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：我们给出了指标 $\alpha \in (1,2)$ 的分数阶半线性椭圆型偏微分方程在 $d$ 维球（$d\geq 2$）上具有多项式梯度非线性项的黏性解存在的充分条件。该方法基于 $\alpha$-稳定分支过程的树状概率表示，能够处理迄今确定性有限差分方法尚未覆盖的梯度非线性项。数值算例表明，该方法在 $d=10$ 维空间中具有准确性，解决了高维背景下确定性有限差分方法所面临的挑战。