The stochastic diffusive behavior conjecture for the generalized Arnold example

Let $H_\varepsilon=H_0+\varepsilon H_1$ be the Hamiltonian under consideration, let $I^*\in\mathbb{R}$, and let $\mu^\varepsilon(I^*)$ be normalized Lebesgue measure on the prescribed neighborhood of $(p,q,I^*,\varphi,t)$. Write $\Pi_I$ for projection onto the $I$-component and set

$$t_\varepsilon=s\,\varepsilon^{-2}\log\frac{1}{\varepsilon}.$$

For a diffusion process $I_\bullet$, denote its drift by $b(I)$ and its variance by $\sigma(I)$.

The stochastic diffusive behavior conjecture. For a generic perturbation $\varepsilon H_1(\cdot)$, there are smooth functions $b(I)$ and $\sigma(I)>0$, depending only on $H_1$ and $H_0$, such that for every $s>0$ the distribution

$$\Pi_I\bigl(\phi^{t_\varepsilon}_*\mu^\varepsilon(I^*)\bigr)$$

converges weakly as $\varepsilon\to0$ to the distribution of $I_s$, where $I_\bullet$ is the diffusion process with drift $b$ and variance $\sigma$, starting from $I_0=I^*$.

This conjecture predicts that the slow action variable exhibits diffusive behavior on the logarithmically corrected time scale $\varepsilon^{-2}\log(1/\varepsilon)$ for generic perturbations. The diffusion coefficients are expected to depend only on the unperturbed and perturbing Hamiltonians; the statement remains unproved in the source.