Researchers analyze hypoelliptic heat kernels on nilpotent Lie groups.
problem Analyzing hypoelliptic heat kernels on nilpotent Lie groups.
method Using generalized Fourier transform and Kirillov's orbit method to describe unitary irreducible representations and write hypoelliptic heat kernels.
result Explicit formula for hypoelliptic heat kernel on Gn. Study describes heat kernel expansion for hypoelliptic operators.
problem Characterize coefficients in small time heat kernel expansion.
method Geometric characterization of coefficients using drift field and curvature-like invariants.
result Geometric characterization of coefficients in terms of drift field and curvature-like invariants.
The paper analyzes hypoelliptic heat kernels near a manifold's cut locus.
problem Analyzing hypoelliptic heat kernels near a manifold's cut locus.
method Probabilistic approach using S. Watanabe's distributional Malliavin calculus and T. Lyons' rough path theory.
result Obtained a short time asymptotic expansion of hypoelliptic heat kernels up to any order.
Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.
problem Properties of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
method Construction and study of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
result Hypoelliptic logarithmic Sobolev inequalities on the space.
We study inequalities related to the heat kernel for the hypoelliptic sublaplacian on an H-type Lie group. Specifically, we obtain precise pointwise upper and lower bounds on the heat kernel function itself. We then apply these bounds to derive an estimate on the gradient of solutions of the heat equation, which is kno…
Researchers compute heat kernel coefficients for 2D diffusion operators.
problem Analyzing heat kernel coefficients for 2D hypoelliptic operators.
method Explicit computation of heat kernel coefficients and interpretation in terms of curvature.
result Interpretation of heat kernel asymptotics for non-sub-Riemannian operators.
This paper discusses the existence of gradient estimates for second order hypoelliptic heat kernels on manifolds. It is now standard that such inequalities, in the elliptic case, are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associated "Ricci curvature" ta…
This paper is part of a series papers devoted to geometric and spectral theoretic applications of the hypoelliptic calculus on Heisenberg manifolds. More specifically, in this paper we make use of the Heisenberg calculus of Beals-Greiner and Taylor to analyze the spectral theory of hypoelliptic operators on Heisenberg …
The book explores stochastic areas and heat kernels on manifolds.
problem Understanding stochastic area functionals and heat kernels on manifolds.
method Study of Brownian motions and heat kernels on Lie groups and Riemannian manifolds.
result Rich interactions between stochastic calculus, geometry, and random matrices.
Study on diffusion in non-complete sub-Riemannian manifolds with specific conditions.
problem Analyzing diffusion in incomplete sub-Riemannian manifolds.
method Identifying conditions for Gaussian-type upper bounds and logarithmic asymptotics of heat kernels.
result Optimal constant in exponent for Gaussian-type upper bounds and concentration of diffusion bridge measures.
Study spectral properties of sub-Riemannian Laplacians, proving quantum ergodicity and heat kernel asymptotics.
problem Spectral properties of sub-Riemannian Laplacians.
method Quantum ergodicity results, small-time asymptotics of sub-Riemannian heat kernels, Weyl law.
result Weyl law and spectral concentration on Lie brackets of length r-1.
The paper develops a heat kernel expansion for Rockland operators on filtered manifolds.
problem Analyzing heat kernel expansions for non-commutative geometries.
method Established a universal heat kernel expansion for Rockland operators on closed filtered manifolds using a new calculus.
result Implications of the heat expansion for complex powers, heat trace asymptotics, and eigenvalue asymptotics are generalized to this new calculus.
Formula for twisted orbital integrals using hypoelliptic Laplacian.
problem Evaluate equivariant trace of Laplacians on compact locally symmetric spaces.
method Using the hypoelliptic Laplacian method and twisted trace formula.
result Explicit geometric formula for twisted orbital integrals.
Density expansions for hypoelliptic diffusions (X1,...,Xd) are revisited. In particular, we are interested in density expansions of the projection (XT1,...,XTl), at time T>0, with l≤d. Global conditions are found which replace the well-known "not-in-cutlocus" condition known from heat-kernel asymptot…
Study spectral properties of sub-Laplacians in Carnot groups.
problem Spectral properties of sub-Laplacians in Carnot groups.
method Proved pure point spectrum and spectral gap; applied to small ball problem and heat content.
result Proved existence of spectral gap and pure point spectrum.
