Heat kernel estimates, Harnack inequalities, and quassisymmetry

The heat kernel is the fundamental solution to a parabolic partial differential equation. From a probabilistic perspective, the heat kernel is the transition probability density of a stochastic process. Harnack inequalities and functional inequalities such as Poincare and Sobolev inequalities provide tools to understand the relationship between the behavior of the heat kernel and the geometry of the underlying space. An important feature of the approach using functional inequalities is its robustness under perturbations.

The study of the heat kernel and its estimates has produced fruitful interactions between the fields of Analysis, Geometry, and Probability. One of the goals of this course is to illustrate these interactions of heat kernel estimates with functional inequalities, boundary trace processes, quasisymmetric maps, circle packings, the time change of Markov processes, Doob's h-transform, and estimates of harmonic measure or exit distribution.

The setting for this course is a symmetric Markov process which is equivalently described using a Dirichlet form. This course will contain an introduction to the theory of Dirichlet forms. This theory will be used to construct and analyze Markov processes. This course will survey both classical results and recent progress in our understanding of heat kernel estimates and Harnack inequalities.