The Green's function for an LTV Itô diffusion with quadratic state killing is a Gaussian in the endpoints whose quadratic form comes from a Riccati ODE.
On the contraction co efficient of the Schr¨ odinger bridge for stochastic linear systems,
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Markov Kernels, Distances and Optimal Control: A Parable of Linear Quadratic Non-Gaussian Distribution Steering
The Green's function for an LTV Itô diffusion with quadratic state killing is a Gaussian in the endpoints whose quadratic form comes from a Riccati ODE.