We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove convergence along subsequences for (i) degenerate equations in one space dimension and (ii) nondegenerate equations in arbitrary dimensions. We also give results on full convergence and convergence to classical solutions. Numerical tests are implemented for a range of different nonlocal diffusions and support our analytical findings.

中文翻译：

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关于分数和非局部平均场游戏的数值近似

我们为具有分数或非局部扩散的平均场游戏构建数值近似。这些方案基于底层控制问题/博弈的半拉格朗日近似以及代理分布的双重近似。这些方法是单调的、稳定的和一致的，我们证明了 (i) 一维空间中的退化方程和 (ii) 任意维中的非退化方程的子序列收敛。我们还给出了完全收敛和收敛到经典解决方案的结果。对一系列不同的非局部扩散进行了数值测试，并支持我们的分析结果。