This proposal studies problems that lie at the interface of probability and other fields of mathematics, or in more contemporary language, problems in stochastic analysis. These include (1) problems related to the hot-spots conjecture of Jeff Rauch for Neumann eigenfunctions and counterparts originally raised by the PI for conditioned Brownian motion and (2) problems related to the sharp spectral gap bounds conjecture for Schrodinger operators. The problems in (1) and (2) are formulated in terms of survival time probabilities for Brownian motion which are then reduced to the study of finite dimensional distributions functions. The probabilistic formulations are more general than the original conjectures but much more natural from the point of view of the stochastic analysis ideas and techniques suggested in this proposal. The conjectures that motivate these problems are well known and very difficult. Any progress on these problems will likely lead to other applications and will be of considerable interest to researchers working on several different areas of mathematics, including harmonic analysis, partial differential equations, geometry and probability. Versions of these problems will also be investigated for processes other than Brownian motion such as symmetric stable processes and more general Levy processes.
It has been known since the beginning of the last century that many physical and biological phenomena can be modeled by random stochastic processes. Indeed, the classical model of Brownian motion which is at the heart of the problems discussed above was discovered by the British botanist Robert Brown (even earlier) from physical observations of pollen particles suspended in fluids. More recently, stochastic models have been used in many other settings in the physical, biological and social sciences with many sophisticated applications to economic models, and particularly to financial markets. Many of these applications involve stochastic processes which, unlike Brownian motion, do not have continuous trajectories. The finite dimensional distribution techniques that arise in the problems above came about because of the connections to these more general stochastic processes.
