Unique measurable loop-erasure of Brownian trajectories

Establish whether a Brownian trajectory in two or three dimensions almost surely admits a unique measurable loop-erasure, up to increasing reparametrization.

Background

The paper characterizes the three-dimensional loop-erased random walk scaling limit through a Brownian loop-soup recovery property and uses the term loop-erased Brownian motion for the resulting continuum object. This characterization concerns uniqueness in distribution of the limiting law, not the existence of a canonical pathwise operation that measurably erases loops from an individual Brownian trajectory.

The authors explicitly distinguish these two notions and state that pathwise uniqueness of measurable loop-erasure remains unresolved in both two and three dimensions.

References

This is uniqueness in distribution. Whether a Brownian trajectory almost surely has a unique measurable loop-erasure, up to increasing reparametrization, remains open in both two and three dimensions.

— Characterization and scaling limit of three-dimensional loop-erased random walk and random-walk loop-catchers  (2610.12047 - Cai et al., 8 Oct 2026) in Remark “Loop-erased Brownian motion,” Section 1, Introduction and main results

The problem is to define the normalized first-power harmonic weights at the random tip, prove independence from the tip regularization, and establish existence and uniqueness of the resulting growth law. One possible route to identification is to derive the intensity-one recovery property eq:lerw-recovery from this local rule and then apply \cref{thm:main}. Let us remark that the comparison of harmonic functions in the proof of the boundary Harnack principle Theorem~1.5 and Remark~1.3 may help control the normalized weights near the tip. This question is left for our future work.

eq:lerw-recovery:

D1(K)=dW[0,τB],\mathcal D_1(K)\stackrel d=W[0,\tau_B],

— Characterization and scaling limit of three-dimensional loop-erased random walk and random-walk loop-catchers  (2610.12047 - Cai et al., 8 Oct 2026) in Remark “Continuum Laplacian-1 motion,” Section 1, Introduction and main results