Long time behaviour and stationary regime of memory gradient diffusions

Gadat, Sébastien; Panloup, Fabien

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 564-601.

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.

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.

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