  We consider the problem to identify the most likely flow in phase space, of
(inertial) particles under stochastic forcing, that is in agreement with
spatial (marginal) distributions that are specified at a set of points in time.
The question raised generalizes the classical Schrodinger Bridge Problem (SBP)
which seeks to interpolate two specified end-point marginal distributions of
overdamped particles driven by stochastic excitation. While we restrict our
analysis to second-order dynamics for the particles, the data represents
partial (i.e., only positional) information on the flow at {\em multiple}
time-points. The solution sought, as in SBP, represents a probability law on
the space of paths this closest to a uniform prior while consistent with the
given marginals. We approach this problem as an optimal control problem to
minimize an action integral a la Benamou-Brenier, and derive a time-symmetric
formulation that includes a Fisher information term on the velocity field. We
underscore the relation of our problem to recent measure-valued splines in
Wasserstein space, which is akin to that between SBP and Optimal Mass Transport
(OMT). The connection between the two provides a Sinkhorn-like approach to
computing measure-valued splines. We envision that interpolation between
measures as sought herein will have a wide range of applications in
signal/images processing as well as in data science in cases where data have a
temporal dimension.
