Abstract
For a class of Laplace exponents, we consider the transition density of the subordinator and the heat kernel of the corresponding subordinate Brownian motion. We derive explicit approximate expressions for these objects in the form of asymptotic expansions: via the saddle point method for the subordinator’s transition density and via the Mellin transform for the subordinate heat kernel. The latter builds on ideas from index theory using zeta functions. In either case, we highlight the role played by the analyticity of the Laplace exponent for the qualitative properties of the asymptotics.
Highlights
Classical results link the short-time asymptotics of the heat kernel on a closed manifold to the geometry of the manifold [33], and it plays a significant role in index theory [2,5]
One is interested in the properties of the associated heat kernels on Euclidean space or Riemannian manifolds with various types of boundary conditions. Such operators and the heat kernels are important from a practical point of view since they appear naturally in physics [4] or financial mathematics [16]
The asymptotics of the subordinate heat kernel are obtained using ideas from index theory: we introduce a function (“zeta function”) that is the Mellin transform of the heat kernel
Summary
We introduce some notation and collect various prerequisites. Landau O-notation Let f and g be two functions (0, ∞) → C. Any Bernstein function can be represented in Lévy–Khintchin form as This means in particular that it is smooth on (0, ∞), can be extended to an analytic function on the half-plane {z ∈ C|Re z > σ0} for some σ0 ≤ 0 and is continuous on the axis σ0 + iR. The transform exchanges growth of f at 0 and ∞ with complex differentiability in the following sense, cf [8, Chapter 4] for details. For our purposes, it suffices to consider the case where f has the following asymptotic expansions:. In the cases of finite α, β one can meromorphically extend the Mellin transform with simple poles that are given in terms of the asymptotic expansions of f , cf [8, Lemmas 4.4.3 and 4.4.6]. M[ f ; z]M[g; 1 − z]dz, c+i R with c ∈ R in the intersection of the strips of analyticity of the transforms, cf. [41]
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