Entropy-Regularized Finite-State Mean Field Control: Regularity and Stability
Abstract
We study infinite-horizon discounted mean field control of finite-state continuous-time Markov chains with unbounded transition rates and a relative-entropy rate penalty. Under mild assumptions, the value function is the unique bounded viscosity solution of the stationary Hamilton--Jacobi--Bellman equation on the closed probability simplex. Joint convexity of a quadratically corrected flux Lagrangian and smoothness of the data yield global semiconvexity, intrinsic 𝒞1,1 regularity, and optimal Gibbs feedback. For positive symmetric reference rates, an entropy-plus-value free energy gives global exponential convergence under a quantitative curvature condition. We also establish inverse-temperature limits for values and, under parameter-uniform hypotheses, for feedbacks and closed-loop trajectories. With population-independent reference rates and a potential state-cost structure, optimal planner trajectories induce discounted finite-state mean field game equilibria. As an application, size-dependent market-making quotes with exponential demand admit an exact revenue-to-entropy reformulation; a convex inventory-crowding potential connects optimal quoting to stabilization of the inventory distribution.
Related
- Entropy-Regularized Optimal Transport for Time-Varying Multi-Agent Coverage Control
- Finite attainment and efficient computation of regularized quantum channel relative entropy
- Orbits in stability manifolds and categorical entropy: applications to varieties with finite Albanese morphisms
- New Tools for Finite-Time Fuzzy Control of Fractional-Order Systems: Stability Lemmas, Composite Disturbance Observers, and Singularity-Free Designs
- Accelerating Sinkhorn for Entropy-Regularized Optimal Transport
BibTeX
@misc{cohen2026entropyregularized,
title = {{Entropy-Regularized Finite-State Mean Field Control: Regularity and Stability}},
author = {Asaf Cohen and Aditya Priya},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24883},
eprint = {2610.24883},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24883},
}
RIS
TY - UNPB
TI - Entropy-Regularized Finite-State Mean Field Control: Regularity and Stability
AU - Asaf Cohen
AU - Aditya Priya
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24883
UR - https://arxiv.org/abs/2610.24883
AB - We study infinite-horizon discounted mean field control of finite-state continuous-time Markov chains with unbounded transition rates and a relative-entropy rate penalty. Under mild assumptions, the value function is the unique bounded viscosity solution of the stationary Hamilton--Jacobi--Bellman equation on the closed probability simplex. Joint convexity of a quadratically corrected flux Lagrangian and smoothness of the data yield global semiconvexity, intrinsic 𝒞^1,1 regularity, and optimal Gibbs feedback. For positive symmetric reference rates, an entropy-plus-value free energy gives global exponential convergence under a quantitative curvature condition. We also establish inverse-temperature limits for values and, under parameter-uniform hypotheses, for feedbacks and closed-loop trajectories. With population-independent reference rates and a potential state-cost structure, optimal planner trajectories induce discounted finite-state mean field game equilibria. As an application, size-dependent market-making quotes with exponential demand admit an exact revenue-to-entropy reformulation; a convex inventory-crowding potential connects optimal quoting to stabilization of the inventory distribution.
ER -