Heat Kernel Research Articles

Nonlocal scalar fields in static spacetimes via heat kernels

We solve the non-local equation ${-e^{-\ell^2\square}\square{\phi}=J}$ for i) static scalar fields in static spacetimes and ii) time-dependent scalar fields in ultrastatic spacetimes. Corresponding equations are rewritten as non-local Poisson/inhomogeneous Helmholtz equations in compact and non-compact weighted/Riemannian manifolds using static/frequency-domain Green's functions, which can be computed from the heat kernels in the respective manifolds. With the help of the heat kernel estimates, we derive the static Green's function estimates and use them to discuss the regularity. We also present several examples of exact and estimated static/frequency-domain Green's functions.

3D Shape Analysis Through a Quantum Lens: the Average Mixing Kernel Signature

The Average Mixing Kernel Signature is a novel spectral signature for points on non-rigid three-dimensional shapes. It is based on a quantum exploration process of the shape surface, where the average transition probabilities between the points of the shape are summarised in the finite-time average mixing kernel. A band-filtered spectral analysis of this kernel then yields the AMKS. Crucially, we show that opting for a finite time-evolution allows the signature to account for a mixing of the Laplacian eigenspaces, similar to what is observed in the presence of noise, explaining the increased noise robustness of this signature when compared to alternative signatures. We perform an extensive experimental analysis of the AMKS under a wide range of problem scenarios, evaluating the performance of our descriptor under different sources of noise (vertex jitter and topological), shape representations (mesh and point clouds), as well as when only a partial view of the shape is available. Our experiments show that the AMKS consistently outperforms two of the most widely used spectral signatures, the Heat Kernel Signature and the Wave Kernel Signature, and suggest that the AMKS should be the signature of choice for various compute vision problems, including as input of deep convolutional architectures for shape analysis.

Logarithmic Sobolev inequalities on non-isotropic Heisenberg groups

We study logarithmic Sobolev inequalities with respect to a heat kernel measure on finite-dimensional and infinite-dimensional Heisenberg groups. Such a group is the simplest non-trivial example of a sub-Riemannian manifold. First we consider logarithmic Sobolev inequalities on non-isotropic Heisenberg groups. These inequalities are considered with respect to the hypoelliptic heat kernel measure, and we show that the logarithmic Sobolev constants can be chosen to be independent of the dimension of the underlying space. In this setting, a natural Laplacian is not an elliptic but a hypoelliptic operator. The argument relies on comparing logarithmic Sobolev constants for the three-dimensional non-isotropic and isotropic Heisenberg groups, and tensorization of logarithmic Sobolev inequalities in the sub-Riemannian setting. Furthermore, we apply these results in an infinite-dimensional setting and prove a logarithmic Sobolev inequality on an infinite-dimensional Heisenberg group modeled on an abstract Wiener space.

On the Biharmonic Heat Equation on Complete Riemannian Manifolds

We study entire solutions of the biharmonic heat equation on complete Riemannian manifolds without boundary. We provide exponential decay estimates for the biharmonic heat kernel under assumptions on the lower bound of Ricci curvature and noncollapsing of unit balls. And we prove a uniqueness criteria for the Cauchy problem. As corollaries, we prove the conservation law for the biharmonic heat kernel and a uniform \(L^\infty \) estimate for entire solutions starting with bounded initial data.

Spectral geometry on manifolds with fibred boundary metrics II: heat kernel asymptotics

In this paper we continue with the analysis of spectral problems in the setting of complete manifolds with fibred boundary metrics, also referred to as phi -metrics, as initiated in our previous work (Grieser et al. in Spectral geometry on manifolds with fibred boundary metrics I: Low energy resolvent, 2020). We consider the Hodge Laplacian for a phi -metric and construct the corresponding heat kernel as a polyhomogeneous conormal distribution on an appropriate manifold with corners. Our discussion is a generalization of an earlier work by Albin and Sher, and provides a fundamental first step towards analysis of Ray–Singer torsion, eta-invariants and index theorems in the setting.

