Large Deviation Principle for invariant distributions of Memory Gradient Diffusions

Sébastien Gadat (1) , Fabien Panloup (1) , Clément Pellegrini (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

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 past of 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 minima of U . Here, except in the linear case, we have no closed formula for the invariant distribution but we show that a LDP can still 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 .

Mots clés: hypoelliptic diffusions; Large Deviation Principle; Hamilton-Jacobi Equations; Freidlin and Wentzell Theory; small stochastic perturbations

Domaines: Systèmes dynamiques [math.DS]; Probabilités [math.PR]

Sébastien Gadat, Fabien Panloup, Clément Pellegrini. Large Deviation Principle for invariant distributions of Memory Gradient Diffusions. Electronic Journal of Probability, 2013, vol. 18, paper n° 81, 34 p. ⟨hal-00759188⟩