Heat Kernel Estimates Research Articles

Sharp Estimates of Radial Dunkl and Heat Kernels in the Complex Case A n

In this article, we consider the radial Dunkl geometric case k = 1 corresponding to flat Riemannian symmetric spaces in the complex case and we prove exact estimates for the positive valued Dunkl kernel and for the radial heat kernel. Dans cet article, nous considerons le cas geometrique radial de Dunkl k = 1 correspondant aux espaces symetriques riemanniens plats dans le cas complexe et nous prouvons des estimations exactes pour le noyau de Dunkla valeur positive et pour le noyau de chaleur radial.

$$L^p$$-Kato class measures for symmetric Markov processes under heat kernel estimates

In this paper, we establish the coincidence of two classes of $$L^p$$ -Kato class measures in the framework of symmetric Markov processes admitting upper and lower estimates of heat kernel under mild conditions. One class of $$L^p$$ -Kato class measures is defined by the pth power of positive order resolvent kernel, another is defined in terms of the pth power of Green kernel depending on some exponents related to the heat kernel estimates. We also prove that functions u such that $$\sup _{x\in E}\int _{B_1(x)}|u|^q\mathrm {d}\mathfrak {m} <\infty $$ are of $$L^p$$ -Kato class if q is greater than a constant related to p and the constants appeared in the upper and lower estimates of the heat kernel. These are complete extensions of some results by Aizenman–Simon and the recent results by the second named author in the framework of Brownian motions on Euclidean space. We further give necessary and sufficient conditions for a Radon measure with Ahlfors regularity to belong to $$L^p$$ -Kato class. Our results 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.

Genuinely sharp heat kernel estimates on compact rank-one symmetric spaces, for Jacobi expansions, on a ball and on a simplex

We prove genuinely sharp two-sided global estimates for heat kernels on all compact rank-one symmetric spaces. This generalizes the authors' recent result obtained for a Euclidean sphere of arbitrary dimension. Furthermore, similar heat kernel bounds are shown in the context of classical Jacobi expansions, on a ball and on a simplex. These results are more precise than the qualitatively sharp Gaussian estimates proved recently by several authors.

Stability of heat kernel estimates for symmetric non-local Dirichlet forms

In this paper, we consider symmetric jump processes of mixed-type on metric measure spaces under general volume doubling condition, and establish stability of two-sided heat kernel estimates and heat kernel upper bounds. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cut-off Sobolev inequalities, and the Faber-Krahn inequalities. In particular, we establish stability of heat kernel estimates for α \alpha -stable-like processes even with α ≥ 2 \alpha \ge 2 when the underlying spaces have walk dimensions larger than 2 2 , which has been one of the major open problems in this area.

Stochastic PDEs on graphs as scaling limits of discrete interacting systems

Stochastic partial differential equations (SPDE) on graphs were recently introduced by Cerrai and Freidlin (Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 865–899). This class of stochastic equations in infinite dimensions provides a minimal framework for the study of the effective dynamics of much more complex systems. However, how they emerge from microscopic individual-based models is still poorly understood, partly due to complications near vertex singularities. In this work, motivated by the study of the dynamics and the genealogies of expanding populations in spatially structured environments, we obtain a new class of SPDE on graphs of Wright–Fisher type which have nontrivial boundary conditions on the vertex set. We show that these SPDE arise as scaling limits of suitably defined biased voter models (BVM), which extends the scaling limits of Durrett and Fan (Ann. Appl. Probab. 26 (2016) 3456–3490). We further obtain a convergent simulation scheme for each of these SPDE in terms of a system of Itô SDEs, which is useful when the size of the BVM is too large for stochastic simulations. These give the first rigorous connection between SPDE on graphs and more discrete models, specifically, interacting particle systems and interacting SDEs. Uniform heat kernel estimates for symmetric random walks approximating diffusions on graphs are the keys to our proofs. Some open problems are provided as further motivations of our study.

Scaling limit of a directed polymer among a Poisson field of independent walks

We consider a directed polymer model in dimension 1+1, where the disorder is given by the occupation field of a Poisson system of independent random walks on Z. In a suitable continuum and weak disorder limit, we show that the family of quenched partition functions of the directed polymer converges to the Stratonovich solution of a multiplicative stochastic heat equation (SHE) with a Gaussian noise, whose space-time covariance is given by the heat kernel.In contrast to the case with space-time white noise where the solution of the SHE admits a Wiener-Itô chaos expansion, we establish an L1-convergent chaos expansions of iterated integrals generated by Picard iterations. Using this expansion and its discrete counterpart for the polymer partition functions, the convergence of the terms in the expansion is proved via functional analytic arguments and heat kernel estimates. The Poisson random walk system is amenable to careful moment analysis, which is an important input to our arguments.

