  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 $0<\alpha<\frac{1}{2}$.
