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We provide sufficient conditions for the existence of viscosity solutions offractional semilinear elliptic PDEs of index $\alpha \in (1,2)$ with polynomialgradient nonlinearities on $d$-dimensional balls, $d\geq 2$. Our approach usesa tree-based probabilistic representation based on $\alpha$-stable branchingprocesses, and allows us to take into account gradient nonlinearities notcovered by deterministic finite difference methods so far. Numericalillustrations demonstrate the accuracy of the method in dimension $d=10$,solving a challenge encountered with the use of deterministic finite differencemethods in high-dimensional settings.

Semilinear fractional elliptic PDEs with gradient nonlinearities on open balls: existence of solutions and probabilistic representation