- Title Information
- Title
- The Loewner Equation with Branching and the Continuum Random Tree
- Name:
Personal
- Name Part
- Healey, Vivian Olsiewski
- Role
- Role Term:
Text
- creator
- Name:
Personal
- Name Part
- Menon, Govind
- Role
- Role Term:
Text
- Advisor
- Name:
Personal
- Name Part
- Kenyon, Richard
- Role
- Role Term:
Text
- Reader
- Name:
Personal
- Name Part
- Rohde, Steffen
- Role
- Role Term:
Text
- Reader
- Name:
Corporate
- Name Part
- Brown University. Department of Mathematics
- Role
- Role Term:
Text
- sponsor
- Origin Information
- Copyright Date
- 2017
- Physical Description
- Extent
- xi, 107 p.
- digitalOrigin
- born digital
- Note:
thesis
- Thesis (Ph. D.)--Brown University, 2017
- Genre (aat)
- theses
- Abstract
- The present work brings together the fields of random maps and Loewner evolution by constructing explicit embeddings of critical Galton-Watson trees in the upper half-plane via the Loewner equation and considering the scaling limit of the associated time-dependent random driving measures as the finite trees converge to the continuum random tree. Chapter 2 addresses the (deterministic) conformal mapping problem of incorporating branching into the Loewner equation. We identify sufficient conditions on the driving measure for the Loewner equation to generate a union of two simple curves that meet at a fixed nontrivial angle on the real line, which is the fundamental step in generating graph embeddings of trees. Chapter 3 identifies a specific repulsive force (the deterministic part of Dyson Brownian motion) that, when used to describe the evolution of a random discrete measure whose atoms represent the particles of a Galton-Watson branching process, satisfies the conditions for tree embedding given in Chapter 2. Chapter 4 investigates the scaling limit of these time-dependent driving measures through the lens of superprocesses. In the setting when the critical Galton-Watson trees are conditioned to converge to the continuum random tree, the sequence of measure-valued processes is shown to be tight. In order to identify the limit, the question of convergence of the sequence of measures is reframed as a question concerning the associated sequence of Stieltjes transforms. For each measure-valued process in the sequence, the flow of the associated Stieltjes transform is shown to satisfy a particular SPDE that is related to the complex Burgers equation. Finally, in the unconditioned case, the density ρ of the limiting superprocess is conjectured to satisfy the equation ∂tρ + ∂x (ρ · Hρ) = σ√ρ · W , where H is the Hilbert transform, W is space-time white noise, and σ is a positive constant.
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/00940832")
- Topic
- Geometric function theory
- Subject
- Topic
- Loewner evolution
- Subject
- Topic
- Continuum random tree
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/00875031")
- Topic
- Conformal mapping
- Subject
- Topic
- Probability
- Language
- Language Term (ISO639-2B)
- English
- Record Information
- Record Content Source (marcorg)
- RPB
- Record Creation Date
(encoding="iso8601")
- 20170616
- Identifier:
DOI
- 10.7301/Z0V69H1Z
- Access Condition:
rights statement
(href="http://rightsstatements.org/vocab/InC/1.0/")
- In Copyright
- Access Condition:
restriction on access
- Collection is open for research.
- Type of Resource (primo)
- dissertations