Continuity of Zero-Hitting Times of Bessel Processes and Welding Homeomorphisms of SLE_κ

2020/04/21 by Dmitry Beliaev, Beliaev, Dmitry, Atul Shekhar +3 · 1 citation
Mathematics · Economics, Econometrics and Finance · Biochemistry, Genetics and Molecular Biology · #Stochastic processes and statistical mechanics #Stochastic processes and financial applications #Diffusion and Search Dynamics

paper · pdf · doi:10.48550/arxiv.2004.10262

Abstract

We consider a family of Bessel Processes that depend on the starting point x and dimension δ, but are driven by the same Brownian motion. Our main result is that almost surely the first time a process hits 0 is jointly continuous in x and δ, provided δ≤ 0. As an application, we show that the SLE(κ) welding homeomorphism is continuous in κ for κ∈ [0,4]. Our motivation behind this is to study the well known problem of the continuity of SLE_κ in κ. The main tool in our proofs is random walks with increments distributed as infinite mean Inverse-Gamma laws.

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