Arnold diffusion conjecture for a priori unstable systems

Consider the near-integrable, time-periodic a priori unstable Hamiltonian

$$H(y,x,v,u,t,\eps)=H_0(y,v,u)+\eps H_1(y,x,v,u,t)+O(\eps^2),$$

where $y\in\overline{\mathcal D}\subset\mathbb{R}^n$, $x\in\mathbb{T}^n$, $(v,u)\in D\subset\mathbb{R}^2$, $\mathcal D$ is an open domain with compact closure, and $H_0(y,v,u)=F(y,f(v,u))$. Assume that $f$ has a unique nondegenerate saddle point $(0,0)$ on a compact connected component of $\{(v,u)\in D:f(v,u)=f(0,0)\}$, that $E(y)=H_0(y,0,0)$ has nondegenerate Hessian, and that $H_1\in C^{\mathbf r}(\mathcal D\times\mathbb{T}^n\times D\times\mathbb{T})$ with $\mathbf r>n+3$. A priori unstable Arnold diffusion conjecture. The diffusion exists for an open and dense set of $C^{\mathbf r}$-perturbations; the slow variables $y$ evolve along any smooth curve $\chi\subset\mathcal D$; and there are fast diffusion trajectories whose average velocity along $\chi$ is of order $\eps/|\log\eps|$. This formulation specifies the expected genericity, path-following, and optimal diffusion-speed properties in a priori unstable systems. The cited conjectural program concerns establishing these properties for perturbations with only finite differentiability.