arXiv subjects
Publications and source records attributed to Laurent Saloff-Coste.
In classical PDE theory, Widder's theorem gives a representation for nonnegative solutions of the heat equation on $\mathbb{R}^n$. 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.
This paper introduces certain elliptic Harnack inequalities for harmonic functions in the setting of the product space $M \times X$, where $M$ is a (weighted) Riemannian Manifold and $X$ is a countable graph. Since some standard arguments for the elliptic case fail in this "mixed" setting, we adapt ideas introduced by Thierry Delmotte for the discrete parabolic case. We then present some useful applications of this inequality, namely, a kernel estimate for functions of the Laplacian on a graph. This application in turn provides sharp estimates for certain Markov kernels on graphs. We then close this paper with an application to convolution power estimates on finitely generated groups of polynomial growth.
We study the decay of convolution powers of a large family $μ_{S,a}$ of measures on finitely generated nilpotent groups. Here, $S=(s_1,...,s_k)$ is a generating $k$-tuple of group elements and $a= (α_1,...,α_k)$ is a $k$-tuple of reals in the interval $(0,2)$. The symmetric measure $μ_{S,a}$ is supported by $S^*=\{s_i^{m}, 1\le i\le k,\,m\in \mathbb Z\}$ and gives probability proportional to $$(1+m)^{-α_i-1}$$ to $s_i^{\pm m}$, $i=1,...,k,$ $m\in \mathbb N$. We determine the behavior of the probability of return $μ_{S,a}^{(n)}(e)$ as $n$ tends to infinity. This behavior depends in somewhat subtle ways on interactions between the $k$-tuple $a$ and the positions of the generators $s_i$ within the lower central series $G_{j}=[G_{j-1},G]$, $G_1=G$.
We study the decay of convolution powers of probability measures without second moment but satisfying some weaker finite moment condition. For any locally compact unimodular group G and any positive function $ρ:G \rightarrow [0,+\infty]$, we introduce a function $Φ_{G,ρ}$ which describes the fastest possible decay of $n \mapsto ϕ^{(2n)}(e)$ when ϕis a symmetric continuous probability density such that $\intρϕ$ is finite. We estimate $Φ_{G,ρ}$ for a variety of groups G and functions ρ. When ρis of the form $ρ=ρ\circ δ$ with $ρ:[0,+\infty) \rightarrow [0,+\infty)$, a fixed increasing function, and $δ:G \rightarrow [0,+\infty)$, a natural word length measuring the distance to the identity element in G, $Φ_{G,ρ}$ can be thought of as a group invariant.
We prove a scale-invariant boundary Harnack principle in inner uniform domains in the context of local regular Dirichlet spaces. For inner uniform Euclidean domains, our results apply to divergence form operators that are not necessarily symmetric, and complement earlier results by H. Aikawa and A. Ancona.
Repeated convolution of a probability measure on Z leads to the central limit theorem and other limit theorems. This paper investigates what kinds of results remain without positivity. It reviews theorems due to Schoenberg, Greville, and Thomee which are motivated by applications to data smoothing (Schoenberg and Greville) and finite difference schemes (Thomee). Using Fourier transform arguments, we prove detailed decay bounds for convolution powers of finitely supported complex functions on Z. If M is an hermitian contraction, an estimate for the off-diagonal entries of the powers M^n k of M^k = I -(I-M)^k is obtained. This generalizes the Carne-Varopoulos Markov chain estimate.
We discuss problems posed by the quantitative study of time inhomogeneous Markov chains. The two main notions for our purpose are merging and stability. Merging (also called weak ergodicity) occurs when the chain asymptotically forgets where it started. It is a loss of memory property. Stability relates to the question of whether or not, despite temporary variations, there is a rough shape describing the long time behavior of the chain. For instance, we will discuss an example where the long time behavior is roughly described by a binomial, with temporal variations.
This paper is concerned with properties of maximal solutions of the Ricci and cross curvature flows on locally homogeneous three-manifolds of type SL(2,R). We prove that, generically, a maximal solution originates at a sub-Riemannian geometry of Heisenberg type. This solves a problem left open in earlier work by two of the authors.
We show that each integral Borel cohomology class of a connected Lie group G can be represented by a Borel bounded cocycle if and only if the radical of G is linear. This leads to a generalization of Gromov's boundedness theorem on characteristic classes of flat bundles.
We introduce the notion of strip complex. A strip complex is a special type of complex obtained by gluing "strips" along their natural boundaries according to a given graph structure. The most familiar example is the one dimensional complex classically associated with a graph, in which case the strips are simply copies of the unit interval (our setup actually allows for variable edge length). A leading key example is treebolic space, a geometric object studied in a number of recent articles, which arises as a horocyclic product of a metric tree with the hyperbolic plane. In this case, the graph is a regular tree, the strips are the closed unit interval times the real line, and each strip is equipped with the hyperbolic geometry of a specific strip in upper half plane. We consider natural families of Dirichlet forms on a general strip complex and show that the associated heat kernels and harmonic functions have very strong smoothness properties. We study questions such as essential selfadjointness of the underlying differential operator acting on a suitable space of smooth functions satisfying a Kirchoff type condition at points where the strip complex bifurcates. Compatibility with projections that arise from proper group actions is also considered.
In this paper, we study the backward Ricci flow on locally homogeneous 3-manifolds. We describe the long time behavior and show that, typically and after a proper re-scaling, there is convergence to a sub-Riemannian geometry. A similar behavior was observed by the authors in the case of the cross curvature flow.
We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions.
We are thankful to the discussants for their hard, interesting work. The main purpose of our paper was to give reasonably sharp rates of convergence for some simple examples of the Gibbs sampler. We chose examples from expository accounts where direct use of available techniques gave practically useless answers. Careful treatment of these simple examples grew into bivariate modeling and Lancaster families. Since bounding rates of convergence is our primary focus, let us begin there. [arXiv:0808.3852]
In this paper, we study the positive cross curvature flow on locally homogeneous 3-manifolds. We describe the long time behavior of these flows. We combine this with earlier results concerning the asymptotic behavior of the negative cross curvature flow to describe the two sided behavior of maximal solutions of the cross curvature flow on locally homogeneous 3-manifolds. We show that, typically, the positive cross curvature flow on locally homogeneous 3-manifold produce an Heisenberg type sub-Riemannian geometry.
Chow and Hamilton introduced the cross curvature flow on closed 3-manifolds with negative or positive sectional curvature. In this paper, we study the negative cross curvature flow in the case of locally homogenous metrics on 3-manifolds. In each case, we describe the long time behavior of the solutions of the corresponding ODE system.
This paper gives a necessary and sufficient condition for a sequence of birth and death chains to converge abruptly to stationarity, that is, to present a cut-off. The condition involves the notions of spectral gap and mixing time. Y. Peres has observed that for many families of Markov chains, there is a cut-off if and only if the product of spectral gap and mixing time tends to infinity. We establish this for arbitrary birth and death chains in continuous time when the convergence is measured in separation and the chains all start at 0.
We study the cutoff phenomenon for generalized riffle shuffles where, at each step, the deck of cards is cut into a random number of packs of multinomial sizes which are then riffled together.
We develop a new approach to formulate and prove the weak uncertainty inequality which was recently introduced by Okoudjou and Strichartz. We assume either an appropriate measure growth condition with respect to the effective resistance metric, or, in the absence of such a metric, we assume the Poincare inequality and reverse volume doubling property. We also consider the weak uncertainty inequality in the context of Nash-type inequalities. Our results can be applied to a wide variety of metric measure spaces, including graphs, fractals and manifolds.