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The subject of this work has its roots in the so called Schroedginer Bridge Problem (SBP) which asks for the most likely distribution of Brownian particles in their passage between observed empirical marginal distributions at two distinct points in time.
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Y. Chen, T. T. Georgiou, and M. Pavon, “Optimal steering of a linear stochastic system to a final probability distribution, Part I,” IEEE Trans. on Automatic Control , vol. 61, no. 5, pp. 1158–1169, 2016
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Y. Chen, T. T. Georgiou, and M. Pavon, “On the relation between optimal transport and Schrödinger bridges: A stochastic control viewpoint,” Journal of Optimization Theory and Applications , vol. 169, no. 2, pp. 671–691, 2016
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