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 
$-2$
 .


Introduction
Asymptotic behaviour as well as estimates of heat kernels have been intensively studied in the last decades. The first results obtained by Pòlya [44] and Blumenthal and Getoor [4] for isotropic α-stable process in R d provided the basis for studies of more complicated processes; for example, subordinated Brownian motions [40,49], isotropic unimodal Lévy processes [6,15,21] and even more general symmetric Markov processes [11,14]. One may, among others, list the articles on heat kernel estimates for jump processes of finite range [10] or with lower intensity of higher jumps [39,51]. While a great many articles with explicit results are devoted to symmetric processes or those which are, in appropriate sense, similar to the symmetric ones, the nonsymmetric case is in general harder to handle due to lack of a familiar structure. This problem was approached in many different ways; see, for example, [7,27,28,31,32,35,36,42,43,50]. For a more specific class of stable processes, see [25,46,53]. Overall, one has to impose some control on the nonsymmetry in order to obtain estimates in an easy-to-handle form. This idea was applied in the recent paper [22] where the authors considered the case of the Lévy measure being comparable to some unimodal Lévy measure. The methods developed in [22,23] contributed significantly to this article. See also [32,42] and the references therein.
In this article the central object is a subordinator; that is, a one-dimensional Lévy process with nondecreasing paths starting at 0; see Section 2 for the precise definition. The abstract introduction of the subordination dates back to 1950s and is due to Bochner [5] and Philips [41]. In the language of the semigroup theory, for a Bernstein function φ and a bounded C 0 -semigroup e −tA : t ≥ 0 with −A being its generator on some Banach space X , via Bochner integral one can define an operator B = φ(A) such that −B also generates a bounded C 0 -semigroup e −tB : t ≥ 0 on X . The semigroup e −tB : t ≥ 0 is then said to be subordinated to e −tA : t ≥ 0 , and although it may be very different from the original one, its properties clearly follow from properties of both the parent semigroup and the involved Bernstein function. See, for example, [18] and the references therein. From a probabilistic point of view, due to positivity and monotonicity, subordinators naturally appear as random time change functions of Lévy processes or, more generally, Markov processes. Namely, if (X t : t ≥ 0) is a Markov process and (T t : t ≥ 0) is an independent subordinator, then Y t = X Tt is again a Markov process with a transition function given by The procedure just described is called a subordination of a Markov process and can be interpreted as a probabilistic form of the equality B = φ(A). Here A and B are (minus) generators of semigroups associated to processes X t and Y t , respectively. From an analytical point of view, the transition density of Y t (the integral kernel of e −tB ) can be obtained as a time average of transition density of X t with respect to distribution of T t . Yet another approach is driven by partial differential equations, as the transition density is a heat kernel of a generalised heat equation. The generalisation can be twofold: either by replacing the Laplace operator with another, possibly nonlocal operator or by introducing a more general fractional-time derivative instead of the classical one. The latter case was recently considered in [9,12,38]. Here the solutions are expressed in terms of corresponding (inverse) subordinators and thus their analysis is essential.
By taking A = −Δ and changing the time of (i.e., subordinating) Brownian motion one can obtain a large class of subordinated Brownian motions. A principal example here is an α-stable subordinator with the Laplace exponent φ(λ) = λ α , α ∈ (0,1), which gives rise to the symmetric, rotation-invariant α-stable process and corresponds to the special case of fractional powers of semigroup e −tA α : t ≥ 0 . For this reason, distributional properties of subordinators were often studied with reference to heat kernel estimates of subordinated Brownian motions (see, e.g., [16,33]). In [24] Hawkes investigated the growth of sample paths of a stable subordinator and obtained the asymptotic behaviour of its distribution function. Jain and Pruitt [30] considered tail probability estimates for subordinators and, in the discrete case, nondecreasing random walks. In a more general setting some related results were obtained in [17,26,42,52]. In [8] new examples of families of subordinators with explicit transition densities were given. Finally, in the recent paper [16], the author derived explicit approximate expressions for the transition density of approximately stable subordinators under very restrictive assumptions.