Gradient and Eigenvalue Estimates on the Canonical Bundle of Kähler Manifolds

We prove certain gradient and eigenvalue estimates, as well as the heat kernel estimates, for the Hodge Laplacian on (m, 0) forms, i.e., sections of the canonical bundle of Kähler manifolds, where m is the complex dimension of the manifold. Instead of the usual dependence on curvature tensor, our condition depends only on the Ricci curvature bound. The proof is based on a new Bochner type formula for the gradient of (m, 0) forms, which involves only the Ricci curvature and the gradient of the scalar curvature.

Estimates of Dirichlet heat kernels for unimodal Lévy processes with low intensity of small jumps

In this paper, we study transition density functions for pure jump unimodal L\'evy processes killed upon leaving an open set $D$. Under some mild assumptions on the L\'evy density, we establish two-sided Dirichlet heat kernel estimates when the open set $D$ is $C^{1, 1}$. Our result covers the case that the L\'evy densities of unimodal L\'evy processes are regularly varying functions whose indices are equal to the Euclidean dimension. This is the first results on two-sided Dirichlet heat kernel estimates for L\'evy processes such that the weak lower scaling index of the L\'evy densities is not necessarily strictly bigger than the Euclidean dimension.

Formula omitted] bounds of eigenfunctions with an indefinite weight function

This paper is concerned with the L∞ bounds of eigenfunctions to the eigenvalue problem Δu=−λm(x)uin Ω,u=0on ∂Ω,where Ω is a bounded domain and m(x) is an indefinite weight function. For the Dirichlet eigenvalue problem when m(x)≡1, the L∞ bound of eigenfunction is obtained by classical heat kernel estimate. However, the case is quite different for the weighted problem, additional difficulty appears if m(x) is indefinite. In this paper, we investigate the L∞ bounds of eigenfunctions by the means of comparison argument and fundamental solution of heat equation.

Time-decay Estimates for Keller-Segel System

For the Cauchy problem to Keller-Segel system, we show well-posedness and time-decay estimates in the critical scaling-invariant Besov spaces by using Littlewood-Paley analysis together with the decay estimates of heat kernels.

Construction and heat kernel estimates of generalstable-like Markov processes

Construction and heat kernel estimates of generalstable-like Markov processes

Characterizations of heat kernel estimates for symmetric non-local Dirichlet forms via resistance forms

Motivated by [5], we obtain new equivalent conditions for two-sided heat kernel estimates of symmetric non-local Dirichlet forms in terms of resistance forms. Characterizations for upper bounds of heat kernel estimates as well as near diagonal lower bounds of Dirichlet heat kernel estimates are also established. These results can be seen as a complement of the recent studies on heat kernel estimates and parabolic Harnack inequalities for symmetric non-local Dirichlet forms in [10, 11].

Heat Kernel Estimates for Non-symmetric Finite Range Jump Processes

In this paper, we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation $${{\cal S}^b}: = {\overline {\rm{\Delta }} ^{\alpha /2}} + b \cdot \nabla $$ where $${\overline {\rm{\Delta }} ^{\alpha /2}}$$ is the truncated fractional Laplacian, α ∈ (1, 2) and b ∈ K−1 . In the second part, for a more general finite range jump process, we present some sufficient conditions to allow that the two sided estimates of the heat kernel are comparable to the Poisson type function for large distance ∣x − y∣ in short time.

Some local maximum principles along Ricci flows

AbstractIn this work, we obtain a local maximum principle along the Ricci flow $g(t)$ under the condition that $\mathrm {Ric}(g(t))\le {\alpha } t^{-1}$ for $t&gt;0$ for some constant ${\alpha }&gt;0$ . As an application, we will prove that under this condition, various kinds of curvatures will still be nonnegative for $t&gt;0$ , provided they are non-negative initially. These extend the corresponding known results for Ricci flows on compact manifolds or on complete noncompact manifolds with bounded curvature. By combining the above maximum principle with the Dirichlet heat kernel estimates, we also give a more direct proof of Hochard’s [15] localized version of a maximum principle by Bamler et al. [1] on the lower bound of different kinds of curvatures along the Ricci flows for $t&gt;0$ .

