Summer Graduate School

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.

In Bernoulli bond percolation, we delete or retain each edge of a graph independently at random with some retention parameter pand study the geometry of the connected components (clusters) of the resulting subgraph. For lattices of dimension d>1, percolation has a phase transition, with a infinite cluster emerging at a critical probability p_c(d). It is believed that critical percolation at and near the critical probability exhibits rich, fractal-like geometry that is expected to be approximately independent of the choice of lattice but highly dependent on the dimension d. In particular, various qualitative distinctions are expected between the low dimensional case d<6, the high-dimensional case  d>6, and the critical case d=6, but this remains poorly understood particularly in dimensions d=3,4,5,6.

In this course, I will give an overview of of what is known about critical percolation, focussing on the non-planar models and including a detailed treatment of recent advances in long-range and hierarchical models for which various aspects of intermediate-dimensional critical phenomena can now be understood rigorously.

No prior knowledge of percolation will be assumed.

Consider two independent and identically distributed continuous-time Markov chains $X$ and $Y$ on a graph with vertex set $V$. The collision measure $\Pi$, which describes the times and positions of collisions of $X$ and $Y$, is given as a random measure on $\mathbb{R}_{\geq 0} \times V$ by  $\Pi([0,t] \times E) = \int_{0}^{t} 1_{\{X(s) = Y(s)\}} ds.$. We have shown that if graphs converge to a metric space in an appropriate Gromov-Hausdorff-type topology with respect to the effective resistance metric, and if there is a suitable upper bound on the heat kernels of the Markov chains on the graphs, then the two independent Markov chains on the graph and its collision measure converge to stochastic processes on the limiting space and its collision measure. The collision measure of the limiting stochastic processes is constructed using positive continuous additive functionals (PCAFs) in Dirichlet form theory. This applies to many low-dimensional spaces, such as the Sierpi\'{n}ski gasket and the Brownian random tree.

Stein’s Method is a powerful tool for studying distances between distributions of random variables. It has been used widely in the context of statistical mechanics to obtain approximations for various thermodynamic quantities. We will discuss an application of Stein’s Method to the $O(N)$ model, a model of magnetism, and show how recent advances in Stein’s theory allow us to study the limiting behaviour of the magnetisation at the phase transition.

The principal Dirichlet Laplacian eigenfunction $\varphi_U$ describes the invariant measure of a Brownian motion conditioned to remain inside a bounded domain $U\subseteq \mathbb{R}^n$. In this talk, I will present results about how domain perturbations affect $\varphi_U$. These results lead to explicit expressions for $\varphi_U$ in certain Euclidean domains, thus implying improved Dirichlet heat kernel estimates. This is joint work with Laurent Saloff-Coste.