The result of the article is asymptotic behaviour as well as upper and lower estimates of transition densities of subordinators satisfying scaling condition imposed on the second derivative of the Laplace exponent φ. Our standing assumption on −φ is the weak lower scaling condition at infinity with scaling parameter α − 2, for some α > 0 (see (2.7) for definition). It is worth highlighting that we do not state our assumptions and results in terms of the Laplace exponent φ, as one could suspect, but in terms of its second derivative and related function ϕ(x) = x 2 (−φ (x)) (see Theorems 3.3,4.7 and 4.8). Usually the transition density of a Lévy process is described by the generalised inverse of the real part of the characteristic exponent ψ −1 (x) (e.g., [23], [36]), but in our setting one can show that the lower scaling property implies that ϕ −1 (x) ≈ ψ −1 (x) for x sufficient large (see Proposition 4.3). In some cases, however, ϕ may be significantly different from the Laplace exponent φ. However, if one assumes additional upper scaling condition with scaling parameter β − 2 for β strictly between 0 and 1, then these two objects are comparable (see Proposition 4.6).
The main results of this article are covered by Theorems 3.3, 4.7, 4.8, 4.11 and 4.17. Theorem 3.3 is essential for the whole article because it provides not only the existence of the transition density but also its asymptotic behaviour, which is later used in derivation of upper and lower estimates. The key argument in the proof is the lower estimate on the holomorphic extension of the Laplace exponent φ (see Lemma 3.1), which justifies the inversion of the Laplace transform and allows us to perform the saddle point type approximation. In Theorem 3.3 we only use the weak lower scaling property on the second derivative of the Laplace exponent. In particular, we do not assume the absolute continuity of ν(dx). 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. Clearly, putting some restrictions on ν(dx) results in sharper estimates (Theorem 4.8), but it is interesting that the scaling property alone is enough to get some information. Our starting point and the main object to work with is the Laplace exponent φ. However, in many cases the primary object is the Lévy measure ν(dx) and results are presented in terms of or require its tail decay. Therefore, it would be convenient to have a connection between those two objects. In Proposition 3.8 we prove that one can impose scaling conditions on the tail of the Lévy measure ν((x,∞)) instead, as they imply the scaling condition on −φ .
We also note that the main results of the article hold true when −φ is a function regularly varying at infinity with regularity index α−2, where α > 0. This follows easily by Potter bounds for regularly varying functions (see [3,Theorem 1.5.6]), which immediately imply both lower and upper scaling properties. Moreover, if additionally α < 1, then, by Karamata's theorem and monotone density theorem, regular variation of −φ with index α − 2 is equivalent to regular variation of φ with index α. This is not the case for the case α = 1 where, in general, only the first direction holds true.
Below we present the special case when global upper and lower scaling conditions are imposed with 0 < α ≤ β < 1; see Theorem 4.17.
Theorem A. Let T be a subordinator with the Laplace exponent φ. Suppose that for some 0 < α ≤ β < 1, the functions are almost increasing and almost decreasing, respectively. We also assume that the Lévy measure ν(dx) has an almost monotone density ν(x). Then the probability distribution of T t has a density p(t, · ). Moreover, for all t ∈ (0,∞) and x > 0, We note that a similar result to Theorem A appeared in [13] in around the same time as our preprint. Our assumptions, however, are weaker, as we assume almost monotonicity of the Lévy density instead of monotonicity of the function t → tν(t). Moreover, our estimates are genuinely sharp; that is, the constants appearing in the exponential factors are the same on both sides of the estimate, while estimates obtained in [13] are qualitatively sharp; that is, the constants in the exponential factors are different.
As a corollary, under the assumption of Theorem A, we obtain a global two-sided estimate on the Green function. Namely, for all x > 0, See Section 5 and Theorem 5.8 for details. The article is organised as follows: In Section 2 we introduce our framework and collect some facts concerning Bernstein functions and their scaling properties. Section 3 is devoted to the proof of Theorem 3.3 and its consequences. In Section 4 we provide both upper and lower estimates on the transition density and discuss when these estimates coincide. Some applications of our results to subordination beyond the familiar R d setting and Green function estimates are presented in Section 5.

