We provide sufficient conditions for the existence of viscosity solutions of fractional semilinear elliptic PDEs of index $\alpha \in (1,2)$ with polynomial gradient nonlinearities on $d$-dimensional balls, $d\geq 2$. Our approach uses a tree-based probabilistic representation based on $\alpha$-stable branching processes, and allows us to take into account gradient nonlinearities not covered by deterministic finite difference methods so far. Numerical illustrations demonstrate the accuracy of the method in dimension $d=10$, solving a challenge encountered with the use of deterministic finite difference methods in high-dimensional settings.

翻译：本文给出了指数$\alpha \in (1,2)$的分数阶半线性椭圆偏微分方程在$d$维球（$d\geq 2$）上具有多项式梯度非线性项时粘性解存在的充分条件。我们的方法基于$\alpha$稳定分枝过程的树状概率表示，能够处理迄今尚未被确定性有限差分方法覆盖的梯度非线性类型。数值实例表明，该方法在维度$d=10$时具有较高的精度，解决了高维背景下确定性有限差分方法面临的计算挑战。