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Laurent Saloff-Coste

Publications and source records attributed to Laurent Saloff-Coste.

At least 37 records · Page 2Linked to original sources
arXiv2022

On-diagonal asymptotics for heat kernels of a class of inhomogeneous partial differential operators

We consider certain constant-coefficient differential operators on $\mathbb{R}^d$ with positive-definite symbols. Each such operator $Λ$ with symbol $P$ defines a semigroup $e^{-tΛ}$ , $t>0$ , admitting a convolution kernel $H^t_P$ for which the large-time behavior of $H_P^t(0)$ cannot be deduced by basic scaling arguments. The simplest example has symbol $P(ξ)=(η+ζ^2)^2+η^4$ , $ξ=(η,ζ)\in \mathbb{R}^2$ . We devise a method to establish large-time asymptotics of $H^t_P(0)$ for several classes of examples of this type and we show that these asymptotics are preserved by perturbations by certain higher-order differential operators. For the $P$ just given, it turns out that $H^t_P(0)\sim c_Pt^{-5/8}$ as $t\to\infty$ . We show how such results are relevant to understand the convolution powers of certain complex functions on $\mathbb Z^d$ . Our work represents a first basic step towards a good understanding of the semigroups associated with these operators. Obtaining meaningful off-diagonal upper bounds for $H_P^t$ remains an interesting challenge.

math.AP↗
arXiv2022

Multidimensional examples of the Metropolis algorithm

Consider the problem of approximating a given probability distribution on the cube $[0,1]^n$ via the use of a square lattice discretization with mesh-size $1/N$ and the Metropolis algorithm. Here the dimension $n$ is fixed and we focus for the most part on the case $n=2$. In order to understand the speed of convergence of such a procedure, one needs to control the spectral gap, $λ$, of the associated finite Markov chain, and how it depends on the parameter $N$. In this work, we study basic examples for which good upper-bounds and lower-bounds on $λ$ can be obtained via appropriate application of path techniques.

math.PR↗
arXiv2022

Random walks on finite nilpotent groups driven by long-jump measures

We consider a variant of random walks on finite groups. At each step, we choose an element from a set of generators ("directions") uniformly, and an integer from a power law ("speed") distribution associated with the chosen direction. We show that if the finite group is nilpotent, the mixing time of this walk is of the same order of magnitude as the diameter of a suitable pseudo-metric, D(S, a), on the group, which depends only on the generators and speeds. Additionally, we give sharp bounds on the $\ell^2$-distance between the distribution of the position of the walker and the stationary distribution, and compute D(S,a) for some examples.

math.PR↗
arXiv2021

Perturbation results concerning Gaussian estimates and hypoellipticity for left-invariant Laplacians on compact groups

In this paper we study left-invariant Laplacians on compact connected groups that are form-comparable perturbations of bi-invariant Laplacians. Our results show that Gaussian bounds for derivatives of heat kernels enjoyed by certain bi-invariant Laplacians hold for their form-comparable perturbations. We further show that the parabolic operators associated with such left-invariant Laplacians, in particular, with the bi-invariant Laplacians, are hypoelliptic in various senses.

math.PR↗
arXiv2021

The hit-and-run version of top-to-random

We study an example of a {\em hit-and-run} random walk on the symmetric group $\mathbf S_n$. Our starting point is the well understood {\em top-to-random} shuffle. In the hit-and-run version, at each {\em single step}, after picking the point of insertion, $j$, uniformly at random in $\{1,\dots,n\}$, the top card is inserted in the $j$-th position $k$ times in a row where $k$ is uniform in $\{0,1,\dots,j-1\}$. The question is, does this accelerate mixing significantly or not? We show that, in $L^2$ and sup-norm, this accelerates mixing at most by a constant factor (independent of $n$). Analyzing this problem in total variation is an interesting open question.

math.PR↗
arXiv2020

Time regularity for local weak solutions of the heat equation on local Dirichlet spaces

