In this project the principal investigator proposes to develop further the theory of analysis on abstract metric spaces by studying Green functions on such spaces. The principal investigator will initially focus on four problems. The first question seeks to find out whether in the framework of abstract metric spaces the corresponding harmonic functions satisfy the strong maximum principle. As a long range goal the principal investigator also hopes to determine whether the strong maximum principle holds for p-harmonic functions for general values of p. The second issue is to construct the Martin boundary for Gromov hyperbolic metric spaces that admit Green functions. It is also proposed to study some properties of Green functions, including the uniqueness property and the boundary Harnack principle. These two problems will be studied for two different constructions of Green functions, one - using the upper gradients, and the other - the Cheeger derivative. The fourth problem is to construct and study Brownian motion and the heat equation on abstract metric spaces.
Potential applications of the research proposed in this project include connections between probability theory and analysis on abstract metric spaces. Such spaces arise in applications in physics and engineering, and hence the questions addressed in the project have a potential impact in physics and engineering as well.