On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates

We show that, for a strongly local, regular symmetric Dirichlet form over a complete, locally compact geodesic metric space, full off-diagonal heat kernel estimates with walk dimension strictly larger than two (sub-Gaussian estimates) imply the singularity of the energy measures with respect to the symmetric measure, verifying a conjecture by M. T. Barlow in (Contemp. Math. 338 (2003) 11–40). We also prove that in the contrary case of walk dimension two, that is, where full off-diagonal Gaussian estimates of the heat kernel hold, the symmetric measure and the energy measures are mutually absolutely continuous in the sense that a Borel subset of the state space has measure zero for the symmetric measure if and only if it has measure zero for the energy measures of all functions in the domain of the Dirichlet form.

Factorization and estimates of Dirichlet heat kernels for non-local operators with critical killings

In this paper we discuss non-local operators with killing potentials, which may not be in the standard Kato class. We first discuss factorization of their Dirichlet heat kernels in metric measure spaces. Then we establish explicit estimates of the Dirichlet heat kernels under critical killings in C1,1 open subsets of Rd or in Rd∖{0}. The decay rates of our explicit estimates come from the values of the multiplicative constants in the killing potentials. Our method also provides an alternative and unified proof of the main results of [18–20].

Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms

In this paper, we establish stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces under general volume doubling condition. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cutoff Sobolev inequalities, and Poincar\'e inequalities. In particular, we establish the connection between parabolic Harnack inequalities and two-sided heat kernel estimates, as well as with the H\"older regularity of parabolic functions for symmetric non-local Dirichlet forms.

Heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms

In this paper, we consider the following symmetric Dirichlet forms on a metric measure space (M,d,μ):E(f,g)=E(c)(f,g)+∫M×M(f(x)−f(y))(g(x)−g(y))J(dx,dy), where E(c) is a strongly local symmetric bilinear form and J(dx,dy) is a symmetric Radon measure on M×M. Under general volume doubling condition on (M,d,μ) and some mild assumptions on scaling functions, we establish stability results for upper bounds of heat kernel (resp. two-sided heat kernel estimates) in terms of the jumping kernels, the cut-off Sobolev inequalities, and the Faber-Krahn inequalities (resp. the Poincaré inequalities). We also obtain characterizations of parabolic Harnack inequalities. Our results apply to symmetric diffusions with jumps even when the underlying spaces have walk dimensions larger than 2.

Schauder estimates for nonlocal kinetic equations and applications

In this paper we develop a new method based on Littlewood-Paley's decomposition and heat kernel estimates in integral form, to establish Schauder's estimate for the following degenerate nonlocal equation in R2d with Hölder coefficients:∂tu=Lκ;v(α)u+b⋅∇u+f,u0=0, where u=u(t,x,v) and Lκ;v(α) is a nonlocal α-stable-like operator with α∈(1,2) and kernel function κ, which acts on the variable v. As an application, we show the strong well-posedness to the following degenerate stochastic differential equation with Hölder drift b:dZt=b(t,Zt)dt+(0,σ(t,Zt)dLt(α)),Z0=(x,v)∈R2d, where Lt(α) is a d-dimensional rotationally invariant and symmetric α-stable process with α∈(1,2), and b:R+×R2d→R2d is a (γ,β)-order Hölder continuous function in (x,v) with γ∈(2+α2(1+α),1) and β∈(1−α2,1), σ:R+×R2d→Rd⊗Rd is a Lipschitz function. Moreover, we also show that for almost all ω, the following random transport equation has a unique Cb1-solution:∂tu(t,x,ω)+(b(t,x)+Lt(α)(ω))⋅∇xu(t,x,ω)=0,u(0,x)=φ(x), where φ∈Cb1(Rd) and b:R+×Rd→Rd is a bounded continuous function of (t,x) and γ-order Hölder continuous in x uniformly in t with γ∈(2+α2(1+α),1).

Sharp heat kernel estimates for spectral fractional Laplacian perturbed by gradients

Using Duhamel’s formula, we prove sharp two-sided estimates for the spectral fractional Laplacian’s heat kernel with time-dependent gradient perturbation in bounded C1,1 domains. In addition, we obtain a gradient estimate as well as the Holder continuity of the heat kernel’s gradient.