In this paper we study multi-dimensional mean-field backward doubly stochastic differential equations (BDSDEs), that is, BDSDEs whose coefficients depend not only on the solution processes but also on their law. The first part of the paper is devoted to the comparison theorem for multi-dimensional mean-field BDSDEs with Lipschitz conditions. With the help of the comparison result for the Lipschitz case we prove the existence of a solution for multi-dimensional mean-field BDSDEs with an only continuous drift coefficient of linear growth, and we also extend the comparison theorem to such BDSDEs with a continuous coefficient.

中文翻译：

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多维通用平均场BDSDES的比较定理

在本文中，我们研究了多维平均场后向双随机微分方程（BDSDE），即其系数不仅取决于求解过程而且取决于其定律的BDSDE。本文的第一部分致力于具有Lipschitz条件的多维平均场BDSDE的比较定理。借助Lipschitz案例的比较结果，我们证明了仅具有线性增长连续漂移系数的多维平均场BDSDE的解的存在，并且我们还将比较定理扩展到了具有连续系数的此类BDSDE 。