Note on short-time behavior of semigroups associated to self-adjoint operators (original) (raw)

Manifolds and graphs with slow heat kernel decay

Inventiones Mathematicae, 2001

We give upper estimates on the long time behavior of the heat kernel on noncompact Riemannian manifolds and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are sharp.

The heat kernel on the diagonal for a compact metric graph

2022

We analyze the heat kernel associated to the Laplacian on a compact metric graph, with standard Kirchoff-Neumann vertex conditions. An explicit formula for the heat kernel as a sum over loops, developed by Roth and Kostrykin-Potthoff-Schrader, allows for a straightforward analysis of small-time asymptotics. We show that the restriction of the heat kernel to the diagonal satisfies a modified version of the heat equation. This observation leads to an "edge" heat trace formula, expressing the a sum over eigenfunction amplitudes on a single edge as a sum over closed loops containing that edge. The proof of this formula relies on a modified heat equation satisfied by the diagonal restriction of the heat kernel. Further study of this equation leads to explicit formulas for completely symmetric graphs.

Semigroups for dynamical processes on metric graphs

Philosophical Transactions of the Royal Society A, 2020

We present the operator semigroups approach to first-and secondorder dynamical systems taking place on metric graphs. We briefly survey the existing results and focus on the well-posedness of the problems with standard vertex conditions. Finally, we show two applications to biological models.

On the large time behavior of heat kernels on Lie groups

Duke Mathematical Journal, 2003

1. Introduction Let G be a connected noncompact semisimple Lie group with finite cen-ter, and let E1,..., En be left-invariant vector fields that satisfy Hörmander's condition; that is, they generate together with their successive Lie brackets [Ei1 ,[Ei2 ,[...,[Eik−1 , Eik ],...]]] the tangent ...

Large time behavior of the heat kernel

Journal of Functional Analysis, 2004

In this paper we study the large time behavior of the (minimal) heat kernel k M P (x, y, t) of a general time independent parabolic operator L = u t +P (x, ∂ x ) which is defined on a noncompact manifold M . More precisely, we prove that lim t→∞ e λ0t k M P (x, y, t) always exists. Here λ 0 is the generalized principal eigenvalue of the operator P in M .

A heat semigroup version of Bernstein's theorem on Lie groups

Monatshefte f�r Mathematik, 1990

This paper establishes a heat semigroup version of Bernstein's theorem, applicable to any unimodular Lie group. The result has an intrinsic geometric content, involving estimates for the norms of the heat kernels for small time and large time. The theorem is stated in terms of certain Lipschitz spaces whose definition incorporates these two geometric features of the group in question. The geometric content is further underlined by showing that, in a certain sence, the theorem is best-possible.

Heat Kernel and Green Kernel Comparison Theorems for Infinite Graphs

Journal of Functional Analysis, 1997

For an infinite graph, general lower and upper estimates of the Green kernel and the heat kernel are given. The estimates are optimal in the case of the homogeneous regular trees. As their applications, solvability of Dirichlet problem for the end compactification is shown and the sharp estimates of several infinite graphs including the distance regular graphs and the free products of finite complete graphs are given. 1997 Academic Press 0. INTRODUCTION In the theory of infinite graphs, the Green kernel would be very important. It expresses the probabilistic properties of the graph, for instance, the transience or the recurrence properties, and also the solvability of the Dirichlet problem for the end compactification. Even its importances and many works about it (see for instance, [W]), it is still difficult to determine its explicit estimates or the explicit form, except the homogeneous regular tree T d of degree d due to the famous work of P. Cartier [C]. On the other hand, there are many works on the Green kernel for a complete Riemannian manifold, for examples, [DGM], [CY], [AS], [I], [K], and [LT]. Especially, it was shown (cf. [DGM], [CY], [K]) that, roughly speaking, if the sectional curvature K satisfies that K &a 0 (resp. 0 K &a), then the heat kernel p(t, x, y) and the Green kernel G(x, y) of a simply connected complete manifold satisfy p(t, x, y) p a (t, x, y) (resp. p(t, x, y) p a (t, x, y)), G(x, y) G a (x, y) (resp. G(x, y) G a (x, y)), where p a (t, x, y) and G a (x, y) are the heat kernel and the Green kernel of the space form of curvature &a 0, respectively.

The heat semigroup on weighted Sobolev and asymptotic spaces

arXiv: Analysis of PDEs, 2019

We prove that the heat equation on Rd\R^dRd is well-posed in weighted Sobolev spaces and in certain spaces of functions allowing spatial asymptotic expansions as ∣x∣toinfty|x|\to\infty∣x∣toinfty of any a priori given order. In fact, we show that the Laplacian on such function spaces generates an analytic semigroup of angle pi/2\pi/2pi/2 with polynomial growth as ttoinftyt\to\inftyttoinfty. We apply these results to nonlinear heat equations on Rd\R^dRd, including global existence in time.