1. Introduction
Asymptotic behaviour as well as estimates of heat kernels have been intensively studied in the last decades. The first results obtained by Pòlya [Reference Pólya44] and Blumenthal and Getoor [Reference Blumenthal and Getoor4] for isotropic $\alpha $ -stable process in $\mathbb {R}^d$ provided the basis for studies of more complicated processes; for example, subordinated Brownian motions [Reference Mimica40, Reference Song49], isotropic unimodal Lévy processes [Reference Bogdan, Grzywny and Ryznar6, Reference Cygan, Grzywny and Trojan15, Reference Grzywny, Ryznar and Trojan21] and even more general symmetric Markov processes [Reference Chen, Kim and Kumagai11, Reference Chen and Kumagai14]. One may, among others, list the articles on heat kernel estimates for jump processes of finite range [Reference Chen, Kim and Kumagai10] or with lower intensity of higher jumps [Reference Mimica39, Reference Sztonyk51]. 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, [Reference Bogdan, Sztonyk and Knopova7, Reference Ishikawa27, Reference Ishikawa28, Reference Kaleta and Sztonyk31, Reference Kaleta and Sztonyk32, Reference Knopova35, Reference Knopova and Kulik36, Reference Picard42, Reference Picard43, Reference Sztonyk50]. For a more specific class of stable processes, see [Reference Hiraba25, Reference Pruitt and Taylor46, Reference Watanabe53]. 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 [Reference Grzywny and Szczypkowski22] where the authors considered the case of the Lévy measure being comparable to some unimodal Lévy measure. The methods developed in [Reference Grzywny and Szczypkowski22, Reference Grzywny and Szczypkowski23] contributed significantly to this article. See also [Reference Kaleta and Sztonyk32, Reference Picard42] 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 [Reference Bochner5] and Philips [Reference Phillips41]. In the language of the semigroup theory, for a Bernstein function $\phi $ and a bounded $C_0$ -semigroup $\big ( e^{-t\mathcal {A}} \colon t \geq 0\big )$ with $-\mathcal {A}$ being its generator on some Banach space $\mathscr {X}$ , via Bochner integral one can define an operator $\mathcal {B}=\phi (\mathcal {A})$ such that $-\mathcal {B}$ also generates a bounded $C_0$ -semigroup $\big ( e^{-t\mathcal {B}} \colon t \geq 0\big )$ on $\mathscr {X}$ . The semigroup $\big ( e^{-t\mathcal {B}} \colon t \geq 0\big )$ is then said to be subordinated to $\big ( e^{-t\mathcal {A}} \colon t \geq 0\big )$ , 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, [Reference Gomilko and Tomilov18] 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 $(X_t \colon t \geq 0)$ is a Markov process and $(T_t \colon t \geq 0)$ is an independent subordinator, then $Y_t=X_{T_t}$ 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 $\mathcal {B}=\phi (\mathcal {A})$ . Here $\mathcal {A}$ and $\mathcal {B}$ 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 $e^{-t\mathcal {B}}$ ) 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 [Reference Chen9, Reference Chen, Kim, Kumagai and Wang12, Reference Meerschaert and Scheffler38]. Here the solutions are expressed in terms of corresponding (inverse) subordinators and thus their analysis is essential.
By taking $\mathcal {A}=-\Delta $ 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 $\alpha $ -stable subordinator with the Laplace exponent $\phi (\lambda )=\lambda ^{\alpha }$ , $\alpha \in (0,1)$ , which gives rise to the symmetric, rotation-invariant $\alpha $ -stable process and corresponds to the special case of fractional powers of semigroup $\big (e^{-t\mathcal {A}^{\alpha }} \colon t \geq 0\big )$ . For this reason, distributional properties of subordinators were often studied with reference to heat kernel estimates of subordinated Brownian motions (see, e.g., [Reference Fahrenwaldt16, Reference Kim and Mimica33]). In [Reference Hawkes24] Hawkes investigated the growth of sample paths of a stable subordinator and obtained the asymptotic behaviour of its distribution function. Jain and Pruitt [Reference Jain and Pruitt30] 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 [Reference Fristedt and Pruitt17, Reference Iksanov, Kabluchko, Marynych and Shevchenko26, Reference Picard42, Reference Vasudeva and Divanji52]. In [Reference Burridge, Kuznetsov, Kwaśnicki and Kyprianou8] new examples of families of subordinators with explicit transition densities were given. Finally, in the recent paper [Reference Fahrenwaldt16], 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 $\phi $ . Our standing assumption on $-\phi ^{\prime \prime }$ is the weak lower scaling condition at infinity with scaling parameter $\alpha -2$ , for some $\alpha>0$ (see (2.7) for definition). It is worth highlighting that we do not state our assumptions and results in terms of the Laplace exponent $\phi $ , as one could suspect, but in terms of its second derivative and related function $\varphi (x)=x^2(-\phi ^{\prime \prime }(x))$ (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 $\psi ^{-1}(x)$ (e.g., [Reference Grzywny and Szczypkowski23], [Reference Knopova and Kulik36]), but in our setting one can show that the lower scaling property implies that $\varphi ^{-1}(x) \approx \psi ^{-1}(x)$ for x sufficient large (see Proposition 4.3). In some cases, however, $\varphi $ may be significantly different from the Laplace exponent $\phi $ . However, if one assumes additional upper scaling condition with scaling parameter $\beta -2$ for $\beta $ 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 $\phi $ (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 $\nu ({\textrm {d}} x)$ . 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 [Reference Fahrenwaldt16]; 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 $\nu ({\textrm {d}} x)$ ; see Theorem 4.7. Clearly, putting some restrictions on $\nu ({\textrm {d}} x)$ 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 $\phi $ . However, in many cases the primary object is the Lévy measure $\nu ({\textrm {d}}x)$ 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 $\nu ((x,\infty ))$ instead, as they imply the scaling condition on $-\phi ^{\prime \prime }$ .
We also note that the main results of the article hold true when $-\phi ^{\prime \prime }$ is a function regularly varying at infinity with regularity index $\alpha -2$ , where $\alpha>0$ . This follows easily by Potter bounds for regularly varying functions (see [Reference Bingham, Goldie and Teugels3, Theorem 1.5.6]), which immediately imply both lower and upper scaling properties. Moreover, if additionally $\alpha <1$ , then, by Karamata’s theorem and monotone density theorem, regular variation of $-\phi ^{\prime \prime }$ with index $\alpha -2$ is equivalent to regular variation of $\phi $ with index $\alpha $ . This is not the case for the case $\alpha =1$ 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 $0 < \alpha \leq \beta < 1$ ; see Theorem 4.17.
Theorem A. Let $\mathbf {T}$ be a subordinator with the Laplace exponent $\phi $ . Suppose that for some $0 < \alpha \leq \beta < 1$ , the functions
are almost increasing and almost decreasing, respectively. We also assume that the Lévy measure $\nu ({\mathrm{d}}x)$ has an almost monotone density $\nu (x)$ . Then the probability distribution of $T_t$ has a density $p(t, \:\cdot \:)$ . Moreover, for all $t \in (0, \infty )$ and $x> 0$ ,
where $w=(\phi ')^{-1}(x/t)$ .
We note that a similar result to Theorem A appeared in [Reference Chen, Kim, Kumagai and Wang13] 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 $t \mapsto t\nu (t)$ . 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 [Reference Chen, Kim, Kumagai and Wang13] 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 $\mathbb {R}^d$ setting and Green function estimates are presented in Section 5.
Notation
By $C_1, c_1, C_2, c_2, \ldots $ we denote positive constants which may change from line to line. For two functions $f,g \colon (0,\infty ) \rightarrow [0,\infty )$ we write $f(x) \gtrsim g(x)$ if there is $c>0$ such that $f(x) \geq cg(x)$ for all $x>0$ . An analogous rule is applied to the symbol $\lesssim $ . We also have $f(x) \approx g(x)$ if there exists $C \geq 1$ such that $C^{-1} f(x) \leq g(x) \leq C f(x)$ for all $x> 0$ . Finally, we set $a \wedge b=\min \{ a,b \}$ and $a \vee b = \max \{a,b\}$ .
2. Preliminaries
Let $(\Omega , \mathcal {F}, \mathbb {P})$ be a probability space. Let $\mathbf {T} = (T_t \colon t \geq 0)$ be a subordinator; that is, a Lévy process in $\mathbb {R}$ 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 $\psi \colon \mathbb {R} \rightarrow \mathbb {C}$ , called the Lévy–Khintchine exponent of $\mathbf {T}$ , such that for all $t \geq 0$ and $\xi \in \mathbb {R}$ ,
Moreover, there are $b \geq 0$ and $\sigma $ -finite measure $\nu $ on $(0, \infty )$ satisfying
such that for all $\xi \in \mathbb {R}$ ,
which is valid thanks to (2.1). By $\phi \colon [0, \infty ) \rightarrow [0, \infty )$ we denote the Laplace exponent of $\mathbf {T}$ , namely,
for all $t \geq 0$ and $\lambda \geq 0$ . Let $\psi ^*$ be the symmetric continuous and nondecreasing majorant of $\Re \psi $ ; that is,
Notice that
where $\psi ^{-1}$ is the generalised inverse function defined as
To study the distribution function of the subordinator $\mathbf {T}$ , it is convenient to introduce two concentration functions K and h. They are defined as
and
Notice that $h(r) \geq K(r)$ . Moreover, by the Fubini–Tonelli theorem, we get
In view of [Reference Grzywny20, Lemma 4], we have
A function $f\colon [0, \infty ) \rightarrow [0, \infty )$ is regularly varying at infinity of index $\alpha $ , if for all $\lambda \geq 1$ ,
Analogously, f is regular varying at the origin of index $\alpha $ if for all $\lambda \geq 1$ ,
If $\alpha = 0$ , 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 $f \colon [0,\infty ) \rightarrow [0,\infty )$ has the weak lower scaling property at infinity if there are $\alpha \in \mathbb {R}$ , $c \in (0,1]$ and $x_0 \geq 0$ such that for all $\lambda \geq 1$ and $x> x_0$ ,
We denote it briefly as $f \in \textrm {WLSC}(\alpha ,c,x_0)$ . Observe that if $\alpha> \alpha '$ then $\textrm {WLSC}(\alpha ,c,x_0) \subsetneq \textrm {WLSC}(\alpha ',c,x_0)$ . Analogously, f has the weak upper scaling property at infinity if there are $\beta \in \mathbb {R}$ , $C \geq 1$ , and $x_0 \geq 0$ such that for all $\lambda \geq 1$ and $x> x_0$ ,
In this case we write $f \in \textrm {WUSC}(\beta ,C,x_0)$ .
