  We characterize the Schr\"odinger bridge problems by a family of
Mckean-Vlasov stochastic control problems with no terminal time distribution
constraint. In doing so, we use the theory of Hilbert space embeddings of
probability measures and then describe the constraint as penalty terms defined
by the maximum mean discrepancy in the control problems. A sequence of the
probability laws of the state processes resulting from $\epsilon$-optimal
controls converges to a unique solution of the Schr\"odinger's problem under
mild conditions on given initial and terminal time distributions and an
underlying diffusion process. We propose a neural SDE based deep learning
algorithm for the Mckean-Vlasov stochastic control problems. Several numerical
experiments validate our methods.