Heat kernel expansion for higher order minimal and nonminimal operators

We build a systematic calculational method for the covariant expansion of the two-point heat kernel $\hat K(\tau|x,x')$ for generic minimal and non-minimal differential operators of any order. This is the expansion in powers of dimensional background field objects -- the coefficients of the operator and the corresponding spacetime and vector bundle curvatures, suitable in renormalization and effective field theory applications. For minimal operators whose principal symbol is given by an arbitrary power of the covariant Laplacian $(-\Box)^M$, $M>1$, this result generalizes the well-known Schwinger--DeWitt (or Seeley--Gilkey) expansion to the infinite series of positive and negative fractional powers of the proper time $\tau^{1/M}$, weighted by the generalized exponential functions of the dimensionless argument $-\sigma(x,x')/2\tau^{1/M}$ depending on the Synge world function $\sigma(x,x')$. The coefficients of this series are determined by the chain of auxiliary differential operators acting on the two-point parallel transport tensor, which in their turn follow from the solution of special recursive equations. The derivation of these operators and their recursive equations are based on the covariant Fourier transform in curved spacetime. The series of negative fractional powers of $\tau$ vanishes in the coincidence limit $x'=x$, which makes the proposed method consistent with the heat kernel theory of Seeley--Gilkey and generalizes it beyond the heat kernel diagonal in the form of the asymptotic expansion in the domain $\sigma(x,x')\ll\tau^{1/M}$, $\tau\to 0$.$\nabla^a\sigma(x,x')\ll\tau^{1/2M}$, $\tau\to 0$.

Bismut-Stroock Hessian formulas and local Hessian estimates for heat semigroups and harmonic functions on Riemannian manifolds

In this article, we develop a martingale approach to localized Bismut-type Hessian formulas for heat semigroups on Riemannian manifolds. Our approach extends the Hessian formulas established by Stroock (An estimate on the Hessian of the heat kernel: 355–371, 1996) and removes in particular the compact manifold restriction. To demonstrate the potential of these formulas, we give as application explicit quantitative local estimates for the Hessian of the heat semigroup, as well as for harmonic functions on regular domains in Riemannian manifolds.

Adaptive stochastic morphology simulation and mesh generation of high-quality 3D particulate composite microstructures with complex surface texture

Particulate composite materials have a broad range of potential applications in engineering and other disciplines. Accurate modeling of their microstructures and fast generation of the finite element meshes play a vital role in investigating many micromechanical phenomena and improving understanding of the underlying failure mechanisms. Due to the exceedingly intricate multiscale internal structures that they possess, the modeling and meshing of their microstructures still remain difficult in general. In this work, we present a computational framework and methodology for the representation, simulation, and mesh generation of 3D stochastic microstructures of particulate composites. Towards this goal, we propose a multi-level multiscale scheme that allows for capturing the multiscale structures of particulate composite materials at both the coarse and fine scales. A briging scale approach based on heat kernel smoothing is also presented to seamlessly link the coarse and fine scales. In addition to the microstructural modeling of particulate composite materials, we also develop an adaptive curvature-based surface and volume mesh generation algorithm for particulate composite microstructures with complex surface texture. Following the implementation of the morphology and mesh generation algorithm, a series of numerical examples are presented to demonstrate the capability and potential of the proposed method.

Littlewood–Paley Characterization for Musielak–Orlicz–Hardy Spaces Associated with Self-Adjoint Operators

Let X , d , μ be a metric measure space endowed with a metric d and a non-negative Borel doubling measure μ . Let L be a non-negative self-adjoint operator on L 2 X . Assume that the (heat) kernel associated to the semigroup e − t L satisfies a Gaussian upper bound. In this paper, we prove that the Musielak–Orlicz–Hardy space H φ , L X associated with L in terms of the Lusin-area function and the Musielak–Orlicz–Hardy space H L , G , φ X associated with L in terms of the Littlewood–Paley function coincide and their norms are equivalent. To do this, we first establish the discrete characterization of these two spaces. It improves the known results in the literature.

Diffusion in small time in incomplete sub-Riemannian manifolds

For incomplete sub-Riemannian manifolds, and for an associated second-order hypoelliptic operator, which need not be symmetric, we identify two alternative conditions for the validity of Gaussian-type upper bounds on heat kernels and transition probabilities, with optimal constant in the exponent. Under similar conditions, we obtain the small-time logarithmic asymptotics of the heat kernel, and show concentration of diffusion bridge measures near a path of minimal energy. The first condition requires that we consider points whose distance apart is no greater than the sum of their distances to infinity. The second condition requires only that the operator not be too asymmetric.

On some generalizations of non-archimedean pseudo-differential operators and their applications

In this article we define and study various generalizations non-archimedean pseudo-differential operators and its connections with probability theory. We provide explicit and sufficient conditions to ensure that these operators can satisfy the positive maximum principle, the heat Kernel or fundamental solution of the Cauchy problem (or heat equation) naturally associated with these operators determine a Feller semigroup and a transition function of some strong Markov processes X with state space the p-adic numbers. Also, we will study the property of Conservation of mass, the Comparison principle and some aspects of these processes including the first passage time problem and the probability of survival. These equations are proposed as mathematical models (connected with energy landscapes) to study the spread of an infectious or contagious disease that takes into account social clusters in a situation of extreme social isolation.