We say that a function $f\colon [0, \infty ) \rightarrow [0, \infty )$ has doubling property on $(x_0, \infty )$ for some $x_0 \geq 0$ if there is $C \geq 1$ 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.
A function $f\colon [0, \infty ) \rightarrow [0, \infty )$ is almost increasing on $(x_0, \infty )$ for some $x_0 \geq 0$ if there is $c \in (0, 1]$ such that for all $y \geq x> x_0$ ,
It is almost decreasing on $(x_0, \infty )$ if there is $C \geq 1$ such that for all $y \geq x> x_0$ ,
In view of [Reference Bogdan, Grzywny and Ryznar6, Lemma 11], $f \in \textrm {WLSC}(\alpha ,c,x_0)$ if and only if the function
is almost increasing. Similarly, $f \in \textrm {WUSC}(\beta ,C,x_0)$ if and only if the function
is almost decreasing. For a function $f\colon [0,\infty ) \rightarrow \mathbb {C}$ its Laplace transform is defined as
2.1. Bernstein functions
In this section we recall some basic facts about Bernstein functions. A general reference here is the book [Reference Schilling, Song and Vondraček48].
A function $f\colon (0, \infty ) \rightarrow [0,\infty )$ is completely monotone if it is smooth and
for all $n \in \mathbb {N}_0$ . Next, a function $\phi $ is a Bernstein function if it is a nonnegative smooth function such that $\phi '$ is completely monotone.
Let $\phi $ be a Bernstein function. In view of [Reference Jacob29, Lemma 3.9.34], for all $n \in \mathbb {N}$ we have
Since $\phi $ is concave, for each $\lambda \geq 1$ and $x> 0$ we have
thus, by (2.9),
By [Reference Schilling, Song and Vondraček48, Theorem 3.2], there are two nonnegative numbers a and b and a Radon measure $\mu $ on $(0, \infty )$ satisfying
and such that
A Bernstein function $\phi $ is called a complete Bernstein function if the measure $\mu $ has a completely monotone density with respect to the Lebesgue measure.
Proposition 2.1. Let f be a completely monotone function. Suppose that f has a doubling property on $(x_0,\infty )$ for some $x_0\geq 0$ . Then there is $C> 0$ such that for all $x> x_0$ ,
Proof. Without loss of generality, we can assume $f \not \equiv 0$ . Clearly,
Since f is completely monotone, it is a positive function and
which, together with the doubling property, gives
for $x> 2 x_0$ . 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 $-f'\in \textrm {WLSC}(\tau ,c,x_0)$ for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\tau \leq -1$ . Then $f\in \textrm {WLSC}(1 + \tau ,c,x_0)$ .
Analogously, if $-f' \in \textrm {WUSC}(\tau ,C,x_0)$ for some $C \geq 1$ , $x_0 \geq 0$ and $\tau \leq -1$ , then $(f-f(\infty )) \in \textrm {WUSC}(\tau ,C,x_0)$ .
Proof. Let $\lambda>1$ . For $y> x > x_0$ , we have
thus,
Since f is nonnegative and nonincreasing, we can take y approaching infinity to get
where in the last inequality we have also used that $1 \geq c \lambda ^{1+\tau }$ . The second part of the proposition can be proved in much the same way.
Proposition 2.3. Let $\phi $ be a Bernstein function with $\phi (0)=0$ . Then $\phi \in \textrm {WLSC}(\alpha ,c,x_0)$ for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\alpha>0$ if and only if $\phi ' \in \textrm {WLSC}(\alpha -1,c',x_0)$ for some $c' \in (0,1]$ . Furthermore, if $\phi \in \textrm {WLSC}(\alpha ,c,x_0)$ , then there is $C \geq 1$ such that for all $x> x_0$ ,
Proof. Assume first that $\phi ^{\prime }\in \textrm {WLSC}(\alpha -1,c,x_0)$ . Without loss of generality, we can assume $\phi ^{\prime } \not \equiv 0$ . We claim that (2.12) holds true. In view of (2.9), it is enough to show that there is $C \geq 1$ such that for all $x> x_0$ ,
First, let us observe that, by the weak lower scaling property of $\phi ^{\prime }$ ,
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 $x> x_0$ ,
Since $\phi ' \in \textrm {WLSC}(\alpha -1,c,x_0)$ , the function
is almost increasing. Hence, for $x \geq 2 x_0$ we have
To conclude (2.14), we notice that $\phi '(x)$ 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 $\phi $ follows.
Now assume that $\phi \in \textrm {WLSC}(\alpha ,c,x_0)$ . By monotonicity of $\phi '$ , for $0 < s < t$ ,
For $s=1$ , by the lower scaling we get
for all $x>x_0$ . Thus, for $t = 2^{1/\alpha } c^{-1/\alpha }$ , we obtain that $x\phi '(x) \gtrsim \phi (x)$ for all $x>x_0$ . Invoking (2.9), we conclude (2.12). In particular, $\phi '$ has the weak lower scaling property. This completes the proof.
Proposition 2.4. Let $\phi $ be a Bernstein function. Suppose that $-\phi ^{\prime \prime } \in \textrm {WUSC}(\beta -2,C,x_0)$ for some $C \geq 1$ , $x_0 \geq 0$ and $\beta <1$ . Then for all $x> x_0$ ,
where b is the drift term from the integral representation (2.11) of $\phi $ .
Proof. Without loss of generality, we can assume $\phi ^{\prime \prime } \not \equiv 0$ . By the scaling property, for $x> x_0$ we have
which concludes the proof.
Remark 2.5. Let $\phi $ be a Bernstein function such that $\phi (0) = 0$ . Suppose that $-\phi ^{\prime \prime } \in \textrm {WLSC}(\alpha -2,c,x_0)$ , for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\alpha \in (0, 1]$ . Since $\phi '$ is completely monotone, by Proposition 2.2, $\phi ' \in \textrm {WLSC}(\alpha -1,c,x_0)$ . Therefore, by Proposition 2.3, we conclude that $\phi \in \textrm {WLSC}(\alpha ,c_1,x_0)$ for some $c_1 \in (0, 1]$ .
Proposition 2.6. Let f be a completely monotone function. Suppose that
for some $c \in (0,1]$ , $C \geq 1$ , $x_0 \geq 0$ and $0<\alpha \leq \beta <1$ . Then
for some $c' \in (0,1]$ and $C' \geq 1$ .
Proof. By monotonicity of $-f'$ , for $0 < s < t$ ,
Taking $s=1$ in the second inequality, the weak upper scaling property yields
for all $x>x_0$ . By selecting $t> 1$ such that $ct^{\beta -1} \leq \frac 12$ , we obtain $x\big (-f'(x)\big ) \gtrsim f(x)$ for $x>x_0$ . 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 $s^{\alpha -1}\geq 2c$ , we obtain $x\big ( -f'(x)\big ) \lesssim f(x)$ for $x>x_0/s$ . Hence,
for all $x>x_0/s$ . Therefore, lower and upper scaling properties follow from (2.16) and the scaling properties of f. This finishes the proof for the case $x_0 = 0$ . If $x_0> 0$ , we notice that both f ad $-f'$ are positive and continuous; thus, at the possible expense of worsening the constants, we get (2.16) for all $x>x_0$ .
Now, by combining Propositions 2.3 and 2.6, we immediately get the following corollary.
Corollary 2.7. Let $\phi $ be a Bernstein function such that $\phi (0)=0$ . Suppose that
for some $c \in (0,1]$ , $C \geq 1$ , $x_0 \geq 0$ and $0<\alpha \leq \beta <1$ . Then
for some $c' \in (0,1]$ and $C' \geq 1$ .
Lemma 2.8. Let $\phi $ be a Bernstein function. Suppose that $-\phi ^{\prime \prime } \in \textrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\alpha>0$ . There is a constant $C> 0$ such that for all $x> x_0$ ,
Moreover, the constant C depends only on $\alpha $ and c.
Proof. Let $f\colon (0, \infty ) \rightarrow \mathbb {R}$ 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 $x = \lambda u> 2 x_0$ , by the weak lower scaling property of $-\phi ^{\prime \prime }$ ,
At this stage, we select $u> 2$ such that
Then again, by the weak lower scaling property of $-\phi ^{\prime \prime }$ , for $\lambda>x_0$ ,
which ends the proof.
Lemma 2.9. Let $\phi $ be a Bernstein function. Suppose that $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\alpha>0$ . Then there exists a complete Bernstein function f such that $f\approx \phi $ for $x>0$ and $-f^{\prime \prime }\approx -\phi ^{\prime \prime }$ for $x>x_0$ .
Proof. Let us define
By [Reference Schilling, Song and Vondraček48, Theorem 6.2 (vi)] the function f is a complete Bernstein function. Since for $y> 0$ ,
we get $f(\lambda ) \approx \phi (\lambda )$ . Moreover,
Hence, by Lemma 2.8 we obtain $-f^{\prime \prime }(\lambda )\approx -\phi ^{\prime \prime }(\lambda )$ for $\lambda>x_0$ .
3. Asymptotic behaviour of densities
Let $\mathbf {T} = (T_t \colon t \geq 0)$ be a subordinator with the Lévy–Khintchine exponent $\psi $ and the Laplace exponent $\phi $ . Since $\phi $ is a Bernstein function, it admits the integral representation (2.11). As it may be easily checked (see, e.g., [Reference Schilling, Song and Vondraček48, Proposition 3.6]), we have $\mu = \nu $ , $a = 0$ and $\psi (\xi ) = \phi (-i\xi )$ . In particular, $\phi (0)=0$ .
