dc.contributor.author Houston-Edwards, Kelsey dc.date.accessioned 2018-10-23T13:33:10Z dc.date.available 2020-08-22T06:00:49Z dc.date.issued 2018-08-30 dc.identifier.other HoustonEdwards_cornellgrad_0058F_10948 dc.identifier.other http://dissertations.umi.com/cornellgrad:10948 dc.identifier.other bibid: 10489628 dc.identifier.uri https://hdl.handle.net/1813/59532 dc.description.abstract We prove a variety of estimates for the heat kernel on domains with discrete space and discrete time. First, we give a novel proof of the fact that, for a fixed graph $G$ with heat semigroup generator $P$ and measure $m$, the following are equivalent: (1) $(G,P,m)$ satisfies volume doubling and the Poincar\'e inequality; (2) the heat kernel on $(G,P,m)$ satisfies two-sided Gaussian bounds; and (3) solutions to the heat equation on $(G,P,m)$ satisfy the Harnack inequality. This useful equivalence---which connects two geometric conditions to stochastic bounds---was first shown in the continuous case by L. Saloff-Coste and A. Grigor'yan and later, in the discrete case, by T. Delmotte. Our proof avoids complications of previous discrete proof. Given a graph $(G,P,m)$ that satisfies the three equivalent conditions above, which subgraphs $U \subseteq G$ also satisfy these conditions? We prove that $(U,P_N,m)$ does, where $U$ is an inner uniform domain and $P_N$ is the Neumann heat semigroup. Then, we prove that $(U,P_h,m_{h^2})$ does, where $U$ is an inner uniform domain and $P_h$ is the heat semigroup obtained through Doob's $h$-transformation. The kernel for $P_h$ is directly related to the kernel for $P_D$, the heat semigroup on $U$ with Dirichlet boundary conditions, and therefore, we are able to derive bounds for the Dirichlet heat kernel in inner uniform domains. All work in this thesis is joint with Laurent Saloff-Coste. dc.language.iso en_US dc.rights Attribution-NonCommercial 4.0 International * dc.rights.uri https://creativecommons.org/licenses/by-nc/4.0/ * dc.subject Mathematics dc.title Discrete Heat Kernel Estimates on Inner Uniform Domains dc.type dissertation or thesis thesis.degree.discipline Mathematics thesis.degree.grantor Cornell University thesis.degree.level Doctor of Philosophy thesis.degree.name Ph. D., Mathematics dc.contributor.chair Saloff-Coste, Laurent Pascal dc.contributor.committeeMember Strichartz, Robert Stephen dc.contributor.committeeMember Levine, Lionel dcterms.license https://hdl.handle.net/1813/59810 dc.identifier.doi https://doi.org/10.7298/X4ZP44CZ
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