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Title: Optimal diffusion time compared with the Nekhoroshev stability bound
Plain: 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: 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.
Settled by: A construction of diffusing orbits with time $\operatorname{exp}(C \epsilon^{-1/(2n)})$ for an open set of perturbations.
Kind: well-posed
Answer type: proof
References: (empty)
Literature status: open
Status note: 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.
Posed since: (empty)