# exterior problem for fractional Laplacian

Does there exist a theory of fractional laplacian on exterior domains such as \ \ \left\{\begin{aligned} (-\Delta)^{s} u&= 0 &&\text{in } \mathbb R^N\setminus \mathbb B \\ u & =g &&\text{ in } \mathbb B \end{aligned} \right. where $$\mathbb B$$ is a unit ball, $$N\geq 1$$ and $$s\in (0, 1).$$ Does there exist a Poisson kernel for this type of problem. Is the solution for the problem be expressible as the combination of the Poisson kernel and $$g.$$