Dirichlet forms and associated heat kernels on the Cantor set induced by random walks on trees

Abstract

Transient random walk on a tree induces a Dirichlet form on its Martin boundary, which is the Cantor set. The procedure of the inducement is analogous to that of the Douglas integral on S 1 associated with the Brownian motion on the unit disk. In this paper, those Dirichlet forms on the Cantor set induced by random walks on trees are investigated. Explicit expressions of the hitting distribution (harmonic measure) ν and the induced Dirichlet form on the Cantor set are given in terms of the effective resistances. An intrinsic metric on the Cantor set associated with the random walk is constructed. Under the volume doubling property of ν with respect to the intrinsic metric, asymptotic behaviors of the heat kernel, the jump kernel and moments of displacements of the process associated with the induced Dirichlet form are obtained. Furthermore, relation to the noncommutative Riemannian geometry is discussed.