On weighted heights of random trees

Consider the family treeT of a branching process starting from a single progenitor and conditioned to havev=v(T) edges (total progeny). To each edge <e> we associate a weightW(e). The weights are i.i.d. random variables and independent ofT. The weighted height of a self-avoiding path inT starting at the root is the sum of the weights associated with the path. We are interested in the asymptotic distribution of the maximum weighted path height in the limit asv=n. Depending on the tail of the weight distribution, we obtain the limit in three cases. In particular ify ² P(W(e)> y)0, then the limit distribution depends strongly on the tree and, in fact, is the distribution of the maximum of a Brownian excursion. If the tail of the weight distribution is regularly varying with exponent 0<2, then the weight swamps the tree and the answer is the asymptotic distribution of the maximum edge weight in the tree. There is a borderline case, namely,P(W(e)> y)cy ^–2 asy, in which the limit distribution exists but involves both the tree and the weights in a more complicated way.     

Gradient estimates for the subelliptic heat kernel on H-type groups

We prove the following gradient inequality for the subelliptic heat kernel on nilpotent Lie groups G of H-type:

|∇Ptf|?KPt(|∇f|),     

The continuum random tree is the scaling limit of unlabeled unrooted trees

We show that the uniform unlabeled unrooted tree with n vertices and vertex degrees in a fixed set converges in the Gromov‐Hausdorff sense after a suitable rescaling to the Brownian continuum random tree. This confirms a conjecture by Aldous (1991). We also establish Benjamini‐Schramm convergence of this model of random trees and provide a general approximation result, that allows for a transfer of a wide range of asymptotic properties of extremal and additive graph parameters from Pólya trees to unrooted trees.     

Rayleigh processes, real trees, and root growth with re-grafting

The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous's Brownian continuum random tree, the random tree-like object naturally associated with a standard Brownian excursion, may be thought of as a random compact real tree. The continuum random tree is a scaling limit as N→∞ of both a critical Galton-Watson tree conditioned to have total population size N as well as a uniform random rooted combinatorial tree with N vertices. The Aldous–Broder algorithm is a Markov chain on the space of rooted combinatorial trees with N vertices that has the uniform tree as its stationary distribution. We construct and study a Markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as N→∞ of the Aldous–Broder chain. A key technical ingredient in this work is the use of a pointed Gromov–Hausdorff distance to metrize the space of rooted compact real trees. Berkeley Statistics Technical Report No. 654 (February 2004), revised October 2004. To appear in Probability Theory and Related Fields. SNE supported in part by NSF grants DMS-0071468 and DMS-0405778, and a Miller Institute for Basic Research in Science research professorship JP supported in part by NSF grants DMS-0071448 and DMS-0405779 AW supported by a DFG Forchungsstipendium     

On heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces

In this paper, we discuss necessary and sufficient conditions on jumping kernels for a class of jump-type Markov processes on metric measure spaces to have scale-invariant finite range parabolic Harnack inequality.     

Symmetric jump processes and their heat kernel estimates

We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes (or equivalently, a class of symmetric integro-differential operators). We focus on the sharp two-sided estimates for the transition density functions (or heat kernels) of the processes, a priori Hölder estimate and parabolic Harnack inequalities for their parabolic functions. In contrast to the second order elliptic differential operator case, the methods to establish these properties for symmetric integro-differential operators are mainly probabilistic.     

On the profile of random trees

Let T be a plane rooted tree with n nodes which is regarded as family tree of a Galton-Watson branching process conditioned on the total progeny. The profile of the tree may be described by the number of nodes or the number of leaves in layer , respectively. It is shown that these two processes converge weakly to Brownian excursion local time. This is done via characteristic functions obtained by means of generating functions arising from the combinatorial setup and complex contour integration. Besides, an integral representation for the two-dimensional density of Brownian excursion local time is derived. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10 , 421–451, 1997     

Subexponential behaviour of the Dirichlet heat kernel

We obtain a formula for the asymptotic behaviour of the Dirichlet heat kernel for large time in terms of the survival probability of a Brownian motion, under the assumption that the latter decays subexponentially for large time. We also obtain a lower bound for the Dirichlet heat kernel for arbitrary open and connected sets in Euclidean space.     

Finite propagation speed and kernel estimates for Schrödinger operators

We point out finite propagation speed phenomena for discrete and continuous Schrödinger operators and discuss various types of kernel estimates from this point of view.

     

Kato class measures of symmetric Markov processes under heat kernel estimates

We establish the coincidence of two classes of Kato class measures in the framework of symmetric Markov processes admitting upper and lower estimates of heat kernel under mild conditions. One class of Kato class measures is defined by way of the heat kernel, another is defined in terms of the Green kernel depending on some exponents related to the heat kernel estimates. We also prove that pth integrable functions on balls with radius 1 having a uniformity of its norm with respect to centers are of Kato class if p is greater than a constant related to the estimate under the same conditions. These are complete extensions of some results for the Brownian motion on Euclidean space by Aizenman and Simon. Our result can be applicable to many examples, for instance, symmetric (relativistic) stable processes, jump processes on d-sets, Brownian motions on Riemannian manifolds, diffusions on fractals and so on.     

