Continuous LERW Started from Interior Points

Boundary behaviour of RW's on planar graphs and convergence of LERW to chordal SLE$_2$

This paper concerns a random walk on a planar graph and presents certain estimates concerning the harmonic measures for the walk in a grid domain which estimates are useful for showing the convergence of a LERW (loop-erased random walk) to an SLE (stochastic Loewner evolution). We assume that the walk started at a fixed vertex of the graph satisfies the invariance principle as in Yadin and Yehudayoff [16] in which the convergence of LERW to a radial SLE is established in this setting. Our main concern is chordal case, where a random walk is started at a boundary vertex of a simply connected gr…

The scaling limits of planar LERW in finitely connected domains

We define a family of stochastic Loewner evolution-type processes in finitely connected domains, which are called continuous LERW (loop-erased random walk). A continuous LERW describes a random curve in a finitely connected domain that starts from a prime end and ends at a certain target set, which could be an interior point, or a prime end, or a side arc. It is defined using the usual chordal Loewner equation with the driving function being $\sqrt{2}B(t)$ plus a drift term. The distributions of continuous LERW are conformally invariant. A continuous LERW preserves a family of local martingale…

Convergence of radial loop-erased random walk in the natural parametrization

In recent work we have shown that loop-erased random walk (LERW) connecting two boundary points of a domain converges to the chordal Schramm-Loewner evolution (SLE(2)) in the sense of curves parametrized by Minkowski content. In this note we explain how to derive the analogous result for LERW from a boundary point to an interior point, converging towards radial SLE(2).

Scaling limit of the loop-erased random walk Green's function

We consider loop-erased random walk (LERW) running between two boundary points of a square grid approximation of a planar simply connected domain. The LERW Green's function is the probability that the LERW passes through a given edge in the domain. We prove that this probability, multiplied by the inverse mesh size to the power 3/4, converges in the lattice size scaling limit to (a constant times) an explicit conformally covariant quantity which coincides with the SLE(2) Green's function. The proof does not use SLE techniques and is based on a combinatorial identity which reduces the problem t…

Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees

We establish existence and uniqueness for Gaussian free field flow lines started at {\em interior} points of a planar domain. We interpret these as rays of a random geometry with imaginary curvature and describe the way distinct rays intersect each other and the boundary. Previous works in this series treat rays started at {\em boundary} points and use Gaussian free field machinery to determine which chordal \SLE_κ(ρ_1; ρ_2) processes are time-reversible when κ< 8. Here we extend these results to whole-plane \SLE_κ(ρ) and establish continuity and transience of these paths. In particular, we ex…

关于一维一般扩散半鞅的表示性质

On the representation property for 1D general diffusion semimartingales

本文研究一维一般扩散半鞅的表示性质，证明其成立当且仅当尺度函数在状态空间内部局部绝对连续，并应用于解决半鞅问题的极点性与强马尔可夫解极性。