Heat kernel asymptotics of the subordinator and subordinate Brownian motion

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.Mathematics Subject Classification: 35K08, 60J55, 41A60

Transition densities of subordinators of positive order

“…Furthermore, the asymptotic is valid in some region described in terms of both space and time variable. By freezing one of them as corollaries we get the results similar to [8], see e.g. Corollary 3.6.…”

“…For this reason, distributional properties of subordinators were often studied with reference to heat kernel estimates of subordinated Brownian motions (see e.g. [21,8]). In [13] Hawkes investigated the growth of sample paths of a stable subordinator and obtained the asymptotic behavior of its distribution function.…”

“…In [6] new examples of families of subordinators with explicit transition densities were given. Finally, in the recent paper [8] the author under very restrictive assumptions derives explicit approximate expressions for the transition density of approximately stable subordinators.…”

Transition Densities of Subordinators of Positive Order

“…Furthermore, the asymptotic is valid in some region described in terms of both space and time variables. By freezing one of them, we obtain as corollaries the results similar to [16]; see, for example, Corollary 3.6. It is also worth highlighting that we obtain a version of the upper estimate on the transition density with no additional assumptions on the Lévy measure ν(dx); see Theorem 4.7.…”

“…In [8] new examples of families of subordinators with explicit transition densities were given. Finally, in the recent paper [16], the author Transition Densities of Subordinators of Positive Order 3 derived explicit approximate expressions for the transition density of approximately stable subordinators under very restrictive assumptions.…”