The chordal Loewner equation driven by Brownian motion with linear drift

Abstract

Schramm-Loewner evolution (SLE_κ) is an important contemporary tool for identifying critical scaling limits of two-dimensional statistical systems. The SLE_κ one-parameter family of processes can be viewed as a special case of a more general, two-parameter family of processes we denote [special characters omitted]. The [special characters omitted] process is defined for κ > 0 and μ ε [special characters omitted]; it represents the solution of the chordal Loewner equations under special conditions on the driving function parameter which require that it is a Brownian motion with drift μ and variance κ. We derive properties of this process by use of methods applied to SLE_κ and application of Girsanov's Theorem. In contrast to SLE_κ, we identify stationary asymptotic behavior of [special characters omitted]. For κ ε (0, 4] and μ ≠ 0, we present a pathwise construction of a process, [special characters omitted], with stationary temporal increments and stationary imaginary component and relate [special characters omitted] to the limiting behavior of the [special characters omitted] generating curve. Our main result is a spatial invariance property of [special characters omitted] achieved by defining a top-crossing probability for points z ε [special characters omitted] with respect to the generating curve.