Phase transitions of McKean-Vlasov SDEs in Multi-well Landscapes

TL;DR

This paper investigates phase transitions in McKean-Vlasov SDEs within multi-well landscapes, revealing how stationary measures change with potential landscapes and confirming conjectures about critical thresholds.

Contribution

It provides a general analysis of phase transitions in MV-SDEs, establishing conditions for multiple stationary measures and confirming conjectures for symmetric multi-well potentials.

Findings

Below a critical threshold, the number of stationary measures equals the potential's extrema.

Above the threshold, the stationary measure is unique.

Critical thresholds are equal for symmetric bistable potentials and increase with the aggregation parameter.

Abstract

Phase transitions and critical behaviour of a class of MV-SDEs, whose concomitant non-local Fokker-Planck equation includes the Granular Media equation with quadratic interaction potential as a special case, is studied. By careful analysis of an implicit auxiliary integral equation, it is shown for a wide class of potentials that below a certain `critical threshold' there are exactly as many stationary measures as extrema of the potential, while above another the stationary measure is unique, and consequently phase transition(s) between. For symmetric bistable potentials, these critical thresholds are proven to be equal and a strictly increasing function of the aggregation parameter. Additionally, a simple condition is provided for symmetric multi-well potentials with an arbitrary number of extrema to demonstrate analogous behaviour. This answers, with considerably more generality, a conjecture of Tugaut [Stochastics, 86:2, 257-284]. To the best of our knowledge many of these results are novel. Others simplify the proofs of known results whilst greatly increasing their applicability.