Transition Densities of Subordinators of Positive Order

Abstract

We prove the existence and asymptotic behaviour of the transition density for a large class of subordinators whose Laplace exponents satisfy lower scaling condition at infinity. Furthermore, we present lower and upper bounds for the density. Sharp estimates are provided if an additional upper scaling condition on the Laplace exponent is imposed. In particular, we cover the case when the (minus) second derivative of the Laplace exponent is a function regularly varying at infinity with regularity index bigger than $-2$ .

Spectral heat content for time-changed killed Brownian motions

“…We first establish an upper bound for the heat kernel for (D t ), which is global in space and is valid for all φ ∈ R β (∞) with β ∈ (0, 1). The result essentially follows from [9], but we state it here in a form that is convenient for our purpose. Then the transition density p(t, x) of (D t ) exists.…”

“…In the notations used in [9], conditions (3.3) and (3.4) are expressed as φ It follows from [9, Theorem 4.15] that there exist x 1 > 0, t 0 > 0 and c > 0 such that (3.2) holds for all 0 < t < t 0 and 0 < x < x 1 with 1 < xφ −1 (1/t). On the other hand, for 0 < t < t 0 and 0 < x < x 1 with 1 ≥ xφ −1 (1/t), the exponential decay of the heat kernel in [9, Theorem 4.15] implies p(t, x) ≤ C 3 for some constant…”

Subordinated Markov processes: sharp estimates for heat kernels and Green functions

“…A standard procedure to obtain such results uses bounds for the heat kernel of the original process together with estimates for transition density of subordinator (see e.g. [11,28,31]). In our approach we only use properties of the Laplace exponent of subordinator.…”