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Gibbs principle with infinitely many constraints: optimality conditions and stability

Louis-Pierre Chaintron, Giovanni Conforti, Julien Reygner

Abstract

We extend the Gibbs conditioning principle to an abstract setting combining infinitely many linear equality constraints and non-linear inequality constraints, which need not be convex. A conditional large large deviation principle (LDP) is proved in a Wassersteintype topology, and optimality conditions are written in this abstract setting. This setting encompasses versions of the Schr{ö}dinger bridge problem with marginal non-linear inequality constraints at every time. In the case of convex constraints, stability results for perturbations both in the constraints and the reference measure are proved. We then specify our results when the reference measure is the path-law of a continuous diffusion process, whose law is constrained at each time. We obtain a complete description of the constrained process through an atypical mean-field PDE system involving a Lagrange multiplier.

Gibbs principle with infinitely many constraints: optimality conditions and stability

Abstract

We extend the Gibbs conditioning principle to an abstract setting combining infinitely many linear equality constraints and non-linear inequality constraints, which need not be convex. A conditional large large deviation principle (LDP) is proved in a Wassersteintype topology, and optimality conditions are written in this abstract setting. This setting encompasses versions of the Schr{ö}dinger bridge problem with marginal non-linear inequality constraints at every time. In the case of convex constraints, stability results for perturbations both in the constraints and the reference measure are proved. We then specify our results when the reference measure is the path-law of a continuous diffusion process, whose law is constrained at each time. We obtain a complete description of the constrained process through an atypical mean-field PDE system involving a Lagrange multiplier.

Paper Structure

This paper contains 35 sections, 30 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. The Gibbs conditioning principle
  4. Schrödinger bridge with additional constraints
  5. Outline
  6. Notations
  7. Abstract setting and main results
  8. Global setting
  9. The rate function
  10. Constraints
  11. Conditional LDP for nonlinear inequality constraints
  12. Statement of the result
  13. Checking \ref{['ass:psi-ldp']} for differentiable constraints
  14. Application: the conditional Sanov Theorem with linear constraints
  15. Gibbs density for minimisers
  16. ...and 20 more sections

Key Result

Lemma 2.2

Let $\phi: E \rightarrow \mathbb{R}_+$ be a continuous function and $\nu \in \mathcal{P}(E)$. If there exists $\alpha>0$ such that $\langle \nu, e^{\alpha \phi}\rangle < \infty$, then any probability measure $\mu \in \mathcal{P}(E)$ such that $H(\mu|\nu)<+\infty$ is in $\mathcal{P}_\phi(E)$.

Theorems & Definitions (64)

  • Definition 2.1: Weak convergence in $\mathcal{P}_\phi(E)$
  • Lemma 2.2: Integrability condition
  • Lemma 2.3: Relative entropy as a good rate function on $\mathcal{P}_\phi(E)$
  • proof
  • Lemma 2.4: Sufficient condition for $\mathcal{I}$ to be a good rate function
  • Remark 2.5: Linear inequality constraints
  • Lemma 2.6
  • Theorem 2.7: Conditional LDP
  • Proposition 2.8: Assumption \ref{['ass:psi-ldp']} for differentiable constraints
  • Remark 2.9: Linear constraints
  • ...and 54 more