Notation
By C 1 ,c 1 ,C 2 ,c 2 , . . . we denote positive constants which may change from line to line.
for all x > 0. An analogous rule is applied to the symbol . We also have for all x > 0. Finally, we set a ∧ b = min{a,b} and a ∨ b = max{a,b}.

Preliminaries
Let (Ω,F,P) be a probability space. Let T = (T t : t ≥ 0) be a subordinator; that is, a Lévy process in R with nondecreasing paths. Recall that a Lévy process is a càdlàg stochastic process with stationary and independent increments such that T 0 = 0 almost surely. There is a function ψ : R → C, called the Lévy-Khintchine exponent of T, such that for all t ≥ 0 and ξ ∈ R, E e iξTt = e −tψ(ξ) .
Moreover, there are b ≥ 0 and σ-finite measure ν on (0,∞) satisfying such that for all ξ ∈ R, for all t ≥ 0 and λ ≥ 0. Let ψ * be the symmetric continuous and nondecreasing majorant of ψ; that is, Notice that where ψ −1 is the generalised inverse function defined as To study the distribution function of the subordinator T, it is convenient to introduce two concentration functions K and h. They are defined as Analogously, f is regular varying at the origin of index α if for all λ ≥ 1, If α = 0, the function f is called slowly varying. We next introduce a notion of scaling conditions frequently used in this article. We say that a function f : [0,∞) → [0,∞) has the weak lower scaling property at infinity if there are α ∈ R, c ∈ (0,1] and x 0 ≥ 0 such that for all λ ≥ 1 and x > x 0 , We denote it briefly as f ∈ WLSC(α,c,x 0 ). Observe that if α > α then WLSC(α,c,x 0 ) WLSC(α ,c,x 0 ). Analogously, f has the weak upper scaling property at infinity if there are β ∈ R, C ≥ 1, and x 0 ≥ 0 such that for all λ ≥ 1 and x > x 0 , In this case we write f ∈ WUSC(β,C,x 0 ). We say that a function f : [0,∞) → [0,∞) has doubling property on (x 0 ,∞) for some x 0 ≥ 0 if there is C ≥ 1 such that for all x > x 0 , Notice that a nonincreasing function with the weak lower scaling has a doubling property. Analogously, a nondecreasing function with the weak upper scaling.
In view of [6,Lemma 11], f ∈ WLSC(α,c,x 0 ) if and only if the function is almost increasing. Similarly, f ∈ WUSC(β,C,x 0 ) if and only if the function is almost decreasing. For a function f : [0,∞) → C its Laplace transform is defined as

Bernstein functions
In this section we recall some basic facts about Bernstein functions. A general reference here is the book [48].
Since φ is concave, for each λ ≥ 1 and x > 0 we have thus, by (2.9), 1 − e −λs μ(ds). (2.11) which, together with the doubling property, gives Hence, we obtain our assertion in the case x 0 = 0. If x 0 > 0, we observe that the function is continuous and positive and thus bounded. This completes the proof.

Proposition 2.2. Let f be a completely monotone function. Suppose that
Since f is nonnegative and nonincreasing, we can take y approaching infinity to get where in the last inequality we have also used that 1 ≥ cλ 1+τ . The second part of the proposition can be proved in much the same way. (2.12) Proof. Assume first that φ ∈ WLSC(α − 1,c,x 0 ). Without loss of generality, we can assume φ ≡ 0. We claim that (2.12) holds true. In view of (2.9), it is enough to show that there is C ≥ 1 such that for all x > x 0 , First, let us observe that, by the weak lower scaling property of φ , Thus, we get the assertion in the case x 0 = 0. If x 0 > 0, it is enough to show that there is C > 0 such that for all x > x 0 , (2.14) Since φ ∈ WLSC(α − 1,c,x 0 ), the function To conclude (2.14), we notice that φ (x) is positive and continuous in [x 0 ,2x 0 ]. Now, by (2.14) we get for all x > x 0 , which, together with (2.13), implies (2.12) and the scaling property of φ follows. Now assume that φ ∈ WLSC(α,c,x 0 ). By monotonicity of φ , for 0 < s < t, For s = 1, by the lower scaling we get for all x > x 0 . Thus, for t = 2 1/α c −1/α , we obtain that xφ (x) φ(x) for all x > x 0 . Invoking (2.9), we conclude (2.12). In particular, φ has the weak lower scaling property. This completes the proof.
where b is the drift term from the integral representation (2.11) of φ.
Proof. Without loss of generality, we can assume φ ≡ 0. By the scaling property, for x > x 0 we have which concludes the proof.
Proposition 2.6. Let f be a completely monotone function. Suppose that for some c ∈ (0,1] and C ≥ 1.