We prove a geometrically meaningful stochastic representation of the derivative of the heat semigroup on sub-Riemannian manifolds with tranverse symmetries. This representation is obtained from the study of Bochner-Weitzenbock type formulas for sub-Laplacians on 1-forms. As a consequence, we prove new hypoelliptic heat…
Short proof of heat kernel asymptotics and convolution approximation.
problem Short time asymptotics and heat kernel approximation for Laplace type operators.
method Short time asymptotic expansion and convolution approximation of heat kernels.
result Approximation of heat kernel using repeated convolutions.
Study Kleinian groups using orbital functions and heat kernels.
problem Understanding Kleinian groups through orbital functions and heat kernels.
method Using the heat kernel approach developed in \cite{artmoiheatcounting1}.
result Developed a method to study Kleinian groups using orbital functions and heat kernels.
The paper compares heat kernels on manifolds with Robin boundary conditions.
problem Comparing heat kernels on manifolds with different boundary conditions.
method Proving comparison theorems for heat kernels on geodesic balls and minimal submanifolds.
result Eigenvalue comparison theorem for the first Robin eigenvalues on minimal submanifolds.
The paper proves recurrence relations for heat kernels on hyperbolic and spherical spaces.
problem Understanding recurrence relations of heat kernels on different space forms.
method Direct proof and computation of recurrence relations for heat kernels on hyperbolic and spherical spaces.
result Computed diagonal of heat kernels for odd dimensional hyperbolic spaces and heat trace asymptotic expansions for odd dimensional spheres.
The heat kernel for the Cauchy-Riemann subLaplacian on S(2n+1) is derived in a manner which is completely analogous to the classical derivation of elliptic heat kernels. This suggests that the classical hamiltonian construction of elliptic heat kernels, with appropriate modifications, does yield heat kernels for subell…
The paper studies heat kernel asymptotics and proves Morse inequalities.
problem Analyzing the asymptotic behavior of heat kernels near critical points.
method Localization and scaling techniques in semi-classical analysis.
result The heat kernel near critical points is approximated by harmonic oscillator kernels, leading to Morse inequalities.
Formulae connect heat kernels on glued manifolds.
problem Connecting heat kernels on joined manifolds.
method Proved gluing formulae for Laplacian heat kernels.
result Formulae linking heat kernels on joined manifolds.
Study the heat kernel on quaternionic anti-de Sitter spaces and related spaces.
problem Understanding the heat kernel on quaternionic anti-de Sitter spaces and related spaces.
method Detailed study of the geometry, derivation of the horizontal Laplacian and subelliptic heat kernel formulas, derivation of small time asymptotics.
result Explicit formulas for the horizontal Laplacian and subelliptic heat kernel of the quaternionic anti-de Sitter fibration.
We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We show explicitly that the obtaine…
Proves upper bounds for heat kernels evolving on manifolds.
problem Bounding heat kernels on evolving manifolds.
method Logarithmic Sobolev inequalities and ultracontractivity estimates.
result Gaussian upper bounds for heat kernels are derived.
The study examines heat kernel bounds for manifolds with Ricci curvature in the Kato class.
problem Heat kernel estimates for manifolds with Ricci curvature in the Kato class.
method Kato conditions on the negative part of the Ricci curvature.
result Recent results on heat kernel estimates.
Paper presents exact heat kernel on hypersphere for SVM improvements.
problem Improving SVM performance with non-Euclidean feature spaces.
method Higher order parametrix expansion of hyperspherical heat kernel.
result Exact kernel often shows superior performance in SVM applications.
From the uniformization theorem, we know that every Riemann surface has a simply-connected covering space. Moreover, there are only three simply-connected Riemann surfaces: the sphere, the Euclidean plane, and the hyperbolic plane. In this paper, we collect the known heat kernels, or Green's functions, for these three …
In a 1991 paper by Buttig and Eichhorn, the existence and uniqueness of a differential forms heat kernel on open manifolds of bounded geometry was proven. In that paper, it was shown that the heat kernel obeyed certain properties, one of which was a relationship between the derivative of heat kernel of different degree…
Survey on heat kernels and path integrals.
problem Approximating Wiener measure on compact manifolds.
method Review of recent results on approximating Wiener measure.
result Approximation of Wiener measure by measures on spaces of piece-wise geodesics.
