Lyapunov stability of the equilibrium of the non-local continuity equation

The paper is concerned with the development of Lyapunov methods for the analysis of equilibrium stability in a dynamical system on the space of probability measures driven by a non-local continuity equation. We derive sufficient conditions of stability of an equilibrium distribution relying on an analysis of a non-smooth Lyapunov function. For the linear dynamics we reduce the stability analysis to a study of a quadratic form on a tangent space to the space of probability measures. These results are illustrated by the studies of the stability of the equilibrium measure for gradient flow in the space of probability measures and Gibbs measure for a system of coupled mathematical pendulums.