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A weak convergence approach to the large deviations of the dynamic Schrödinger problem

Viktor Nilsson, Pierre Nyquist

TL;DR

The paper advances large deviation theory for dynamical Schrödinger bridges by formulating a weak convergence/variational approach that works beyond Brownian references. It proves a uniform Laplace principle for conditioned bridge processes and uses it, alongside static SB LDPs, to establish a full LDP for dynamic Schrödinger bridges with general reference dynamics; the key rate function is $I_D(\varphi)= I_S(\varphi_0,\varphi_1) + I_B^{\varphi_0\varphi_1}(\varphi)$, with $I_B^{xy}$ defined via a controlled-ODE framework. The results are complemented by explicit examples with scaled Brownian motion and Ornstein–Uhlenbeck references, demonstrating applicability to non-Brownian dynamics and paving the way for extensions to reflected Schrödinger bridges. Overall, the work integrates Schrödinger bridges, entropic OT, and weak convergence methods to provide a robust large deviations toolkit for dynamical SBs and related transport problems.

Abstract

In this paper, we consider the large deviations for dynamical Schrödinger problems, using the variational approach developed by Dupuis, Ellis, Budhiraja, and others. Recent results on scaled families of Schrödinger problems, in particular by Bernton, Ghosal, and Nutz, and the authors, have established large deviation principles for the static problem. For the dynamic problem, only the case with a scaled Brownian motion reference process has been explored by Kato. Here, we derive large deviations results using the variational approach, with the aim of going beyond the Brownian reference dynamics considered by Kato. Specifically, we develop a uniform Laplace principle for bridge processes conditioned on their endpoints. When combined with existing results for the static problem, this leads to a large deviation principle for the corresponding (dynamic) Schrödinger bridge. In addition to the specific results of the paper, our work puts such large deviation questions into the weak convergence framework, and we conjecture that the results can be extended to cover also more involved types of reference dynamics. Specifically, we provide an outlook on applying the result to reflected Schrödinger bridges.

A weak convergence approach to the large deviations of the dynamic Schrödinger problem

TL;DR

The paper advances large deviation theory for dynamical Schrödinger bridges by formulating a weak convergence/variational approach that works beyond Brownian references. It proves a uniform Laplace principle for conditioned bridge processes and uses it, alongside static SB LDPs, to establish a full LDP for dynamic Schrödinger bridges with general reference dynamics; the key rate function is , with defined via a controlled-ODE framework. The results are complemented by explicit examples with scaled Brownian motion and Ornstein–Uhlenbeck references, demonstrating applicability to non-Brownian dynamics and paving the way for extensions to reflected Schrödinger bridges. Overall, the work integrates Schrödinger bridges, entropic OT, and weak convergence methods to provide a robust large deviations toolkit for dynamical SBs and related transport problems.

Abstract

In this paper, we consider the large deviations for dynamical Schrödinger problems, using the variational approach developed by Dupuis, Ellis, Budhiraja, and others. Recent results on scaled families of Schrödinger problems, in particular by Bernton, Ghosal, and Nutz, and the authors, have established large deviation principles for the static problem. For the dynamic problem, only the case with a scaled Brownian motion reference process has been explored by Kato. Here, we derive large deviations results using the variational approach, with the aim of going beyond the Brownian reference dynamics considered by Kato. Specifically, we develop a uniform Laplace principle for bridge processes conditioned on their endpoints. When combined with existing results for the static problem, this leads to a large deviation principle for the corresponding (dynamic) Schrödinger bridge. In addition to the specific results of the paper, our work puts such large deviation questions into the weak convergence framework, and we conjecture that the results can be extended to cover also more involved types of reference dynamics. Specifically, we provide an outlook on applying the result to reflected Schrödinger bridges.

Paper Structure

This paper contains 15 sections, 11 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Preliminaries
  4. Large deviation principles
  5. A uniform Laplace principle for bridge processes
  6. Tightness of controlled processes
  7. Convergence to controlled ODE
  8. Laplace principle for $\{X ^{\eta, xy}\} _{\eta}$
  9. Uniformity of Laplace principle on compacts
  10. Laplace principle of the dynamic Schrödinger problem
  11. Laplace upper bound
  12. Laplace lower bound
  13. Examples of reference dynamics
  14. Scaled Brownian motion
  15. Ornstein-Uhlenbeck processes

Key Result

Theorem 1.2

Under Assumptions ass:b-sigma-local-properties--ass:log-gradient-convergence, for any compact set $K \Subset D$, where $X^{\eta, xy}$ is given by the dynamics eq:R-SDE-main-results, conditioned on starting in $x$ at $t=0$ and ending in $y$ at $t=1$. The good rate function $I_B$ is given by where $U^{xy}_\varphi \coloneq \{\nu \in L^2([0,1]:\mathbb R^d): \varphi(\mathbin{\vcenter{\hbox{$\m@th\bul

Theorems & Definitions (17)

  • Theorem 1.2: Uniform Laplace principle of bridges
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 3.3
  • Proposition 3.4: Variational representation for bridges
  • Proposition 3.5
  • Remark 3.10
  • Lemma 3.11
  • proof
  • Proposition 3.12: Laplace upper bound
  • ...and 7 more