Self-stabilizing processes in multi-wells landscape in $\mathbb{R}^d$ - Convergence

Julian Tugaut (1)

1 Universität Bielefeld = Bielefeld University

Self-stabilizing processes are non-markovian diffusions. The own law of the process intervenes in the drift. When the non-interacting part of the drift corresponds to the gradient of a convex potential, it has been proved in some different ways that such processes converge weakly towards the unique stationary measure when the time goes to infinity. However, in the one-dimensional case, it has been pointed out that there are several stationary measures under easy to verify conditions. The convergence is then much more difficult to get. It has been obtained in a previous paper by using the free-energy. In superior dimension, we also have the non-uniqueness of the stationary measures. The aim of this paper is to study the long-time behaviour of the self-stabilizing processes even if the set of stationary measures is not finite.

Mots clés: Self-interacting diffuion; Free-energy; McKean-Vlasov stochastic differential equations; Multi-wells potential; Granular media equation

Domaines: Probabilités [math.PR]

Julian Tugaut. Self-stabilizing processes in multi-wells landscape in $\mathbb{R}^d$ - Convergence. 2011. ⟨hal-00628086v1⟩