We study the time regularity of local weak solutions of the heat equation in the context of local regular symmetric Dirichlet spaces. Under two basic and rather minimal assumptions, namely, the existence of certain cut-off functions and a very weak $L^2$ Gaussian type upper-bound for the heat semigroup, we prove that the time derivatives of a local weak solution of the heat equation are themselves local weak solutions. This applies, for instance, to local weak solutions of parabolic equations with uniformly elliptic symmetric divergence form second order operators with measurable coefficients. We describe some applications to the structure of ancient local weak solutions of such equations which generalize recent results of [8] and [33].

math.AP↗
arXiv2020

Rough Hypoellipticity for the Heat Equation in Dirichlet Spaces

This paper aims at proving the local boundedness and continuity of solutions of the heat equation in the context of Dirichlet spaces under some rather weak additional assumptions. We consider symmetric local regular Dirichlet forms which satisfy mild assumptions concerning (a) the existence of cut-off functions, (b) a local ultracontractivity hypothesis, and (c) a weak off-diagonal upper bound. In this setting, local weak solutions of the heat equation, and their time derivatives, are shown to be locally bounded; they are further locally continuous, if the semigroup admits a locally continuous density function. Applications of the results are provided including discussion on the existence of locally bounded heat kernel; $L^\infty$ structure results for ancient solutions of the heat equation. The last section presents a special case where the $L^\infty$ off-diagonal upper bound follows from the ultracontractivity property of the semigroup. This paper is a continuation of [7].

math.AP↗
arXiv2020

Geometric analysis on manifolds with ends

In this survey article, we discuss some recent progress on geometric analysis on manifold with ends. In the final section, we construct manifolds with ends with oscillating volume functions which may turn out to have a different heat kernel estimates from those provided by known results.

math.DG↗
arXiv2020

Davies' method for heat-kernel estimates: An extension to the semi-elliptic setting

We consider a class of constant-coefficient partial differential operators on a finite-dimensional real vector space which exhibit a natural dilation invariance. Typically, these operators are anisotropic, allowing for different degrees in different directions. The heat kernels associated to these so-called positive-homogeneous operators are seen to arise naturally as the limits of convolution powers of complex-valued measures, just as the classical heat kernel appears in the central limit theorem. Building on the functional-analytic approach developed by E. B. Davies for higher-order uniformly elliptic operators with measurable coefficients, we formulate a general theory for (anisotropic) self-adjoint variable-coefficient operators, each comparable to a positive-homogeneous operator, and study their associated heat kernels. Specifically, under three abstract hypotheses, we show that the heat kernels satisfy off-diagonal (Gaussian type) estimates involving the Legendre-Fenchel transform of the operator's principle symbol. Our results extend those of E. B. Davies and G. Barbatis and partially extend results of A. F. M. ter Elst and D. Robinson.

math.AP↗
arXiv2019

Analytic-geometric methods for finite Markov chains with applications to quasi-stationarity

For a relatively large class of well-behaved absorbing (or killed) finite Markov chains, we give detailed quantitative estimates regarding the behavior of the chain before it is absorbed (or killed). Typical examples are random walks on box-like finite subsets of the square lattice $\mathbb Z^d$ absorbed (or killed) at the boundary. The analysis is based on Poincaré, Nash, and Harnack inequalities, moderate growth, and on the notions of John and inner-uniform domains.

math.PR↗
arXiv2019

Gambler's ruin estimates on finite inner uniform domains

Gambler's ruin estimates can be viewed as harmonic measure estimates for finite Markov chains which are absorbed (or killed) at boundary points. We relate such estimates to properties of the underlying chain and its Doob transform. Precisely, we show that gambler's ruin estimates reduce to a good understanding of the Perron-Frobenius eigenfunction and eigenvalue whenever the underlying chain and its Doob transform are Harnack Markov chains. Finite inner-uniform domains (say, in the square grid $\mathbb Z^n$) provide a large class of examples where these ideas apply and lead to detailed estimates. In general, understanding the behavior of the Perron-Frobenius eigenfunction remains a challenge.

