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Large deviation principle for invariant distributions of memory gradient diffusions

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1.	Title	Title of document	Large deviation principle for invariant distributions of memory gradient diffusions
2.	Creator	Author's name, affiliation, country	Sébastien Gadat; Université de Toulouse; France
2.	Creator	Author's name, affiliation, country	Fabien Panloup; Université de Toulouse; France
2.	Creator	Author's name, affiliation, country	Clément Pellegrini; Université de Toulouse; France
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Large Deviation Principle; Hamilton-Jacobi Equations; Freidlin and Wentzell The- ory; small stochastic perturbations; hypoelliptic diffusions.
3.	Subject	Subject classification	60F10, 60G10, 60J60, 60K35, 35H10, 93D30
4.	Description	Abstract	In this paper, we consider a class of diffusion processes based on a memory gradient descent, i.e. whose drift term is built as the average all along the trajectory of the gradient of a coercive function U. Under some classical assumptions on U, this type of diffusion is ergodic and admits a unique invariant distribution. In view to optimization applications, we want to understand the behaviour of the invariant distribution when the diffusion coefficient goes to 0. In the non-memory case, the invariant distribution is explicit and the so-called Laplace method shows that a Large Deviation Principle (LDP) holds with an explicit rate function, that leads to a concentration of the invariant distribution around the global minimums of U. Here, excepted in the linear case, we have no closed formula for the invariant distribution but we show that a LDP can be obtained. Then, in the one-dimensional case, we get some bounds for the rate function that lead to the concentration around the global minimum under some assumptions on the second derivative of U.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2013-09-06
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/2031
10.	Identifier	Digital Object Identifier	10.1214/EJP.v18-2031
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 18
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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