Related papers: Logarithmic heat kernels estimates without curvatu…

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…

The recent analysis on noncommutative geometry, showing quantization of the volume for the Riemannian manifold entering the geometry, can support a view of quantum mechanics as arising by a stochastic process on it. A class of stochastic…

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…

We propose a novel derivation of the non-local heat kernel expansion, first studied by Barvinsky, Vilkovisky and Avramidi, based on simple diagrammatic equations satisfied by the heat kernel. For Laplace-type differential operators we…

In this paper we provide the small-time heat kernel asymptotics at the cut locus in three relevant cases: generic low-dimensional Riemannian manifolds, generic 3D contact sub-Riemannian manifolds (close to the starting point) and generic 4D…

We consider a class of constant-coefficient partial differential operators on a finite-dimensional real vector space which exhibit a natural dilation invariance. Typically, these operators are anisotropic, allowing for different degrees in…

Using our recently proposed covariant algebraic approach the heat kernel for a Laplace-like differential operator in low-energy approximation is studied. Neglecting all the covariant derivatives of the gauge field strength (Yang-Mills…

In the first part of this paper, we get new Li-Yau type gradient estimates for positive solutions of heat equation on Riemmannian manifolds with $Ricci(M)\ge -k$, $k\in \mathbb R$. As applications, several parabolic Harnack inequalities are…

In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group. The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk. Here the non…

We study pointwise and $L^p$ gradient estimates of the heat kernel, on manifolds that may have some amount of negative Ricci curvature, provided it is not too negative (in an integral sense) at infinity. We also prove uniform boundedness…

In this article, we establish Gaussian decay for the Box_b-heat kernel on polynomial models in C^2. Our technique attains the exponential decay via a partial Fourier transform. On the transform side, the problem becomes finding quantitative…

Computer algebra methods are applied to investigation of spectral asymptotics of elliptic differential operators on curved manifolds with torsion and in the presence of a gauge field. In this paper we present complete expressions for the…

This paper introduces a novel, non-deterministic method for embedding data in low-dimensional Euclidean space based on computing realizations of a Gaussian process depending on the geometry of the data. This type of embedding first appeared…

For $d\geq 2$, we establish the existence and uniqueness of heat kernels for a large class of time-dependent second order diffusion operator with jumps, which is the sum of time-dependent of a second order elliptic differential operators…

In this paper, we study the heat kernel associated to the intrinsic sublaplacian on a quaternionic contact manifold considered as a subriemannian manifold. More precisely, we explicitly compute the first two coefficients $c_0$ and $c_1$…

The trace of the heat kernel and the one-loop effective action for the generic differential operator are calculated to third order in the background curvatures: the Riemann curvature, the commutator curvature and the potential. In the case…

This paper presents a unified geometric framework for Brownian motion on manifolds, encompassing intrinsic Riemannian manifolds, embedded submanifolds, and Lie groups. The approach constructs the stochastic differential equation by…

Gaussian processes provide a flexible, non-parametric framework for the approximation of functions in high-dimensional spaces. The covariance kernel is the main engine of Gaussian processes, incorporating correlations that underpin the…

In this paper, we employ probabilistic techniques to derive sharp, explicit two-sided estimates for the heat kernel of the nonlocal kinetic operator $$ \Delta^{\alpha/2}_v + v \cdot \nabla_x, \quad \alpha \in (0, 2),\ (x,v)\in {\mathbb…

This paper proposes a novel framework for manifold-valued regression and establishes its consistency as well as its contraction rate. It assumes a predictor with values in the interval $[0,1]$ and response with values in a compact…