 Boundary-point Bessel flow for SLE

For $\kappa >0$ and a fixed $x>0$, write $V_t=g_t(x)-\sqrt{\kappa }{B}_t$ before the boundary-point swallowing time for a Loewner chain. The Chordal Loewner equation givesThus $R_t=V_t/\sqrt{\kappa }$ satisfies $d{R}_t=d{W}_t+2/(\kappa R_t)dt$, where $W=-B$ is another Brownian motion. Comparing with the Bessel process drift $(d-1)/(2R_t)$ identifiesIts first zero is the point's swallowing time. A Bessel process hits zero exactly for $d<2$, so a fixed nonzero boundary point is swallowed in finite time when $\kappa >4$. Swallowing need not be a Loewner trace visit.^[ref]

For $x<0$, use $|V_t|/\sqrt{\kappa }$ and reflected Brownian motion; the signed ratio itself is negative. At $\kappa =0$ no division by $\sqrt{\kappa }$ is allowed: the deterministic solution is $g_t(x)=\mathrm{sgn}(x)\sqrt{x^2+4t}$, so no nonzero boundary point is swallowed.