When single optimal paths dominate extreme events
In plain words
For very rare large events in noisy or chaotic systems, theory often assumes they occur along one most likely route, called an instanton. Whether and when that holds in high-dimensional systems like turbulence decides whether extremes can be computed rather than waited for.
Precise statement
For stochastically forced systems (Burgers, 2D and 3D Navier-Stokes with Gaussian forcing of strength $\epsilon$) and for deterministic chaotic systems, determine the observables $O$ and thresholds $z$ for which $P(O > z)$ is given asymptotically ($z \to \infty \text{ or } \epsilon \to 0$) by a single minimizer of the Freidlin-Wentzell action, including the prefactor, and characterize failures due to fluctuations around the instanton or several competing paths. An answer is a classification with verified prefactors.
What would settle it
Comparison of instanton predictions, including prefactors, with rare-event sampling of tails over many orders of magnitude for a set of flows and observables.
Status in the literature
Instanton predictions match velocity-gradient tails in Burgers turbulence and some 2D flows; extremes in 3D Navier-Stokes turbulence are not settled.
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