Gibbs principle with infinitely many constraints: optimality conditions and stability

TL;DR

This paper extends the Gibbs conditioning principle to settings with infinitely many constraints, providing optimality conditions and stability results, especially for non-linear and non-convex constraints, with applications to diffusion processes.

Contribution

It introduces a general framework for Gibbs principles with infinitely many constraints, including non-linear and non-convex cases, and establishes stability and optimality conditions in this setting.

Findings

Proved a large deviation principle in Wasserstein topology.

Derived optimality conditions for the constrained Gibbs measures.

Applied results to diffusion processes with path constraints.

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{\"o}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.