In this section we study the asymptotic behaviour of the probability density of $T_t$ . In the whole section we assume that $\phi ^{\prime \prime }\not \equiv 0$ ; otherwise, $T_t=b t$ is deterministic. The main result is Theorem 3.3. Let us start by showing an estimate on the real part of the complex extension $\phi $ .
Lemma 3.1. Suppose that $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\alpha>0$ . Then there exists $C> 0$ such that for all $w> x_0$ and $\lambda \in \mathbb {R}$ ,
Proof. By the integral representation (2.11), for $\lambda \in \mathbb {R}$ we have
In particular,
Thus, it is sufficient to consider $\lambda> 0$ . We can estimate
Due to Lemma 2.8, we obtain, for $\lambda \geq w$ ,
If $w> \lambda > 0$ , then, by (3.1), we have
which, by Lemma 2.8, completes the proof.
Remark 3.2. Suppose that $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\alpha>0$ . Since
by Lemma 2.8 we obtain
for all $x> x_0$ .
Theorem 3.3. Let $\mathbf {T}$ be a subordinator with the Laplace exponent $\phi $ . Suppose that $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\alpha>0$ . Then the probability distribution of $T_t$ is absolutely continuous for all $t>0$ . If we denote its density by $p(t, \: \cdot \:)$ , then for each $\epsilon> 0$ there is $M_0> 0$ such that
provided that $w> x_0$ and $t w^2 (-\phi ^{\prime \prime }(w))> M_0$ .
Proof. Let $x=t \phi '(w)$ and $M> 0$ . We first show that
provided that $w> x_0$ and $t w^2 (-\phi ^{\prime \prime }(w))> M$ , where for $\lambda \in \mathbb {R}$ we have set
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 $p(t, \: \cdot \:)$ and
Therefore, our task is to justify the statement (3.4). For $L> 0$ , we write
By the change of variables
we obtain
Let us note here that $-\phi ^{\prime \prime }$ is nonincreasing and integrable at infinity; thus, we in fact have $\alpha \leq 1$ . We claim that there is $C> 0$ not depending on M, such that for all $u \in \mathbb {R}$ ,
provided that $w> x_0$ and $t w^2 (-\phi ^{\prime \prime }(w))> M$ . 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 ${\lvert {u} \rvert } \leq w \sqrt {t (-\phi ^{\prime \prime }(w))}$ , then
Otherwise, since $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ , we obtain
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 $\epsilon> 0$ there is $M_0> 0$ such that
provided that $w> x_0$ and $t w^2 (- \phi ^{\prime \prime }(w))> M_0$ . In view of (3.5), by taking $M_0>1$ sufficiently large, we get
and
Next, we claim that there is $C> 0$ such that
Indeed, since
by Taylor’s formula, we get
where $\xi $ is some number satisfying
Observe that
Since $-\phi ^{\prime \prime }$ 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 $t w^2 (-\phi ^{\prime \prime }(w))> M_0$ . Now, (3.10) easily follows by (3.13) and (3.11).
Finally, since for any $z \in \mathbb {C}$ ,
by (3.10) we obtain
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.
Remark 3.4. If $x_0 = 0$ , then the constant $M_0$ in Theorem 3.3 depends only on $\alpha $ and c. If $x_0> 0$ , it also depends on
By Theorem 3.3, we immediately get the following corollaries.
Corollary 3.5. Suppose that $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\alpha>0$ . Then there is $M_0> 0$ such that
uniformly on the set
where $w = (\phi ')^{-1}(x/t)$ .
Corollary 3.6. Suppose that $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\alpha>0$ . Assume also that $b = 0$ . Then for any $x>0$ ,
where $w = (\phi ')^{-1}(x/t)$ .
By imposing on $-\phi ^{\prime \prime }$ 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.
Corollary 3.7. Suppose that $\phi \in \mathrm {WLSC}(\alpha ,c,x_0) \cap \mathrm {WUSC}(\beta ,C,x_0)$ for some $c \in (0, 1]$ , $C \geq 1$ , $x_0 \geq 0$ and $0 < \alpha \leq \beta < 1$ . Assume also that $b = 0$ . Then there is $\delta> 0$ such that
uniformly on the set
where $w = (\phi ')^{-1}(x/t)$ .
Proof. By Proposition 2.3, there is $C_1 \geq 1$ such that for all $u> x_0$ ,
thus, for $(t, x)$ belonging to the set (3.14),
By Proposition 2.3, $\phi ' \in \mathrm {WLSC}(-1+\alpha ,c,x_0)$ ; hence, for all $D \geq 1$ ,
By taking $\delta $ sufficiently small, we get
thus, by (3.15), we obtain
which implies that
In particular, $w> x_0$ . On the other hand, by Propositions 2.3 and 2.4, there is $c_1 \in (0,1]$ such that
By Remark 2.5, $\phi \in \mathrm {WLSC}(\alpha ,c_2,x_0)$ for some $c_2 \in (0,1]$ . Therefore,
which, together with (3.16), gives
for $\delta $ sufficiently small. Hence, by Corollary 3.5, we conclude the proof.
The following proposition provides a sufficient condition on the measure $\nu $ that entails the weak lower scaling property of $-\phi ^{\prime \prime }$ and allows us to apply Theorem 3.3.
Proposition 3.8. Suppose that there are $x_0 \geq 0$ , $C \geq 1$ and $\alpha> 0$ such that for all $0 < r < 1/x_0$ and $0 < \lambda \leq 1$ ,
Then $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0, 1]$ .
Proof. Let us first notice that by the Fubini–Tonelli theorem,
Thus, by (3.17), for all $0 < r < 1/x_0$ and $0 < \lambda \leq 1$ ,
Hence, by [Reference Grzywny and Szczypkowski23, Lemma 2.3], there is $C' \geq 1$ such that for all $0 < r <1/x_0$ ,
The integral representation of $\phi $ entails that
thus, by (3.19), we obtain
for all $x> x_0$ . Now, the weak lower scaling property of $-\phi ^{\prime \prime }$ is a consequence of (3.18).
4. Estimates on the density
Let $\mathbf {T} = (T_t \colon t \geq 0)$ be a subordinator with the Lévy–Khintchine exponent $\psi $ and the Laplace exponent $\phi $ . In this section we always assume that $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\alpha \in (0, 1]$ . In particular, by Theorem 3.3, the probability distribution of $T_t$ has a density $p(t, \: \cdot \:)$ . To express the majorant on $p(t, \: \cdot \:)$ , it is convenient to set
Obviously, $\varphi \in \mathrm {WLSC}(\alpha ,c,x_0)$ . Let $\varphi ^{-1}$ denote the generalised inverse function defined as
where
We start by showing comparability between the two concentration functions K and h defined in (2.3) and (2.4), respectively.
Proposition 4.1. Suppose that $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\alpha>0$ . Then there is $C \geq 1$ such that for all $0 < r < 1/x_0$ ,
Proof. Since $h(r) \geq K(r)$ , it is enough to show that for some $C \geq 1$ and $0 < r < 1/x_0$ ,
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 $K(r) \approx \varphi (1/r)$ , for $0<r<1/x_0$ , which implies
This finishes the proof in the case $x_0 = 0$ . If $x_0> 0$ , then, for $1/(2x_0)\leq r<1/x_0$ , we have
Hence, $K(r) \gtrsim 1$ 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.
Let us notice that by (2.6), Proposition 4.1 and Remark 3.2, we have
for all $x> x_0$ . In particular, there is $c_1 \in (0, 1]$ such that $\psi ^* \in \mathrm {WLSC}(\alpha ,c_1,x_0)$ . Moreover,
thus, for all $x> x_0$ ,
Since for $\lambda \geq 1$ and $x> 0$ ,
we get
Proposition 4.2. Suppose that $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\alpha>0$ . Then for all $r> 2 h(1/x_0)$ ,
Furthermore, there is $C \geq 1$ such that for all $\lambda \geq 1$ and $r> 2 h(1/x_0)$ ,
Proof. Using (2.6), we immediately get
for all $r>0$ . On the other hand, by Proposition 4.1 and [Reference Grzywny and Szczypkowski23, Lemma 2.3], there is $C \geq 1$ such that for all $\lambda \geq 1$ and $r> h(1/x_0)$ ,
Hence, for $r> 2 h(1/x_0)$ ,
proving (4.6). The weak upper scaling property of $\psi ^{-1}$ is a consequence of (4.7) and (4.8).
Proposition 4.3. Suppose that $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\alpha>0$ . Then for all $x> x_0$ ,
and for all $r> \varphi (x_0)$ ,
Furthermore, there is $C \geq 1$ such that for all $\lambda \geq 1$ and $r> \varphi (x_0)$ ,
Proof. We start by showing that there is $C \geq 1$ 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 $r> C \psi ^*(x_0)$ . Hence, by Proposition 4.2,
for $r> C \max {\big \{\psi ^*(x_0), 2 h(1/x_0)\big \}}$ . Finally, since both $\psi ^{-1}$ and $\varphi ^{-1}$ are positive and continuous, at the possible expense of worsening the constant, we can extend the area of comparability to conclude (4.10). Now, the scaling property of $\varphi ^{-1}$ follows by (4.10) and Proposition 4.2.
Remark 4.4. Note that, alternatively, one can define the (left-sided) generalised inverse
where
In such a case we have
Clearly, for all $x>0$ ,
Let $u>x_0$ and set
By Proposition 4.3, $\varphi ^* \in \mathrm {WLSC}(\alpha ,c,x_0)$ for some $c \in (0,1]$ and $x_0 \geq 0$ . Thus, for $\lambda> c^{-1/\alpha }$ , we get $\varphi ^*(\lambda r_0)> \varphi ^*(r_0)$ . It follows that for all $u>x_0$ ,
Thus, for all $r>x_0$ ,
Corollary 4.5. Suppose that $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\alpha>0$ . Then there is $C> 0$ such that for all $x> x_0$ ,
Proof. Suppose $x_0>0$ . We have
where
By the weak lower scaling property of $\varphi $ , for any $x_0/x < u \leq 1$ , we have
thus,
We denote $c_1 = \phi (x_0)-x_0 \phi '(x_0^+)$ . Using the scaling property of $\varphi $ , we conclude that
provided that $x>x_0$ , which proves (4.12) if $x_0>0$ . For $x_0=0$ it is enough to observe that (2.9) implies that $\lim _{x \to 0^+}x \phi '(x)=0$ , and the claim follows.
