Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weyl Calculus and Exactly Solvable Schrödinger Bridges with Quadratic State Cost

Alexis M. H. Teter, Wenqing Wang, Abhishek Halder

TL;DR

The paper tackles Schrödinger bridge problems with quadratic state costs by deriving the Markov kernel for the associated reaction-diffusion PDE using Weyl calculus, providing a systematic PDE->Weyl-operator->Weyl-symbol->kernel workflow.Starting from a linear PDE, the authors construct the Weyl operator, obtain its Weyl symbol, and then extract the kernel via an explicit integral transform, illustrated first with the heat equation and then for quadratic costs.A principal result is a closed-form kernel κ_Λ for the diagonalized quadratic case, along with a general affine extension κ_(Λ, r, s); the isotropic choice recovers the classical Mehler kernel, linking quantum-mechanical kernels to SB problems.The work demonstrates a practical, scalable alternative to Hermite-polynomial computations, broadening the toolbox for diffusion and control practitioners and suggesting wide applicability to SB, diffusion models, and related linear PDE IVPs.

Abstract

Schrödinger bridge--a stochastic dynamical generalization of optimal mass transport--exhibits a learning-control duality. Viewed as a stochastic control problem, the Schrödinger 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 Schrödinger 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 Schrödinger 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.

Weyl Calculus and Exactly Solvable Schrödinger Bridges with Quadratic State Cost

TL;DR

The paper tackles Schrödinger bridge problems with quadratic state costs by deriving the Markov kernel for the associated reaction-diffusion PDE using Weyl calculus, providing a systematic PDE->Weyl-operator->Weyl-symbol->kernel workflow.Starting from a linear PDE, the authors construct the Weyl operator, obtain its Weyl symbol, and then extract the kernel via an explicit integral transform, illustrated first with the heat equation and then for quadratic costs.A principal result is a closed-form kernel κ_Λ for the diagonalized quadratic case, along with a general affine extension κ_(Λ, r, s); the isotropic choice recovers the classical Mehler kernel, linking quantum-mechanical kernels to SB problems.The work demonstrates a practical, scalable alternative to Hermite-polynomial computations, broadening the toolbox for diffusion and control practitioners and suggesting wide applicability to SB, diffusion models, and related linear PDE IVPs.

Abstract

Schrödinger bridge--a stochastic dynamical generalization of optimal mass transport--exhibits a learning-control duality. Viewed as a stochastic control problem, the Schrödinger 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 Schrödinger 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 Schrödinger 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.
Paper Structure (14 sections, 1 theorem, 74 equations)

This paper contains 14 sections, 1 theorem, 74 equations.

Table of Contents

  1. Introduction
  2. Notations and Background
  3. Weyl Calculus
  4. From PDE to Weyl Operator
  5. From Weyl Operator to Weyl Symbol
  6. From Weyl Symbol to Kernel
  7. Product Rule of Weyl Calculus
  8. Kernel for the Schrödinger Bridge with Quadratic State Cost
  9. The Weyl Operator $H_{\bm{\Lambda}}$
  10. Deriving the PDE for the Weyl Symbol $h_{\bm{\Lambda}}$
  11. Solving the PDE for the Weyl Symbol $h_{\bm{\Lambda}}$
  12. The Kernel $\kappa_{\bm{\Lambda}}$
  13. Generalization
  14. Conclusions

Key Result

Theorem 1

Given $\bm{Q}\succeq\bm{0}$, $\bm{r}\in\mathbb{R}^{n}$, $s\in\mathbb{R}$, consider the eigen-decomposition $\frac{1}{2}\bm{Q} = \bm{V}^{\top}\bm{\Lambda}\bm{V}$, where the main diagonal entries of the diagonal matrix $\Lambda$ are $\{\lambda_k\}_{k=1}^{n}$. Let $\bm{v}_{k}$ be the $k$th column of $\ where the kernel and the constants

Theorems & Definitions (8)

  • Example 1: Reaction-diffusion PDE IVP
  • Example 2: Weyl operator for the heat PDE
  • Example 3: Weyl symbol for the heat PDE
  • Example 4: Kernel for the heat PDE
  • Remark 1
  • Remark 2
  • Theorem 1
  • proof