Discrete Heat Kernel Estimates on Inner Uniform Domains

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