Proposition 4.6. Suppose that $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0) \cap \mathrm {WUSC}(\beta -2,C,x_0)$ for some $c \in (0, 1]$ , $C \geq 1$ , $x_0 \geq 0$ and $0 < \alpha \leq \beta < 1$ . Assume also that $b = 0$ . Then for all $x> x_0$ ,
and for all $r>\varphi (x_0)$ ,
Furthermore, there is $c' \in (0, 1]$ such that for all $\lambda \geq 1$ and $r> \varphi (x_0)$ ,
Proof. Let us observe that, by (2.9), Proposition 2.3 and Proposition 2.4, there is $c_1 \in (0, 1]$ such that for all $x> x_0$ ,
Now the proof of the lemma is similar to the proof of Proposition 4.3 and is therefore omitted.
4.1. Estimates from above
In this section we show the upper estimates on $p(t, \: \cdot \:)$ . Before embarking on the proof, let us introduce some notation. Given a set $B \subset \mathbb {R}$ , we define
and
Let
In view of (2.2), the above definition of $b_r$ is in line with the usual one (see, e.g., [Reference Kaleta and Sztonyk32, formula (4)] or [Reference Grzywny and Szczypkowski23, formula (1.2)]). Let us define $\zeta \colon [0, \infty ) \rightarrow [0, \infty ]$ ,
where $A = \varphi ^*(x_0)/\phi (x_0) \in (0, 2]$ .
Theorem 4.7. Let $\mathbf {T}$ be a subordinator with the Lévy–Khintchine exponent $\psi $ and the Laplace exponent $\phi $ . Suppose that $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\alpha>0$ . Then the probability distribution of $T_t$ has a density $p(t, \: \cdot \:)$ . Moreover, there is $C> 0$ such that for all $t \in (0, 1/\varphi (x_0))$ and $x \in \mathbb {R}$ ,
In particular, for all $t \in (0, 1/\varphi (x_0))$ and $x \geq 2e t\phi '(\psi ^{-1}(1/t))$ ,
Proof. Without loss of generality, we can assume $b = 0$ . Indeed, otherwise it is enough to consider a shifted process $\widetilde {T_t} = T_t - tb$ . Next, let us observe that for any Borel set $B \subset \mathbb {R}$ , we have
Furthermore, for $\delta (B)<1/x_0$ , by Proposition 4.1 and Remark 3.2,
Thus, $\nu (B) \lesssim \zeta (\delta (B))$ . We claim that $\zeta $ has doubling property on $(0, \infty )$ . Indeed, since $-\phi ^{\prime \prime }$ is nonincreasing function with the weak lower scaling property, it has doubling property on $(x_0, \infty )$ ; thus, for $0 < s < x_0^{-1}$ ,
This completes the argument in the case $x_0 = 0$ . If $x_0> 0$ , then by (2.10), for $s> 2 x_0^{-1}$ we have
Lastly, the function
is continuous and thus it is bounded.
Next, for $s> 0$ and $x \in \mathbb {R}$ ,
thus, by motonicity and the doubling property of $\zeta $ , we get
Hence, by (2.4) and (4.2), for $r> 0$ ,
Since $\psi ^*$ has the weak lower scaling property and satisfies (4.3), by [Reference Grzywny and Szczypkowski23, Proposition 3.4] together with Proposition 4.2, there are $C> 0$ and $t_1 \in (0, \infty ]$ such that for all $t \in (0, t_1)$ ,
If $x_0 = 0$ , then $t_1 = \infty $ . If $t_1 < 1/\varphi (x_0)$ , we can expand the above estimate for $t_1\leq t< 1/\varphi (x_0)$ using positivity of the right-hand side and monotonicity of the left-hand side.
In view of (4.19), (4.20) and (4.21), by [Reference Kaleta and Sztonyk32, Theorem 1] with $\gamma =\,0$ , there are $C_1,C_2,C_3> 0$ such that for all $t \in (0 , 1/\varphi (x_0))$ and $x \in \mathbb {R}$ ,
Let us consider $x> 0$ and $t \in (0, 1/\varphi (x_0))$ such that $t \zeta (x) \leq 1$ . We claim that
First suppose that $x>x_0^{-1}$ . Let us observe that the function
is bounded. Therefore,
Since $x \phi ^{-1}(1/t) \geq 1$ , by (2.10), we have
Next, in light of (2.9), for all $y> 0$ ,
hence, by the monotonicity of $\phi ^{-1}$ ,
where in the last step we have used Proposition 4.3. Putting (4.23), (4.24) and (4.25) together, we obtain (4.22) as claimed.
Now let $0<x \leq x_0^{-1}$ . Observe that the function
is also bounded. Hence,
Since $x \varphi ^{-1}(1/t) \geq 1$ , using (4.5) we get
Hence, putting together (4.26) and (4.27) and invoking Proposition 4.2, we again obtain (4.22).
Finally, using doubling property of $\zeta $ we get
thus, another application of Proposition 4.3 leads to (4.17).
For the proof of (4.18), we observe that
Thus,
Hence, by monotonicity and the doubling property of $\zeta $ , for $x> 2e t \phi '(\psi ^{-1}(1/t))$ , we obtain
and the theorem follows.
Now we define $\eta \colon [0, \infty ) \rightarrow [0, \infty ]$ ,
where $A = \varphi ^*(x_0)/\phi (x_0) \in (0, 2]$ . Notice that, by (2.9), if $2 t \zeta ({\lvert {x} \rvert }) \leq 1$ , then $t \varphi ^*(1/{\lvert {x} \rvert }) \leq 1$ , and so
Therefore,
Theorem 4.8. Let $\mathbf {T}$ be a subordinator with the Lévy–Khintchine exponent $\psi $ and the Laplace exponent $\phi $ . Suppose that $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0,1]$ , $x_0 \geq 0$ and $\alpha>0$ . We also assume that the Lévy measure $\nu $ has an almost monotone density $\nu (x)$ . Then the probability distribution of $T_t$ has a density $p(t, \: \cdot \:)$ . Moreover, there is $C> 0$ such that for all $t\in (0,1/\varphi (x_0))$ and $x \in \mathbb {R}$ ,
In particular, for all $t \in (0, 1/\varphi (x_0))$ and $x \geq 2e t\phi '(\psi ^{-1}(1/t))$ ,
Proof. Without loss of generality, we can assume $b = 0$ . Let us observe that for any $\lambda> 0$ ,
and
Hence,
Since $\eta $ is nonincreasing, for any Borel subset $B \subset \mathbb {R}$ ,
Arguing as in the proof of Theorem 4.7, we conclude that $\eta $ has a doubling property on $(0,\infty )$ . Using that and monotonicity of $\eta $ , for $s> 0$ and $x \in \mathbb {R}$ ,
Therefore, by (4.2), for $r> 0$ ,
Since $\psi ^*$ has the weak lower scaling property and satisfies (4.3), by [Reference Grzywny and Szczypkowski23, Theorem 3.1] and Proposition 4.2, there are $C> 0$ and $t_1 \in (0, \infty ]$ such that for all $t \in (0, t_1)$ ,
If $x_0 = 0$ , then $t_1 = \infty $ . If $t_1 < 48/\varphi (x_0)$ , we can expand the above estimate for $t_1\leq t< 48/\varphi (x_0)$ using positivity of the right-hand side and monotonicity of the left-hand side.
In view of (4.31), (4.32) and (4.33), by [Reference Grzywny and Szczypkowski22, Theorem 2.1], there is $C> 0$ such that for all $t \in (0, 1/\varphi (x_0))$ and $x \in \mathbb {R}$ ,
We claim that
whenever $t \eta ({\lvert {x} \rvert }) \leq \tfrac {A}{2} \varphi ^{-1}(1/t)$ .
First, let us show that for any $\epsilon \in (0, 1]$ , the condition $t \eta ({\lvert {x} \rvert }) \leq \tfrac {A \epsilon }{2} \varphi ^{-1}(1/t)$ implies that
Indeed, by (2.9), we have ${\lvert {x} \rvert } \eta ({\lvert {x} \rvert }) \geq \tfrac {A}{2} \varphi ^*(1/{\lvert {x} \rvert })$ ; thus,
Notice also that $\epsilon ^{1/3} {\lvert {x} \rvert } \varphi ^{-1}(1/t) \geq 1$ since otherwise, by (4.5),
which entails that $\epsilon ^{2/3} < t \varphi ^*(1/{\lvert {x} \rvert })$ ; that is, $\epsilon ^{1/3} {\lvert {x} \rvert } \varphi ^{-1}(1/t) < \epsilon ^{-2/3} t \varphi ^*(1/{\lvert {x} \rvert })$ , contrary to (4.35).
To show (4.34), let us suppose that $t \eta ({\lvert {x} \rvert }) \leq \tfrac {A}{2} \varphi ^{-1}(1/t)$ ; thus, ${\lvert {x} \rvert } \varphi ^{-1}(1/t) \geq 1$ . By (4.5), we have
which, by Proposition 4.3, gives
proving (4.34), and (4.28) follows. The inequality (4.29) holds by the same argument as in the proof of Theorem 4.7.
