Hitting Times of Points and Intervals for Symmetric Lévy Processes

For one-dimensional symmetric Lévy processes, which hit every point with positive probability, we give sharp bounds for the tail function Px(TB>t), where TB is the first hitting time of B which is either a single point or an interval. The estimates are obtained under some weak type scaling assumptions on the characteristic exponent of the process. We apply these results to prove sharp two-sided estimates of the transition density of the process killed after hitting B.


Introduction
The purpose of this paper is to investigate the distribution of the first hitting time of a point or an interval by a symmetric Lévy process such that {0} is regular for itself. Such processes hit points with positive probability. Our main results, under certain regularity assumptions, provide sharp estimates of the tail function P x (T 0 > t), t > 0, where T 0 is the first hitting time of the point 0 by the process starting from x. We further derive similar estimates for the first hitting time of an interval of a given width, under some weak scaling assumption on the characteristic exponent ψ of the process. We also find the asymptotic behaviour of the tail function either for the first hitting time of a point or a compact set under the assumption that the characteristic exponent is regularly varying at zero with index δ ≥ 1. The estimates or asymptotics obtained in the paper are expressed in terms of the generalized inverse ψ −1 of the characteristic exponent and the compensated potential kernel Here p s (x), s > 0, x ∈ R, is the transition density of the process, which must exist for processes we study. If ψ is comparable with a non-decreasing function we are able to provide sharp estimates of K in terms of the characteristic exponent, so in these cases the estimates become quite explicit and are given in terms of the characteristic exponent and its generalized inverse. For example we show that if ψ has the weak lower scaling property with index α > 1 (see Preliminaries for the definition) then Moreover, we find a similar estimate in the case when P x (T 0 > t) is replaced by the tail function of the first hitting time of an interval (see Theorem 5.3). While in principle, for starting points x far away from the interval, such estimates should follow from the estimates of P x (T 0 > t), but for points close to the boundary of the interval the behaviour of the tail function is not clear. In order to overcome this difficulty we proved and then applied the global Harnack inequality under the weak scaling assumption for ψ (see Theorem 4.5). The Harnack inequality is one of the central topics in the potential theory and the present paper contributes to these studies. Usually the Harnack inequality for Lévy or generally Markov processes is proved under the assumptions which enforce the transience of the process and absolute continuity of its Lévy measure [1,9,12,21,38]. In our case the process is not only recurrent but point recurrent.
Finally, under the assumption the process is unimodal and ψ has the lower and upper weak scaling property we apply the estimates of the hitting times and derive sharp estimates of p D , the transition density (heat kernel) of the process killed after hitting an interval. We show that for D = (−∞, −R) ∪ (R, ∞), R > 0 we have with the comparability constant independent of R and where τ D denotes the first exit time from D. The problem of estimating the heat kernel for symmetric Lévy processes has brought a lot of attention recently; see e.g. [6,8,14,15,27]. Most of the results for unbounded open sets are derived under the assumptions implying the transience of the process. The recurrent processes, except isotropic stable [6], were not explored with regard to heat kernel estimates for exterior sets, and to the best of our knowledge our result is the first one with such generality. Our approach follows general framework used in most of the works on this subject and requires some exit time estimates. While in the transient case some needed hitting or exit time estimates are quite obvious, here they present the main difficulty. This is done in Propositions 6.1, 6.2 and 6.4, by applying the results from Sections 4 and 5. To see the main difference between the transient and recurrent case note that, for D = (−∞, −R) ∪ (R, ∞) and |x|, |y| > 2R, we have p D t (x, y) ≈ p t (x − y), t > 0, in the stable transient case [6,16]. See also [8,26] for further generalizations if the weak scaling properties imply transience. In the case studied in this paper it is certainly not true since P x (τ D > t) decays to 0 while t → ∞.
One of the drawbacks is that we deal with one-dimensional processes which are point recurrent. It would be desirable to provide such optimal estimates for one or twodimensional recurrent symmetric Lévy processes, which do not hit points. Unfortunately our approach, based on the nice behaviour of the compensated kernel K, will not work in this case.
The distribution of the hitting time of points or compact sets for one-dimensional α-stable processes was the subject of studies in several papers [18,25,28,29,31,32,37,42]. Let T B be the first hitting time of a set B. Port in [32] found the asymptotics of P x (T B > t), t → ∞ for a compact set B if 1 < α < 2, and for not necessarily symmetric stable processes. The density f x (t) of T x for the symmetric α-stable process, 1 < α < 2, was found in [42]. For spectrally positive (no negative jumps) α-stable process, 1 < α < 2, Peskir [31] and Simon [37] found the density f x (t), x > 0,in a form of a series from which one can derive the asymptotics of f x (t) as t → 0 + or t → ∞. In a recent paper [28] this type of result was extended to α-stable processes, 1 < α < 2, having both negative and positive jumps. In [28] the authors derived the Mellin transform of the distribution of T x and then successfully inverted it to obtain the series representation of the density of T x .
Relatively little is known about the distribution of hitting times of single points for general Lévy processes. To our best knowledge such explicit results as mentioned above do not exist. Only recently Kwaśnicki [29] studied the distribution of T x for symmetric Lévy processes under certain regularity assumptions on the characteristic exponent of the process. The main result of [29] provides an integral representation of the distribution function of T x in terms of generalized eigenfunctions for the killed semigroup upon hitting {0}. This representation was then successfully applied in [25] to obtain various asymptotics and estimates of the tail function of T x and its derivatives under further additional regularity assumptions on characteristic exponent and the Lévy measure. Namely it is assumed that the process has completely monotone Lévy density. Comparing our results with those obtained in [25] we remark that our assumptions are much less restrictive, however our approach does not allow to treat the estimates of the density or the higher derivatives of the distribution functions. In a forthcoming paper we provide sharp estimates of the density under the weak upper and lower scaling property for the characteristic exponent for unimodal Lévy processes. Moreover we also treat the hitting distribution of intervals and provide sharp estimates and asymptotics of the tail function, which was not investigated in [25]. We also mention that our methods are more elementary and are based on the estimates of the Laplace transforms of the hitting distributions and various estimates of exit probabilities.
The paper is composed as follows. In Section 2 we recall some basic material regarding one-dimensional symmetric Lévy processes and present some auxiliary results which we use in the sequel. In Section 3 we obtain estimates and asymptotics of the tail function P x (T 0 > t). Section 4 is devoted to the uniform Harnack inequality and boundary behaviour of harmonic functions. These tools we use in Section 5 to prove estimates of the function P x (T [−r,r] > t). Section 6 focuses on symmetric unimodal processes with weak global scaling. We use the methods and results of the previous sections to obtain estimates of the Dirichlet heat kernel of a complement of an interval.

