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

Pavlos Tsatsoulis, Hendrik Weber

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30 Citations (SciVal)


We study the long time behavior of the stochastic quantization equation. Extending recent results by Mourrat and Weber (Global well-posedness of the dynamic Φ4 model 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.

Original languageEnglish
Pages (from-to)1204-1249
Number of pages46
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Issue number3
Publication statusPublished - 1 Aug 2018


  • Exponential mixing
  • Singular SPDEs
  • Strong Feller property
  • Support theorem

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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