Some Problems in Stochastic Flows and Random Media

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0450756Cranston The PI will work on problems in the area of stochastic flows and random media. The problems in flows involve rates of dispersion of a set of passive tracers carried by the flow. Of special interest here are the set of ballistic points, those which travel at a linear rate that greatly exceeds the diffusive rate. The PI will examine the Hausdorff dimension of this set of points and the structure of the image of the set of points which have traveled at a superdiffusive rate. Another problem is the study of the length of a curve moving under the flow. The PI will attempt to show, under the assumption that the top Lyapunov exponent is positive, that the length grows exponentially at a rate greater than the top Lyapunov exponent. The PI also proposes to study solutions of the parabolic Anderson equation in both the scalar and vector case. The problems in the vector case involve establishing exponential growth rates for magnetic fields generated by turbulent velocity fields. In the scalar case, the problems involve gaining more insight into the properties of solutions. The proposed problems on stochastic flows are motivated by the motion of pollutants on the ocean surface or in ground water. A stochastic flow is a convenient model for the motion of particles under the effect of ocean currents. The proposed work on the parabolic Anderson model arises from a problem in astrophysics called the dynamo problem. The solution of a vector version of the parabolic Anderson equation models the magnetic field in a young star. The basic open question in astrophysics which will be approached by the PI is whether the magnetic field grows exponentially fast and to determine the exponential growth rate.