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One precise relation is to radial SLE with parameter $2$. Its orientation is from the boundary point $1$ to the interior target $0$. With conformal-radius time it is defined byThere is a coupling of the conditioned Brownian path $B$ with this simple radial curve $\eta$ such that, for each initial curve segment, the first point at which $B$ encounters it is its tip. More precisely, if $s_t=\inf \{s:B_s\in \eta [0,t]\}$, then $B_{s_t}=\eta (t)$. Reversing the Brownian path turns these first-hit times into the last-visit times that characterize loop-erasure of planar Brownian motion. ThusThis is an existence statement for a coupling, not a pointwise equality of the two processes or a claim that a naive finite-loop deletion algorithm applies to Brownian motion.

Here is the martingale mechanism behind the coupling. Put $a_t=e^{i{\theta }_t}$ andThis is the positive slit-domain Poisson kernel at the tip, normalized by $Q_t(0)=1$. Writing $q_t=g_t(z)/a_t$, the radial equation givesThe drift of $(1+q_t)/(1-q_t)$ is $(2-\kappa )q_t(1+q_t)/(1-q_t{)}^3$, so $Q_t(z)$ is a local martingale at $\kappa =2$.

Start independent stopped $B$ and radial SLE2, and weight their joint laws, before meeting, byFor fixed $s$ this is an SLE local martingale. For fixed $t$ it is a Brownian-transform local martingale, because ${\mathcal{L}}^h(Q_t/h)=(2h{)}^{-1}\Delta Q_t=0$. Since $L(0,t)=L(s,0)=1$, localized two-parameter weighting preserves both original marginals. For a fixed slit, the changed Brownian law is the transform by $Q_t$, so it hits that slit at the pole, its tip. Compatible stopped couplings, followed by exhaustion, give the first-hit-at-tip property. This completion is the substantive continuum coupling step. Discrete loop-erased random walks and their radial-SLE2 limit give a parallel approximation picture.

For example, in this coupling avoidance by $B$ implies avoidance by $\eta$, since the radial curve is contained in the Brownian trace. The probability from part (b) therefore supplies a lower bound $|{\varphi }^{\prime }(1)|\le \mathbb{P}(\eta \subset U)$. The two avoidance laws are not identical: unweighted simple radial ${\mathrm{SLE}}_2$ does not have the Brownian radial-restriction property. This is a way to transfer path-containment information while retaining the distinct geometry and orientation of the two random curves.

For radial SLE at parameter two, the normalized slit-domain Poisson kernel at the growing tip is a local martingale. This observable provides the coupling to loop-erasure of planar Brownian motion, oriented from the boundary point to the interior target. Its discrete counterpart is a loop-erased random walk.