Markov Kernels, Distances and Optimal Control: A Parable of Linear Quadratic Non-Gaussian Distribution Steering

For a controllable linear time-varying (LTV) pair (boldsymbolAt,boldsymbolBt) and boldsymbolQt positive semidefinite, we derive the Markov kernel for the Ito diffusion dboldsymbolxt=boldsymbolAtboldsymbolxt d t + √2boldsymbolBtdboldsymbolwt with an accompanying killing of probability mass at rate frac12boldsymbolxtopboldsymbolQtboldsymbolx. This Markov kernel is the Green's function for an associated linear reaction-advection-diffusion partial differential equation. Our result generalizes the recently derived kernel for the special case left(boldsymbolAt,boldsymbolBtright)=left(boldsymbol0,boldsymbolIright), and depends on the solution of an associated Riccati matrix ODE. A consequence of this result is that the linear quadratic non-Gaussian Schr"odinger bridge is exactly solvable. This means that the problem of steering a controlled LTV diffusion from a given non-Gaussian distribution to another over a fixed deadline while minimizing an expected quadratic cost can be solved using dynamic Sinkhorn recursions performed with the derived kernel. Our derivation for the left(boldsymbolAt,boldsymbolBt,boldsymbolQtright)-parametrized kernel pursues a new idea that relies on finding a state-time dependent distance-like functional given by the solution of a deterministic optimal control problem. This technique breaks away from existing methods, such as generalizing Hermite polynomials or Weyl calculus, which have seen limited success in the reaction-diffusion context. Our technique uncovers a new connection between Markov kernels, distances, and optimal control. This connection is of interest beyond its immediate application in solving the linear quadratic Schr"odinger bridge problem.