Convergence to the equilibria for self-stabilizing processes in double-well landscape

TL;DR

This paper proves the global convergence of self-stabilizing McKean-Vlasov diffusions in a double-well landscape, addressing the complex nonconvex case with multiple stationary measures.

Contribution

It introduces a novel approach combining free-energy monotonicity with stationary measure analysis to establish convergence in nonconvex settings.

Findings

Proved global convergence of self-stabilizing processes in a double-well landscape.

Characterized the set of stationary measures for the process.

Extended convergence results beyond classical convex cases.

Abstract

We investigate the convergence of McKean-Vlasov diffusions in a nonconvex landscape. These processes are linked to nonlinear partial differential equations. According to our previous results, there are at least three stationary measures under simple assumptions. Hence, the convergence problem is not classical like in the convex case. By using the method in Benedetto et al. [J. Statist. Phys. 91 (1998) 1261-1271] about the monotonicity of the free-energy, and combining this with a complete description of the set of the stationary measures, we prove the global convergence of the self-stabilizing processes.