Preliminaries
Throughout the paper by c, c 1 . . . we denote nonnegative constants which may depend on other constant parameters only. The value of c or c 1 . . . may change from line to line in a chain of estimates. If we use enumerated C 1 , C 2 . . . then they are fixed constants and usually used in the sequel parts of the paper. Any subsets and real functions considered in the paper are assumed to be Borel measurable. The notion p(u) ≈ q(u), u ∈ A means that the ratio p(u)/q(u), u ∈ A is bounded from below and above by positive (comparability) constants which may depend on other constant parameters only but does not depend on the set A.
We present in this section some basic material regarding one-dimensional symmetric Lévy processes which hit points with non-zero probability. For more detailed information, see [2,11]. For questions regarding the Markov and the strong Markov properties, semigroup properties, Schrödinger operators and basic potential theory, the reader is referred to [17] and [4].
In this paper we assume that a Lévy process X = (X t , t ≥ 0) [34], is symmetric. By ν we denote its Lévy measure and by ψ its Lévy-Khintchine exponent (symbol). Notice that ν and ψ are symmetric as well. Recall that any Lévy measure is a measure such that If the Lévy measure ν is absolutely continuous with respect to the Lebesgue measure, then with a slight abuse of notation, we denote its density by ν as well. Since the process is symmetric there is σ ∈ R such that and E e iξ X t = e −tψ(ξ) , ξ ∈ R.
For x ∈ R, by P x and E x we denote the distribution and the resulting expectation of the process x + X. Obviously P 0 = P and E 0 = E. The process X is called unimodal if for any t > 0 the distribution p t (dx) of X t is unimodal, that is, it is absolutely continuous on R \ {0} and its density p t (x) is symmetric on R and non-increasing on (0, ∞). Unimodal Lévy processes are characterized in [40] by unimodal Lévy measures ν(dx) = ν(x)dx = ν(|x|)dx.
The first exit time of an (open) set D ⊂ R d by the process X t is defined by the formula If F ⊂ R is a closed set we define the first hitting time T F of F as the first exit time from F c . In the case when F = {a}, a ∈ R, we denote T F = T a .
In this paper we consider symmetric Lévy processes which have the property that Note that the above condition implies that ψ is unbounded, so excludes compound Poisson processes and in consequence ψ(x) > 0 for x = 0. From [11, Theoreme 7 and Theoreme 8] we infer that for symmetric Lévy processes, which are not compound Poisson, the condition (1) is equivalent to the fact that 0 is regular for the set {0} that is Moreover (1) guarantees that the distribution of X t , t > 0, is absolutely continuous and its density p t (·) ∈ C ∞ (R).
In general potential theory a very important role is played by λ-potential kernels, λ > 0, which are defined as If the defining integral above is finite for λ = 0 we call u 0 (x) = u(x) the potential kernel and then the underlying process is transient.
Under the above assumptions it follows from [2, Corollary II.18 and Theorem II.19] that h λ (x) = E 0 e −λT x is continuous and and By symmetry and [2,Theorem II.19], The monotone convergence theorem implies For a number of results below we make the assumption that K is non-decreasing on (0, ∞). We do not know any general criterion which guarantees monotonicity, but it is clear that sufficient conditions are: ψ(x)/x is non-decreasing on (0, ∞) or the process X is unimodal. Another interesting problem is the question if monotonicity of K implies some monotonicity properties of ψ.
1 ψ(s) ds < ∞ the process X is transient and it is clear from Lemma 2.1 that its potential kernel satisfies

