Date of Completion
Analysis, Probability, Fractals, Diffusions, Harnack Inequality, Heat Kernel Estimates, Laplacian
Field of Study
Doctor of Philosophy
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.
Andrews, Ulysses A. IV, "Existence of Diffusions on 4N Carpets" (2017). Doctoral Dissertations. 1477.