In a companion paper, the authors have characterized all deterministic semigroups, and all Markov semigroups, whose trajectories are Carathéodory solutions to a given ODE x˙ = f(x) , where f is a possibly discontinuous, regulated function. The present paper establishes two approximation results. Namely, every deterministic semigroup can be obtained as the pointwise limit of the flows generated by a sequence of ODEs x˙ = fn(x) with smooth right hand sides. Moreover, every Markov semigroup can be obtained as limit of a sequence of diffusion processes with smooth drifts and with diffusion coefficients approaching zero.
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