2026-09-19T02:51:24Zhttps://ecommons.cornell.edu/server/oai/requestoai:ecommons.cornell.edu:1813/1159062026-05-15T19:49:42Zcom_1813_35col_1813_47
Dautenhahn, Emily
Saloff-Coste, Laurent
Healey, Timothy
Cao, Xiaodong
2024-11-05T19:46:19Z
2024-11-05T19:46:19Z
2024-05
ProQuest Submission ID: 14221
ProQuest Publication ID: 31239327
https://hdl.handle.net/1813/115906
https://doi.org/10.7298/grb2-5b04
16575592
214 pages
In this thesis, we prove heat kernel estimates in two main contexts: (1) manifolds with ends with mixed Dirichlet and Neumann boundary condition and (2) infinite (countable) graphs satisfying certain properties, which we call book-like graphs. In both of these settings, we start with ''sufficiently nice'' pieces (pieces satisfying two-sided Gaussian heat kernel estimates) that are ''glued'' together in some sufficiently nice way. The results in setting (1) extend previous results of Grigor'yan and Saloff-Coste in the case of manifolds with ends with Neumann (or no) boundary condition. In setting (2), we are in the discrete case, where there is not direct prior work. This thesis extends some of the continuous setting results of Grigor'yan and Saloff-Coste mentioned above to the discrete setting, and the results here are also related to results of Grigor'yan and Ishiwata regarding gluing two copies of n-dimensional real space via a surface of revolution. In both settings, the results of this thesis rely heavily on the h-transform technique and understanding particular harmonic functions and hitting probabilities. In the setting of (1), we show the existence of a global harmonic function satisfying particular properties. In the setting of (2), we give estimates on certain hitting probabilities that naturally arise from considering subgraphs of larger graphs. All work in this thesis is joint with Laurent Saloff-Coste.
en
h-transform
Harnack inequality
heat kernel
manifolds with ends
random walk
Heat kernel estimates on glued spaces
dissertation or thesis
https://newcatalog.library.cornell.edu/catalog/16575592
application/pdf
Mathematics
Cornell University
Doctor of Philosophy
Ph. D., Mathematics
https://hdl.handle.net/1813/59810.2
Publication
Cornell Theses and Dissertations
Dautenhahn, Emily
5893a6ea-7af3-41d7-abc6-04bcd26ab5df
open.access
open.access
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<Abstract>In this thesis, we prove heat kernel estimates in two main contexts: (1) manifolds with ends with mixed Dirichlet and Neumann boundary condition and (2) infinite (countable) graphs satisfying certain properties, which we call book-like graphs. In both of these settings, we start with ''sufficiently nice'' pieces (pieces satisfying two-sided Gaussian heat kernel estimates) that are ''glued'' together in some sufficiently nice way. The results in setting (1) extend previous results of Grigor'yan and Saloff-Coste in the case of manifolds with ends with Neumann (or no) boundary condition. In setting (2), we are in the discrete case, where there is not direct prior work. This thesis extends some of the continuous setting results of Grigor'yan and Saloff-Coste mentioned above to the discrete setting, and the results here are also related to results of Grigor'yan and Ishiwata regarding gluing two copies of n-dimensional real space via a surface of revolution. In both settings, the results of this thesis rely heavily on the h-transform technique and understanding particular harmonic functions and hitting probabilities. In the setting of (1), we show the existence of a global harmonic function satisfying particular properties. In the setting of (2), we give estimates on certain hitting probabilities that naturally arise from considering subgraphs of larger graphs. All work in this thesis is joint with Laurent Saloff-Coste.</Abstract>
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