Remark on the (Non)convergence of Ensemble Densities in Dynamical Systems

TL;DR

This paper investigates the behavior of ensemble densities in dynamical systems, showing that the expectation of the log-density evolves linearly over time under stationary measures, with implications for convergence properties.

Contribution

It establishes a linear evolution law for the expectation of log densities in dynamical systems, highlighting conditions under which convergence of ensemble densities may fail.

Findings

Expectation of log density is linear in time under stationary measures.

The growth rate K depends only on the stationary measure and vanishes for absolutely continuous measures.

The results provide insight into the (non)convergence behavior of ensemble densities in dynamical systems.

Abstract

We consider a dynamical system with state space $M$, a smooth, compact subset of some ${\Bbb R}^n$, and evolution given by $T_t$, $x_t = T_t x$, $x \in M$; $T_t$ is invertible and the time $t$ may be discrete, $t \in {\Bbb Z}$, $T_t = T^t$, or continuous, $t \in {\Bbb R}$. Here we show that starting with a continuous positive initial probability density $\rho(x,0) > 0$, with respect to $dx$, the smooth volume measure induced on $M$ by Lebesgue measure on ${\Bbb R}^n$, the expectation value of $\log \rho(x,t)$, with respect to any stationary (i.e. time invariant) measure $\nu(dx)$, is linear in $t$, $\nu(\log \rho(x,t)) = \nu(\log \rho(x,0)) + Kt$. $K$ depends only on $\nu$ and vanishes when $\nu$ is absolutely continuous wrt $dx$.