Gibbs Principle with Infinitely Many Constraints: Optimality Conditions and Stability

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 deviation principle (LDP) is proved in a Wasserstein-type topology, and optimality conditions are written in this setting. This setting encompasses new versions of the Schrödinger bridge problem with marginal non-linear inequality constraints at every time. In the case of convex constraints, novel 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.