Heat Kernel Estimates Research Articles

Conformally invariant random fields, Liouville quantum gravity measures, and random Paneitz operators on Riemannian manifolds of even dimension

AbstractFor large classes of even‐dimensional Riemannian manifolds , we construct and analyze conformally invariant random fields. These centered Gaussian fields , called co‐polyharmonic Gaussian fields, are characterized by their covariance kernels k which exhibit a precise logarithmic divergence: . They share a fundamental quasi‐invariance property under conformal transformations. In terms of the co‐polyharmonic Gaussian field , we define the Liouville Quantum Gravity measure, a random measure on , heuristically given as and rigorously obtained as almost sure weak limit of the right‐hand side with replaced by suitable regular approximations . In terms on the Liouville Quantum Gravity measure, we define the Liouville Brownian motion on and the random GJMS operators. Finally, we present an approach to a conformal field theory in arbitrary even dimension with an ansatz based on Branson's ‐curvature: we give a rigorous meaning to the Polyakov–Liouville measure and we derive the corresponding conformal anomaly. The set of admissible manifolds is conformally invariant. It includes all compact 2‐dimensional Riemannian manifolds, all compact non‐negatively curved Einstein manifolds of even dimension, and large classes of compact hyperbolic manifolds of even dimension. However, not every compact even‐dimensional Riemannian manifold is admissible. Our results concerning the logarithmic divergence of the kernel rely on new sharp estimates for heat kernels and higher order Green kernels on arbitrary closed manifolds.

On a nonhomogeneous heat equation on the complex plane

In this paper, we investigate the existence, uniqueness, and asymptotic behaviors of mild solutions of parabolic evolution equations on complex plane, in which the diffusion operator has the form [Formula: see text], where [Formula: see text], the function [Formula: see text] is smooth and subharmonic on [Formula: see text], and [Formula: see text] is the formal adjoint of [Formula: see text]. Our method combines certain estimates of heat kernel associating with the homogeneous linear equation of Raich [Heat equations in [Formula: see text], J. Funct. Anal. 240(1) (2006) 1–35] and a fixed point argument.

Heat kernel estimates on local and non-local Dirichlet spaces satisfying a weak chain condition

In this paper, we focus on the heat kernel estimates for diffusions and jump processes on metric measure spaces satisfying a weak chain condition, where the length of a nearly shortest ε-chain between two points x,y is comparable with a function of d(x,y) and ε. For a diffusion, the best estimate is already given by Grigor'yan and Telcs, and we make it explicit in our particular case. For jump processes, especially those where the scale of the process is different with that of the jump kernel, we improve the results by Bae, Kang, Kim and Lee. Uniformity of the coefficients (or parameters) in the known estimates and metric transforms play the key role in our proof. We also show by examples how the weak chain condition is valid in practice.

Two-sided heat kernel estimates for Schrödinger operators with unbounded potentials

Two-sided heat kernel estimates for Schrödinger operators with unbounded potentials

Heat kernel fluctuations and quantitative homogenization for the one-dimensional Bouchaud trap model

We present on-diagonal heat kernel estimates and quantitative homogenization statements for the one-dimensional Bouchaud trap model. The heat kernel estimates are obtained using standard techniques, with key inputs coming from a careful analysis of the volume growth of the invariant measure of the process under study. As for the quantitative homogenization results, these include both quenched and annealed Berry–Esseen-type theorems, as well as a quantitative quenched local limit theorem. Whilst the model we study here is a particularly simple example of a random walk in a random environment, we believe the roadmap we provide for establishing the latter result in particular will be useful for deriving quantitative local limit theorems in other, more challenging, settings.

Asymptotic Log-Harnack inequality for the 2D Ericksen–Leslie model system with degenerate noise and damping

In this paper, we consider a 2D liquid crystal model with damping and examine some asymptotic behaviors of the strong solution. More precisely, we establish the asymptotic log-Harnack inequality for the transition semigroup associated with a simplified liquid crystal model driven by an additive degenerate noise via the asymptotic coupling method. As consequences of the asymptotic log-Harnack inequality, we derive the gradient estimate, the asymptotic irreducibility, the asymptotic strong Feller property, the asymptotic heat kernel estimate and the ergodicity.

