We consider the problem of minimizing a generalized relative entropy, with respect to a reference diffusion law, over the set of path-measures with fully prescribed marginal distributions. When dealing with the actual relative entropy, problems of this kind have appeared in the stochastic mechanics literature, and minimizers go under the name of Nelson Processes.
Through convex duality and stochastic control techniques, we obtain in our main result the full characterization of minimizers, containing the related results in the pioneering works of Cattiaux \& Léonard and Mikami as particular cases. We also establish that minimizers need not be Markovian in general, and may depend on the form of the generalized relative entropy if the state space has dimension greater or equal than two. Finally, we illustrate how generalized relative entropy minimization problems of this kind may prove useful beyond stochastic mechanics, by means of two applications: the analysis of certain mean-field games, and the study of scaling limits for a class of backwards SDEs.
An entropy minimization approach to second-order variational mean-field games
INRIA-Paris, MOKAPLAN（法国国家信息与自动化研究所巴黎分部，MOKAPLAN项目组）；CEREMADE, UMR CNRS 7534, Université Paris IX Dauphine（巴黎第九大学，CNRS 7534联合研究实验室（CEREMADE））；Indam, Unità SNS, Pisa, Italy（意大利比萨，意大利国家高等研究院数学与应用研究所（SNS））；CNRS（法国国家科学研究中心）
Three Laws of Multivariable Feedback Systems, Extended Spectral Flatness (Extended Wiener Entropy), 'Uncertainty Principles' in Variance Minimization, and Performance Limitations in Minimum Variance Estimation/Filtering
Nonlocal Onsager Operators and Entropy Dissipation for Finite-State Schrödinger Bridges
Higher Institute of Computer Science and Multimedia of Sfax, University of Sfax（萨法克斯多媒体与计算机科学高等学院，萨法克斯大学）；Qassim University（卡西姆大学）；Sfax Preparatory Engineering Institute, University of Sfax（萨法克斯预备工程学院，萨法克斯大学）