Remark 4.9. In statements of Theorems 4.7 and 4.8, we can replace $b_{1/\psi ^{-1}(1/t)}$ by $b_{1/\varphi ^{-1}(1/t)}$ . Indeed, let us observe that if $0 < r_1 \leq r_2 < 1/x_0$ , then
where in the last estimate we have used (4.2). Hence, by (4.9), we get
Therefore, by (4.36), (4.37) and Proposition 4.3, there is $C \geq 1$ such that
provided that $0 < t < 1/\varphi (x_0)$ . Now, let us suppose that $8 C^2 t \zeta \big ({\lvert {x} \rvert }\big ) \leq 1$ . Then, by (2.10) and (4.5),
that is,
Hence, by (4.38),
which, together with monotonicity and the doubling property of $\zeta $ , gives
Similarly, if $t \eta ({\lvert {x} \rvert }) \leq \tfrac {A \epsilon }{2} \varphi ^{-1}(1/t)$ , then
thus, by taking $\epsilon = (2C)^{-3}$ , we obtain (4.39). Hence, by monotonicity and the doubling property of $\eta $ , we again obtain
4.2. Estimates from below
In this section we develop estimates from below on the density $p(t, \: \cdot \:)$ . The main result is Theorem 4.11. Its proof is inspired by the ideas from [Reference Picard42], see also [Reference Grzywny and Szczypkowski23]. Thanks to Theorem 3.3, we can generalise results obtained in [Reference Picard42] to the case when $-\phi ^{\prime \prime }$ satisfies the weak lower scaling of index $\alpha -2$ for $\alpha>0$ together with a certain additional condition. We use the following variant of the celebrated Pruitt’s result [Reference Pruitt45, Section 3] adapted to subordinators.
Proposition 4.10. Let $\mathbf {T}$ be a subordinator with the Lévy–Khintchine exponent
Then there is an absolute constant $c>0$ such that for all $\lambda> 0$ and $t> 0$ ,
Proof. We are going to apply the estimates [Reference Pruitt45, (3.2)]. To do so, we need to express the Lévy–Khintchine exponent of $T_s - s b_\lambda $ in the form used in [Reference Pruitt45, Section 3], namely,
Since
we have
Hence, by [Reference Pruitt45, (3.2)],
as desired.
Theorem 4.11. Let $\mathbf {T}$ be a subordinator with the Laplace exponent $\phi $ . Suppose that $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\alpha>0$ , and assume that one of the following conditions holds true:
-
(i) $-\phi ^{\prime \prime } \in \mathrm {WUSC}(\beta -2,C,x_0)$ for some $C\geq 1$ and $\alpha \leq \beta <1$ , or
-
(ii) $-\phi ^{\prime \prime }$ is a function regularly varying at infinity with index $-1$ . If $x_0 = 0$ we also assume that $-\phi ^{\prime \prime }$ is regularly varying at zero with index $-1$ .
Then there is $M_0>1$ such that for each $M \geq M_0$ there exists $\rho _0> 0$ , so that for all $0 < \rho _1 < \rho _0$ , $0 < \rho _2$ there is $C>0$ such that for all $t \in (0, 1/\varphi (x_0))$ 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 $-\phi ^{\prime \prime }$ is $(-1)$ -regular at infinity and satisfies upper scaling at $0$ with $\alpha \leq \beta < 1$ . Alternatively, one can assume that $-\phi ^{\prime \prime }$ satisfies upper scaling at infinity with $\alpha \leq \beta < 1$ 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 $t \in (0,1/\varphi (x_0))$ and all $x> 0$ satisfying
Indeed, since $\varphi ^{-1}$ 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 $\rho _0$ .
Without loss of generality, we can assume that $b = 0$ . Let $\lambda> 0$ , whose value will be specified later. We decompose the Lévy measure $\nu ({\mathrm {d}} x)$ as follows: Let $\nu _1({\mathrm {d}} x)$ be the restriction of $\tfrac 12 \nu ({\mathrm {d}} x)$ to the interval $(0,\lambda ]$ and
We set
Let us denote by $\mathbf {T}^{(j)}$ the subordinator having the Laplace exponent $\phi _j$ , for $j \in \{ 1, 2 \}$ . Let $\psi _j(\xi ) = \phi _j(-i\xi )$ . Notice that $\tfrac 12 \nu \leq \nu _2 \leq \nu $ ; thus,
and for every $n \in \mathbb {N}$ ,
Therefore, for all $u> 0$ ,
Next, by Theorem 3.3, the random variables $T_t^{(2)}$ and $T_t$ are absolutely continuous. Let us denote by $p^{(2)}(t, \:\cdot \:)$ and $p(t, \:\cdot \:)$ the densities of $T^{(2)}_t$ and $T_t$ , respectively.
Let $M \geq 2 M_0+1$ , where $M_0$ is determined in Corollary 3.5 for the process $\mathbf {T}^{(2)}$ . For $0 < t < 1/\varphi (x_0)$ , we set
Since $\varphi ^{-1}(M/t)> x_0$ , we have
Let
Then, by (4.42) we get
Moreover, by Corollary 4.5 together with (4.42) we get
Hence, by Corollary 3.5,
Notice that, by (4.41) and Remark 3.4, the implied constant in (4.43) is independent of t and $\lambda $ . Since
by (4.43) and monotonicity of $\varphi ^{-1}$ , we get
for some constant $C_1> 0$ .
Next, by the Fourier inversion formula
thus, by [Reference Grzywny and Szczypkowski23, Proposition 3.4] and Propositions 4.2 and 4.3 we see that there is $C_2> 0$ such that for all $t \in (0, 1/\varphi (x_0))$ ,
By the mean value theorem, for $y \in \mathbb {R}$ , we get
Hence, for $y \in \mathbb {R}$ satisfying
by (4.44), we get
Therefore,
where we have set $C_0=C_1(2C_2)^{-1}$ and
Let $\rho _0 = \tfrac {1}{2} C_0$ and
We have
Thus, $\tfrac 12 tb_{\lambda } - (\tilde {x}_t - x_t)$ is nonnegative, and in view of (4.2) and (4.45),
for some constant $C_3>0$ . Next, setting
we get
Hence, the problem is reduced to showing that the infimum above is positive. Let us consider a collection $\{Y_t \colon t \in (0, 1/\varphi (x_0)) \}$ of infinitely divisible nonnegative random variables $Y_t = \lambda ^{-1}\big ( T^{(1)}_t-\tfrac 12 tb_{\lambda }\big )$ . The Lévy measure corresponding to $Y_t$ is
for any Borel subset $B \subset \mathbb {R}$ . Since for each $R> 1$ ,
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 $\big ((Y_{t_n}, y_n) \colon n \in \mathbb {N}\big )$ be a sequence realising the infimum in (4.47). By the Prokhorov theorem, we can assume that $(Y_{t_n} \colon n \in \mathbb {N})$ is weakly convergent to the random variable $Y_0$ . We note that $Y_{t_n}$ has the probability distribution supported in $\big [-\tfrac 12t_n\lambda _n^{-1}b_{\lambda _n}, \infty \big )$ where $\lambda _n$ is defined as $\lambda $ corresponding to $t_n$ .
Suppose that $(t_n : n \in \mathbb {N})$ contains a subsequence convergent to $t_0> 0$ . Then $Y_0 = Y_{t_0}$ and the support of its probability distribution equals $\big [-\tfrac 12t_0\lambda _0^{-1}b_{\lambda _0}, \infty \big )$ . Since $\rho (t_0) \leq \tfrac 12t_0\lambda _0^{-1}b_{\lambda _0}$ , we easily conclude that the infimum in (4.47) is positive.
Hence, it remains to investigate the case when $(t_n \colon n \in \mathbb {N})$ has no positive accumulation points. If zero is the only accumulation point, then $(\lambda _n \colon n \in \mathbb {N})$ has a subsequence convergent to zero. Otherwise, $(t_n)$ diverges to infinity; thus, $x_0 = 0$ and $(\lambda _n)$ contains a subsequence diverging to infinity. In view of (4.46), $\rho (t)$ is uniformly bounded in t. Thus, after taking a subsequence, we may and do assume that there exists a limit
By compactness we can also assume that $(y_n \colon n \in \mathbb {N})$ converges to $y_0 \in [-\rho _1-\tilde{\rho},\, \rho _2]$ . 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 [Reference Sato47, Theorem 8.7], it is also infinitely divisible.
Given $w \colon \mathbb {R} \rightarrow \mathbb {R}$ a continuous function satisfying
we write the Lévy–Khintchine exponent of $Y_{t_n}$ in the form
where
Since $(Y_{t_n} : n \in \mathbb {N})$ converges weakly to $Y_0$ , there are $\gamma _0 \in \mathbb {R}$ and $\sigma $ -finite measure $\mu _0$ on $(0, \infty )$ satisfying
such that the Lévy–Khintchine exponent of $Y_0$ is
where
Moreover, for any bounded continuous function $f\colon \mathbb {R} \rightarrow \mathbb {R}$ vanishing in a neighbourhood of zero, we have
Next, let us fix w satisfying (4.50) which equals $1$ on $[0,1]$ . In view of (4.48) and the definition of $\nu _1$ , the support of $\mu _{t_n}$ is contained in $[0, 1]$ . Hence, $\gamma _n=0$ for every $n \in \mathbb {N}$ and, consequently, $\gamma _0=0$ . We also conclude that $\operatorname {\mathrm {supp}} \mu _0 \subset [0,1]$ .
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 $(\lambda _n)$ also approaches zero, and we impose that $-\phi ^{\prime \prime }$ is a function regularly varying at infinity with index $-1$ ; otherwise, $(t_n)$ tends toward infinity as well as $(\lambda _n)$ , and thus $x_0 = 0$ , and we additionally assume that $-\phi ^{\prime \prime }$ 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 [Reference Sato47, Theorem 24.10], the latter can be deduced from
Since $\operatorname {\mathrm {supp}} \mu _0 \subset [0, 1]$ , for each $\epsilon \in (0,1)$ we can write
thus, to conclude (4.53), it is enough to show that
For the proof, for any $\epsilon \in (0,1)$ we define the following bounded continuous function:
We have, in view of (4.52),
Let us estimate the last integral. We write
By the Fubini–Tonelli theorem, we get
Thus,
Setting $z = 1/\lambda $ , by (4.2) and (4.45), we obtain
Moreover, since $\varphi $ is a $1$ -regularly varying function at infinity, we have
as z tends to infinity. Therefore, it remains to estimate the integral in (4.57). Using (4.2) we get
Since $-\phi ^{\prime \prime }(s)=s^{-1} \ell (s)$ for a certain function $\ell $ slowly varying at infinity, by [Reference Bingham, Goldie and Teugels3, Theorem 1.5.6],
as z tends to infinity. Hence,
which by (4.56) implies (4.54).
