Off-Diagonal Heat Kernel Asymptotics of Pseudodifferential Operators on Closed Manifolds and Subordinate Brownian Motion

Abstract

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.Mathematics Subject Classification. Primary 35K08; Secondary 58J65, 35S05.

Heat kernel asymptotics of the subordinator and subordinate Brownian motion

“…as t → 0 in lowest orders in t and the Euclidean distance d. It is instructive to compare this result with the literature: it agrees with [21] translated to the case of a flat manifold. Moreover, the bounds for the relativistic stable process (α = 1/2) obtained in […”

“…This paper continues the theme of [21] albeit on Euclidean space. To summarise our approach and results in a non-technical way, denote by B t a standard Brownian motion on R n and let X t be a subordinator with corresponding Laplace exponent f , i.e., X t is an almost surely increasing Lévy process that takes values in the nonnegative reals.…”

“…Moving from the trace of the complex powers A −s to their integral kernel yields the above definition of the zeta function ζ(s; x, y). We refer the reader also to [21] where this is done in the context of pseudodifferential operators and ζ(s; x, y) is indeed the integral kernel of the complex powers A −s . (ii) It is not obvious that Definition 3.10 makes sense as the integral may not converge.…”

“…This is independent of the choice of a. (10) is then merely the exponential series for the integral kernel of the operator e −t[ f ( )+a] with the constant term omitted, see also Remark 3.6 in [21]. (ii) The lowest order heat kernel coefficient is given as…”

“…We note, however, that the lowest orders in z are unaffected by this change. The reader is referred to the proof of Theorem 3.4 in [21] for detailed arguments.…”