This project deals with several problems and conjectures which lie at the interface of probability, harmonic analysis and the spectral theory for the Laplacian, the fractional Laplacian, and its relativistic versions. It aims to find new sharp inequalities for a class of Fourier multipliers that result from functional modifications of Levy processes via the Levy-Khintchine formula. It proposes to give universal bounds for the trace of the semigroup of Brownian motion, stable and relativistic stable processes and to explore the second order asymptotics for the spectral counting function for these processes. The unifying theme in these investigations is the use of the symbols of the underlying Levy process. Martingale theory, heat kernel estimates and techniques (perhaps even yet to be discovered) from the theory of the wave equations for fractional Laplacians, will play a key role in these investigations.
Martingales, which were invented more than a hundred years ago to explain the theory of "fair" games, play a fundamental role in many branches of mathematics and their applications in countless areas of the physical, biological and social sciences. They are intimately related to a second order differential operator knows as the Laplacian. This operator, named after the mathematical astronomer Marquis Pierre-Simon De Laplace more than two hundred years ago, gives the matheamtical foundation for the theory of heat, waves, electricity, magnetism and fluids. Both martingales and the Laplace operator interact with Brownian motion which models physical, biological and economic systems, and even with more general Levy processes which are widely used to model various financial systems which are subject to more instantaneous changes or jumps. At the root of the interactions between martingales, the Laplace operator and Brownian motion, is a deep mathematical theory which makes these applications scientifically sound. This project deals with several of these core questions and relates them to several other branches of mathematics including more general partial differential equations and their applications to the diffusion of heat, propagation of waves and their connections with random phenomenon.
These projects will involve graduate students. The results will be disseminated through publications in professional journals, lectures and on the web. A sincere effort will be made to expose (and involve) students and young Ph.D.'s from underrepresented groups to this research and to increase their participation in mathematics.
During this period (07/01/2013 to 06/30/2014) the PI, in joint work with collaborators and students, produce several research articles as described in the file attached with the final report. These papers include (1) A paper with Dr. D. Applebaum of Sheffield University, UK, in which techniques of probability and Lie groups are combined to study boundedness properties of operators on Lie groups that arise from martingale transforms, (2) a paper with Drs. J. Mijena and E. Nane of Auburn University which combines techniques from probability and partial differential equations to study problems in spectral theory arising from a model involving the relativistic Brownian motion and (3) a paper with Dr. Osekowski of Warsaw University, Poland, in which novel techniques are used to obtain estimates related to a celebrated conjecture (open for more than 30 years) concerning the norm of the Beurling-Ahlfors operator. These publications addressed, directly, problems listed in the original proposal. For this purpose, tools and techniques were developed that (it is hoped) will lead to new applications within the fields of probability, harmonic analysis and spectral theory. During this period (as described in the attached file) the PI delivered several lectures at professional meetings and departments of mathematics colloquia, including lectures in Europe. Mexico, and the US. During this period the PI worked in the training of three graduate students and advising many other graduate students who come from the traditionally underrepresented groups in mathematics.
