Sato's conjecture on recurrence conditions for multidimensional processes of Ornstein-Uhlenbeck type

摘要

A stochastic process of Ornstein-Uhlenbeck type (OU type process) {X_t} was introduced in one dimension by Wolfe and in multidimension by Sato and Yamazato. It is a Markov process (Ω,F,F_t,P~x,X_t) on the d-dimensional Euclidean space R~d obtained from a spatially homogeneous Markov process undergoing a linear drift force determined by a matrix -Q. The purpose of this paper is to give an integral condition of recurrence and transience for OU type processes. Let {Z_t} be a Levy process on R~d, that is, a stochastically continuous process with stationary independent increments, starting at the origin. Let Q be a real d x d matrix of which all eigenvalues have positive real parts.