Next, let us consider the case (i); that is, when $-\phi ^{\prime \prime } \in \mathrm {WUSC}(\beta -2,C,x_0)$ with $C \geq 1$ and $\alpha \leq \beta <1$ . We claim that for all $\epsilon \in (0, 1)$ ,
To see this, it is enough to show that there is $C> 0$ such that for all $\epsilon \in (0, 1]$ and $t \in (0, 1/\varphi (x_0))$ ,
For the proof, we select a continuous function on $\mathbb {R}$ such that
and for each $\tau> 0$ set
Since for $0 < 2 \tau < \epsilon $ ,
Since $Y_{t_n}$ and $Y_0$ are purely non-Gaussian, by [Reference Sato47, Theorem 8.7(2)],
thus,
which entails (4.58).
We now turn to showing (4.59). We have
thus, by (4.2) and the weak lower scaling property of $\varphi $ ,
which, together with the definition of $\lambda $ , implies (4.59).
Since the support of the probability distribution of $Y_0$ is not the whole real line, by [Reference Picard42, Lemma 2.5], the inequality (4.58) implies that
and the support of $Y_0$ equals $[\chi , \infty )$ where
To conclude (4.49), it is enough to show that $\chi \leq -\tilde {\rho }$ . Since $\rho (t_n) \leq \frac {1}{2}t_n \lambda _n^{-1} b_{\lambda _n}$ , the latter can be deduced from
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
and
where $f_{\epsilon }$ 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 [Reference Graves19, Chapter VII], it is enough to show that the limit in (4.65) is uniform with respect to $n \in \mathbb {N}$ .
We write
By (4.63) and the Fubini–Tonelli theorem, we have
By almost monotonicity of $\varphi $ ,
Now, setting $z = \varphi ^{-1}(M/t)$ , by (4.45), we get
In view of Proposition 2.4, by the upper scaling of $-\phi ^{\prime \prime }$ , there is $c> 0$ such that for all $z> x_0$ ,
Hence, the limit in (4.65) is uniform with respect to $n \in \mathbb {N}$ , which justifies (4.62). This completes the proof of (4.49) and the theorem follows.
Theorem 4.13. Let $\mathbf {T}$ be a subordinator with the Laplace exponent $\phi $ . Suppose that $\phi \in \mathrm {WLSC}(\alpha ,c,x_0) \cap \mathrm {WUSC}(\beta ,C,x_0)$ for some $c \in (0, 1]$ , $C \geq 1$ , $x_0 \geq 0$ and $0 < \alpha \leq \beta < 1$ . We also assume that $b=0$ . Then for all $0<\chi _1<\chi _2$ there is $C' \geq 1$ such that for all $t \in (0, 1/\varphi (x_0))$ and $x> 0$ satisfying
we have
Proof. First let us notice that Corollary 2.7 implies that $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0) \cap \mathrm {WUSC}(\beta -2,C,x_0)$ . Therefore, the hypothesis of Theorem 4.11 is satisfied.
It is enough to show the first inequality in (4.68) since the latter is an easy consequence of (4.28) and Proposition 4.6. For $t \in (0, 1/\varphi (x_0))$ and $M \geq 1$ , we set
By Proposition 4.6, the function $\varphi ^{-1}$ possesses the weak lower scaling property. Moreover, there is $C_1 \geq 1$ such that for all $r> \varphi (x_0)$ ,
Hence, by Proposition 2.4, there is $C_2 \geq 1$ , such that
We select $M \geq 1$ satisfying
Let $\rho _1 = \rho _0/2$ where $\rho _0$ is determined in Theorem 4.11. Then, by (4.69) and (4.70), we have
Now set $\rho _2 = C_1 \chi _2$ . Then, by (4.69), we have
Putting (4.72) and (4.71) together, we conclude that
Therefore, by Theorem 4.11, for all $t \in (0, 1/\varphi (x_0))$ and $x> 0$ satisfying
we have
In view of (4.69), this completes the proof of the theorem.
Proposition 4.14. Let $\mathbf {T}$ be a subordinator with the Laplace exponent $\phi $ . Suppose that $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0, 1]$ , $x_0 \geq 0$ and $\alpha>0$ , and assume that one of the following conditions holds true:
-
(i) $-\phi ^{\prime \prime } \in \mathrm {WUSC}(\beta -2,C,x_0)$ for some $C\geq 1$ and $\alpha \leq \beta <1$ , or
-
(ii) $-\phi ^{\prime \prime }$ is a function regularly varying at infinity with index $-1$ . If $x_0 = 0$ , we also assume that $-\phi ^{\prime \prime }$ is regularly varying at zero with index $-1$ .
We also assume that the Lévy measure $\nu ({\mathrm {d}} x)$ has an almost monotone density $\nu (x)$ . Then the probability distribution of $T_t$ has a density $p(t, \: \cdot \:)$ . Moreover, there are $M_0>1$ , $\rho _0> 0$ and $C> 0$ such that for all $t \in (0, 1/\varphi (x_0))$ and
we have
Proof. Let $\lambda> 0$ . We begin by decomposing the Lévy measure $\nu ({\mathrm {d}} x)$ . Let $\nu _1({\mathrm {d}} x) = \nu _1(x) {\: \rm d} x$ and $\nu _2({\mathrm {d}}x) = \nu _2(x) {\: \rm d} x$ where
For $u> 0$ , we set
Let $\mathbf {T}^{(j)}$ be the Lévy process having the Laplace exponent $\phi _j$ , for $j \in \{1, 2\}$ . Since $\tfrac {1}{2} \nu \leq \nu _1 \leq \nu $ , we have
and for all $n \in \mathbb {N}$ ,
Thus,
and so for all $u> 0$ ,
In particular, $-\phi _1^{\prime \prime }$ has the weak lower scaling property. Therefore, by Theorem 3.3, $T_t^{(1)}$ and $T_t$ are absolutely continuous. Let us denote by $p(t, \:\cdot \:)$ and $p^{(1)}(t, \:\cdot \:)$ the densities of $T_t$ and $T_t^{(1)}$ , respectively. Observe that $\mathbf {T}^{(2)}$ is a compound Poisson process with the probability distribution denoted by $P_t({\mathrm {d}} x)$ . By [Reference Sato47, Remark 27.3],
We apply Theorem 4.11 to the process $\mathbf {T}^{(1)}$ . For $t> 0$ , we set
Then there are $C> 0$ and $\rho _0> 0$ , such that for all $t \in (0, 1/\varphi (x_0))$ and $x \geq 0$ satisfying
we have
Therefore, if
then
Next, if $\lambda \geq \rho _0/\varphi ^{-1}(1/t)$ , then, by (4.2),
where the penultimate inequality follows either by monotonicity of h or by [Reference Grzywny and Szczypkowski23, Lemma 2.1 (4)]. Finally, by (4.75) and (4.77), for $x \geq 2 \lambda $ we can compute
Hence, by the monotonicity of $\nu $ , 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.
4.3. Sharp two-sided estimates
In this section we present sharp two-sided estimates on the density $p(t, \: \cdot \:)$ assuming both the weak lower and upper scaling properties on $-\phi ^{\prime \prime }$ . First, following [Reference Bogdan, Grzywny and Ryznar6, Lemma 13], we prove an auxiliary result.
Proposition 4.15. Assume that the Lévy measure $\nu ({\mathrm {d}}x)$ has an almost monotone density $\nu (x)$ . Suppose that $-\phi ^{\prime \prime } \in \mathrm {WUSC}(\gamma ,C,x_0)$ for some $C \geq 1$ , $x_0 \geq 0$ and $\gamma <0$ . Then there are $a \in (0,1]$ and $c \in (0,1]$ such that for all $0 < x < a/x_0$ ,
Proof. Let $a \in (0,1]$ . Recall that by (4.30) we have $\nu (s) \leq C_1 s^{-3} \big ( -\phi ^{\prime \prime }(1/s) \big )$ for any $s>0$ . Hence, for any $u>0$ ,
where $C_2$ is a constant from the almost monotonicity of $\nu $ . If $u> x_0$ , then by the scaling property of $-\phi ^{\prime \prime }$ we obtain
By selecting $a \in (0, 1]$ such that
we get
Since
by (4.78) we obtain
provided that $u> x_0$ . Now, by the monotonicity of $-\phi ^{\prime \prime }$ we conclude the proof.
In view of Propositions 2.3 and 2.4, we immediately obtain the following corollary.
Corollary 4.16. Assume that the Lévy measure $\nu ({\mathrm {d}}x)$ has an almost monotone density $\nu (x)$ . Suppose that $b=0$ and $\phi \in \mathrm {WLSC}(\alpha ,c,x_0) \cap \mathrm {WUSC}(\beta ,C,x_0)$ for some $c \in (0,1]$ , $C \geq 1$ , $x_0 \geq 0$ and $0<\alpha \leq \beta < 1$ . Then there are $a \in (0,1]$ and $c' \in (0,1]$ such that for all $0 < x<a/x_0$ ,
We are now ready to prove our main result in this section.
Theorem 4.17. Let $\mathbf {T}$ be a subordinator with the Laplace exponent $\phi $ . Suppose that $\phi \in \mathrm {WLSC}(\alpha ,c,x_0) \cap \mathrm {WUSC}(\beta ,C,x_0)$ for some $c \in (0, 1]$ , $C \geq 1$ , $x_0 \geq 0$ and $0 < \alpha \leq \beta < 1$ . We also assume that $b=0$ and that the Lévy measure $\nu ({\mathrm {d}} x)$ has an almost monotone density $\nu (x)$ . Then there is $x_1 \in (0, \infty ]$ such that for all $t \in (0, 1/\varphi (x_0))$ and $x \in (0,x_1)$ ,
where $w = (\phi ')^{-1}(x/t)$ . If $x_0 = 0$ then $x_1 = \infty $ .