Proof.
By monotonicity of −f , for 0 < s < t, Taking s = 1 in the second inequality, the weak upper scaling property yields Similarly, taking t = 1 in the first inequality in (2.15), by the weak lower scaling property we get for all x > x 0 /s. By selecting 0 < s < 1 such that for some c ∈ (0,1], C ≥ 1, x 0 ≥ 0 and 0 < α ≤ β < 1. Then for some c ∈ (0,1] and C ≥ 1.

Lemma 2.8. Let φ be a Bernstein function. Suppose that
Moreover, the constant C depends only on α and c.
Proof. Let f : (0,∞) → R be a function defined as We observe that, by the Fubini-Tonelli theorem, for x > 0 we have ).
Since f is nondecreasing, for any s > 0, Hence, for any u > 2, Therefore, setting x = λu > 2x 0 , by the weak lower scaling property of −φ , At this stage, we select u > 2 such that Then again, by the weak lower scaling property of −φ , for λ > x 0 , which ends the proof.

Lemma 2.9. Let φ be a Bernstein function. Suppose that
Proof. Let us define By [48,Theorem 6.2 (vi)] the function f is a complete Bernstein function. Since for y > 0,
In this section we study the asymptotic behaviour of the probability density of T t . In the whole section we assume that φ ≡ 0; otherwise, T t = bt is deterministic. The main result is Theorem 3.3. Let us start by showing an estimate on the real part of the complex extension φ.
Then there exists C > 0 such that for all w > x 0 and λ ∈ R, Proof. By the integral representation (2.11), for λ ∈ R we have In particular, Thus, it is sufficient to consider λ > 0. We can estimate (3.1) Due to Lemma 2.8, we obtain, for λ ≥ w, If w > λ > 0, then, by (3.1), we have which, by Lemma 2.8, completes the proof.
by Lemma 2.8 we obtain for all x > x 0 .
Then the probability distribution of T t is absolutely continuous for all t > 0. If we denote its density by p(t, · ), then for Proof. Let x = tφ (w) and M > 0. We first show that To do so, let us recall that Thus, by Mellin's inversion formula, if the limit then the probability distribution of T t has a density p(t, · ) and Therefore, our task is to justify the statement (3.4). For L > 0, we write By the change of variables Let us note here that −φ is nonincreasing and integrable at infinity; thus, we in fact have α ≤ 1. We claim that there is C > 0 not depending on M, such that for all u ∈ R, provided that w > x 0 and tw 2 (−φ (w)) > M. Indeed, by (3.3) and Lemma 3.1, for w > x 0 we get We next estimate the right-hand side of (3.6). If |u| ≤ w t(−φ (w)), then Hence, we deduce (3.5). To finish the proof of (3.4), we invoke the dominated convergence theorem. Consequently, by Mellin's inversion formula we obtain (3.2). Our next task is to show that for each > 0 there is provided that w > x 0 and tw 2 (−φ (w)) > M 0 . In view of (3.5), by taking M 0 > 1 sufficiently large, we get where ξ is some number satisfying Observe that Since −φ is a nonincreasing function with the weak lower scaling property, it is doubling. Thus, by Proposition 2.1, for w > x 0 , which together with (3.12) give whenever tw 2 (−φ (w)) > M 0 . Now, (3.10) easily follows by (3.13) and (3.11). Finally, since for any z ∈ C, provided that M 0 is sufficiently large, which, together with (3.8) and (3.9), completes the proof of (3.7) and the theorem follows.
By Theorem 3.3, we immediately get the following corollaries.
uniformly on the set By imposing on −φ an additional condition of the weak upper scaling, we can further simplify the description of the set where the sharp estimates on p(t,x) hold.
uniformly on the set thus, for (t,x) belonging to the set (3.14), By taking δ sufficiently small, we get which implies that In particular, w > x 0 . On the other hand, by Propositions 2.3 and 2.4, there is c 1 ∈ (0,1] such that which, together with (3.16), gives for δ sufficiently small. Hence, by Corollary 3.5, we conclude the proof.
The following proposition provides a sufficient condition on the measure ν that entails the weak lower scaling property of −φ and allows us to apply Theorem 3.3.
Proof. Let us first notice that by the Fubini-Tonelli theorem, Thus, by (3.17), for all 0 < r < 1/x 0 and 0 < λ ≤ 1, Hence, by [23,Lemma 2.3], there is C ≥ 1 such that for all 0 < r < 1/x 0 , The integral representation of φ entails that for all x > x 0 . Now, the weak lower scaling property of −φ is a consequence of (3.18).

