Brownian motion with killing and reflection, stable processes and projections of martingales

Purdue University, West Lafayette IN

Investigators

Abstract

0303259 Banuelos This project is to study a number of open problems which lie at the interface of probability theory and other fields of mathematics. These include: (1) Problems concerning `hot-spots' properties for the survival time probabilities of Brownian motion which is killed on part of the boundary of a domain and reflected on the rest; (2) Problems concerning the `fine' spectral theoretic properties of symmetric stable processes, such as variational characterization for eigenvalues, properties of nodal lines, geometric, analytic and probabilistic properties of finite dimensional distributions as a function of the starting point; and (3) Optimal problems that arise from studying the Beurling-Ahlfors singular integral operator using space-time Brownian motion. Solutions to the 'hot-spots' problems will lead to further progress on the celebrated 'hot-spots' conjecture of J. Rauch which asserts that the maximum and minimum of ground state Neumann eigenfunctions are attained only at boundary points. Many of the proposed problems for stable processes are motivated by their well known counterparts for Brownian motion. However, some of the questions for stable processes in turn lead to problems which will provide new information even for Brownian motion. Since the techniques for the Brownian motion results do not apply, new techniques must be developed for these problems. Such techniques are likely to lead to new and unexpected applications in other areas particularly for other stochastic processes with jumps but whose transition probabilities still retain certain rotational invariance properties. Martingales have played an important role in the study of singular integral operators in general and in the study of Beurling-Ahlfors operator in particular. A new probabilistic approach, based on heat martingales, is proposed. This will give a better understanding of a celebrated conjecture of T. Iwaniec on the size of the norm of this operator. A remarkable aspect of contemporary mathematics is the unexpected yet deep connections between previously very different fields. The problems in the proposed project are all interdisciplinary in nature and as such have the potential to impact different fields in mathematics, engineering, science and economics. Symmetric stable processes have been used to model many physical and economic phenomena, particularly for certain stocks where the Brownian motion models are not adequate. Estimates on the Beurling-Ahlfors operator have many applications to nonlinear problems in elasticity. The 'hot-spots' conjecture, when formulated in terms of the theory of heat conduction, asserts that if one begins with an initial heat distribution on a plate which is insulated around its boundary and waits for the initial transients to settle down, then the hottest and coldest regions will be found on the boundary of the plate. Since the mathematical description of heat is given by a second order partial differential operator, introduced by the Marquis Pierre-Simon De Laplace (1749-1827) more than 200 years ago, the conjecture has been of interest to researchers in mathematics, physics and chemistry. The connection to probability arises from the description by Norbert Wiener (1894-1964) of heat flow in terms of Brownian motion.