Continuity of Zero-Hitting Times of Bessel Processes and Welding Homeomorphisms of SLE$_\kappa$
Abstract
We consider a family of Bessel Processes that depend on the starting point $x$ and dimension $\delta$, 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 $\delta$, provided $\delta\le 0$. As an application, we show that the SLE($\kappa$) welding homeomorphism is continuous in $\kappa$ for $\kappa\in [0,4]$. Our motivation behind this is to study the well known problem of the continuity of SLE$_\kappa$ in $\kappa$. The main tool in our proofs is random walks with increments distributed as infinite mean Inverse-Gamma laws.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2020
- DOI:
- 10.48550/arXiv.2004.10262
- arXiv:
- arXiv:2004.10262
- Bibcode:
- 2020arXiv200410262B
- Keywords:
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- Mathematics - Probability
- E-Print:
- 12 pages