Regularity and stability for the Gibbs conditioning principle on path space via McKean-Vlasov control
Louis-Pierre Chaintron, Giovanni Conforti
TL;DR
The paper addresses the Gibbs conditioning principle for diffusion systems with empirical interaction by recasting the conditional law as an entropic projection on path space and connecting it to a constrained McKean–Vlasov control problem. It proves existence and characterisation of the minimiser via a coupled forward–backward FP–HJB system with a distributional Lagrange multiplier and shows that, under regularity, the multiplier has a density on $(0,T)$ with end-point atoms allowed. The authors develop a smoothing approach to obtain quantitative stability: entropic, transport, and multiplier-stability bounds with explicit rates, and establish enhanced regularity for the value functions and log-densities via novel Hessian and time-reversal estimates. Together, these results yield a rigorous, quantitative version of the Gibbs conditioning principle in a dynamic mean-field setting with distributional constraints, with potential applications in stochastic control and statistical physics.
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.