论文标题
在完全边缘约束下的广义熵最小化
Generalized entropy minimization under full marginal constraints
论文作者
论文摘要
我们考虑了在具有完全处方的边缘分布的一组路径测量的集合，将广义相对熵与参考扩散定律相对于参考扩散定律的问题。在处理实际的相对熵时，这种问题出现在随机力学文献中，而最小化的人以尼尔森过程的名义出现。 通过凸双重性和随机控制技术，我们在主要结果中获得了最小化器的全部表征，其中包含Cattiaux \＆Léonard和Mikami的开创性作品中的相关结果。我们还确定，最小化的人通常不必是马尔可夫人，如果状态空间的维度大于或等于两个，则可能取决于广义相对熵的形式。最后，我们说明了这种类型的广义相对熵最小化问题如何通过两种应用来证明是有用的：对某些平均场游戏的分析以及一类向后SDE的缩放限制的研究。
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.
