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

For a controllable linear time-varying (LTV) pair $(\boldsymbol{A}_t,\boldsymbol{B}_t)$ and $\boldsymbol{Q}_{t}$ positive semidefinite, we derive the Markov kernel for the It\^{o} diffusion ${\mathrm{d}}\boldsymbol{x}_{t}=\boldsymbol{A}_{t}\boldsymbol{x}_t {\mathrm{d}} t + \sqrt{2}\boldsymbol{B}_{t}{\mathrm{d}}\boldsymbol{w}_{t}$ with an accompanying killing of probability mass at rate $\frac{1}{2}\boldsymbol{x}^{\top}\boldsymbol{Q}_{t}\boldsymbol{x}$. 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 $\left(\boldsymbol{A}_t,\boldsymbol{B}_t\right)=\left(\boldsymbol{0},\boldsymbol{I}\right)$, and depends on the solution of an associated Riccati matrix ODE. A consequence of this result is that the linear quadratic non-Gaussian Schr\"{o}dinger 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 $\left(\boldsymbol{A}_t,\boldsymbol{B}_t,\boldsymbol{Q}_t\right)$-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\"{o}dinger bridge problem.