Continuity of Zero-Hitting Times of Bessel Processes and Welding Homeomorphisms of SLE$_kappa$

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 $deltale 0$. As an application, we show that the SLE($kappa$) welding homeomorphism is continuous in $kappa$ for $kappain [0,4]$. 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.