adaptés de Chouk et Friz (Support theorem for a singular SPDE: the case of gPAM (2016) Preprint). Enfin, en combinant tous ces résultats, nous montrons que le semigroupe de transition satisfait le critère de Doeblin, ce qui entraine une convergence exponentielle vers l’équilibre.

Nous obtenons également au passage une preuve directe de la propriété de Markov pour les solutions, ainsi qu’un argument indépendant pour l’existence de mesures invariantes en utilisant le théorème d’existence de Krylov–Bogoliubov. Notre méthode n’utilise pas le caractère réversible de la dynamique ni la connaissance explicite de la mesure invariante, et peut donc en théorie s’appliquer dans des cas où ces propriétés ne sont pas connues, par exemple le cas vectoriel.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 3 (2018), 1204-1249.

Received: 27 September 2016
Revised: 24 March 2017
Accepted: 10 April 2017
First available in Project Euclid: 11 July 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37A25: Ergodicity, mixing, rates of mixing 60H15: Stochastic partial differential equations [See also 35R60] 81T08: Constructive quantum field theory

Singular SPDEs Strong Feller property Support theorem Exponential mixing


Tsatsoulis, Pavlos; Weber, Hendrik. Spectral gap for the stochastic quantization equation on the 2-dimensional torus. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 3, 1204--1249. doi:10.1214/17-AIHP837.

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