Cuplength estimates for time-periodic measures of Hamiltonian systems with diffusion

TL;DR

This paper employs Hamiltonian Floer theory and probabilistic techniques to establish lower bounds on the number of time-periodic measures in Hamiltonian systems with diffusion, demonstrating convergence of solutions as the diffusion parameter varies.

Contribution

It introduces a novel approach combining Floer theory and probabilistic methods to analyze time-periodic measures in Hamiltonian systems with diffusion.

Findings

Existence of closed random periodic solutions.

Convergence of Floer curves in probability distribution.

Lower bounds on the number of time-periodic measures.

Abstract

We show how methods from Hamiltonian Floer theory can be used to establish lower bounds for the number of different time-periodic measures of time-periodic Hamiltonian systems with diffusion. After proving the existence of closed random periodic solutions and of the corresponding Floer curves for Hamiltonian systems with random walks with step width $1/n$ for every $n\in\mathbb{N}$, we show that, after passing to a subsequence, they converge in probability distribution as $n\to\infty$. Besides using standard results from Hamiltonian Floer theory and about convergence of tame probability measures, we crucially use that sample paths of Brownian motion are almost surely H\"older continuous with H\"older exponent $0<\alpha<\frac{1}{2}$.