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.     

Harnack inequality for some classes of Markov processes

In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps. Mathematics Subject Classification (2000): Primary 60J45, 60J75, Secondary 60J25.This work was completed while the authors were in the Research in Pairs program at the Mathematisches Forschungsinstitut Oberwolfach. We thank the Institute for the hospitality.The research of this author is supported in part by NSF Grant DMS-9803240.The research of this author is supported in part by MZT grant 0037107 of the Republic of Croatia.     

Comparison inequalities for heat semigroups and heat kernels on metric measure spaces

We prove a certain inequality for a subsolution of the heat equation associated with a regular Dirichlet form. As a consequence of this inequality, we obtain various interesting comparison inequalities for heat semigroups and heat kernels, which can be used for obtaining pointwise estimates of heat kernels. As an example of application, we present a new method of deducing sub-Gaussian upper bounds of the heat kernel from on-diagonal bounds and tail estimates.     

Heat kernel estimates for jump processes of mixed types on metric measure spaces

In this paper, we investigate symmetric jump-type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors d-regular sets, which is a class of fractal sets that contains geometrically self-similar sets. A typical example of our jump-type processes is the symmetric jump process with jumping intensity where ν is a probability measure on , c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c ₀(x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between γ₁ and γ₂, where either γ₂ ≥ γ₁ > 0 or γ₁ = γ₂ = 0. This example contains mixed symmetric stable processes on as well as mixed relativistic symmetric stable processes on . We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes. Dedicated to Professor Masatoshi Fukushima on the occasion of his 70th birthday. The research of Zhen-Qing Chen is supported in part by NSF Grants DMS-0303310 and DMS-06000206. The research of Takashi Kumagai is supported in part by the Grant-in-Aid for Scientific Research (B) 18340027.     

Harnack inequality for the linearized parabolic Monge-Ampère equation

In this paper we prove the Harnack inequality for nonnegative solutions of the linearized parabolic Monge-Ampère equation

on parabolic sections associated with , under the assumption that the Monge-Ampère measure generated by satisfies the doubling condition on sections and the uniform continuity condition with respect to Lebesgue measure. The theory established is invariant under the group , where denotes the group of all invertible affine transformations on .

     

Lower inequalities of heat semigroups by using parabolic maximum principle

Using parabolic maximum principle, we apply the analytic method to obtain lower comparison inequalities for non-negative weak supersolutions of the heat equation associated with a regular strongly ρ-local Dirichle form on the abstract metric measure space. As an application, we obtain lower estimates for heat kernels on some Riemannian manifolds.     

Ratio limit theorem for parabolic horn-shaped domains

We prove that for horn-shaped domains of parabolic type, the ratio of the heat kernel at different fixed points has a limit when the time tends to infinity. We also give an explicit formula for the limit in terms of the harmonic functions.

     

A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces

On doubling metric measure spaces endowed with a strongly local regular Dirichlet form, we show some characterisations of pointwise upper bounds of the heat kernel in terms of global scale-invariant inequalities that correspond respectively to the Nash inequality and to a Gagliardo–Nirenberg type inequality when the volume growth is polynomial. This yields a new proof and a generalisation of the well-known equivalence between classical heat kernel upper bounds and relative Faber–Krahn inequalities or localised Sobolev or Nash inequalities. We are able to treat more general pointwise estimates, where the heat kernel rate of decay is not necessarily governed by the volume growth. A crucial role is played by the finite propagation speed property for the associated wave equation, and our main result holds for an abstract semigroup of operators satisfying the Davies–Gaffney estimates.     

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.     

Hamilton-type gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Let (M,g) be a complete noncompact Riemannian manifold. In this note, we derive a local Hamilton-type gradient estimate for positive solution to a simple nonlinear parabolic equation

Harnack type inequality and a priori estimates for solutions of a class of semilinear elliptic equations

For dimensions 3?n?6, we derive here the Harnack type inequality

     

Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below

Using the coupling by parallel translation, along with Girsanov's theorem, a new version of a dimension-free Harnack inequality is established for diffusion semigroups on Riemannian manifolds with Ricci curvature bounded below by , where c>0 is a constant and ρo is the Riemannian distance function to a fixed point o on the manifold. As an application, in the symmetric case, a Li-Yau type heat kernel bound is presented for such semigroups.     

The Dirichlet heat kernel in inner uniform domains: Local results,compact domains and non-symmetric forms

This paper provides sharp Dirichlet heat kernel estimates in inner uniform domains, including bounded inner uniform domains, in the context of certain (possibly non-symmetric) bilinear forms resembling Dirichlet forms. For instance, the results apply to the Dirichlet heat kernel associated with a uniformly elliptic divergence form operator with symmetric second order part and bounded measurable real coefficients in inner uniform domains in Rⁿ${\mathbb{R}}^n$. The results are applicable to any convex domain, to the complement of any convex domain, and to more exotic examples such as the interior and exterior of the snowflake.     

Volume growth and heat kernel estimates for the continuum random tree

In this article, we prove global and local (point-wise) volume and heat kernel bounds for the continuum random tree. We demonstrate that there are almost–surely logarithmic global fluctuations and log–logarithmic local fluctuations in the volume of balls of radius r about the leading order polynomial term as r → 0. We also show that the on-diagonal part of the heat kernel exhibits corresponding global and local fluctuations as t → 0 almost–surely. Finally, we prove that this quenched (almost–sure) behaviour contrasts with the local annealed (averaged over all realisations of the tree) volume and heat kernel behaviour, which is smooth.      

Equivalence conditions for on-diagonal upper bounds of heat kernels on self-similar spaces

We obtain the equivalence conditions for an on-diagonal upper bound of heat kernels on self-similar measure energy spaces. In particular, this upper bound of the heat kernel is equivalent to the discreteness of the spectrum of the generator of the Dirichlet form, and to the global Poincaré inequality. The key ingredient of the proof is to obtain the Nash inequality from the global Poincaré inequality. We give two examples of families of spaces where the global Poincaré inequality is easily derived. They are the post-critically finite (p.c.f.) self-similar sets with harmonic structure and the products of self-similar measure energy spaces.     

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     

Cowling-price theorem and characterization of heat kernel on symmetric spaces

We extend the uncertainty principle, the Cowling-Price theorem, on noncompact Riemannian symmetric spacesX. We establish a characterization of the heat kernel of the Laplace-Beltrami operator onX from integral estimates of the Cowling-Price type.     

Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces

Starting with a regular symmetric Dirichlet form on a locally compact separable metric space X$X$, our paper studies elements of vector analysis, _Lp$L_p$-spaces of vector fields and related Sobolev spaces. These tools are then employed to obtain existence and uniqueness results for some quasilinear elliptic PDE and SPDE in variational form on X$X$ by standard methods. For many of our results locality is not assumed, but most interesting applications involve local regular Dirichlet forms on fractal spaces such as nested fractals and Sierpinski carpets.     

On r-quick limit sets for empirical and related processes based on mixing random variables

The r-quick limit points of normalized sample paths and empirical distribution functions of mixing processes are characterized. An r-quick version of Bahadur-Kiefer-type representation for sample quantiles is established, which yields the r-quick limit points of quantile processes. These results are applied to linear functions of order statistics. Some results on r-quick convergence of certain Gaussian processes are also established.