Estimates on the density
Let T = (T t : t ≥ 0) be a subordinator with the Lévy-Khintchine exponent ψ and the Laplace exponent φ. In this section we always assume that −φ ∈ WLSC(α − 2,c,x 0 ) for some c ∈ (0,1], x 0 ≥ 0 and α ∈ (0,1]. In particular, by Theorem 3.3, the probability distribution of T t has a density p(t, · ). To express the majorant on p(t, · ), it is convenient to set Obviously, ϕ ∈ WLSC(α,c,x 0 ). Let ϕ −1 denote the generalised inverse function defined as We start by showing comparability between the two concentration functions K and h defined in (2.3) and (2.4), respectively.
In view of (2.5), we have Let us consider the first term on the right-hand side of (4.1). By Remark 3.2 we have K(r) ≈ ϕ(1/r), for 0 < r < 1/x 0 , which implies This finishes the proof in the case x 0 = 0. If x 0 > 0, then, for 1/(2x 0 ) ≤ r < 1/x 0 , we have Hence, K(r) 1 for all 0 < r < 1/x 0 . Since the second term on the right-hand side of (4.1) is constant, the proof is completed.

Proof.
We start by showing that there is C ≥ 1 such that for all x > x 0 , The first inequality in (4.11) immediately follows from (4.2). If x 0 = 0, then the second inequality is also the consequence of (4.2). In the case x 0 > 0, we observe that for x > x 0 , we have proving (4.11). Now, using (4.11), we easily get for all r > Cψ * (x 0 ). Hence, by Proposition 4.2,  In such a case we have Clearly, for all x > 0, Let u > x 0 and set r 0 = inf{r > 0: ϕ * (r) = u}.
and for all r > ϕ(x 0 ), Furthermore, there is c ∈ (0,1] such that for all λ ≥ 1 and r > ϕ(x 0 ), Now the proof of the lemma is similar to the proof of Proposition 4.3 and is therefore omitted.

Estimates from below
In this section we develop estimates from below on the density p(t, · ). The main result is Theorem 4.11. Its proof is inspired by the ideas from [42], see also [23]. Thanks to Theorem 3.3, we can generalise results obtained in [42] to the case when −φ satisfies the weak lower scaling of index α − 2 for α > 0 together with a certain additional condition. We use the following variant of the celebrated Pruitt's result [45, Section 3] adapted to subordinators.

