Ruin probability with certain stationary stable claims generated by conservative flows

We study the ruin probability where the claim sizes are modeled by a stationary ergodic symmetric .-stable process. We exploit the flow representation of such processes, and we consider the processes generated by conservative flows. We focus on two classes of conservative .-stable processes (one discrete-time and one continuous-time), and give results for the order of magnitude of the ruin probability as the initial capital goes to infinity. We also prove a solidarity property for null-recurrent Markov chains as an auxiliary result, which might be of independent interest.