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

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 $1\mathrm{/}n$ for every $n\in \mathbb{N}$, we show that, after passing to a subsequence, they converge in probability distribution as $n\to \mathrm{\infty }$. 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 Hölder continuous with Hölder exponent $0<\alpha <\frac{1}{2}$.