  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.
