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

Gradient Flows in Uncertainty Propagation and Filtering of Linear Gaussian Systems