Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Mean-field interaction of Brownian occupation measures, I: Uniform tube property of the Coulomb functional

Wolfgang König and Chiranjib Mukherjee

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Abstract

We study the transformed path measure arising from the self-interaction of a three-dimensional rownian motion via an exponential tilt with the Coulomb energy of the occupation measures of the motion by time $t$. The logarithmic asymptotics of the partition function were identified in the 1980s by Donsker and Varadhan (Comm. Pure Appl. Math. 505 (1983) 505–528) in terms of a variational formula. Recently (Brownian occupations measures, compactness and large deviations (2014) Preprint) a new technique for studying the path measure itself was introduced, which allows for proving that the normalized occupation measure asymptotically concentrates around the set of all maximizers of the formula. In the present paper, we show that likewise the Coulomb functional of the occupation measure concentrates around the set of corresponding Coulomb functionals of the maximizers in the uniform topology. This is a decisive step on the way to a rigorous proof of the convergence of the normalized occupation measures towards an explicit mixture of the maximizers, derived in (Mean-field interaction of Brownian occupation measures, II: A rigorous construction of the Pekar process. Preprint). Our methods rely on deriving Hölder-continuity of the Coulomb functional of the occupation measure with exponentially small deviation probabilities and invoking the large deviation theory developed in (Brownian occupations measures, compactness and large deviations (2014) Preprint) to a certain shift-invariant functional of the occupation measures.

Résumé

Nous étudions la mesure de trajectoire transformée engendrée par l’auto-interaction d’un mouvement Brownien tridimensionnel en utilisant un biais exponentiel par l’énergie de Coulomb des mesures d’occupation de ce mouvement au temps $t$. Les asymptotes logarithmiques de la fonction de partition ont été identifiées dans les années 1980 par Donsker et Varadhan [Comm. Pure Appl. Math. 505 (1983) 505–528] au moyen d’une formule variationnelle. Récemment, dans (Brownian occupations measures, compactness and large deviations (2014) Preprint), une nouvelle technique pour étudier la mesure de chemins elle-même a été introduite. Elle permet de prouver que la mesure d’occupation normalisée se concentre asymptotiquement autour de l’ensemble des maximums de la formule.

Dans le présent article, nous prouvons que la fonctionnelle de Coulomb de la mesure d’occupation se concentre elle aussi, dans la topologie uniforme, autour de l’ensemble des fonctionnelles de Coulomb correspondant aux maximums. Ceci représente une étape décisive vers une preuve rigoureuse de la convergence des mesures d’occupation normalisées vers un mélange explicite des maximums, dérivée dans (Mean-field interaction of Brownian occupation measures, II: A rigorous construction of the Pekar process. Preprint). Nos méthodes reposent sur l’obtention de la continuité hölderienne de la fonctionnelle de Coulomb de la mesure d’occupation avec des probabilités de déviations exponentiellement petites, en invoquant la théorie des grandes déviations développée dans (Brownian occupations measures, compactness and large deviations (2014) Preprint) pour une certaine fonctionnelle, invariante par dćalage, des mesures d’occupation.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 2214-2228.

Dates
Received: 3 February 2016
Revised: 26 July 2016
Accepted: 22 August 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1511773743

Digital Object Identifier
doi:10.1214/16-AIHP788

Mathematical Reviews number (MathSciNet)
MR3729652

Zentralblatt MATH identifier
1382.60107

Subjects
Primary: 60J65: Brownian motion [See also 58J65] 60J55: Local time and additive functionals 60F10: Large deviations

Keywords
Gibbs measures Interacting Brownian motions Coulomb functional Polaron problem

Citation

König, Wolfgang; Mukherjee, Chiranjib. Mean-field interaction of Brownian occupation measures, I: Uniform tube property of the Coulomb functional. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 2214--2228. doi:10.1214/16-AIHP788. https://projecteuclid.org/euclid.aihp/1511773743


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References

  • [1] E. Bolthausen and U. Schmock. On self-attracting $d$-dimensional random walks. Ann. Probab. 25 (1997) 531–572.
  • [2] E. Bolthausen, W. König and C. Mukherjee. Mean-field interaction of Brownian occupation measures, II: A rigorous construction of the Pekar process. Comm. Pure. Appl. Math. Preprint. Available at http://arxiv.org/abs/1511.05921.
  • [3] M. D. Donsker and S. R. S. Varadhan. On laws of iterated logarithm for local times. Comm. Pure Appl. Math. XXX (1977) 707–753.
  • [4] M. D. Donsker and S. R. S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time, I–IV. Comm. Pure Appl. Math. 28 (29) (1983) 1–47.
  • [5] M. D. Donsker and S. R. S. Varadhan. Asymptotics for the polaron. Comm. Pure Appl. Math. 36 (1983) 505–528.
  • [6] E. H. Lieb. Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57 (1976) 93–105.
  • [7] C. Mukherjee and S. R. S. Varadhan. Brownian occupations measures, compactness and large deviations. Ann. Probab. (2014). Preprint. Available at http://arxiv.org/abs/1404.5259.
  • [8] N. I. Portenko. Diffusion processes with unbounded drift coefficient. Theoret. Probability Appl. 20 (1976) 27–31.
  • [9] H. Spohn. The effective mass of the polaron: A functional integral approach. Ann. Physics 175 (1987) 278–318.
  • [10] D. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Processes. Springer, Berlin, 1979.
  • [11] A.-S. Sznitman. Brownian Motion, Obstacles and Random Media. Springer, Berlin, 1998.