math.PR↗
arXiv2018

Left-invariant geometries on $\mathrm{SU}(2)$ are uniformly doubling

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 $\mathrm{SU}(2)$. We show that left-invariant geometries on $\mathrm{SU}(2)$ 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.

math.DG↗
arXiv2018

Heat kernel estimates for anomalous heavy-tailed random walks

Sub-Gaussian estimates for the natural random walk is typical of many regular fractal graphs. Subordination shows that there exist heavy tailed jump processes whose jump indices are greater than or equal to two. However, the existing machinery used to prove heat kernel bounds for such heavy tailed random walks fail in this case. In this work we extend Davies' perturbation method to obtain transition probability bounds for these anomalous heavy tailed random walks. We prove global upper and lower bounds on the transition probability density that are sharp up to constants. An important feature of our work is that the methods we develop are robust to small perturbations of the symmetric jump kernel.

math.PR↗
arXiv2017

Parabolic Harnack inequality for time-dependent non-symmetric Dirichlet forms

In the context of a metric measure Dirichlet space satisfying volume doubling and Poincaré inequality, we prove the parabolic Harnack inequality for weak solutions of the heat equation associated with local nonsymmetric bilinear forms. In particular, we show that these weak solutions are locally bounded.

math.PR↗
arXiv2016

Positive-homogeneous operators, heat kernel estimates and the Legendre-Fenchel transform

We consider a class of homogeneous partial differential operators on a finite-dimensional vector space and study their associated heat kernels. The heat kernels for this general class of operators are seen to arise naturally as the limiting objects of the convolution powers of complex-valued functions on the square lattice in the way that the classical heat kernel arises in the (local) central limit theorem. These so-called positive-homogeneous operators generalize the class of semi-elliptic operators in the sense that the definition is coordinate-free. More generally, we introduce a class of variable-coefficient operators, each of which is uniformly comparable to a positive-homogeneous operator, and we study the corresponding Cauchy problem for the heat equation. Under the assumption that such an operator has Hölder continuous coefficients, we construct a fundamental solution to its heat equation by the method of E. E. Levi, adapted to parabolic systems by A. Friedman and S. D. Eidelman. Though our results in this direction are implied by the long-known results of S. D. Eidelman for $2\vec{b}$-parabolic systems, our focus is to highlight the role played by the Legendre-Fenchel transform in heat kernel estimates. Specifically, we show that the fundamental solution satisfies an off-diagonal estimate, i.e., a heat kernel estimate, written in terms of the Legendre-Fenchel transform of the operator's principal symbol--an estimate which is seen to be sharp in many cases.

math.AP↗
arXiv2016

Heat kernel estimates on connected sums of parabolic manifolds

We obtain matching two sided estimates of the heat kernel on a connected sum of parabolic manifolds, each of them satisfying the Li-Yau estimate. The key result is the on-diagonal upper bound of the heat kernel at a central point. Contrary to the nonparabolic case (which was settled in [15]), the on-diagonal behavior of the heat kernel in our case is determined by the end with the maximal volume growth function. As examples, we give explicit heat kernel bounds on the connected sums $R^2#R^2$ and $R^1#R^2$ where $R^1 = R_+\timesS^1$.

math.PR↗
arXiv2016

Davies' method for anomalous diffusions

Davies' method of perturbed semigroups is a classical technique to obtain off-diagonal upper bounds on the heat kernel. However Davies' method does not apply to anomalous diffusions due to the singularity of energy measures. In this note, we overcome the difficulty by modifying the Davies' perturbation method to obtain sub-Gaussian upper bounds on the heat kernel. Our computations closely follow the seminal work of Carlen, Kusuoka and Stroock \cite{CKS}. However, a cutoff Sobolev inequality due to Andres and Barlow \cite{AB} is used to bound the energy measure.

math.PR↗