Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein–Uhlenbeck processes

Abstract

Properties of the law μ of the integral ∫₀^∞c^−N_t− dY_t are studied, where c>1 and {(N_t, Y_t), t≥0} is a bivariate Lévy process such that {N_t} and {Y_t} are Poisson processes with parameters a and b, respectively. This is the stationary distribution of some generalized Ornstein–Uhlenbeck process. The law μ is parametrized by c, q and r, where p=1−q−r, q, and r are the normalized Lévy measure of {(N_t, Y_t)} at the points (1, 0), (0, 1) and (1, 1), respectively. It is shown that, under the condition that p>0 and q>0, μ_c, q, r is infinitely divisible if and only if r≤pq. The infinite divisibility of the symmetrization of μ is also characterized. The law μ is either continuous-singular or absolutely continuous, unless r=1. It is shown that if c is in the set of Pisot–Vijayaraghavan numbers, which includes all integers bigger than 1, then μ is continuous-singular under the condition q>0. On the other hand, for Lebesgue almost every c>1, there are positive constants C₁ and C₂ such that μ is absolutely continuous whenever q≥C₁p≥C₂r. For any c>1 there is a positive constant C₃ such that μ is continuous-singular whenever q>0 and max {q, r}≤C₃p. Here, if {N_t} and {Y_t} are independent, then r=0 and q=b/(a+b).