Date of Completion

5-5-2017

Embargo Period

5-5-2017

Keywords

Analysis, Probability, Fractals, Diffusions, Harnack Inequality, Heat Kernel Estimates, Laplacian

Major Advisor

Alexander Teplyaev

Associate Advisor

Luke Rogers

Associate Advisor

Maria Gordina

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

Following the methods used by Barlow and Bass to prove the existence of a diffusion on the Sierpinski Carpet, we establish the existence of a diffusion for a class of planar fractals which are not post critically finite. We conjecture that specific resistance estimates hold on our class of fractals. We further conjecture that these resistance estimates imply the existence of the spectral dimension for our class of fractals. Under these assumptions we use the methods of Barlow and Bass to establish heat kernel asymptotics. From there, we can use the techniques of Barlow, Bass, Kumagai, and Teplyaev to show the uniqueness of the diffusion up to scalar multiples. As a corollary, we conjecture the existence and uniqueness of a diffusion on the Octagasket, thus partially answering a question posed by Strichartz.

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