Regularity and stability for the Gibbs conditioning principle on path space via McKean-Vlasov control

TL;DR

This paper investigates the regularity and stability of the Gibbs conditioning principle for systems of interacting diffusion processes, using McKean-Vlasov control techniques and new estimates for related PDEs.

Contribution

It introduces a novel approach to analyze the Gibbs conditioning principle on path space via McKean-Vlasov control with distributional constraints, providing new regularity and stability results.

Findings

Established regularity results for the Gibbs conditioning principle.

Derived quantitative stability estimates for optimal solutions.

Developed new estimates for Hamilton-Jacobi-Bellman equations and Hessians.

Abstract

We consider a system of diffusion processes interacting through their empirical distribution. Assuming that the empirical average of a given observable can be observed at any time, we derive regularity and quantitative stability results for the optimal solutions in the associated version of the Gibbs conditioning principle. The proofs rely on the analysis of a McKean-Vlasov control problem with distributional constraints. Some new estimates are derived for Hamilton-Jacobi-Bellman equations and the Hessian of the log-density of diffusion processes, which are of independent interest.