Proof. First let us note that by Corollary 2.7, $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0) \cap \mathrm {WUSC}(\beta -2,C,x_0)$ . Therefore, we are in position to apply Proposition 4.14. By Corollary 3.7, for $\chi _1 = \min {\{1, \delta \}}$ , we have
whenever $0 < x \phi ^{-1}(1/t) \leq \chi _1$ . Next, let $M_0'$ be $M_0$ determined by Proposition 4.14. By Proposition 2.4, (4.2) and monotonicity of $\varphi ^{-1}$ , for $t \in (0, 1/\varphi (x_0))$ , we get
thus, by Propositions 4.3 and 4.6, there is $C_1> 0$ such that
and
where $\rho _0'$ is the value of $\rho _0$ determined in Proposition 4.14. Let $\chi _2 = \max {\{1, C_1, \chi _1\}}$ . By Proposition 4.14 and Corollary 4.16, there is $a \in (0,1]$ such that if $x \phi ^{-1}(1/t)> \chi _2$ and $0 < x< a/x_0$ , then
Furthermore, by (4.29), if $x \phi ^{-1}(1/t)> \chi _2$ , then
where in the last step we have also used (4.13). Lastly, by Theorem 4.13 there is $C_2 \geq 1$ such that for all $t \in (0, 1/\varphi (x_0))$ and $x> 0$ satisfying
we have
We next claim that the following holds true.
Claim 4.18. There exist $0< c_1 \leq 1 \leq c_2$ such that for all $t \in (0,c_1/\varphi (x_0))$ and $x>0$ satisfying
we have
By Proposition 4.6, there is $C_3 \geq 1$ such that for $r> \varphi (x_0)$ ,
Let $c_2 = (\chi _1 c' C_3^{-2})^{-\beta /(1-\beta )} \in [1, \infty )$ , where $c'$ is taken from (4.15). Then
Consequently, by Proposition 2.3,
Moreover, there is $C_4 \geq 1$ such that $C_4 x\phi '(x) \geq \phi (x)$ provided that $x> x_0$ . Therefore, if $\chi _2 \leq C_4^{-1}$ , then
which yields (4.80) with $c_1 = 1$ . Otherwise, if $\chi _2> C_4^{-1}$ , then we set $c_1 =\big (C_4 \chi _2 C_3^2 (c')^{-1}\big )^{-\beta /(1-\beta )} \in (0,1]$ . Hence, by Proposition 4.6, for all $t \in (0,c_1/\varphi (x_0))$ ,
Therefore,
which, combined with (4.81) and (4.82), implies (4.80).
Now, using Claim 4.18 and Propositions 4.3 and 4.6, we deduce that for $t \in (0,c_1/\varphi (x_0))$ and $\chi _1 \leq x \phi ^{-1}(1/t) \leq \chi _2$ ,
and
Hence, $tw\phi '(w) \approx 1$ and
Next, by Propositions 2.4 and 2.1,
thus, by (4.83) and (4.84), we obtain
which, together with (4.85), implies that
for $t \in (0,c_1/\varphi (x_0))$ and $\chi _1 \leq x\phi ^{-1}(1/t) \leq \chi _2$ . In view of (4.79), the theorem follows in the case $x_0=0$ . Now, it remains to observe that in the case $x_0>0$ we may use positivity and continuity to conclude the claim for all $t \in (0,1/\varphi (x_0))$ .
5. Applications
5.1. Subordination
Let $(\mathscr {X}, \tau )$ be a locally compact separable metric space with a Radon measure $\mu $ having full support on $\mathscr {X}$ . Assume that $(X_t \colon t \geq 0)$ is a homogeneous in time Markov process on $\mathscr {X}$ with density $h(t, \: \cdot \:,\: \cdot \:)$ ; that is,
for any Borel set $B \subset \mathscr {X}$ , $x \in \mathscr {X}$ and $t> 0$ . Assume that for all $t> 0$ and $x, y \in \mathscr {X}$ ,
where n and $\gamma $ are some positive constants, $\Phi _1$ and $\Phi _2$ are nonnegative nonincreasing function on $[0, \infty )$ such that $\Phi _1(1)> 0$ and
By $H(t, x, y)$ we denote the heat kernel for the subordinate process $\left (X_{T_t} \colon t \geq 0\right )$ ; that is,
where
Suppose that $\phi \in \mathrm {WLSC}(\alpha ,c,x_0) \cap \mathrm {WUSC}(\beta ,C,x_0)$ for some $c \in (0, 1]$ , $C \geq 1$ , $x_0> 0$ and $0 < \alpha \leq \beta < 1$ . We also assume that
and that the Lévy measure $\nu ({\mathrm {d}} x)$ has an almost monotone density $\nu (x)$ .
Claim 5.1. For all $x, y \in \mathscr {X}$ satisfying $\tau (x, y)^{-\gamma }> x_0$ and any $t \in (0, 1/\varphi (x_0))$ ,
By Proposition 2.3, $\phi ' \in \mathrm {WLSC}(\alpha -1,c,x_0) \cap \mathrm {WUSC}(\beta -1,C,x_0)$ . Let $0 < r < \phi '(x_0^+)$ . If $0 < \lambda \leq C$ , then by setting
the weak upper scaling property of $\phi '$ implies that
Therefore,
Analogously, we can prove the lower estimate: If $0 < \lambda \leq c$ , then by setting
we obtain
and, consequently,
Since $(\phi ')^{-1}$ is nonincreasing, the last inequality is valid for all $0 < \lambda \leq 1$ . Let
By Theorem 4.17,
where
Recall that, by Proposition 2.3, for all $r> x_0$ we have
We can assume that
By (5.6) and the weak upper scaling of $\phi '$ , we get
thus,
Hence, by (5.3) and (5.4), we obtain
Moreover, since $w> x_0$ , by (5.6) and Proposition 4.6,
Thus, (5.7) entails that
Next, by Proposition 4.6 and (5.6), we get
Therefore, by (5.7),
Now, by (5.5) and (5.1) together with (5.8) and (5.9), we can estimate
and
where
Suppose that $A \leq 1$ . Since $\Phi _1$ and $\Phi _2$ are nonincreasing, by (5.10) and (5.11), we easily see that
We also have
Therefore,
We now turn to the case $A> 1$ . By (5.2) and (5.10),
It remains to estimate $I_2$ . Let us observe that for all $r> x_0$ , if $u \geq 1$ , then by the weak upper scaling of $\phi $ , we have
On the other hand, if $0 < u \leq 1$ , then by (2.10) and the monotonicity of $\phi $ , we get
Therefore, for all $u> 0$ and $r> x_0$ ,
Since $\tau (x, y)^{-\gamma }> x_0$ , by Theorem 4.17, (5.1) and estimates (5.13), we get
and
By (5.2), we have
thus,
Finally, since $A> 1$ , by (2.10), we have
hence, by (5.12),
proving the claim.
Example 5.2. Let $(\mathscr {X}, \tau )$ be a nested fractal with the geodesic metric on $\mathscr {X}$ . Let $d_w$ and $d_f$ be the walk dimension and the Hausdorff dimension of $\mathscr {X}$ , respectively. Let $(X_t \colon t \geq 0)$ be the diffusion on $\mathscr {X}$ constructed in [Reference Barlow2, Section 7]. By [Reference Barlow2, Theorem 8.18], the corresponding heat kernel satisfies (5.1) with $n = d_f$ , $\gamma = d_w$ , and
Let $\mathbf {T}$ be a subordinator with the Laplace exponent
where $\alpha \in (0, 1)$ and $\sigma \in \mathbb {R}$ . Then, by Claim 5.1, the process $(X_{T_t} \colon t \geq 0)$ has density $H(t, x, y)$ such that for all $x, y \in \mathscr {X}$ and $t> 0$ ,
-
•. if $t> \tau (x, y)^{\alpha \gamma } \log ^{-\sigma }\left (2 + \tau (x, y)^{-\gamma }\right )$ , then
$$ \begin{align*} H(t, x, y) \approx t^{-\frac{n}{\alpha\gamma}} \log^{-\frac{\sigma n}{\alpha\gamma}}\big(2+t^{-1}\big), \end{align*} $$ -
•. if $t < \tau (x, y)^{\alpha \gamma } \log ^{-\sigma }\left (2 + \tau (x, y)^{-\gamma }\right )$ , then
$$ \begin{align*} H(t, x, y) \approx t \tau(x, y)^{-\alpha \gamma - n} \log^{\sigma}\big(2 + \tau(x, y)^{-\gamma}\big). \end{align*} $$
Example 5.3. Let $(\mathscr {X}, \tau )$ be a complete manifold without boundary, having nonnegative Ricci curvature. Then by [Reference Li and Yau37], the heat kernel corresponding to the Laplace–Beltrami operator on $\mathscr {X}$ satisfies estimates (5.1) with
Now, one can take $\mathbf {T}$ with a Lévy–Khintchine exponent regularly varying at infinity with index $\alpha \in (0,1)$ and apply Claim 5.1 to obtain the asymptotic behaviour of the subordinate process.
5.2. Green function estimates
Let $\mathbf {T} = (T_t \colon t \geq 0)$ be a subordinator with the Laplace exponent $\phi $ . If $-\phi ^{\prime \prime }$ has the weak lower scaling property of index $\alpha -2$ for some $\alpha> 0$ , then the probability distribution of $T_t$ has a density $p(t, \: \cdot \:)$ ; 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,
where
Notice that by (2.9) and Proposition 2.3, for all $x> x_0$ ,
In view of (4.2) and Proposition 4.3, the function $\varphi $ is almost increasing; thus, by monotonicity of $\phi '$ , f is almost increasing as well. Therefore, there is $c_0 \in (0, 1]$ such that for all $x> x_0$ ,
Moreover, f has the doubling property on $(x_0,\infty )$ . Since $\varphi $ belongs to $\mathrm {WLSC}(\alpha ,c,x_0)$ , by monotonicity of $\phi '$ , we conclude that f belongs to $\mathrm {WLSC}(\alpha ,c,x_0)$ . It follows that $f^{-1} \in \mathrm {WUSC}(1/\alpha ,C,f^*(x_0))$ for some $C \geq 1$ and since $f^{-1}$ is increasing, we infer that $f^{-1}$ also has doubling property on $(f^*(x_0),\infty )$ .
