We establish precise upper and lower bounds for the subelliptic heat kernel on nilpotent Lie groups $G$ of H-type. Specifically, we show that there exist positive constants $C_1$, $C_2$ and a polynomial correction function $Q_t$ on $G$ such that $$C_1 Q_t e^{-\frac{d^2}{4t}} \le p_t \le C_2 Q_t e^{-\frac{d^2}{4t}}$$ where $p_t$ is the heat kernel, and $d$ the Carnot-Carath\'eodory distance on $G$. We also obtain similar bounds on the norm of its subelliptic gradient $|\nabla p_t|$. Along the way, we record explicit formulas for the distance function $d$ and the subriemannian geodesics of H-type groups.Comment: 35 pages. Identical to published version except that some typos are fixed her

We prove the following gradient inequality for the subelliptic heat kernel on nilpotent Lie groups $G$ of H-type: $$|\nabla P_t f| \le K P_t(|\nabla f|)$$ where $P_t$ is the heat semigroup corresponding to the sublaplacian on $G$, $\nabla$ is the subelliptic gradient, and $K$ is a constant. This extends a result of H.-Q. Li for the Heisenberg group. The proof is based on pointwise heat kernel estimates, and follows an approach used by Bakry, Baudoin, Bonnefont, and Chafa\"i.Comment: 23 pages; updated with peer-review revision

A classical aspect of Riemannian geometry is the study of estimates that hold uniformly over some class of metrics. The best known examples are eigenvalue bounds under curvature assumptions. In this paper, we study the family of all left-invariant geometries on SU(2). We show that left-invariant geometries on SU(2) are uniformly doubling and give a detailed estimate of the volume of balls that is valid for any of these geometries and any radius. We discuss a number of consequences concerning the spectrum of the associated Laplacians and the corresponding heat kernels.1991 Mathematics Subject Classification. Primary 53C21; Secondary 35K08, 53C17, 58J35, 58J60, 22C05, 22E30.

I wrote these lecture notes for a graduate topics course I taught at Cornell University in Fall 2011 (Math 7770). The ostensible primary goal of the course was for the students to learn some of the fundamental results and techniques in the study of probability on infinite-dimensional spaces, particularly Gaussian measures on Banach spaces (also known as abstract Wiener spaces). As others who have taught such courses will understand, a nontrivial secondary goal of the course was for the instructor (i.e., me) to do the same. These notes only scratch the very surface of the subject, but I tried to use them to work through some of the basics and see how they fit together into a bigger picture. In addition to theorems and proofs, I've left in some more informal discussions that attempt to develop intuition.Most of the material here comes from the books [14, 16, 2], and the lecture notes prepared by Bruce Driver for the 2010 Cornell Probability Summer School [4,5]. If you are looking to learn more, these are great places to look. 1 Any text marked Question N is something that I found myself wondering while writing this, but didn't ever resolve. I'm not proposing them as open problems; the answers could be well-known, just not by me. If you know the answer to any of them, I'd be happy to hear about it! There are also still a few places where proofs are rough or have some gaps that I never got around to filling in.On the other hand, something marked Exercise N is really meant as an exercise. I would like to take this opportunity to thank the graduate students who attended the course. These notes were much improved by their questions and contributions. I'd also like to thank several colleagues who sat in on the course or otherwise contributed to these notes, particularly Clinton Conley, Bruce Driver, Leonard Gross, Ambar Sengupta, and Benjamin Steinhurst. Obviously, the many deficiencies in these notes are my responsibility and not theirs.Questions and comments on these notes are most welcome. I am now at the University of Northern Colorado, and you can email me at

In classical PDE theory, Widder's theorem gives a representation for nonnegative solutions of the heat equation on R n . We show that an analogous theorem holds for local weak solutions of the canonical "heat equation" on a symmetric local Dirichlet space satisfying a local parabolic Harnack inequality. B Cutoff Sobolev inequalities 27 C Maximum principle 28