  We develop a new computational framework to solve the partial differential
equations (PDEs) governing the flow of the joint probability density functions
(PDFs) in continuous-time stochastic nonlinear systems. The need for computing
the transient joint PDFs subject to prior dynamics arises in uncertainty
propagation, nonlinear filtering and stochastic control. Our methodology breaks
away from the traditional approach of spatial discretization or function
approximation -- both of which, in general, suffer from the
"curse-of-dimensionality". In the proposed framework, we discretize time but
not the state space. We solve infinite dimensional proximal recursions in the
manifold of joint PDFs, which in the small time-step limit, is theoretically
equivalent to solving the underlying transport PDEs. The resulting computation
has the geometric interpretation of gradient flow of certain free energy
functional with respect to the Wasserstein metric arising from the theory of
optimal mass transport. We show that dualization along with an entropic
regularization, leads to a cone-preserving fixed point recursion that is proved
to be contractive in Thompson metric. A block co-ordinate iteration scheme is
proposed to solve the resulting nonlinear recursions with guaranteed
convergence. This approach enables remarkably fast computation for
non-parametric transient joint PDF propagation. Numerical examples and various
extensions are provided to illustrate the scope and efficacy of the proposed
approach.
