The principal investigator will study whether the analog of the strong law of large numbers holds for reinforced random walk on the integers, whenever the initial weightings are equal, under arbitrary past dependent reinforcement. He also will investigate recurrence of two dimensional, and n dimensional, reinforced random walk. In addition, he proposes to find necessary and sufficient geometrical conditions for certain classes of domains, including the class of those domains in R**2 with boundary in a sense given locally by the graph of a function, for the semigroup connected with the Dirichlet Laplacian to be intrinsically ultracontractive. The principal investigator will conduct research at the interface between probability and analysis. He will work on two classes of problems: reinforced random walk, and intrinsic ultracontractivity.
