Boundary-point Bessel flow for SLE ($d=1+4/\kappa$)

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 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.

For $0<\kappa <4$ and $x>0$, stop the simple chordal SLE trace on first entering the radius-$r$ half-disc about $x$, with $r<x/4$. A crosscut from its tip to $x$ separates the positive-side boundary arc containing $x/2$ from infinity. Harmonic measure comparison with the half-disc bounds the length of that arc's mapped interval, $g_t(x/2)-{\xi }_t$, by a universal constant times $r$. This centered boundary image is a positive Bessel process of dimension $1+4/\kappa >2$ and has a strictly positive all-time minimum. Hence the full trace stays a positive distance from each fixed nonzero real point. Reflection gives the negative-side version.

For $0<x<y$, let $D_t=g_t(y)-g_t(x)$. Before the first boundary-point swallowing time for a Loewner chain, the ratio obeysThe clock $u=\int D_t^{-2}dt$ removes the denominator. An increasing scale function of a one-dimensional diffusion is $s_{\kappa }(z)={\int }_1^z[v(v-1){]}^{-4/\kappa }dv$. For $\kappa >4$ the endpoint $1$ represents strict swallowing; escape at infinite clock represents simultaneous swallowing.

For $\kappa \ge 8$, almost surely the positive-boundary swallowing times satisfy $T(x)<T(y)$ simultaneously for every $0<x<y$. The SLE two-boundary-point ratio diffusion has unbounded scale, ruling out escape before hitting $1$. For $4<\kappa <8$, its total scale $S_{\kappa }$ is finite, and the simultaneous-swallowing probability is $s_{\kappa }(y/(y-x))/S_{\kappa }$. At $\kappa =4$, fixed positive points have infinite swallowing times.

A continuous capacity-parameterized Loewner trace with finite, strictly ordered positive swallowing times visits every positive real point. An unvisited swallowed point has an interval disjoint from the compact trace up to its swallowing time; that interval is swallowed simultaneously, contradicting strict order. Reflection yields the negative-axis conclusion for chordal SLE at $\kappa \ge 8$.

For two boundary points $1<r$, let $V_t^x=g_t(x)-U_t$ and set $Z_t=\log (V_t^r-V_t^1)-\log V_t^1$. After the clock $q(u)={\int }_0^u(V_s^1{)}^{-2}ds$, the Dambis-Dubins-Schwarz theorem givesThis one-dimensional diffusion process compares the swallowing times of nearby boundary points.

For a fixed $x\in \mathbb{R}\setminus \{0\}$, a chordal SLE trace in $(\mathbb{H},0,\infty )$ satisfiesFor $\kappa \le 4$ the trace avoids the entire nonzero real boundary. For $4<\kappa <8$ it meets a random boundary subset but misses each predetermined point. A fixed point is nevertheless swallowed almost surely in this latter range: visiting and disconnecting from infinity are different events.^[ref]

The restriction $\kappa <8$ is essential. At $\kappa \ge 8$, the strict ordering threshold for SLE boundary swallowing and real-line visitation from strict SLE swallowing order give visits to every real boundary point. The SLE boundary-point logarithmic separation diffusion proves the below-eight fixed-point assertion through the probability of swallowing before a nearby point; its limiting behavior changes at eight.

For $1<d<2$, couple Bessel flows $X>0>Y$ with the same Brownian motion and let $v=x/y<0$. ThenIt is the probability that the SLE hull swallows the chosen point on the negative side before the point on the positive side.