### Abstract

We study the long time behavior of the stochastic quantization equation. Extending recent results by Mourrat and Weber (Global well-posedness of the dynamic Φ^{4} 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 language | English |
---|---|

Pages (from-to) | 1204-1249 |

Number of pages | 46 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 54 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Aug 2018 |

### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

**Spectral gap for the stochastic quantization equation on the 2-dimensional torus.** / Tsatsoulis, Pavlos; Weber, Hendrik.

Research output: Contribution to journal › Article

*Annales de l'institut Henri Poincare (B) Probability and Statistics*, vol. 54, no. 3, pp. 1204-1249. https://doi.org/10.1214/17-AIHP837

}

TY - JOUR

T1 - Spectral gap for the stochastic quantization equation on the 2-dimensional torus

AU - Tsatsoulis, Pavlos

AU - Weber, Hendrik

PY - 2018/8/1

Y1 - 2018/8/1

N2 - 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.

AB - 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.

KW - Exponential mixing

KW - Singular SPDEs

KW - Strong Feller property

KW - Support theorem

UR - http://www.scopus.com/inward/record.url?scp=85049748740&partnerID=8YFLogxK

U2 - 10.1214/17-AIHP837

DO - 10.1214/17-AIHP837

M3 - Article

AN - SCOPUS:85049748740

VL - 54

SP - 1204

EP - 1249

JO - Annales de l'Institut Henri Poincaré: Probabilités et Statistiques

JF - Annales de l'Institut Henri Poincaré: Probabilités et Statistiques

SN - 0246-0203

IS - 3

ER -