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In this paper we investigate a non-linear and non-local one dimensionaltransport equation under random perturbations on the real line. We firstestablish a local-in-time theory, i.e., existence, uniqueness and blow-upcriterion for pathwise solutions in Sobolev spaces $H^{s}$ with $s>3$.Thereafter, we give a complete picture of the long time behavior of thesolutions based on the type of noise we consider. On one hand, we identify afamily of noises such that blow-up can be prevented with probability $1$,guaranteeing the existence and uniqueness of global solutions almost surely. Onthe other hand, in the particular linear noise case, we show that singularitiesoccur in finite time with positive probability, and we derive lower bounds ofthese probabilities. To conclude, we introduce the notion of stability ofexiting times and show that one cannot improve the stability of the exitingtime and simultaneously improve the continuity of the dependence on initialdata.

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