Heat kernel estimates on manifolds with mixed boundary conditions.
problem Estimating heat kernels on manifolds with ends and mixed boundary conditions.
method Global harmonic function construction and h-transform technique. result Two-sided heat kernel estimates for Riemannian manifolds with mixed boundary conditions.
Proves heat kernel superconvexity in hyperbolic space.
problem Heat kernel superconvexity in hyperbolic space.
method Proves conjecture by Bernstein in all dimensions.
result Analog of Huisken's monotonicity formula for mean curvature flow.
Proposes learning manifold implicitly via heat kernel.
problem Direct manifold learning methods lack flexibility for down-stream applications.
method Implicit manifold learning using heat kernel.
result Framework achieves state-of-the-art results for data generation and Bayesian inference.
Derives properties of heat kernel for Rumin complex on Heisenberg groups.
problem Analyzing heat kernel properties for Rumin complex.
method Derives properties of heat equation with Hodge operator on Heisenberg groups.
result Constructs Calderón reproducing formula using heat kernel for Rumin forms.
The paper studies local heat kernel properties on smooth manifolds.
problem Understanding heat kernel properties in open convex sets of smooth Riemannian manifolds.
method Utilizes path integral formulation to investigate properties like uniqueness, symmetry, and asymptotics.
result Uniqueness and symmetry of Seeley-DeWitt coefficients are established.
In this paper we give Hamilton's Laplacian estimates for the heat equation on complete noncompact manifolds with nonnegative Ricci curvature. As an application, combining Li-Yau's lower and upper bounds of the heat kernel, we give an estimate on Laplacian form of the heat kernel on complete manifolds with nonnegative R…
The paper studies heat kernel behavior in RCD spaces and initiates Weyl's law study.
problem Understanding heat kernel behavior in RCD spaces.
method Proved pointwise convergence of heat kernels for mGH-convergent sequences of RCD spaces.
result Initiated study of Weyl's law in RCD spaces.
The paper studies heat kernel asymptotics for Kohn Laplacians on CR manifolds.
problem Analyzing heat kernel asymptotics for Kohn Laplacians on CR manifolds.
method Establishing asymptotics of heat kernels and equivariant heat kernels on CR manifolds.
result Heat kernel asymptotics for Kohn Laplacians on CR manifolds are derived.
Heat kernel resurgent structure from Picard-Lefschetz theory
problem Short-time heat kernel asymptotics
method Picard-Lefschetz theory
result 1-Gevrey small-time expansion
The paper develops heat kernel comparison theorems and applies them to spectral geometry.
problem Developing mathematical tools for spectral geometry.
method Established weighted heat kernel comparison theorems for manifolds with bounded radial curvatures.
result Two eigenvalue comparison theorems for the first Dirichlet eigenvalue of the Witten-Laplacian.
Proves metric spaces with Euclidean heat kernel are isometric to Euclidean space.
problem Characterizing metric measure spaces with specific heat kernels.
method Analyzes Dirichlet forms and heat kernels to prove rigidity.
result Metric measure spaces with Euclidean heat kernel are isometric to Euclidean space.
In this paper, we study the large time behavior of the heat kernel on complete Riemannian manifolds with nonnegative Ricci curvature, which was studied by P. Li with additional maximum volume growth assumption. Following Y. Ding's original strategy, by blowing down the metric, using Cheeger and Colding's theory about l…
The paper provides gradient estimates for heat kernels on manifolds with negative Ricci curvature.
problem Estimating gradients of heat kernels on manifolds with negative Ricci curvature.
method Pointwise and Lp gradient estimates, uniform boundedness results for the heat operator of the Hodge Laplacian. result Uniform boundedness results and gradient estimates for heat kernels and Hodge Laplacian.
We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We argue that the obtained formal s…
Estimates heat kernel gradients on fractal-like cable systems.
problem Bounding gradients of heat kernels on complex fractal structures.
method Pointwise upper estimates for heat kernel gradients.
result Derives Lp-boundedness of quasi-Riesz transforms. Sharp heat kernel estimate on graphs proved.
problem Estimating heat kernels on graphs.
method Proved sharp Davies-Gaffney-Grigor'yan lemma.
result Sharp estimate of heat kernels on graphs.
Sharp gradient estimate for heat kernels on metric measure spaces.
problem Establishing gradient estimates for heat kernels on metric measure spaces.
method Elliptic local Li-Yau gradient estimate for weak solutions of the heat equation.
result Sharp gradient estimate for the logarithm of heat kernels.