Long time behaviour and stationary regime of memory gradient diffusions

Sébastien Gadat; Fabien Panloup

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 2, page 564-601
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Gadat, Sébastien, and Panloup, Fabien. "Long time behaviour and stationary regime of memory gradient diffusions." Annales de l'I.H.P. Probabilités et statistiques 50.2 (2014): 564-601. <http://eudml.org/doc/271993>.

@article{Gadat2014,
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.},
author = {Gadat, Sébastien, Panloup, Fabien},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stochastic differential equation; memory diffusions; ergodic processes; Lyapunov function; Non-Markovian diffusions; long-time behavior; Lyapunov conditions},
language = {eng},
number = {2},
pages = {564-601},
publisher = {Gauthier-Villars},
title = {Long time behaviour and stationary regime of memory gradient diffusions},
url = {http://eudml.org/doc/271993},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Gadat, Sébastien
AU - Panloup, Fabien
TI - Long time behaviour and stationary regime of memory gradient diffusions
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 2
SP - 564
EP - 601
AB - 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.
LA - eng
KW - stochastic differential equation; memory diffusions; ergodic processes; Lyapunov function; Non-Markovian diffusions; long-time behavior; Lyapunov conditions
UR - http://eudml.org/doc/271993
ER -

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