Proposition 5.4. Suppose that $b=0$ and $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0,1]$ , $x_0 \geq 0$ and $\alpha>0$ . Then for each $A>0$ and $M>0$ there is $C \geq 1$ so that for all $x < A/x_0$ ,
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 $C \geq 1$ such that for all $x<A_M/x_0$ ,
Let
where $M_0$ is determined in Corollary 3.5, and $c_0$ is taken from (5.15). We claim that the following holds true.
Claim 5.5. For each $M>0$ there is $C \geq 1$ so that for all $x < A_M/x_0$ ,
Suppose that
with $M_1 = c_0^{-1} M_0$ . Notice that for $x < A_M/x_0$ , we have $x < M_1/f^*(x_0)$ . Hence, $x_0 \leq f^{-1}\big (M_1 /x\big )$ ; thus, by monotonicity of $\phi '$ , we obtain
Moreover, for $w = (\phi ')^{-1}(x/t)$ , the condition (5.18) implies that
which together with (5.15) gives
Now, to justify the claim, let us first consider $M \geq M_1$ . In view of (5.19) and (5.20), we can apply Corollary 3.5 to get
Since by Proposition 4.1 and Remark 3.2, for all $w>x_0$ ,
after the change of variables $t = x / \phi '(s)$ , we can find $C_2 \geq 1$ such that for all $x < A_M/x_0$ ,
Recall that $f^{-1}$ has the doubling property on $(f^*(x_0),\infty )$ . Using now Proposition 2.3 and (5.15), we get
where the implicit constants may depend on M. Therefore, by monotonicity of $f^{-1}$ and $\phi '$ , the estimate (5.22) gives
This proves the first inequality in (5.17).
We next observe that (5.14) entails that $f^{-1}(s) \gtrsim s$ for $s> f^*(x_0)$ ; thus, by (5.23),
where the last estimate follows by Proposition 2.3.
We next show the second inequality in (5.17). By (5.21), Proposition 2.3 and monotonicity of $\phi $ ,
where in the last inequality we have used
Since $\varphi \in \mathrm {WLSC}(\alpha ,c,x_0)$ , by [Reference Bogdan, Grzywny and Ryznar6, Lemma 16],
Finally, the doubling property of $f^{-1}$ , monotonicity of $\phi $ and Proposition 2.3 give
where the implied constant may depend on M. This finishes the proof of (5.17) for $M \geq M_1$ .
We next consider $0 < M < M_1$ . By monotonicity, the lower estimate remains valid for all $M> 0$ . Therefore, it is enough to show that for each $0 < M < M_1$ , there is $C \geq 1$ such that for all $x < A_M/x_0$ ,
By [Reference Grzywny and Szczypkowski23, Theorem 3.1], there is $t_0> 0$ such that for all $0 < t < t_0$ ,
If $x_0=0$ , then $t_0=\infty $ . Since $\varphi $ is almost increasing we have
Hence, by continuity and positivity of $p(t,x)$ and $\varphi ^{-1}(1/t)$ , we can take
Therefore, by the change of variables $t = x/\phi '(s)$ we get
Next, by monotonicity and the doubling property of $f^{-1}$ and $\phi '$ , we obtain
Since by (5.15) for $s \geq f^{-1}(M/x)$ we have
by monotonicity of $\varphi ^{-1}$ , Proposition 4.3, Remark 4.4 and the doubling property of $f^{-1}$ and $\varphi ^{-1}$ , we get
which together with (5.25) gives (5.17) for $0 < M < M_1$ . This completes the proof of Claim 5.5.
Our next task is to deduce (5.16) from Claim 5.5. By Lemma 2.9 and Proposition 2.3, there is a complete Bernstein function $\tilde {\phi }$ such that $\tilde {\phi } \approx \phi $ and
for all $x> x_0$ . Let $\tilde {\mathbf {T}}$ be a subordinator with the Laplace exponent $\tilde {\phi }$ . By $\tilde {p}(t, \: \cdot \: )$ we denote the density of the probability distribution of $\tilde {T}_t$ . We set
and
Fix $M>0$ . By Claim 5.5, there is $A_M> 0$ such that for all $x < A_M/x_0$ ,
On the other hand, since $\tilde {\phi }$ is the complete Bernstein function, by (5.24) and [Reference Kim, Song and Vondraček34, Corollary 2.6], there is $C_3 \geq 1$ such that
Therefore, by (5.26), for $x < A_M/x_0$ ,
and (5.16) follows for all $A \leq A_M$ . Let us now consider $A> A_M$ . Observe that the functions
and
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 $b=0$ , $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0)$ for some $c \in (0,1]$ , $x_0 \geq 0$ and $\alpha>0$ and that the Lévy measure $\nu ({\mathrm {d}}x)$ is absolutely continuous with respect to the Lebesgue measure with a monotone density $\nu (x)$ . Then there is $\epsilon \in (0,1)$ such that for each $A>0$ , there is $C \geq 1$ such that for all $x < A/x_0$ ,
Proof. In view of (5.27), it is enough to show that for some $\epsilon \in (0, 1)$ and all $A>0$ there is $C \geq 1$ , such that for all $x<A/x_0$ ,
Let $\epsilon \in (0, 1)$ and
Suppose that
that is,
Hence, by monotonicity of $\varphi ^{-1}$ and $\phi '$ ,
By Proposition 4.3 and the scaling property of $\phi '$ , there are $c \in (0,1]$ and $C \geq 1$ such that
Therefore, by taking $\epsilon = (2e)^{-1}cC^{\alpha -1}$ , we get
Since $\nu (x)$ is the monotone density of $\nu ({\mathrm {d}} x)$ , by Theorem 4.8 we get
By (5.14), $f^{-1}(s) \gtrsim s$ for $s>f^*(x_0)$ ; thus, using (4.4),
which entails (5.28). The extension to arbitrary A follows by continuity and positivity argument as in the proof of Proposition 5.4.
It is possible to get the same conclusion as in Proposition 5.6 without imposing the existence of the monotone density of $\nu ({\mathrm {d}} x)$ ; however, instead we need to assume the weak upper scaling property in $-\phi ^{\prime \prime }$ .
Proposition 5.7. Suppose that $b=0$ and $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0) \cap \mathrm {WUSC}(\beta -2,C,x_0)$ for some $c \in (0,1]$ , $C \geq 1$ , $x_0 \geq 0$ and $\tfrac {1}{2}<\alpha \leq \beta <1$ . Then there is $\epsilon \in (0, 1)$ such that for each $A>0$ , there is $C_1 \geq 1$ , so that for all $x < A/x_0$ ,
Proof. Let
By repeating the same reasoning as in the proof of Proposition 5.6, we can see that the condition
implies
for $\epsilon = (2e)^{-1} c C^{\alpha -1}$ . Therefore, we can apply Theorem 4.7 to get
where the implied constant may depend on $\epsilon $ . Since $\alpha> \tfrac {1}{2}$ , by Proposition 4.3, [Reference Aljančić and Aranđelović1, Theorem 3] and the doubling property of $\varphi ^{-1}$ , we obtain
In view of (5.15), we have
thus, by Proposition 4.3 and Remark 4.4,
In view of Propositions 2.3 and 4.6, we have $f(s) \approx s$ for $s> x_0$ ; thus, $f^{-1}(s) \approx s$ , for $s> f^*(x_0)$ . Hence,
Therefore, by (5.31) and (5.32), we conclude that
which, by Proposition 5.4 and (5.27), entails (5.30). The extension to arbitrary A follows by positivity and continuity argument.
Theorem 5.8. Let $\mathbf {T}$ be a subordinator with the Laplace exponent $\phi $ . Suppose that
for some $c \in (0,1]$ , $C \geq 1$ , $x_0 \geq 0$ and $0<\alpha \leq \beta <1$ . We assume that one of the following conditions holds:
-
(i) The Lévy measure $\nu ({\mathrm {d}}x)$ is absolutely continuous with respect to the Lebesgue measure with monotone density $\nu (x)$ , or
-
(ii) $\alpha> \tfrac 12$ .
Then for each $A> 0$ there is $C_1 \geq 1$ such that for all $x< A/x_0$ ,
Proof. By Corollary 2.7, $-\phi ^{\prime \prime } \in \mathrm {WLSC}(\alpha -2,c,x_0) \cap \mathrm {WUSC}(\beta -2,C,x_0)$ . Let $p(t, \: \cdot \:)$ be the transition density of $T_t$ . In view of Propositions 5.4, 5.6 and 5.7 and (5.27), it is enough to show that for each $A> 0$ and $\epsilon \in (0, 1)$ there is $C_1> 0$ such that for all $x<A/x_0$ ,
By [Reference Grzywny and Szczypkowski23, Theorem 3.1], there is $t_0> 0$ such that for all $t \in (0, t_0)$ ,
If $x_0 = 0$ , then $t_0 = \infty $ . We can take
Therefore, by monotonicity of $\varphi ^{-1}$ , we get
By the doubling property of $\varphi ^{-1}$ , definition of f and Remark 4.4,
since by the weak upper scaling property of $-\phi ^{\prime \prime }$ , $f(s) \approx s$ for all $s> f^*(x_0)$ . Consequently, we obtain (5.33) and the theorem follows.
Acknowledgements
We thank Professor Jerzy Zabczyk for drawing our attention to the problem considered in this article. The main results of this article were presented at the XV Probability Conference held from 21 to 25 May 2018 in Będlewo, Poland, and at the Semigroups of Operators: Theory and Applications Conference held from 30 September to 5 October 2018 in Kazimierz Dolny, Poland. We thank the organisers for the invitations. The authors were partially supported by the National Science Centre (Poland) (Grant No. 2016/23/B/ST1/01665).
Competing Interest
None.