Markovian Solutions to Discontinuous ODEs

Given a possibly discontinuous, bounded function f: R↦ R, we consider the set of generalized flows, obtained by assigning a probability measure on the set of Carathéodory solutions to the ODE x˙ = f(x). The paper provides a complete characterization of all such flows which have a Markov property in time. This is achieved in terms of (i) a positive, atomless measure supported on the set f^{- 1}(0) where f vanishes, (ii) a countable number of Poisson random variables, determining the waiting times at points in f^{- 1}(0) , and (iii) a countable set of numbers θ_k∈ [0 , 1] , describing the probability of moving up or down, at isolated points where two distinct trajectories can originate.