Professors Marcus and Rosen will continue their research on permanental processes. These are positive real valued processes that include processes that are the squares of Gaussian processes. However, whereas Gaussian processes are defined by a positive definite symmetric covariance function, in the definition of permanental processes, the restrictions to symmetry and positive definiteness are not required. Permanental process are the missing link that can be used to extend the Dynkin Isomorphism Theorem to an isomorphism theorem that contains the local times of Markov processes that do not have symmetric potential densities. They also plan to extend the study of permanental processes to permanental fields and to develop new isomorphism theorems that relate them to continuous additive functionals and intersection local times of general Markov processes. They expect to be able to use them to obtain sample path properties of continuous additive functionals and intersection local times.
Important phenomena in our lives, like weather, financial markets, voting patterns and detection of enemy activity are so complex that they can only be modeled as random events called stochastic processes. Nevertheless, although these events are random, they all have certain structures, usually different, that enables us to make good predictions about how they behave, so that we can exploit them or defend ourselves against them. Many probabilists study stochastic processes. Our specialization is local times which is a measure of what are the outcomes of these processes and which outcomes are more or less likely. Our motivation is twofold. One is esthetic, because the underlying mathematics is very beautiful. The other is practical, to provide tools for engineers and scientists engaged in protecting us from devastating weather, controlling destructive market fluctuations, analyzing voter patterns, protecting us from enemy missles...the list of potential applications is endless. Devices and techniques employing the most advanced mathematics contribute to the best of the new growth industries and will aid in keeping our country competitive
10/01/2013 to 09/30/2014 Jay Rosen, Yves Le Jan and I have written a book Intersection local times, loop soups and permanental Wick powers, relating these three stochastic pro- cesses. We use the loop measure to construct intersection local times and study their continuity. We then use loop soups to establish a general isomorphism theorem for intersection local times. This is used to study their properties. Several stochastic processes related to transient L ?evy processes with potential densities u(x, y) = u(y − x), that need not be symmetric nor bounded on the diagonal, are defined and studied. Specifically, real valued processes on a space of measures V endowed with a metric d, are considered, and sufficient conditions for the continuity of these processes on (V,d) are obtained. The processes defined and studied include n-fold self-intersection local times of tran- sient Levy processes and permanental chaoses. These chaoses are â€˜loop soup n-fold self-intersection local timesâ€™ constructed from the loop soup of the Levy process. Loop soups are also used to define permanental Wick powers, which, when u(x,y) is symmetric, are standard Wick powers, a class of n-th order Gaussian chaoses. I have continued my study of permanental processes by obtaining a rep- resentation of them that gives a concrete description of what they are. The square of a Gaussian process is an example of a permanental process. Using this representation Jay Rosen and I have shown that the sample path behavior of the very large class of permanental processes is similar to the sample path behavior of the square of a Gaussian process, when the kernel that defines the permanental process is symmetric. We can also describe the behavior when the kernel is not symmetric and obtain a Sudakov type inequality for them. This enables us to give good necessary conditions for a permanental process to be bounded. This work is in progress. Book Intersection local times, loop soups and permanental Wick powers, (with Y. Le Jan and M. Marcus), submitted to the American mathematical Society Memoirs series, under consideration. http://arxiv.org/pdf/1308.2701.pdf Article Multivariate gamma distributions, Electronic Journal of Mathematics, to appear.