Domain of attraction of the quasi-stationary distributions for the Ornstein-Uhlenbeck process

Abstract

Let $X=(X_t)$ be a one-dimensional Ornstein-Uhlenbeck process with an initial density function $f$ supported on the positive real-line that is a regularly varying function with exponent $-(1+\eta)$, with $\eta\in (0,1)$. We prove the existence of a probability measure $\nu$ with a Lebesgue density, depending on $\eta$, such that for every Borel set $A$ of the positive real-line: $\lim_{t\to\infty} P_f(X_t\in A | T_0^X>t)=\nu(A)$, where $T_0^X$ is the hitting time of 0 of $X$.