We present results regarding the existence, regularity and uniqueness of solutions of the master equation associated with the mean field planning problem in the finite state space case, in the presence of a common noise. The results hold under monotonicity assumptions, which are used crucially, in the different proofs of the paper. We also make a link with the trajectories induced by the solution of the master equation and start a discussion on the case of boundary conditions.

中文翻译：

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有限状态空间规划问题的主方程

我们给出了在有限状态空间情况下，在存在共同噪声的情况下，与平均场规划问题相关的主方程解的存在性、规律性和唯一性的结果。结果在单调性假设下成立，这在论文的不同证明中至关重要。我们还与由主方程解引起的轨迹建立联系，并开始讨论边界条件的情况。