Related papers: Kolmogorov-Fokker-Planck operators in dimension tw…

We find a Gaussian off-diagonal heat kernel estimate for uniformly elliptic operators with measurable coefficients acting on regions $\Omega\subseteq\real^N$, where the order $2m$ of the operator satisfies $N<2m$. The estimate is expressed…

The first three coefficients in an expansion of the heat kernel of a nonminimal nonabelian kinetic operator taken in an arbitrary background gauge in arbitrary space-time dimension are calculated

In this paper we prove a short time asymptotic expansion of a hypoelliptic heat kernel on an Euclidean space and a compact manifold. We study the "cut locus" case, namely, the case where energy-minimizing paths which join the two points…

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…

We survey the recent progress in the study of heat kernels for a class of non-symmetric non-local operators. We focus on the existence and sharp two-sided estimates of the heat kernels and their connection to jump diffusions.

We construct the heat kernel on curvilinear polygonal domains in arbitrary surfaces for Dirichlet, Neumann, and Robin boundary conditions as well as mixed problems, including those of Zaremba type. We compute the short time asymptotic…

We review the status of covariant methods in quantum field theory and quantum gravity, in particular, some recent progress in the calculation of the effective action via the heat kernel method. We study the heat kernel associated with an…

We calculate the second coefficient of the asymptotic expansion of the Bergman kernel of the Hodge-Dolbeault operator associated to high powers of a Hermitian line bundle with non-degenerate curvature, using the method of formal power…

This paper is an overview on our recent results in the calculation of the heat kernel in quantum field theory and quantum gravity. We introduce a deformation of the background fields (including the metric of a curved spacetime manifold) and…

The high temperature asymptotics of the Helmholtz free energy of electromagnetic field subjected to boundary conditions with spherical and cylindrical symmetries are constructed by making use of a general expansion in terms of heat kernel…

In this paper, we study the geometry associated with Schroedinger operator via Hamiltonian and Lagrangian formalism. Making use of a multiplier technique, we construct the heat kernel with the coefficient matrices of the operator both…

We prove Cameron-Martin type quasi-invariance results for the heat kernel measure of infinite-dimensional Kolmogorov and related diffusions. We first study quantitative functional inequalities for appropriate finite-dimensional…

A non-relativistic quantum model is considered with a point particle carrying a charge $e$ and moving on the plane pierced by two infinitesimally thin Aharonov-Bohm solenoids and subjected to a perpendicular uniform magnetic field of…

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…

Motivated by applications to stochastic differential equations, an extension of H\"{o}rmander's hypoellipticity theorem is proved for second-order degenerate elliptic operators with non-smooth coefficients. The main results are established…

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…

In this paper we analyze the heat kernel of the equation $\partial_tv =\pm\mathcal{L} v$, where $\mathcal{L}=\partial_x^N+u_{N-2}(x)\partial_x^{N-2}+\cdots+u_0(x)$ is an $N$-th order differential operator and the $\pm$ sign on the…

Consider the following nonlocal integro-differential operator: for $\alpha\in(0,2)$, $$ \cal L^{(\alpha)}_{\sigma,b} f(x):=\mbox{p.v.} \int_{\mathbb{R}^d-\{0\}}\frac{f(x+\sigma(x)z)-f(x)}{|z|^{d+\alpha}}d z+b(x)\cdot\nabla f(x), $$ where…

We define and study pseudo-differential operators on a class of fractals that include the post-critically finite self-similar sets and Sierpinski carpets. Using the sub-Gaussian estimates of the heat operator we prove that our operators…

We consider the heat kernel (and the zeta function) associated with Laplace type operators acting on a general irreducible rank 1 locally symmetric space X. The set of Minakshisundaram- Pleijel coefficients {A_k(X)}_{k=0}^{\infty} in the…