A boundary local time for super-Brownian motion and new path properties for superprocess densities
department or school: Science, Faculty of
department or school: Mathematics, Department of
data provider: DSpace
degree campus: UBCV
creator: Hughes, Thomas J.
issued: 2020-08-20T17:29:11Z
issued: 2020
related degree: Doctor of Philosophy - PhD
degree grantor: University of British Columbia
description: We establish new fractal properties for superprocess densities. Consider the density of super-Brownian motion in one dimension at a fixed time. We construct a random measure supported on the boundary of the zero set of the density, which we call the boundary local time, and prove that it is locally non-degenerate, in the sense that it almost surely has positive mass on every neighbourhood of the boundary of the support. We then use the boundary local time to give an almost sure lower bound on the Hausdorff dimension of the boundary of the zero set, confirming a conjecture of Mueller et al. [69], whose work on this problem motivated ours. The Hausdorff dimension is characterized in terms of the lead eigenvalue of a certain killed Ornstein-Uhlenbeck operator which also appears in the construction of the boundary local time. The proof of local non-degeneracy includes a novel analysis of the density near the right endpoint of its support, including a new bound on the expected mass in a neighbourhood of the endpoint. We then consider the density of (α, β)-superprocess in dimension d with α â (0, 2), β â (0, 1], and βd < α. A classical result called instantaneous propagation asserts that, at a fixed time, the density has positive mass on any open set almost surely when conditioned on survival. We give general conditions under which the integral the density against a measure μ is positive almost surely when conditioned on survival. In particular, we show that this is determined by the relationship between the dimension of the measure, which we define in two ways, and a critical parameter determined by α, β and d. When the density does not charge μ almost surely, we obtain upper and lower bounds on the probability that it does so. In both cases our results correspond to new existence and non-existence results for solutions to the partial differential equation which is dual to the process. In the regime where the density is continuous and positive at a fixed point …
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/75612
note: A boundary local time for super-Brownian motion and newpath properties for superprocess densitiesbyThomas J. HughesB.Sc., Queenâs University, 2014M.Sc., University of British Columbia, 2016a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoral studies(Mathematics)The University of British Columbia(Vancouver)August 2020c© Thomas J. Hughes, 2020The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the thesis entitled:A boundary local time for super-Brownian motion and new path propertiesfor superprocess densitiessubmitted by Thomas J. Hughes in partial fulfillment of the requirements for the degreeof Doctor of Philosophy in Mathematics.Examining Committee:Edwin Perkins, MathematicsSupervisorMartin Barlow, MathematicsSupervisory Committee MemberJun-cheng Wei, MathematicsUniversity ExaminerNick Harvey, Computer ScienceUniversity ExaminerAdditional Supervisory Committee Members:Gordon Slade, MathematicsSupervisory Committee MemberiiAbstractWe establish new fractal properties for superprocess densities.Consider the density of super-Brownian motion in one dimension at a fixed time. Weconstruct a random measure supported on the boundary of the zero set of the density, whichwe call the boundary local time, and prove that it is locally non-degenerate, in the sense thatit almost surely has positive mass on every neighbourhood of the boundary of the support.We then use the boundary local time to give an almost sure lower bound on the Hausdorffdimension of the boundary of the zero set, confirming a conjecture of Mueller et al. [69],whose work on this problem motivated ours. The Hausdorff dimension is characterizedin terms of the lead eigenvalue of a certain killed Ornstein-Uhlenbeck operator which alsoappears in the construction of the boundary local time. The proof of local non-degeneracyincludes a novel analysi…
has type: Thesis/Dissertation
date issued: 2020-11
is shown at: 10.14288/1.0392885
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
type: Text
identifier URI: http://hdl.handle.net/2429/75612