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" Eigenvalue Asymptotics, Passive Scalar Intermittency and Small Ball Estimates for Fractional Brownian Motions " Intermittency- the fact that the probability distribution function for a quantity transported by a turbulent flow is asymptotically broad- is an important phenomena in turbulence. We present some work (with R.M. McLaughlin- UNC Chapel Hill) on a model of passive scalar intermittency originally due to Majda: dT/dt=g(t) x dT/dy+DT, where g(t) is a random process and T is a passive scalar (for instance which is advected by the random (shear) flow). Majda was able to explicitly calculate moments of the distribution of the scalar T. McLaughlin and B were able to calculate the large N asymptotics of the moments of the distibution and, by a large deviations/Tauberian type argument calculate the distribution of the quantity T. I will also talk about some recent work on a generalization of this model (originally proposed by E. Vanden-Eijnden). A similar calculation can be done for this generalized model, which involves calculating the asymptotics of a certain compact eigenvalue problem. As a by-product of this calculation one finds the (previously unknown) optimal constants in a certain probabilistic "small ball" estimate for the probability that a fractional Brownian motion stays in a small ball in L2. |