A universality class in Markovian persistence

We consider the class of Markovian processes defined by the equation $\dd x /\dd t = -\beta x + \sum_k z_k \delta (t-t_k)$. Such processes are encountered in systems (like coalescing systems) where dynamics creates discrete upward jumps at random instants $t_k$ and of random height $z_k$. We observe that the probability for these processes to remain above their mean value during an interval of time $T$ decays as $\exp{-\theta T}$ defining $\theta$ as the persistence exponent. We show that $\theta$ takes the value $\beta$ which thereby extends the well known result of the Gaussian noise case to a much larger class of non-Gaussian processes.