Related papers: Heat kernel estimates for nonlocal kinetic operato…

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…

We construct the biharmonic heat kernel for a suitable self-adjoint extension of the bi-Laplacian on a manifold with incomplete edge singularities. We employ a microlocal description of the biharmonic heat kernel to establish mapping…

In this paper, we consider symmetric $\alpha$-stable processes on (unbounded) horn-shaped regions which are non-uniformly $C^{1,1}$ near infinity. By using probabilistic approaches extensively, we establish two-sided Dirichlet heat…

We consider the heat-kernel expansion of the massive Laplace operator on the three dimensional ball with Dirichlet boundary conditions. Using this example, we illustrate a very effective scheme for the calculation of an (in principle)…

In the present paper we establish sharp exponential decay estimates for operator and integral kernels of the (not necessarily self-adjoint) operators $L=-(\nabla-i\mathbf{a})^TA(\nabla-i\mathbf{a})+V$. The latter class includes, in…

We consider fractional Schr\"odinger operators with possibly singular potentials and derive certain spatially averaged estimates for its complex-time heat kernel. The main tool is a Phragm\'en-Lindel\"of theorem for polynomially bounded…

We consider the Dirichlet form given by \sE(f,f)&=&{1/2}\int_{\bR^d}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial f(x)}{\partial x_i} \frac{\partial f(x)}{\partial x_j} dx &+&\int_{\bR^d\times \bR^d} (f(y)-f(x))^2J(x,y)dxdy. Under the assumption…

The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and…

We provide sharp two-sided estimates on the Dirichlet heat kernel $k_1(t,x,y)$ for the Laplacian in a ball. The result accurately describes the exponential behaviour of the kernel for small times and significantly improves the qualitatively…

We generalize the Endo formula originally developed for the computation of the heat kernel asymptotic expansion for non-minimal operators in commutative gauge theories to the noncommutative case. In this way, the first three non-zero heat…

Mean value formulas are of great importance in the theory of partial differential equations: many very useful results are drawn, for instance, from the well known equivalence between harmonic functions and mean value properties. In the…

We study the heat kernel for the Laplace type partial differential operator acting on smooth sections of a complex spin-tensor bundle over a generic $n$-dimensional Riemannian manifold. Assuming that the curvature of the U(1) connection…

We study a spatial asymptotic behaviour at infinity of kernels $p_t(x)$ for convolution semigroups of nonlocal pseudo-differential operators. We give general and sharp sufficient conditions under which the limits $$ \lim_{r \to \infty}…

Given a metric measure space $(\mathcal{X}, d, \mu)$ satisfying the volume doubling condition, we consider a semigroup $\{S_t\}$ and the associated heat operator. We propose general conditions on the heat kernel so that the solutions of the…

With $\vec{\Delta}_j\geq 0$ is the uniquely determined self-adjoint realization of the Laplace operator acting on $j$-forms on a geodesically complete Riemannian manifold $M$ and $\nabla$ the Levi-Civita covariant derivative, we prove…

We consider a complete non-compact Riemannian manifold satisfying the volume doubling property and a Gaussian upper bound for its heat kernel (on functions). Let -- $\rightarrow$ $\Delta$ k be the Hodge-de Rham Laplacian on differential…

In this paper, we establish sharp two-sided heat kernel estimates for a large class of symmetric Markov processes in exterior $C^{1,\eta}$ open sets for all $t> 0$. The processes are symmetric pure jump Markov processes with jumping kernel…

Heat kernel coefficients encode the short distance behavior of propagators in the presence of background fields, and are thus useful in quantum field theory. We present a Mathematica program for computing these coefficients and their…

Let $H:=-\Delta+V$ be a nonnegative Schr\"odinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ and $V$ is an inverse square potential. In this paper we obtain sharp decay estimates of the operator norms of $e^{-tH}$ and $\nabla e^{-tH}$ in…

In this paper we study differential operators of the form \begin{align*} \left[\mathcal{L}_\infty v \right](x) = A\triangle v(x) + \left\langle Sx,\nabla v(x) \right\rangle - Bv(x), \,x \in \mathbb{R}^d, \,d \geqslant 2, \end{align*} for…