Lyapunov stability of an equilibrium of the nonlocal continuity equation

The paper is devoted to developing Lyapunov's methods for analyzing the stability of an equilibrium of a dynamical system in the space of probability measures that is defined by a nonlocal continuity equation. Sufficient stability conditions are obtained based on the basis of an analysis of the behaviour of a nonsmooth Lyapunov function in a neighbourhood of the equilibrium and the investigation of a certain quadratic form defined on the tangent space of the space of probability measures. The general results are illustrated by the study of the stability of an equilibrium for a gradient flow in the space of probability measures and the Gibbs measure for a system of connected simple pendulums. Bibliography: 28 titles.