Asymptotics of Karhunen-Loeve eigenvalues and tight constants for probability distributions of passive scalar transport

In this paper we study the asymptotics of the probability distribution function for a certain model of freely decaying passive scalar transport. In particular we prove rigorous large n, or semiclassical, asymptotics for the eigenvalues of the covariance of a fractional Brownian motion. Using these asymptotics, along with some standard large deviations results, we are able to derive tight asymptotics for the rate of decay of the tails of the probability density for a generalization of the Majda model of scalar intermittency originally due to Vanden Eijnden. We are also able to derive asymptotically tight estimates for the closely related problem of small L₂ ball probabilities for a fractional Brownian motion.