arXiv subjects

Emily Dautenhahn

Publications and source records attributed to Emily Dautenhahn.

4 recordsLinked to original sources

arXiv2026

Heat kernel estimates on book-like graphs

In this paper, we prove two-sided heat kernel estimates on what we call "book-like" graphs. These are graphs consisting of pieces that satisfy the parabolic Harnack inequality that are glued together in a sufficiently nice way over a possibly infinite set of vertices. The prototypical example is gluing a copy of the square four-dimensional lattice $\mathbb{Z}^4,$ a copy of $\mathbb{Z}^5$, and a copy of $\mathbb{Z}^6$ by identifying their $x_1$-axes and taking the lazy simple random walk on this glued graph. Our results are flexible enough to handle perturbations of this example, for instance by adding diagonals to one of the lattices or a few extra vertices/edges.

math.PR↗

arXiv2025

Faber-Krahn inequality and heat kernel estimates on glued graphs

Faber-Krahn functions provide lower bounds on the first Dirichlet eigenvalue of the Laplacian and are useful because they imply heat kernel upper bounds. In this paper, we are interested in Faber-Krahn functions and heat kernel estimates for a certain class of graphs consisting of "sufficiently nice pages" (satisfying a Harnack inequality) glued together via a "sufficiently nice spine." For such graphs, we obtain a relative Faber-Krahn function in terms of the Faber-Krahn functions on the pages. The corresponding heat kernel upper bound involves the volumes on the various pages. In the case our graphs satisfy a property we call "book-like" and the spine is appropriately transient, we provide a matching lower bound for the heat kernel between two points on the gluing spine.

math.PR↗

arXiv2023

Hitting probabilities and uniformly $S$-transient subgraphs

We study the probability that a random walk started inside a subgraph of a larger graph exits that subgraph (or, equivalently, hits the exterior boundary of the subgraph). Considering the chance a random walk started in the subgraph never leaves the subgraph leads to a notion we call "survival" transience, or $S$-transience. In the case where the heat kernel of the larger graph satisfies two-sided Gaussian estimates, we prove an upper bound on the probability of hitting the boundary of the subgraph. Under the additional hypothesis that the subgraph is inner uniform, we prove a two-sided estimate for this probability. The estimate depends upon a harmonic function in the subgraph. We also provide two-sided estimates for related probabilities, such as the harmonic measure (the chance the walk exits the subgraph at a particular point on its boundary).

math.PR↗

arXiv2021

Heat kernel estimates on manifolds with ends with mixed boundary condition

We obtain two-sided heat kernel estimates for Riemannian manifolds with ends with mixed boundary condition, provided that the heat kernels for the ends are well understood. These results extend previous results of Grigor'yan and Saloff-Coste by allowing for Dirichlet boundary condition. The proof requires the construction of a global harmonic function which is then used in the $h$-transform technique.

math.DG↗