  We develop a new method to solve the Fokker-Planck or Kolmogorov's forward
equation that governs the time evolution of the joint probability density
function of a continuous-time stochastic nonlinear system. Numerical solution
of this equation is fundamental for propagating the effect of initial
condition, parametric and forcing uncertainties through a nonlinear dynamical
system, and has applications encompassing but not limited to forecasting, risk
assessment, nonlinear filtering and stochastic control. Our methodology breaks
away from the traditional approach of spatial discretization for solving this
second-order partial differential equation (PDE), which in general, suffers
from the "curse-of-dimensionality". Instead, we numerically solve an infinite
dimensional proximal recursion in the space of probability density functions,
which is theoretically equivalent to solving the Fokker-Planck-Kolmogorov PDE.
We show that the dual formulation along with the introduction of an entropic
regularization, leads to a smooth convex optimization problem that can be
implemented via suitable block co-ordinate iteration and has fast convergence
due to certain contraction property that we establish. This approach enables
meshless implementation leading to remarkably fast computation.
