Abstract

Let (X-t) be a one-dimensional Ornstein-Uhlenbeck process with initial density function f : R+ --> R+, which is a regularly varying function with exponent -(1 + eta), eta is an element of (0, 1). We prove the existence of a probability measure nu with a Lebesgue density, depending on eta, such that for every A is an element of B(R+): lim(t-->infinity) P-f(X-t is an element of A - T-0(X) > t) = nu(A).