Heat Kernel Estimates for Stable-driven SDEs with Distributional Drift

Heat Kernel Estimates for Stable-driven SDEs with Distributional Drift

Uniform complex time heat Kernel estimates without Gaussian bounds

Abstract The aim of this article is twofold. First, we study the uniform complex time heat kernel estimates of e − z ( − Δ ) α 2 {e}^{-z{\left(-\Delta )}^{\frac{\alpha }{2}}} for α > 0 , z ∈ C + \alpha \gt 0,z\in {{\mathbb{C}}}^{+} . To this end, we establish the asymptotic estimates for P ( z , x ) P\left(z,x) with z z satisfying 0 < ω ≤ ∣ θ ∣ < π 2 0\lt \omega \le | \theta | \lt \frac{\pi }{2} followed by the uniform complex time heat kernel estimates. Second, we studied the uniform complex time estimates of the analytic semigroup generated by H = ( − Δ ) α 2 + V H={\left(-\Delta )}^{\tfrac{\alpha }{2}}+V , where V V belongs to higher-order Kato class.

Bulk Behaviour of Ground States for Relativistic Schrödinger Operators with Compactly Supported Potentials

We propose a probabilistic representation of the ground states of massive and massless Schrödinger operators with a potential well in which the behaviour inside the well is described in terms of the moment-generating function of the first exit time from the well and the outside behaviour in terms of the Laplace transform of the first entrance time into the well. This allows an analysis of their behaviour at short to mid-range from the origin. In a first part, we derive precise estimates on these two functionals for stable and relativistic stable processes. Next, by combining scaling properties and heat kernel estimates, we derive explicit local rates of the ground states of the given family of non-local Schrödinger operators both inside and outside the well. We also show how this approach extends to fully supported decaying potentials. By an analysis close-by to the edge of the potential well, we furthermore show that the ground state changes regularity, which depends qualitatively on the fractional power of the non-local operator.

Heat kernel estimate for the Laplace-Beltrami operator under Bakry-Émery Ricci curvature condition and applications

We establish a Gaussian upper bound of the heat kernel for the Laplace-Beltrami operator on complete Riemannian manifolds with Bakry-Émery Ricci curvature bounded below. As applications, we first prove an L1-Liouville property for non-negative subharmonic functions when the potential function of the Bakry-Émery Ricci curvature tensor is of at most quadratic growth. Then we derive lower bounds of the eigenvalues of the Laplace-Beltrami operator on closed manifolds. An upper bound of the bottom spectrum is also obtained.

Heat and Martin Kernel estimates for Schrödinger operators with critical Hardy potentials

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Ergodicity of supercritical SDEs driven by α-stable processes and heavy-tailed sampling

Let α∈(0,2) and d∈N. We consider the stochastic differential equation (SDE) driven by an α-stable process dXt=b(Xt)dt+σ(Xt−)dLtα,X0=x∈Rd, where b: Rd→Rd and σ: Rd→Rd⊗Rd are locally γ-Hölder continuous with γ∈(0∨(1−α)+,1], and Ltα is a d-dimensional symmetric rotationally invariant α-stable process. Under certain dissipative and non-degenerate assumptions on b and σ, we show the V-uniformly exponential ergodicity for the semigroup Pt associated with {Xt(x),t≥0}. Our proofs are mainly based on the heat kernel estimates recently established in (J. Éc. Polytech. Math. 9 (2022) 537–579) to demonstrate the strong Feller property and irreducibility of Pt. Interestingly, when α tends to zero, the diffusion coefficient σ can increase faster than the drift b. As an application, we put forward a new heavy-tailed sampling scheme.

Bound States and Heat Kernels for Fractional-Type Schrödinger Operators with Singular Potentials

We consider non-local Schrödinger operators H=-L-V in L^2(mathbb {R}^d), d geqslant 1, where the kinetic terms L are pseudo-differential operators which are perturbations of the fractional Laplacian by bounded non-local operators and V is the fractional Hardy potential. We prove pointwise estimates of eigenfunctions corresponding to negative eigenvalues and upper finite-time horizon estimates for heat kernels. We also analyze the relation between the matching lower estimates of the heat kernel and the ground state near the origin. Our results cover the relativistic Schrödinger operator with Coulomb potential.