Monotonicity properties of the Neumann heat kernel in the ball

A well-known conjecture of R. Laugesen and C. Morpurgo asserts that the diagonal of the Neumann heat kernel of the unit ball U⊂Rn is a strictly increasing radial function. In this paper we use probabilistic arguments to settle this conjecture and to prove some inequalities for the Neumann heat kernel in the ball.     

On gradient bounds for the heat kernel on the Heisenberg group

It is known that the couple formed by the two-dimensional Brownian motion and its Lévy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is hypoelliptic but not elliptic, which makes difficult the derivation of functional inequalities for the heat kernel. However, Driver and Melcher and more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel on the Heisenberg group. We provide in this paper simple proofs of these bounds, and explore their consequences in terms of functional inequalities, including Cheeger and Bobkov type isoperimetric inequalities for the heat kernel.     

Asymptotics of trees with a prescribed degree sequence and applications

Let t be a rooted tree and n_bi(t) the number of nodes in t having i children. The degree sequence of t satisfies , where denotes the number of nodes in t. In this paper, we consider trees sampled uniformly among all plane trees having the same degree sequence ; we write for the corresponding distribution. Let be a list of degree sequences indexed by κ corresponding to trees with size . We show that under some simple and natural hypotheses on the trees sampled under converge to the Brownian continuum random tree after normalisation by . Some applications concerning Galton–Watson trees and coalescence processes are provided.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 290‐316, 2014     

Revisiting the heat kernel on isotropic and nonisotropic Heisenberg groups*

The aim of this paper is threefold. First, we obtain the precise bounds for the heat kernel on isotropic Heisenberg groups by using well-known results in the three-dimensional case. Second, we study the asymptotic estimates at infinity for the heat kernel on nonisotropic Heisenberg groups. As a consequence, we give uniform upper and lower estimates of the heat kernel, and complete its short-time behavior obtained by Beals–Gaveau–Greiner. Third, we prove that the uniform asymptotic behaviour at infinity (so the small-time asymptotic behaviour) of the heat kernel for Grushin operators, obtained by the first author, are still valid in two and three dimensions.     

Central limit theorem for the heat kernel measure on the unitary group

We prove that for a finite collection of real-valued functions f₁,…,fn on the group of complex numbers of modulus 1 which are derivable with Lipschitz continuous derivative, the distribution of under the properly scaled heat kernel measure at a given time on the unitary group U(N) has Gaussian fluctuations as N tends to infinity, with a covariance for which we give a formula and which is of order N−1. In the limit where the time tends to infinity, we prove that this covariance converges to that obtained by P. Diaconis and S.N. Evans in a previous work on uniformly distributed unitary matrices. Finally, we discuss some combinatorial aspects of our results.     

On the equivalence of heat kernel estimates and logarithmic Sobolev inequalities for the Hodge Laplacian

In this paper we consider the Hodge Laplacian on differential k-forms over smooth open manifolds MN, not necessarily compact. We find sufficient conditions under which the existence of a family of logarithmic Sobolev inequalities for the Hodge Laplacian is equivalent to the ultracontractivity of its heat operator.We will also show how to obtain a logarithmic Sobolev inequality for the Hodge Laplacian when there exists one for the Laplacian on functions. In the particular case of Ricci curvature bounded below, we use the Gaussian type bound for the heat kernel of the Laplacian on functions in order to obtain a similar Gaussian type bound for the heat kernel of the Hodge Laplacian. This is done via logarithmic Sobolev inequalities and under the additional assumption that the volume of balls of radius one is uniformly bounded below.     

Sharp Heat Kernel Estimates in the Fourier–Bessel Setting for a Continuous Range of the Type Parameter

The heat kernel in the setting of classical Fourier–Bessel expansions is defined by an oscillatory series which cannot be computed explicitly.We prove qualitatively sharp estimates of this kernel.Our method relies on establishing a connection with a situation of expansions based on Jacobi polynomials and then transferring known sharp bounds for the related Jacobi heat kernel.Keywords Fourier–Bessel expansions,heat kernel,Bessel process,transition density     

Self-similarity and spectral asymptotics for the continuum random tree

We use the random self-similarity of the continuum random tree to show that it is homeomorphic to a post-critically finite self-similar fractal equipped with a random self-similar metric. As an application, we determine the mean and almost-sure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on the continuum random tree. We also obtain short time asymptotics for the trace of the heat semigroup and the annealed on-diagonal heat kernel associated with this Dirichlet form.     

Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps

We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates. M. T. Barlow’s research was partially supported by NSERC (Canada), the twenty-first century COE Program in Kyoto University (Japan), and by EPSRC (UK). R. F. Bass’s research was partially supported by NSF Grant DMS-0601783. T. Kumagai’s research was partially supported by the Grant-in-Aid for Scientific Research (B) 18340027 (Japan).