Proposition 4.10. Let T be a subordinator with the Lévy-Khintchine exponent
Then there is an absolute constant c > 0 such that for all λ > 0 and t > 0, Proof. We are going to apply the estimates [45, (3.2)]. To do so, we need to express the Lévy-Khintchine exponent of T s − sb λ in the form used in [45, Section 3], namely, Hence, by [45, (3.2)], as desired.
Theorem 4.11. Let T be a subordinator with the Laplace exponent φ. Suppose that −φ ∈ WLSC(α − 2,c,x 0 ) for some c ∈ (0,1], x 0 ≥ 0 and α > 0, and assume that one of the following conditions holds true: (ii) −φ is a function regularly varying at infinity with index −1. If x 0 = 0 we also assume that −φ is regularly varying at zero with index −1.
Then there is M 0 > 1 such that for each M ≥ M 0 there exists ρ 0 > 0, so that for all 0 < ρ 1 < ρ 0 , 0 < ρ 2 there is C > 0 such that for all t ∈ (0,1/ϕ(x 0 )) and all x > 0 satisfying we have Remark 4.12. From the proof of Theorem 4.11 it stems that if x 0 = 0, one can obtain the same statement under the condition that −φ is (−1)-regular at infinity and satisfies upper scaling at 0 with α ≤ β < 1. Alternatively, one can assume that −φ satisfies upper scaling at infinity with α ≤ β < 1 and varies regularly at zero with index −1. The same remark applies to Proposition 4.14.
Proof. First let us observe that it is enough to prove that (4.40) holds true for all t ∈ (0,1/ϕ(x 0 )) and all x > 0 satisfying Indeed, since ϕ −1 is nondecreasing and has upper scaling property (see Proposition 4.3), it has a doubling property. Hence, the lemma will follow immediately with possibly modified ρ 0 . Without loss of generality, we can assume that b = 0. Let λ > 0, whose value will be specified later. We decompose the Lévy measure ν(dx) as follows: Let ν 1 (dx) be the restriction of 1 2 ν(dx) to the interval (0,λ] and We set 1 − e −us ν 2 (ds).
Let us denote by T (j) the subordinator having the Laplace exponent φ j , for j ∈ {1,2}. Let ψ j (ξ) = φ j (−iξ). Notice that 1 2 ν ≤ ν 2 ≤ ν; thus, 1 2 φ ≤ φ 2 ≤ φ, and for every n ∈ N, (4.41) Therefore, for all u > 0, Next, by Theorem 3.3, the random variables T (2) t and T t are absolutely continuous. Let us denote by p (2) (t, · ) and p(t, · ) the densities of T (2) t and T t , respectively. Let M ≥ 2M 0 + 1, where M 0 is determined in Corollary 3.5 for the process T (2) . For 0 < t < 1/ϕ(x 0 ), we set Since Then, by (4.42) we get Moreover, by Corollary 4.5 together with (4.42) we get Hence, by Corollary 3.5, . (4.43) Notice that, by (4.41) and Remark 3.4, the implied constant in (4.43) is independent of t and λ. Since by (4.43) and monotonicity of ϕ −1 , we get for some constant C 1 > 0. Next, by the Fourier inversion formula thus, by [23,Proposition 3.4] and Propositions 4.2 and 4.3 we see that there is C 2 > 0 such that for all t ∈ (0,1/ϕ(x 0 )), By the mean value theorem, for y ∈ R, we get Hence, for y ∈ R satisfying by (4.44), we get Therefore, where we have set C 0 = C 1 (2C 2 ) −1 and Let ρ 0 = 1 2 C 0 and λ = 1 We have Thus, 1 2 tb λ − (x t − x t ) is nonnegative, and in view of (4.2) and (4.45), , (4.46) for some constant C 3 > 0. Next, setting Hence, the problem is reduced to showing that the infimum above is positive. Let us consider a collection {Y t : t ∈ (0,1/ϕ(x 0 ))} of infinitely divisible nonnegative random by Proposition 4.10, thus, where in the last estimate we have used (4.2). Therefore, recalling (4.45), we conclude that the collection is tight. Next, let (Y tn ,y n ) : n ∈ N be a sequence realising the infimum in (4.47). By the Prokhorov theorem, we can assume that (Y tn : n ∈ N) is weakly convergent to the random variable Y 0 . We note that Y tn has the probability distribution supported in − 1 2 t n λ −1 n b λn ,∞ where λ n is defined as λ corresponding to t n . Suppose that (t n : n ∈ N) contains a subsequence convergent to t 0 > 0. Then Y 0 = Y t0 and the support of its probability distribution equals − 1 we easily conclude that the infimum in (4.47) is positive. Hence, it remains to investigate the case when (t n : n ∈ N) has no positive accumulation points. If zero is the only accumulation point, then (λ n : n ∈ N) has a subsequence convergent to zero. Otherwise, (t n ) diverges to infinity; thus, x 0 = 0 and (λ n ) contains a subsequence diverging to infinity. In view of (4.46), ρ(t) is uniformly bounded in t. Thus, after taking a subsequence, we may and do assume that there exists a limit ρ = lim n→∞ ρ(t n ).
