The PI proposes to carry on research on problems in stochastic flows, the parabolic Anderson model, polymer phase transitions and random Schrodinger operators. In the area of stochastic flows, the PI will continue efforts on the distribution of passive tracers under the motion of stochastic flows. Here, the goal will be to investigate the joint asymptotic distribution of two disjoint bodies of passive tracers moving under the flow. This will also be investigated for the case of turbulence where the tracers are carried by Kolmogorov velocity fields. Other research will be carried out involving the behavior of the singular flows called Kraichnan flows. In the area roughly labeled the parabolic Anderson model, the PI will investigate the so-called dynamo problem. This involves a model for the generation of magnetic fields in turbulent media such as on the surface of a star. The dynamo conjecture is that the magnetic field of a star will exhibit exponential growth. The PI will attempt to establish this exponential growth and examine the asymptotics of the exponential constant as the inverse Reynolds number goes to zero.
The general spirit of this proposal is to pursue a mathematical investigation of physical phenomena in the presence of random and chaotic media. Examples of this in the investigations in stochastic flows are provided by the distribution of plankton particles when carried by ocean currents or the spread of oil as in the recent disaster in the Gulf of Mexico. The goal of this study is to give information on the distribution and shape of a body of particles being carried by a random current. Another example of phenomena in chaotic media is the creation of magnetic fields in young stars. The field strength is conjectured to exhibit rapid growth and the mathematical model should also have the high focusing of the magnetic field strength in small regions which are sun spots. This will lead to more understanding of the development of these spots which have an effect on events on earth. Another project relates to behavior of polymer chains. One aspect of this work will be to make a detailed study of the phase transitions of polymers and the effect the relation of length to temperature has on the nature of the transition. Another aspect of the polymer study is to gauge the effect that a random environment has on the shape of a polymer. So far, very noisy environments have been shown to force the polymer into a particular shape, that is intense randomness reduces the degrees of freedom of the polymer in the model. We aim to gain more insight into the nature of this particular shape.
The results obtained in this grant are all motivated by physical problems, such as behavior of magnetic fields generated in turbulent media, polymer behavior in random media, polymer behavior in the presence of an attracting molecule and the spread of disease in tree like networks. The outcomes relied on sophisticated mathematical tools and introduced novel ideas in the resolution of the problems. In one work we consider a version of a classical concentration inequality for sums of independent, isotropic random vectors with a mild restriction on the distribution of the radial part of these vectors. The proof uses a little Fourier analysis, the Laplace asymptotic method and a conditioning idea that traces its roots to some of the original works on concentration inequalities. The results here show that the distribution of a sum of such random vectors cannot become too focused in any resion of space. This is expected to have application to the behavior of magnetic fields in stars. Another work considers large time, (long length) behaviour of typical paths under the Anderson polymer measure. This is a model for polmers interacting with an inhomogeneous fluid. The inhomogeneous fluid is modeled by a Gaussian environment. The model for the polymer involves creating a measure on the space of nice paths onthe integer lattice called a Gibbs measure. These measures are analogs of measures used in the Ising model for magnetization of metals. We establish that the polymer measure gives a macroscopic mass to a small neighbourhood of a typical path as the length becomes large and the temperature goes to zero.The localization becomes complete as in the sense that the mass assigned to a particular special path grows to 1. The proof makes use of the overlap between two independent samples of the polymer drawn under the Gibbs measure. The overlap (meaning time spent in the same place by the two paths) can be estimated by the integration by parts formula for the Gaussian environment. Conditioning this measure on the number of jumps, we obtain a canonical measure which already shows scaling properties, thermodynamic limits, and decoupling of the parameters. In the next work, the PI considers a closed self-adjoint extension of the infinitesimal generator of a symmetric stable process whose domain core is the space of functions which are smooth and vanish in a neighborhood of the origin. Such extensions of the Laplacian have been used in models of the hydrogen atom and more recently have modeled pinning at the origin of polymer models based on Brownian motion. We also outline the construction of pinned polymer models based on the symmetric stable processes, even when the underlying stable process does not possess a local time at the origin. Finally, the PI considers the contact process with infection rate l on a finite homogeneous tree of vertex degree d, and of height n. The contact process is a standard model for the spread of disease in a population where l represents a level of contagion and individuals recover eventually from an infection and may be infected again (like a cold.) It can also be used to model the spread of a computer virus through the internet, where the virus can be eliminated from a computer once it is infected. If l is small, an infection will die out and we study the extinction time, that is, the random time it takes for the infection to disappear when the process is started from full occupancy, i.e. an infection at every site. We prove two conjectures of Stacey. Let l2 denote the upper critical value for the contact process on the infinite d-ary tree. First, if l< l2, then the extinction time divided by the height of the tree converges in probability, as n goes to infinity, to a positive constant. Second, if λ > l2, then the log of the expected extinction time divided by the volume of the tree converges in probability to a positive constant, and the extinction time divided by its expectation converges in distribution to the exponential distribution of mean 1.
