Green function of symmetric stable process in dimension 1 and 2

Question

Are the results in this paper on the Green function of a symmetric stable process available also in space dimension $d =1$ and $d=2$? The main theorems here are stated only for $d \ge 3$.

Answer 1

Short answer: see Corollary 1.2 in by Z.-Q. Chen Heat kernel estimates for the Dirichlet fractional Laplacian, DOI:10.4171/JEMS/231.

---

Long answer: Extension should be automatic as long as $\alpha < d$. When $\alpha \geqslant d = 1$, the estimates are available, too, but have a different form, because the underlying process is point-recurrent (when $\alpha > d = 1$), and because the potential kernel has a different form (when $\alpha = d = 1$).

For a discussion of the case $d \geqslant 2$, see, for example, A note on the Green function estimates for symmetric stable processes by Z.-Q. Chen and R. Song, DOI:10.1142/9789812702241_0008. Note that the original estimate was found independently by T. Kulczycki in the paper Properties of the Green function of symmetric stable processes linked in the question, and by Z.-Q. Chen and R. Song in the paper Estimates on Green functions and Poisson kernels for symmetric stable processes, DOI:10.1007/s002080050232.

Off the top of my head, I do not know a reference that would cover $d = 1$ using purely potential-theoretic methods (although this case should be identical to the above when $\alpha < 1$, and much simpler when $\alpha \geqslant 1$). In the paper linked in the short answer above, the authors estimate the heat kernel first, and then integrate it with respect to time to get a bound for the Green function.

Answer 2

I have obtained the closed form (as the integral of an elementary algebraic function) of the Green function on the unit interval for $\alpha$-stable processes with $1\leq \alpha\leq 2$ and $0<\rho<1$.

I would like to know how much of it is known.