Intrinsic Ultracontractivity and Other Properties of Mixed Barrier Brownian Motions

Abstract

This thesis consists of two main parts. First, we study an asymptotic behavior of Brownian motions with Dirichlet boundary condition on an unbounded domain D above the graph of a bounded Lipschitz function. We discover that the large time behavior of the Dirichlet heat kernel pD = pD(t , x, y), for x, y ∈ D is limt→infinity td+22pD t,x,y=C1u xuy, where u is a minimal harmonic function corresponding to the Martin point at infinity and C1 is a positive constant. Next, we study some properties of mixed barrier Brownian motions on various domains. More specifically, we obtain not only a Gaussian upper bound for the transition density of the mixed barrier Brownian motion, i.e., qt,x,y≤C2 t-d2exp -x-y2 C3t, for some positive constants C2 and C3, but also intrinsic ultracontractivity (IU) for some Schrodinger operator H. IU was introduced by E. B. Davies and B. Simon in 1984 and states that pt,x,y≤Ct 41x4 1y, where 41 is the ground state of H on D with mixed boundary conditions. Logarithmic Sobolev inequality and the theory of Dirichlet forms are the major tools used to prove IU.