Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Path-dependent infinite-dimensional SDE with non-regular drift: An existence result

David Dereudre and Sylvie Rœlly

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Abstract

We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither bounded or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy and a finite second moment.

The originality of our method leads in the use of the specific entropy as a tightness tool and in the description of such infinite-dimensional stochastic process as solution of a variational problem on the path space. Our result clearly improves previous ones obtained for free dynamics with bounded drift.

Résumé

Nous établissons, dans cet article, l’existence de solutions faibles pour un système infini-dimensionnel de diffusions browniennes. Le terme de dérive est véritablement général, au sens où il est supposé n’être ni borné, ni continu, ni Markovien. Nous supposons cependant que la loi initiale admet une entropie spécifique finie.

L’originalité de notre méthode consiste en l’utilisation de la bornitude de l’entropie spécifique comme critère de tension et en l’identification des solutions du système comme solutions d’un problème variationnel sur l’espace des trajectoires. Notre résultat améliore clairement ceux préexistants concernant des dynamiques libres perturbées par des dérives bornées.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 2 (2017), 641-657.

Dates
Received: 19 October 2014
Revised: 31 August 2015
Accepted: 7 November 2015
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1491897739

Digital Object Identifier
doi:10.1214/15-AIHP728

Mathematical Reviews number (MathSciNet)
MR3634268

Zentralblatt MATH identifier
1367.60066

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Infinite-dimensional SDE Non-Markov drift Non-regular drift Variational principle Specific entropy

Citation

Dereudre, David; Rœlly, Sylvie. Path-dependent infinite-dimensional SDE with non-regular drift: An existence result. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 641--657. doi:10.1214/15-AIHP728. https://projecteuclid.org/euclid.aihp/1491897739


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