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The aim of this paper is to construct (explicit) heat kernels for some hybrid evolution equations which arise in physics, conformal geometry and subelliptic PDEs. Hybrid means that the relevant partial differential operator appears in the…
We consider heat kernels on Weyl chambers corresponding to Laplacians subject to mixed Dirichlet-Neumann boundary conditions imposed on the boundary. Using purely analytic tools we prove genuinely sharp two-sided global estimates in the…
We explicitly construct parametrices for magnetic Schr\"odinger operators on R^d and prove that they provide a complete small-t expansion for the corresponding heat kernel, both on and off the diagonal.
A diagramatic heat kernel expansion technique is presented. The method is especially well suited to the small-derivative expansion of the heat kernel, but it can also be used to reproduce the results obtained by the approach known as…
Let $\displaystyle L = -\frac{1}{w} \, \mathrm{div}(A \, \nabla u) + \mu$ be the generalized degenerate Schr\"odinger operator in $L^2_w(\mathbb{R}^d)$ with $d\ge 3$ with suitable weight $w$ and measure $\mu$. The main aim of this paper is…
A functorial derivation is presented of a heat-kernel expansion coefficient on a manifold with a singular fixed point set of codimension two. The existence of an extrinsic curvature term is pointed out.
The worldline formalism has in recent years emerged as a powerful tool for the computation of effective actions and heat kernels. However, implementing nontrivial boundary conditions in this formalism has turned out to be a difficult…
We point out that using the heat kernel on a cone to compute the first quantum correction to the entropy of Rindler space does not yield the correct temperature dependence. In order to obtain the physics at arbitrary temperature one must…
Using the technique of labeled operators, compact explicit expressions are given for all traced heat kernel coefficients containing zero, two, four and six covariant derivatives, and for diagonal coefficients with zero, two and four…
Complex fractal dimensions, defined as poles of appropriate fractal zeta functions, describe the geometric oscillations in fractal sets. In this work, we show that the same possible complex dimensions in the geometric setting also govern…
This paper provides sharp Dirichlet heat kernel estimates in inner uniform domains, including bounded inner uniform domains, in the context of certain (possibly non-symmetric) bilinear forms resembling Dirichlet forms. For instance, the…
We define measures on central extension of current groups in any dimension by using infinite dimensional Brownian motion.
Many contemporary statistical learning methods assume a Euclidean feature space. This paper presents a method for defining similarity based on hyperspherical geometry and shows that it often improves the performance of support vector…
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 $\varepsilon$-chain between two points $x,y$ is…
We provide sharp two-sided estimates of the heat kernel of the Dirichlet fractional Laplacian on the half-line perturbed by the Hardy potential.
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…
We investigate the heat equation corresponding to the Bessel operators on a symmetric cone $\Omega=G/K$. These operators form a one-parameter family of elliptic self-adjoint second order differential operators and occur in the Lie algebra…
In recent years, quantities derived from the heat equation have become popular in shape processing and analysis of triangulated surfaces. Such measures are often robust with respect to different kinds of perturbations, including…
We give the upper and the lower estimates of heat kernels for Schr\"odinger operators $H=-\Delta+V$, with nonnegative and locally bounded potentials $V$ in $\mathbb{R}^d$, $d \geq 1$. We observe a factorization: the contribution of the…
We establish global two-sided heat kernel estimates (for full time and space) of the Schr\"odinger operator $-\frac{1}{2}\Delta+V$ on $\R^d$, where the potential $V(x)$ is locally bounded and behaves like $c|x|^{-\alpha}$ near infinity with…