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.

Questo articolo esplora i giri e le loro implicazioni.

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