Continuity Properties and Infinite Divisibility of Stationary Distributions of Some Generalised Ornstein-uhlenbeck Processes

Properties of the law μ of the integral ∫∞ 0 c −Nt− dYt are studied, where c > 1 and {(Nt, Yt), t ≥ 0} is a bivariate Lévy process such that {Nt} and {Yt} are Poisson processes with parameters a and b, respectively. This is the stationary distribution of some generalised Ornstein-Uhlenbeck process. The law μ is parametrised by c, q and r, where p = 1−q−r, q, and r are the normalised Lévy measure of {(Nt, Yt)} at the points (1, 0), (0, 1) and (1, 1), respectively. It is shown that, under the condition that p > 0 and q > 0, μc,q,r is infinitely divisible if and only if r ≤ pq. The infinite divisibility of the symmetrisation of μ is also characterised. The law μ is either continuoussingular 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 continuoussingular whenever q > 0 and max{q, r} ≤ C3p. Here, if {Nt} and {Yt} are independent, then r = 0 and q = b/(a+ b).