Global existence, blow-up and stability for a stochastic transport equation with non-local velocity

In this paper we investigate a non-linear and non-local one dimensional transport equation under random perturbations on the real line. We first establish local-in-time theory, i.e., existence, uniqueness blow-up criterion for pathwise solutions in Sobolev spaces Hs with s>3. Thereafter, give picture of long time behavior based type noise consider. On hand, identify family noises such that can be prevented probability 1, guaranteeing existence global almost surely. other particular linear case, show singularities occur finite positive probability, derive lower bounds these probabilities. To conclude, introduce notion stability exiting times cannot improve simultaneously continuity dependence initial data.