New Atomic Decomposition for Besov Type Space dot{B}^0_{1,1} Associated with Schrödinger Type Operators

Let (X, d, mu ) be a space of homogeneous type. Let L be a nonnegative self-adjoint operator on L^2(X) satisfying certain conditions on the heat kernel estimates which are motivated from the heat kernel of the Schrödinger operator on mathbb {R}^n. The main aim of this paper is to prove a new atomic decomposition for the Besov space dot{B}^{0, L}_{1,1}(X) associated with the operator L. As a consequence, we prove the boundedness of the Riesz transform associated with L on the Besov space dot{B}^{0, L}_{1,1}(X).

Equivalence of Besov spaces on p.c.f. self-similar sets

AbstractOn post-critically finite self-similar sets, whose walk dimensions of diffusions are in general larger than 2, we find a sharp region where two classes of Besov spaces, the heat Besov spaces $B^{p,q}_\sigma (K)$ and the Lipschitz–Besov spaces $\Lambda ^{p,q}_\sigma (K)$ , are identical. In particular, we provide concrete examples that $B^{p,q}_\sigma (K)=\Lambda ^{p,q}_\sigma (K)$ with $\sigma&gt;1$ . Our method is purely analytical, and does not involve heat kernel estimate.

Sharp estimates for W-invariant Dunkl and heat kernels in the An case

In this article, we prove exact estimates for the W-invariant Dunkl kernel and heat kernel, for the root system of type A with arbitrary positive multiplicities. We apply the estimates of the W-invariant Dunkl heat kernel to compute sharp estimates for the Newton kernel and for the s-stable semigroups generated by a fractional power of the W-invariant Dunkl Laplacian.

Ricci Curvature on Birth-Death Processes

In this paper, we study curvature dimension conditions on birth-death processes which correspond to linear graphs, i.e., weighted graphs supported on the infinite line or the half line. We give a combinatorial characterization of Bakry and Émery’s CD(K,n) condition for linear graphs and prove the triviality of edge weights for every linear graph supported on the infinite line Z with non-negative curvature. Moreover, we show that linear graphs with curvature decaying not faster than −R2 are stochastically complete. We deduce a type of Bishop-Gromov comparison theorem for normalized linear graphs. For normalized linear graphs with non-negative curvature, we obtain the volume doubling property and the Poincaré inequality, which yield Gaussian heat kernel estimates and parabolic Harnack inequality by Delmotte’s result. As applications, we generalize the volume growth and stochastic completeness properties to weakly spherically symmetric graphs. Furthermore, we give examples of infinite graphs with a positive lower curvature bound.

Poincaré constant on manifolds with ends

AbstractWe obtain optimal estimates of the Poincaré constant of central balls on manifolds with finitely many ends. Surprisingly enough, the Poincaré constant is determined by the second largest end. The proof is based on the argument by Kusuoka–Stroock where the heat kernel estimates on the central balls play an essential role. For this purpose, we extend earlier heat kernel estimates obtained by the authors to a larger class of parabolic manifolds with ends.

Brownian motion on the golden ratio Sierpinski gasket

We construct a strongly local regular Dirichlet form on the golden ratio Sierpinski gasket, which is a self-similar set without a finitely ramified cell structure, via a study on the trace of an electrical network on an infinite graph. The Dirichlet form is the unique one that is self-similar in the sense of an infinite iterated function system, and is decimation invariant with respect to a graph-directed construction. The proof is based on a fixed point problem of a renormalization map, inspired by Sabot's celebrated work for finitely ramified fractals. Lastly, the Hunt process associated with the Dirichlet form satisfies a two-sided sub-Gaussian heat kernel estimate.

Neumann boundary value problems for elliptic operators with measure-valued coefficients

In this paper, we prove that there exists a unique weak solution to the Neumann boundary value problem for second order elliptic operators whose coefficients are signed measures. We will give a probabilistic representation of the solution. The heat kernel estimates and reflected diffusion processes play important roles.