FKN Formula and Ground State Energy for the Spin Boson Model with External Magnetic Field

We consider the spin boson model with external magnetic field. We prove a path integral formula for the heat kernel, known as Feynman–Kac–Nelson (FKN) formula. We use this path integral representation to express the ground state energy as a stochastic integral. Based on this connection, we determine the expansion coefficients of the ground state energy with respect to the magnetic field strength and express them in terms of correlation functions of a continuous Ising model. From a recently proven correlation inequality, we can then deduce that the second order derivative is finite. As an application, we show existence of ground states in infrared-singular situations.

Heat kernel of supercritical nonlocal operators with unbounded drifts

Heat kernel of supercritical nonlocal operators with unbounded drifts

Temporal-spatial decays for the Stokes flows in half spaces

In this paper, we prove new estimates on temporal-spatial decays in L^1 for solutions to the Stokes equations in the half spaces. Our approach is based on the Ukai’s solution formula of the Stokes equations and heat kernel estimates.

Scattering Theory for the Hodge Laplacian

We prove using an integral criterion the existence and completeness of the wave operators \(W_{\pm }(\Delta _h^{(k)}, \Delta _g^{(k)}, I_{g,h}^{(k)})\) corresponding to the Hodge Laplacians \(\Delta _\nu ^{(k)}\) acting on differential \(k\)-forms, for \(\nu \in \{g,h\}\), induced by two quasi-isometric Riemannian metrics g and h on a complete open smooth manifold M. In particular, this result provides a criterion for the absolutely continuous spectra \(\sigma _{\mathrm {ac}}(\Delta _g^{(k)}) = \sigma _{\mathrm {ac}}(\Delta _h^{(k)})\) of \(\Delta _\nu ^{(k)}\) to coincide. The proof is based on gradient estimates obtained by probabilistic Bismut-type formulae for the heat semigroup defined by spectral calculus. By these localised formulae, the integral criterion requires local curvature bounds and some upper local control on the heat kernel acting on functions provided the Weitzenböck curvature endomorphism is in the Kato class, but no control on the injectivity radii. A consequence is a stability result of the absolutely continuous spectrum under a Ricci flow. As an application we concentrate on the important case of conformal perturbations.

Estimates on fundamental solutions of parabolic magnetic Schrödinger operators and uniform parabolic equations with nonnegative potentials and their applications

We study the fundamental solutions of parabolic magnetic Schrödinger operators and uniform parabolic operators with nonnegative potentials in the reverse Hölder class. The main aim of the paper is to give pointwise estimates of the heat kernel of the operators above, which improve and generalize the main results by Kurata [J. London Math. Soc. (2) 62 (2000), pp. 885–903].

A note on spectral multipliers on Engel and Cartan groups

The aim of this short note is to give examples of L p L^p - L q L^q bounded spectral multipliers for operators involving left-invariant vector fields and their inverses, in the settings of Engel and Cartan groups. The interest in such examples lies in the fact that a group does not have to have flat co-adjoint orbits, and that the considered operator is not related to the usual sub-Laplacian. The discussed examples show how one can still obtain L p L^p - L q L^q estimates for similar operators in such settings. As immediate consequences, one gets the corresponding Sobolev-type inequalities and heat kernel estimates.

Regularity of Fractional Heat Semigroup Associated with Schrödinger Operators

Let L=−Δ+V be a Schrödinger operator, where the potential V belongs to the reverse Hölder class. By the subordinative formula, we introduce the fractional heat semigroup {e−tLα}t>0, 0<α<1, associated with L. By the aid of the fundamental solution of the heat equation: ∂tu+Lu=∂tu−Δu+Vu=0, we estimate the gradient and the time-fractional derivatives of the fractional heat kernel Kα,tL(·,·), respectively. This method is independent of the Fourier transform, and can be applied to the second-order differential operators whose heat kernels satisfy the Gaussian upper bounds. As an application, we establish a Carleson measure characterization of the Campanato-type space BMOLγ(Rn) via the fractional heat semigroup {e−tLα}t>0.

Convergence rate of the EM algorithm for SDEs with low regular drifts

AbstractIn this paper we employ a Gaussian-type heat kernel estimate to establish Krylov’s estimate and Khasminskii’s estimate for the Euler–Maruyama (EM) algorithm. For applications, by taking Zvonkin’s transformation into account, we investigate the convergence rate of the EM algorithm for a class of multidimensional stochastic differential equations (SDEs) with low regular drifts, which need not be piecewise Lipschitz.

Heat kernel estimates for subordinate Markov processes and their applications

In this paper, we establish sharp two-sided estimates for transition densities of a large class of subordinate Markov processes. As applications, we show that the parabolic Harnack inequality and Hölder regularity hold for parabolic functions of such processes, and derive sharp two-sided Green function estimates.