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June 2022 Central limit theorem for Gibbs measures on path spaces including long range and singular interactions and homogenization of the stochastic heat equation
Chiranjib Mukherjee
Author Affiliations +
Ann. Appl. Probab. 32(3): 2028-2062 (June 2022). DOI: 10.1214/21-AAP1727


We consider a class of Gibbs measures defined with respect to increments {ω(t)ω(s)}s<t of d-dimensional Wiener measure, with the underlying Hamiltonian carrying interactions of the form H(ts,ω(t)ω(s)) that are invariant under uniform translations of paths. In such interactions, we allow long-range dependence in the time variable (including power law decay up to t(1+t)(2+ε) for ε>0) and unbounded (singular) interactions (including singularities of the form x1/|x|p in d3 or xδ0(x) in d=1) attached to the space variables. These assumptions on the interaction seem to be sharp and cover quantum mechanical models like the Nelson model and the polaron problem with ultraviolet cut off (both carrying bounded spatial interactions with power law decay in time) as well as the Fröhlich polaron with a short range interaction in time but carrying Coulomb singularity in space. In this set up, we develop a unified approach for proving a central limit theorem for the rescaled process of increments for any coupling parameter and obtain an explicit expression for the limiting variance, which is strictly positive.

As a further application, we study the solution of the multiplicative-noise stochastic heat equation in spatial dimensions d3. When the noise is mollified both in time and space, we show that the averages of the diffusively rescaled solutions converge pointwise to the solution of a diffusion equation whose coefficients are homogenized in this limit.

Funding Statement

The present work is supported by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure.


It is a pleasure to thank Volker Betz and Herbert Spohn for their encouragement to pursue this work and many valuable discussions on the Nelson model. The author would also like to thank Erwin Bolthausen, Sabine Jansen and S.R.S. Varadhan for useful comments on an earlier version of the manuscript and Ofer Zeitouni for helpful discussions. Finally, the author would like to thank an anonymous referee for a very careful reading of the earlier version and pointing out a number of inaccuracies, which led to a more elaborate version of our manuscript.


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Chiranjib Mukherjee. "Central limit theorem for Gibbs measures on path spaces including long range and singular interactions and homogenization of the stochastic heat equation." Ann. Appl. Probab. 32 (3) 2028 - 2062, June 2022.


Received: 1 June 2020; Revised: 1 May 2021; Published: June 2022
First available in Project Euclid: 29 May 2022

Digital Object Identifier: 10.1214/21-AAP1727

Primary: 60F05 , 60H15 , 60H30 , 60J55 , 60J65

Keywords: annealed central limit theorem , central limit theorem , Fröhlich polaron , Gibbs measures , Homogenization‎ , long-range correlations , Nelson model , singular interactions , Stochastic heat equation

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.32 • No. 3 • June 2022
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