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

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