Description
- 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.
- Notes:
- Thesis (Ph. D.)--Brown University, 2017
Access Conditions
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Citation
Healey, Vivian Olsiewski,
"The Loewner Equation with Branching and the Continuum Random Tree"
(2017).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z0V69H1Z
