Random Loewner Chains in Riemann Surfaces

Abstract

The thesis describes an extension of O. Schramm's SLE processes to complicated plane domains and Riemann surfaces. First, three kinds of new SLEs are defined for simple conformal types. They have properties similar to traditional SLEs. Then harmonic random Loewner chains (HRLC) are defined in finite Riemann surfaces. They are measures on the space of Loewner chains, which are increasing families of closed subsets satisfying certain properties. An HRLC is first defined on local charts using Loewner's equation. Since the definitions in different charts agree with each other, these local HRLCs can be put together to construct a global HRLC. An HRLC in a plane domain can be described by differential equations involving canonical plane domains. Those old and new SLEs are special cases of HRLCs. An HRLC is determined by a parameter [kappa] >= 0, a starting point and a target set. When [kappa] = 6, the HRLC satisfies the locality property. When [kappa] = 2, the HRLC preserves some observable that resembles the observable for the corresponding loop-erased random walk (LERW). So HRLC_2 should be the scaling limit of LERW. With reasonable assumptions, HRLC_{8/3} differs from a restriction measure by a conformally invariant density; for [kappa] in (0,8/3), HRLC_[kappa] differs from a pre-restriction measure by a conformally invariant density. A restriction measure could be constructed from a pre-restriction measure by adding Brownian bubbles.