Generalised Loewner Evolutions and their driving measures
type: CreativeWork
description: In this thesis we study the interplay between Loewner chains and their driving measures. We prove three versions of a Loewner-Kufarev bijection between Loewner chains, i.e. normalised conformal maps f_t for t >= 0, defined on a fixed domain and whose images are continuously shrinking, and their driving measures, i.e. a time-dependent family of locally finite measures defined on the domain boundaries. There are two variants of Loewner chains where the functions f_t are either normalised at an interior or at a boundary point. Radial Loewner Evolutions are usually defined on the unit disc D = {z in C: |z| < 1} and normalised in the origin. Alternatively, chordal Loewner Evolutions are normally defined on the upper halfplane H := {z in C: Im(z) > 0} and are normalised at infinity. The theory of Loewner chains was originally introduced by Charles Loewner in 1923. Dating back to this paper Loewner chains solve a differential equation containing the driving measure as a time-dependent parameter. For radial Loewner Evolutions it is known that there exists a bijection between Loewner chains and driving measures. Our main results regards the reverse direction: We obtain the driving measure as a (weak) limit of absolutely continuous measures with known densities. Despite Loewner chains being a popular tool in Complex Analysis and Probability Theory this question had not previously been addressed. We prove this result three times. Firstly, we obtain a formula to derive for radial driving measures as locally finite measures defined on the product space [0, infinity) x {z in C : |z| = 1}. Secondly, we show an analogous result for chordal Loewner Evolutions. In this case the measures mu are defined on the product space [0, infinity) x (R cup {infinity}) such that mu([0, T] x (R cup {infinity})) < infinity for all T > 0. Thirdly, there exist locally finite measures mu defined on [0, infinity) x (R cup {infinity}) this finiteness condition. Hence, we study the most general case of (…
abstract: In this thesis we study the interplay between Loewner chains and their driving measures. We prove three versions of a Loewner-Kufarev bijection between Loewner chains, i.e. normalised conformal maps f_t for t >= 0, defined on a fixed domain and whose images are continuously shrinking, and their driving measures, i.e. a time-dependent family of locally finite measures defined on the domain boundaries. There are two variants of Loewner chains where the functions f_t are either normalised at an interior or at a boundary point. Radial Loewner Evolutions are usually defined on the unit disc D = {z in C: |z| < 1} and normalised in the origin. Alternatively, chordal Loewner Evolutions are normally defined on the upper halfplane H := {z in C: Im(z) > 0} and are normalised at infinity. The theory of Loewner chains was originally introduced by Charles Loewner in 1923. Dating back to this paper Loewner chains solve a differential equation containing the driving measure as a time-dependent parameter. For radial Loewner Evolutions it is known that there exists a bijection between Loewner chains and driving measures. Our main results regards the reverse direction: We obtain the driving measure as a (weak) limit of absolutely continuous measures with known densities. Despite Loewner chains being a popular tool in Complex Analysis and Probability Theory this question had not previously been addressed. We prove this result three times. Firstly, we obtain a formula to derive for radial driving measures as locally finite measures defined on the product space [0, infinity) x {z in C : |z| = 1}. Secondly, we show an analogous result for chordal Loewner Evolutions. In this case the measures mu are defined on the product space [0, infinity) x (R cup {infinity}) such that mu([0, T] x (R cup {infinity})) < infinity for all T > 0. Thirdly, there exist locally finite measures mu defined on [0, infinity) x (R cup {infinity}) this finiteness condition. Hence, we study the most general case of (…
author: Schreuder, Anne
language: eng
keywords: Loewner chain, Complex Analysis, Probability Theory, Particle Aggregation Models, Hasting-Levitov Models
license: https://www.repository.cam.ac.uk/bitstreams/1db74788-446c-43a0-b4a0-839bc4a6628a/download
license: http://purl.org/NET/rdflicense/allrightsreserved
date published: 2025-08-15
date modified: 2026-07-22T22:24:14Z
provider: Cambridge University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Cambridge

PropertyValue
property ID: DOI
value: 10.17863/CAM.127999