2018/06/25 by Dapeng Zhan, Zhan, Dapeng
Biochemistry, Genetics and Molecular Biology · Economics, Econometrics and Finance · Mathematics · #30C #60D #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:30C #msc:60D
paper · pdf · doi:10.48550/arxiv.1806.09663
42 pages; the previous version was modified according to the referee's comments
openalex publication_date 2018/06/25 · arxiv created 2020/01/31 · arxiv updated 2020/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A 2-SLEκ (κ∈(0,8)) is a pair of random curves (η1,η2) in a simply connected domain D connecting two pairs of boundary points such that conditioning on any curve, the other is a chordal SLEκ curve in a complement domain. In this paper we prove that for any z0∈ D, the limit limr→ 0+r-α0 ℙ[dist(z0,ηj)<r,j=1,2], where α0=((12-κ)(κ+4))/(8κ), exists. Such limit is called a two-curve Green's function. We find the convergence rate and the exact formula of the Green's function in terms of a hypergeometric function up to a multiplicative constant. For κ∈(4,8), we also prove the convergence of limr→ 0+r-α0 ℙ[dist(z0,η1∩ η2)<r], whose limit is a constant times the previous Green's function. To derive these results, we work on two-time-parameter stochastic processes, and use orthogonal polynomials to derive the transition density of a two-dimensional diffusion process that satisfies some system of SDE.