Resolvents and Martin boundaries of product spaces

双曲空间上扩散与随机游走的Martin边界

Martin boundaries of diffusions and random walks on hyperbolic spaces

本文研究双曲空间上特定扩散过程及其在等距群上关联随机游走的Martin边界。

Potential Theory on Minimal Hypersurfaces I: Singularities as Martin Boundaries

This is Part 1 of two papers where we develop the basic potential theory of elliptic operators on posssibly singular almost minimzers using their hyperbolic unfoldings. We can establish surprisingly robust boundary Harnack inequalities along the singular set. We apply them to derive a Martin theory and solve classical boundary value problems and we also consider eigenvalue problems and variational problems on such singular spaces.

Geometric and Martin boundaries of a Cartan-Hadamard surface

We give a general criterion for the Dirichlet problem at infinity (DPI) on a Cartan-Hadamard surface to be solvable, which we primarily use to give the best possible upper radial radial curvature bound for solvability of the DPI, but which is also flexible enough to accommodate flats. In particular, any (upper) radial curvature bound which implies transience also implies solvability of the DPI, which is perhaps surprising. Taking advantage of the structure provided by uniformization, we show that solvability of the DPI implies there is a natural continuous surjection of the Martin boundary ont…

Hinges and geometric constructions of boundaries of symmetric spaces

We give elementary constructions for Satake-Furstenberg, Martin and Karpelevich boundaries of symmetric spaces. We also consruct some "new" boundaries