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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 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.

Steffen Rohde works where complex analysis meets probability theory. His University of Washington profile also identifies interests in numerical analysis.^[ref]

With Oded Schramm, he established basic geometric properties of Schramm–Loewner evolution, in which a Brownian motion drives a conformal growth equation. Their paper proves a change in path behavior with the parameter: simple curves for $0\le \kappa \le 4$, self-intersections for $4<\kappa <8$, and space-filling paths for $\kappa >8$. These regimes connect an analytic differential equation with the geometry of random interfaces.^[ref]