Optimal diffusion time compared with the Nekhoroshev stability bound
In plain words
Nekhoroshev proved that drift in nearly regular systems takes at least an exponentially long time. Whether typical systems actually drift that fast, and with which exponent, is open.
Precise statement
For analytic $H = h(I) + \epsilon f$ with $h$ convex in $n$ degrees of freedom, Nekhoroshev gives $\left|I(t) - I(0)\right| \le C \epsilon^b$ for $\left|t\right| \le \exp(c \epsilon^{-a})$ with optimal $a = 1/(2n)$. Prove that for generic $f$ there are orbits drifting by order one in time $\exp(C \epsilon^{-1/(2n)})$, matching the upper bound. Answer: a proof of the matching lower bound or a counter-argument.
What would settle it
A construction of diffusing orbits with time $\operatorname{exp}(C \epsilon^{-1/(2n)})$ for an open set of perturbations.
Status in the literature
Specific analytic perturbations with diffusion time exponentially long in a power of 1/eps near the Nekhoroshev exponent were built by Ke Zhang (Inventiones Mathematicae, 2011); matching bounds for generic f are open.