By compactness we can also assume that (y n : n ∈ N) converges to y 0 ∈ [−ρ 1 −ρ, ρ 2 ]. Consequently, to prove that the infimum in (4.47) is positive, it is sufficient to show that Observe that (4.49) is trivially satisfied if the support of the probability distribution of Y 0 is the whole real line. Therefore, we can assume that Y 0 is purely non-Gaussian. In view of [47,Theorem 8.7], it is also infinitely divisible. Given w : R → R a continuous function satisfying we write the Lévy-Khintchine exponent of Y tn in the form Since (Y tn : n ∈ N) converges weakly to Y 0 , there are γ 0 ∈ R and σ-finite measure μ 0 on (0,∞) satisfying Next, let us fix w satisfying (4.50) which equals 1 on [0,1]. In view of (4.48) and the definition of ν 1 , the support of μ tn is contained in [0,1]. Hence, γ n = 0 for every n ∈ N and, consequently, γ 0 = 0. We also conclude that supp μ 0 ⊂ [0,1].
At this stage we consider the cases (i) and (ii) separately. In (ii) we need to distinguish two possibilities: if (t n ) tends toward zero, then (λ n ) also approaches zero, and we impose that −φ is a function regularly varying at infinity with index −1; otherwise, (t n ) tends toward infinity as well as (λ n ), and thus x 0 = 0, and we additionally assume that −φ is a function regularly varying at zero with index −1. For the sake of clarity of presentation, we restrict attention to the first possibility only. In the second one the reasoning is analogous. We show that the support of the probability distribution of Y 0 is the whole real line. By [47,Theorem 24.10], the latter can be deduced from For the proof, for any ∈ (0,1) we define the following bounded continuous function: i fs ≥ 1. (4.55) We have, in view of (4.52), 1] s μ tn (ds). (4.56) Let us estimate the last integral. We write ( ,1] s μ t (ds) = tλ −1 s ν(ds).
We now turn to showing (4.59). We have which, together with the definition of λ, implies (4.59).
Since the support of the probability distribution of Y 0 is not the whole real line, by [42, Lemma 2.5], the inequality (4.58) implies that To conclude (4.49), it is enough to show that χ ≤ −ρ. Since ρ(t n ) ≤ 1 2 t n λ −1 n b λn , the latter can be deduced from s μ tn (ds), (4.62) where the last equality is a consequence of (4.48) since Therefore, the problem is reduced to showing (4.62). By the monotone convergence theorem and (4.52), we have where f is as in (4.55). Hence, we just need to justify the change in the order of limits.
In view of the Moore-Osgood theorem [19,Chapter VII], it is enough to show that the limit in (4.65) is uniform with respect to n ∈ N. We write s μ t (ds).
and for all n ∈ N, (4.73) Thus, 1 2 ϕ ≤ ϕ 1 ≤ ϕ, and so for all u > 0, In particular, −φ 1 has the weak lower scaling property. Therefore, by Theorem 3.3, T (1) t and T t are absolutely continuous. Let us denote by p(t, · ) and p (1) (t, · ) the densities of T t and T (1) t , respectively. Observe that T (2) is a compound Poisson process with the probability distribution denoted by P t (dx). By [47,Remark 27.3], (4.75) We apply Theorem 4.11 to the process T (1) . For t > 0, we set Then there are C > 0 and ρ 0 > 0, such that for all t ∈ (0,1/ϕ(x 0 )) and x ≥ 0 satisfying we have Next, if λ ≥ ρ 0 /ϕ −1 (1/t), then, by (4.2), where the penultimate inequality follows either by monotonicity of h or by [23,Lemma 2.1 (4)]. Finally, by (4.75) and (4.77), for x ≥ 2λ we can compute Hence, by the monotonicity of ν, we get where in the last estimate we have used (4.76). Using (4.73) and (4.74), we can easily show that and the proposition follows.

