Kulczycki, Tadeusz Properties of Green function of symmetric stable processes. (English) Zbl 0903.60063 Probab. Math. Stat. 17, No. 2, 339-364 (1997). For \(0<\alpha< 2\) and \(d\geq 3\), let \((X_t, P_x)\) be the symmetric \(\alpha\)-stable process on \(\mathbb{R}^d\). Denote by \(G_D(x, y)\) the Green function for a bounded open set \(D\) in \(\mathbb{R}^d\). The main purpose of this paper is to show the following upper and lower estimates of \(G_D(x,y)\) for a bounded domain \(D\) with \(C^{1,1}\) boundary: There exist constants \(C_1\) and \(C_2\) depending on \(D\), \(d\) and \(\alpha\) such that \[ \begin{aligned} C_1\min(1/| x- y|^{d-\alpha}, \delta^{1/2}(x)\delta^{1/2}(y)/| x-y|^d) & \leq 2^\alpha \pi^{d/2}\Gamma((d-\alpha)/2)^{-1}\Gamma(\alpha/2)G_D(x,y),\\ & \leq\min(1/| x-y|^{d- \alpha}, C_2\delta^{1/2}(x) \delta^{1/2}(y)/| x-y|^d)\end{aligned} \] where \(\delta(x)= \text{dist}(x,\partial D)\). This is first shown by using the explicit expression of the Green function in the case that \(D\) is the ball \(B(0,r)\) of center \(0\) and radius \(r\) and then it is used to get the result for general bounded \(C^{1,1}\) domains. Some approach is taken from Z. Zhao [J. Math. Anal. Appl. 116, 309-334 (1986; Zbl 0608.35012)] in which Brownian motion case is concerned. Hence the essential change of the proof is necessary according to the jumping property of the sample paths. As applications, 3G Theorem and the estimates of the mean exit times are given. Reviewer: Y.Oshima (Kumamoto) Cited in 3 ReviewsCited in 87 Documents MSC: 60J45 Probabilistic potential theory 60E07 Infinitely divisible distributions; stable distributions Keywords:Green function; Brownian motion Citations:Zbl 0608.35012 PDF BibTeX XML Cite \textit{T. Kulczycki}, Probab. Math. Stat. 17, No. 2, 339--364 (1997; Zbl 0903.60063) OpenURL