Phase transitions of McKean-Vlasov processes in double-wells landscape

Julian Tugaut (1)

1 Universität Bielefeld = Bielefeld University

We aim to establish results about a particular class of inhomogeneous processes, the so-called McKean-Vlasov diffusions. Such a diffusion corresponds to the hydrodynamical limit of an interacting particle system, the mean-field one. Existence and uniqueness of the invariant probability are classical results provided that the external force corresponds to the gradient of a convex potential. However, previous results, see [Herrmann, Tugaut|2010], state that the non-convexity of this potential implies the nonuniqueness of the invariant probabilities under easily checked assumptions. Here, we prove that there exists phase transitions, that is under a critical value, there are exactly three invariant probabilities and over another critical value, there is exactly one. Under simple assumptions, these two critical values coincide and it is characterize by a simple implicit equation. We exhibit other cases in which phase transitions occur. Finally, we proceed numerical estimations of the critical values.

Domaines: Probabilités [math.PR]

Julian Tugaut. Phase transitions of McKean-Vlasov processes in double-wells landscape. 2012. ⟨hal-00573046v3⟩