Weyl Calculus and Exactly Solvable Schr\"{o}dinger Bridges with Quadratic State Cost

TL;DR

This paper uses Weyl calculus from quantum mechanics to explicitly solve Schr"odinger bridge problems with quadratic state costs, simplifying the derivation of Markov kernels.

Contribution

It introduces a novel application of Weyl calculus to derive Markov kernels for Schr"odinger bridges with quadratic costs, avoiding complex Hermite polynomial computations.

Findings

Explicit Markov kernel for quadratic state cost derived using Weyl calculus

Simplified approach compared to Hermite polynomial methods

Connections established between quantum mechanics tools and stochastic control

Abstract

Schr\"{o}dinger bridge--a stochastic dynamical generalization of optimal mass transport--exhibits a learning-control duality. Viewed as a stochastic control problem, the Schr\"{o}dinger bridge finds an optimal control policy that steers a given joint state statistics to another while minimizing the total control effort subject to controlled diffusion and deadline constraints. Viewed as a stochastic learning problem, the Schr\"{o}dinger bridge finds the most-likely distribution-valued trajectory connecting endpoint distributional observations, i.e., solves the two point boundary-constrained maximum likelihood problem over the manifold of probability distributions. Recent works have shown that solving the Schr\"{o}dinger bridge problem with state cost requires finding the Markov kernel associated with a reaction-diffusion PDE where the state cost appears as a state-dependent reaction rate. We explain how ideas from Weyl calculus in quantum mechanics, specifically the Weyl operator and the Weyl symbol, can help determine such Markov kernels. We illustrate these ideas by explicitly finding the Markov kernel for the case of quadratic state cost via Weyl calculus, recovering our earlier results but avoiding tedious computation with Hermite polynomials.