Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Long time behaviour and stationary regime of memory gradient diffusions

Sébastien Gadat and Fabien Panloup

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Abstract

In this paper, we are interested in a diffusion process based on a gradient descent. The process is non Markov and has a memory term which is built as a weighted average of the drift term all along the past of the trajectory. For this type of diffusion, we study the long time behaviour of the process in terms of the memory. We exhibit some conditions for the long-time stability of the dynamical system and then provide, when stable, some convergence properties of the occupation measures and of the marginal distribution, to the associated steady regimes. When the memory is too long, we show that in general, the dynamical system has a tendency to explode, and in the particular Gaussian case, we explicitly obtain the rate of divergence.

Résumé

Nous nous intéressons dans ce travail à une diffusion issue d’une descente de gradient, dont le terme de dérive utilise une mémoire sur le passé de la trajectoire. Le processus ainsi introduit est non-Markovien. Nous étudions les propriétés de stabilité et de convergence à l’équilibre des mesures d’occupation des trajectoires. Dans les situations stables, nous donnons des vitesses de convergence à la stationnarité alors que dans les cas où la mémoire possède une longue portée, nous prouvons l’explosion du système dynamique. Nous exhibons enfin des formules précises dans le cas gaussien.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 2 (2014), 564-601.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1395856141

Digital Object Identifier
doi:10.1214/12-AIHP536

Mathematical Reviews number (MathSciNet)
MR3189085

Zentralblatt MATH identifier
1299.60092

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 60G10: Stationary processes 37A25: Ergodicity, mixing, rates of mixing 93D30: Scalar and vector Lyapunov functions 35H10: Hypoelliptic equations

Keywords
Stochastic differential equation Memory diffusions Ergodic processes Lyapunov function

Citation

Gadat, Sébastien; Panloup, Fabien. Long time behaviour and stationary regime of memory gradient diffusions. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 2, 564--601. doi:10.1214/12-AIHP536. https://projecteuclid.org/euclid.aihp/1395856141


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