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 martingales, which are composed of generalized Poisson kernels, normalized by their behaviors near the target set. These local martingales resemble the discrete martingales preserved by the corresponding LERW on the discrete approximation of the domain. For all kinds of targets, if the domain satisfies certain boundary conditions, we use these martingales to prove that when the mesh of the discrete approximation is small enough, the continuous LERW and the corresponding discrete LERW can be coupled together, such that after suitable reparametrization, with probability close to 1, the two curves are uniformly close to each other.Comment: Published in at http://dx.doi.org/10.1214/07-AOP342 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Ergodicity of the tip of an SLE curve

We first prove that, for $\kappa\in(0,4)$, a whole-plane SLE$(\kappa;\kappa+2)$ trace stopped at a fixed capacity time satisfies reversibility. We then use this reversibility result to prove that, for $\kappa\in(0,4)$, a chordal SLE$_\kappa$ curve stopped at a fixed capacity time can be mapped conformally to the initial segment of a whole-plane SLE$(\kappa;\kappa+2)$ trace. A similar but weaker result holds for radial SLE$_\kappa$. These results are then used to study the ergodic behavior of an SLE curve near its tip point at a fixed capacity time. The proofs rely on the symmetry of backward SLE laminations and conformal removability of SLE$_\kappa$ curves for $\kappa\in(0,4)$.Comment: 25 pages. Added a remark after Theorem 6.6; added Corollary B.