Multidimensional Stochastic Analysis

University Of Connecticut, Storrs CT

Investigators

Abstract

0244737 Bass The principal investigator will be working on problems in two areas of probability. The first is concerned with Harnack inequalities. A Harnack inequality asserts that nonnegative solutions to a partial differential equation satisfy certain boundedness estimates at points, and thus allow one to obtain pointwise estimates from global information. They are used in partial differential equations to obtain estimates on heat kernels and to prove regularity properties of solutions. They are used in probability to obtain transition density estimates and regularity properties of certain stochastic processes. The principal investigator will investigate when one can obtain Harnack inequalities for functions related to non-local operators. The operators in question have an integral term and correspond to processes with jumps. The second area of research concerns uniqueness for the solutions of stochastic differential equations arising from population models in mathematical biology. These equations describe the limit of branching diffusion processes as the number of particles increases, the mass of each particle decreases, and the branching rate increases. The branching rate and the diffusion mechanism for a particle are allowed to depend on all other particles in the system. Branching diffusions are used as models of population dynamics for a large variety of species. The equations that result are typically either infinite dimensional, degenerate, or both. The principal investigator will continue his work on proving uniqueness for these equations. It has been known for a long time that many systems in the physical and biological sciences can be modeled by stochastic processes. More recently it has been discovered that many financial and economic systems can also be so modeled. To investigate more complex systems, new types of random processes have arisen. To give an example, stock prices are often viewed as depending on a continuous random process, Brownian motion. Yet the fluctuations of stock prices often have sudden jumps, resulting from wars, new discoveries, etc. Thus it is essential to also study stochastic processes with jumps. When studying population models, one expects that the behavior of the population will be qualitatively different depending on whether the population is large or whether it is small. The research of the principal investigator is primarily concerned with two types of stochastic processes, ones with jumps, as in the stock market example, and ones concerning systems that can degenerate, as in the population example. Some of the questions that are being investigated are whether there is only one solution to the equation and whether the solution has sufficient regularity to be useful in providing new information for the model.