  Lambert's problem concerns with transferring a spacecraft from a given
initial to a given terminal position within prescribed flight time via velocity
control subject to a gravitational force field. We consider a probabilistic
variant of the Lambert problem where the knowledge of the endpoint constraints
in position vectors are replaced by the knowledge of their respective joint
probability density functions. We show that the Lambert problem with endpoint
joint probability density constraints is a generalized optimal mass transport
(OMT) problem, thereby connecting this classical astrodynamics problem with a
burgeoning area of research in modern stochastic control and stochastic machine
learning. This newfound connection allows us to rigorously establish the
existence and uniqueness of solution for the probabilistic Lambert problem. The
same connection also helps to numerically solve the probabilistic Lambert
problem via diffusion regularization, i.e., by leveraging further connection of
the OMT with the Schr\"odinger bridge problem (SBP). This also shows that the
probabilistic Lambert problem with additive dynamic process noise is in fact a
generalized SBP, and can be solved numerically using the so-called
Schr\"odinger factors, as we do in this work. We explain how the resulting
analysis leads to solving a boundary-coupled system of reaction-diffusion PDEs
where the nonlinear gravitational potential appears as the reaction rate. We
propose novel algorithms for the same, and present illustrative numerical
results. Our analysis and the algorithmic framework are nonparametric, i.e., we
make neither statistical (e.g., Gaussian, first few moments, mixture or
exponential family, finite dimensionality of the sufficient statistic) nor
dynamical (e.g., Taylor series) approximations.
