= Boundary-point Bessel flow for SLE
{c}
{title2=$d=1+4/\kappa$}
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> gives
$$
dV_t=\frac2{V_t}\,dt-\sqrt\kappa\,dB_t.
$$
Thus $R_t=V_t/\sqrt\kappa$ satisfies $dR_t=dW_t+2/(\kappa R_t)\,dt$, where $W=-B$ is another <Brownian motion>. Comparing with the <Bessel process> drift $(d-1)/(2R_t)$ identifies
$$
d=1+4/\kappa.
$$
Its 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.<Basic properties of SLE by Steffen Rohde and Oded Schramm>[Section 6, before Lemma 6.2, printed p. 23]{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 $g_t(x)=\operatorname{sgn}(x)\sqrt{x^2+4t}$, so no nonzero boundary <point> is swallowed.
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