Small value probabilities for continuous state branching processes with immigration

We consider the small value probability of supercritical continuous state branching processes with immigration.From Pinsky(1972) it is known that under regularity condition on the branching mechanism and immigration mechanism,the normalized population size converges to a non-degenerate finite and positive limit W as t tends to infinity.We provide sharp estimate on asymptotic behavior of P(W≤ε) as ε → 0+ by studying the Laplace transform of W.Without immigration,we also give a simpler proof for the small value probability in the non-subordinator case via the prolific backbone decomposition.