Related papers: Higher-order heat equation and the Gelfand-Dickey …

In this note I present some properties of sub-Laplaceans associated with a collection of smooth vector fields satisfying H\"ormander's finite rank assumption. One notable aspect of the paper is the development of the fractional powers of…

Let us consider a time-dependent differential operator quadratic with respect to the phase variables. Let us consider a multiplication operator defined with the help of a "small" matrix-valued function. Under suitable conditions, we give an…

Let $L=-\Delta+V$ be a Schr\"odinger operator, where the potential $V$ belongs to the reverse H\"older class. By the subordinative formula, we introduce the fractional heat semigroup $\{e^{-t{L}^\alpha}\}_{t>0}, \alpha>0$, associated with…

Suppose that $d\geq2$ and $\alpha\in(1,2)$. Let D be a bounded $C^{1,1}$ open set in $\mathbb{R}^d$ and b an $\mathbb{R}^d$-valued function on $\mathbb{R}^d$ whose components are in a certain Kato class of the rotationally symmetric…

The differential-geometric and topological structure of Delsarte transmutation operators and associated with them Gelfand-Levitan-Marchenko type eqautions are studied making use of the De Rham-Hodge-Skrypnik differential complex. The…

The covariant technique for calculating the heat kernel asymptotic expansion for an elliptic differential second order operator is generalized to manifolds with boundary. The first boundary coefficients of the asymptotic expansion which are…

By making use of the potentials of the heat conduction equation the integral equations are derived which determine the heat kernel for the Laplace operator $-a^2\Delta$ in the case of compound media. In each of the media the parameter $a^2$…

One of the principal topics of this paper concerns the realization of self-adjoint operators $L_{\Theta, \Om}$ in $L^2(\Om; d^n x)^m$, $m, n \in \bbN$, associated with divergence form elliptic partial differential expressions $L$ with…

The heat kernel expansion is a very convenient tool for studying one-loop divergences, anomalies and various asymptotics of the effective action. The aim of this report is to collect useful information on the heat kernel coefficients…

Let $n\ge2$ and $\Omega$ be a bounded non-tangentially accessible domain (for short, NTA domain) of $\mathbb{R}^n$. Assume that $L_D$ is a second-order divergence form elliptic operator having real-valued, bounded, measurable coefficients…

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…

In this note we consider the nonlinear heat equation associated to the fractional Hermite operator $H^\beta =(-\Delta+|x|^2)^\beta$, $0<\beta\leq 1$. We show the local solvability of the related Cauchy problem in the framework of modulation…

We present a diagram technique used to calculate the Seeley-DeWitt coefficients for a covariant Laplace operator. We use the combinatorial properties of the coefficients to construct a matrix formalism and derive a formula for an arbitrary…

Using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernels of some nonautonomous Kolmogorov operators with possibly unbounded drift and diffusion coefficients.

Let $G$ be a noncompact semisimple Lie group equipped with a certain invariant Riemannian metric. Then, we can consider a heat kernel function on $G$ associated to the Riemannian metric. We give an explicit formula for the heat kernel when…

Let M be a smooth closed (compact without boundary) Riemannian manifold of dimension n and P a q-dimensional smooth submanifold of M. U will denote the tubular neighborhood of P in M. Let E be a smooth vector bundle over M. Here we will…

The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential…

We construct an explicit solution of the Cauchy initial value problem for certain diffusion-type equations with variable coefficients on the entire real line. The corresponding Green function (heat kernel) is given in terms of elementary…

Let $A$ be an elliptic pseudo-differential operator of order $m$ on a closed manifold $\mathcal{X}$ of dimension $n>0$, formally positive self-adjoint with respect to some positive smooth density $d\mu_\mathcal{X}$. Then, the spectrum of…

A method for calculation of the DWSG coefficients for operators in spaces with metric incompatible with connection is suggested based on a generalization of the pseudodifferential operators technique. By using the proposed method, the…