  The purpose of this work is mostly expository and aims to elucidate the
Jordan-Kinderlehrer-Otto (JKO) scheme for uncertainty propagation, and a
variant, the Laugesen-Mehta-Meyn-Raginsky (LMMR) scheme for filtering. We point
out that these variational schemes can be understood as proximal operators in
the space of density functions, realizing gradient flows. These schemes hold
the promise of leading to efficient ways for solving the Fokker-Planck equation
as well as the equations of non-linear filtering. Our aim in this paper is to
develop in detail the underlying ideas in the setting of linear stochastic
systems with Gaussian noise and recover known results.