Sharp two-sided estimates
In this section we present sharp two-sided estimates on the density p(t, · ) assuming both the weak lower and upper scaling properties on −φ . First, following [6, Lemma 13], we prove an auxiliary result.

Proposition 4.15.
Assume that the Lévy measure ν(dx) has an almost monotone density ν(x). Suppose that −φ ∈ WUSC(γ,C,x 0 ) for some C ≥ 1, x 0 ≥ 0 and γ < 0. Then there are a ∈ (0,1] and c ∈ (0,1] such that for all 0 < x < a/x 0 , where C 2 is a constant from the almost monotonicity of ν. If u > x 0 , then by the scaling property of −φ we obtain By selecting a ∈ (0,1] such that by (4.78) we obtain provided that u > x 0 . Now, by the monotonicity of −φ we conclude the proof.
In view of Propositions 2.3 and 2.4, we immediately obtain the following corollary.

Subordination
Let (X ,τ ) be a locally compact separable metric space with a Radon measure μ having full support on X . Assume that (X t : t ≥ 0) is a homogeneous in time Markov process on X with density h(t, · , · ); that is, for any Borel set B ⊂ X , x ∈ X and t > 0. Assume that for all t > 0 and x,y ∈ X , where n and γ are some positive constants, Φ 1 and Φ 2 are nonnegative nonincreasing function on [0,∞) such that Φ 1 (1) > 0 and By H(t,x,y) we denote the heat kernel for the subordinate process (X Tt : t ≥ 0); that is, where G(t,s) = P T t ≥ s .
Now, one can take T with a Lévy-Khintchine exponent regularly varying at infinity with index α ∈ (0,1) and apply Claim 5.1 to obtain the asymptotic behaviour of the subordinate process.

Green function estimates
Let T = (T t : t ≥ 0) be a subordinator with the Laplace exponent φ. If −φ has the weak lower scaling property of index α − 2 for some α > 0, then the probability distribution of T t has a density p(t, · ); see Theorem 3.3. In this section we want to derive sharp estimates on the Green function based on Sections 3 and 4. Let us recall that the Green function is We set Let us denote by f −1 the generalised inverse of f ; that is, Notice that by (2.9) and Proposition 2.3, for all x > x 0 , f * (x) x. (5.14) In view of (4.2) and Proposition 4.3, the function ϕ is almost increasing; thus, by monotonicity of φ , f is almost increasing as well. Therefore, there is c 0 ∈ (0,1] such that for all x > x 0 , Moreover, f has the doubling property on (x 0 ,∞). Since ϕ belongs to WLSC(α,c,x 0 ), by monotonicity of φ , we conclude that f belongs to WLSC(α,c,x 0 ). It follows that f −1 ∈ WUSC(1/α,C,f * (x 0 )) for some C ≥ 1 and since f −1 is increasing, we infer that f −1 also has doubling property on (f * (x 0 ),∞).
In particular, for each A > 0 there is C > 0 such that for all x < A/x 0 , Proof. For M > 0 and x > 0 we set Let us first show that for each M > 0 there are A M > 0 and C ≥ 1 such that for all x < A M /x 0 , where M 0 is determined in Corollary 3.5, and c 0 is taken from (5.15). We claim that the following holds true.
. (5.17) where the implicit constants may depend on M. Therefore, by monotonicity of f −1 and for all x > x 0 . LetT be a subordinator with the Laplace exponentφ. Byp(t, · ) we denote the density of the probability distribution ofT t . We set Fix M > 0. By Claim 5.5, there is A M > 0 such that for all x < A M /x 0 , p(t,x) dt are both positive and continuous; thus, they are bounded for each A. Therefore, at the possible expense of worsening the constant, we can conclude the proof of the proposition. Proposition 5.6. Suppose that b = 0, −φ ∈ WLSC(α − 2,c,x 0 ) for some c ∈ (0,1], x 0 ≥ 0 and α > 0 and that the Lévy measure ν(dx) is absolutely continuous with respect to the Lebesgue measure with a monotone density ν(x). Then there is ∈ (0,1) such that for each A > 0, there is C ≥ 1 such that for all x < A/x 0 , x φ (f −1 (1/x)) 0 p(t,x) dt ≤ C 1 xφ (1/x) .