A Note on the Green Function Estimates for Symmetric Stable Processes

Suppose D is a bounded C domain in R and GD is the Green function of the killed symmetric α-stable process in D. In this note we establish a sharp upper bound estimate for GD in such a way that the explicit dependence of the constant in the estimate on α is also given. This sharp estimate allows us to recover the sharp upper bound estimate for the Green function of killed Brownian motion in D by letting α ↑ 2. One of the most important family of Markov processes is the family of symmetric stable processes. A symmetric α-stable process X on Rn is a Lévy process such that for any ξ ∈ Rn and t ≥ 0, E [ eiξ·(Xt−X0) ] = e−t|ξ| α . Here α must be in the interval (0, 2]. When α = 2, we get a Brownian motion running with a time clock twice as fast as the standard one. In this note, symmetric stable processes refer to the case when α ∈ (0, 2). We always assume that n ≥ 2 in this note. For any Borel measurable set A ⊂ Rn, we define τA = inf{t > 0 : Xt 6∈ A}. The killed symmetric stable process XD in D is defined by X t = { Xt, if t < τD, ∂, if t ≥ τD where ∂ is the cemetery point. The Green function GD(x, y) of X D is a function that is continuous on D ×D except along the diagonal such that for every Borel measurable function f ≥ 0 on D, Ex [∫ τD