We study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $\sigma\in(0,2)$ since they involve fractional Laplace operators $(-\Delta)^{\sigma/2}$. They arise e.g.~in control and game theory as dynamic programming equations, and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of $\sigma$. The accuracy of previous approximations depend on $\sigma$ and are worse when $\sigma$ is close to $2$. We show that the schemes are monotone, consistent, $L^\infty$-stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We present several numerical examples.

翻译：我们研究了通过离散拉普拉斯算子的幂次对分数阶完全非线性方程进行离散化。这些问题为抛物型，阶数为$\sigma\in(0,2)$，因为它们涉及分数阶拉普拉斯算子$(-\Delta)^{\sigma/2}$。这些问题出现在控制论和博弈论中，例如作为动态规划方程，且解通常非光滑，应解释为粘性解。我们的近似方法通过有限差分求积实现，对所有$\sigma$值均具有二阶精度。先前的近似精度依赖于$\sigma$，且在$\sigma$接近2时更差。我们证明了这些格式是单调的、相容的、$L^\infty$稳定的，并通过先验估计、粘性解理论以及半松弛极限方法证明其收敛性。我们给出了若干数值算例。