Green's function for chordal SLE curves

2016/07/13 by Mohammad A. Rezaei, Rezaei, Mohammad A., Dapeng Zhan +1
Mathematics · #30C #60G #FOS: Mathematics #Probability (math.PR) #math.PR #msc:30C #msc:60G

paper · pdf · doi:10.48550/arxiv.1607.03840

59 pages, 4 figures. Added figures and made modifications according to referees' comments

arxiv created 2017/09/04 · arxiv updated 2017/09/05

Abstract

For a chordal SLE_κ (κ∈(0,8)) curve in a domain D, the n-point Green's function valued at distinct points z₁,…,z_n∈ D is defined to be G(z₁,…,z_n)=lim_{r1,…,rn\downarrow 0} ∏_k=1ⁿ r_k^d-2 ℙ[dist(γ,z_k)<r_k,1≤ k≤ n], where d=1+\fracκ8 is the Hausdorff dimension of SLE_κ, provided that the limit converges. In this paper, we will show that such Green's functions exist for any finite number of points. Along the way we provide the rate of convergence and modulus of continuity for Green's functions as well. Finally, we give up-to-constant bounds for them.

Related

• Two-curve Green's function for 2-SLE: the interior case
• Green's function for cut points of chordal SLE attached with boundary arcs
• Up-to-constants bounds on the two-point Green's function for SLE curves
• Two-curve Green's function for 2-SLE: the boundary case
• Existence of multi-point boundary Green's function for chordal Schramm-Loewner evolution (SLE)