Renyi's Relative Entropy Lyapunov Functional for Convergence to Invariant Measure in Dissipative Stochastic Hamiltonian Systems

Abstract

This paper is concerned with multivariable stochastic Hamiltonian systems governed by an ordinary differential equation for the position and an Ito stochastic differential equation for the momentum. The momentum dynamics involve both conservative and nonconservative forcing including Langevin viscous damping and the Wiener process as a random force. The setting allows for rotational degrees of freedom, a non-quadratic potential energy function and position-dependent mass, diffusion and damping matrices. In addition to modelling nonlinear physical dynamics subject to random forcing, the system is motivated by a stochastic optimisation viewpoint, where the search for global minima of the potential energy (for example, using the gradient descent) is carried out through convergence to a Boltzmann equilibrium measure which favours smaller values of this function. Under certain regularity conditions, we show that the $χ^2$-divergence or, equivalently, Renyi's second-order relative entropy with respect to the invariant measure has a nonpositive time derivative and thus provides a Lyapunov functional candidate for this convergence. However, this derivative vanishes at some moments of time, which is related to the behaviour of the probability distribution of the momentum conditioned on the position and has a link to a family of auxiliary quantum harmonic oscillator Hamiltonians parameterised by the position space of the underlying classical system. This position-momentum conditioning is used in order to prove that such pre-equilibrium break times in the entropy dissipation are isolated, thus making the $χ^2$-divergence a strictly decreasing Lyapunov functional for the Fokker-Planck-Kolmogorov equation which governs the joint position-momentum distribution in such systems.