Abstract
We derive the off-diagonal short-time asymptotics of the heat kernels of functions of generalised Laplacians on a closed manifold. As an intermediate step we give an explicit asymptotic series for the kernels of the complex powers of generalised Laplacians. Each asymptotic series is formulated in terms of the geodesic distance. The key application concerns upper bounds for the transition density of subordinate Brownian motion. The approach is highly explicit and tractable.
Highlights
Let M be a closed Riemannian manifold of dimension n and let A be a generalised Laplacian acting on smooth functions on M, i.e. A is a second-order differential operator whose principal symbol is the metric tensor
Given the short-time asymptotics of the heat kernel K(e−tA; x, y) of A we represent the off-diagonal kernel of the complex powers K(A−z; x, y) as an asymptotic series in powers of the geodesic distance d(x, y)
The present paper considers subordinate Brownian motion using the calculus of classical pseudodifferential operators
Summary
Let M be a closed Riemannian manifold of dimension n and let A be a generalised Laplacian acting on smooth functions on M , i.e. A is a second-order differential operator whose principal symbol is the metric tensor. We consider the operator f (A) for suitable functions f and give short-time asymptotics of the heat kernel of f (A) denoted by K(e−f(A)t; x, y) and Aronson-type upper bounds. This can be illustrated as follows where d denotes the geodesic distance on M. These articles contain further references concerning the application of pseudodifferential operators in the context of certain stochastic processes Noteworthy examples of this are the calculus developed in [20] and in the context of Weyl-Hormander operators the paper [2] which allows variable-order subordination.
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