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

Properties of the law $\mu$ of the integral $\int_0^{\infty}c^{-N_{t-}}\,dY_t$ are studied, where $c>1$ and $\{(N_t,Y_t),t\geq0\}$ is a bivariate L\'{e}vy 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 $\mu$ is parametrized by $c$, $q$ and $r$, where $p=1-q-r$, $q$, and $r$ are the normalized L\'{e}vy 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 $q>0$, $\mu_{c,q,r}$ is infinitely divisible if and only if $r\leq pq$. The infinite divisibility of the symmetrization of $\mu$ is also characterized. The law $\mu$ 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 $\mu$ is continuous-singular under the condition $q>0$. On the other hand, for Lebesgue almost every $c>1$, there are positive constants $C_1$ and $C_2$ such that $\mu$ is absolutely continuous whenever $q\geq C_1p\geq C_2r$. For any $c>1$ there is a positive constant $C_3$ such that $\mu$ is continuous-singular whenever $q>0$ and $\max\{q,r\}\leq C_3p$. Here, if $\{N_t\}$ and $\{Y_t\}$ are independent, then $r=0$ and $q=b/(a+b)$.