Proposition 2.2 K is subadditive on R.
Proof Observe that T x+y ≤ T x + T x+y • T x , where • denotes the usual shift operation. By the strong Markov property, for λ > 0, The fundamental object of the potential theory is the killed process X D t when exiting the set D. It is defined in terms of sample paths up to time τ D . More precisely, , t > 0 . The density function of the transition probability of the process X D t is denoted by p D t . We have t>0 is a strongly contractive semigroup (under composition) and shares most of properties of the semigroup p t . In particular, it is strongly Feller and symmetric: p D t (x, y) = p D t (y, x). The λ-potential measure of the process X D t started from x is a Borel measure defined as for any Borel subset A of R. For the Lévy processes explored in the paper their potential measures are absolutely continuous and the corresponding density is λ-potential kernel of the process X D t and is called λ-Green function of the set D. It is denoted by G λ D and we have If λ = 0 the corresponding 0-Green function we simply call the Green function of D and denote G D (x, y). Another important object in the potential theory of X t is the harmonic measure of the set D. It is defined by the formula: The density kernel (with respect to the Lebesgue measure) of the measure P D (x, A) (if it exists) is called the Poisson kernel of the set D. The relationship between the Green function of D and the harmonic measure is provided by the Ikeda-Watanabe formula [24], Now we define harmonic and regular harmonic functions with respect to the process X. Let u be a Borel measurable function on R. We say that u is harmonic function in an open for every bounded open set B with the closure B ⊂ D. We say that u is regular harmonic We note that for any open D the Green function G D (x, y)(if exists) is harmonic in D \ {y} as a function of x. This follows from the strong Markov property and is frequently used in the paper. The following formula for the Green function of the complement of a point can be found in [41,Lemma 4.1], [19,Theorem 6.1] for recurrent processes and [10, Lemma 4] for stable processes.
Proof Recall that for λ > 0 we define Hence by the monotone convergence theorem By the dominated convergence theorem we get continuity of K and G {0} c as well.
The following observation plays a crucial role in the sequel.

Proposition 2.4
For any x, y ∈ R we have If additionally K(·) is non-decreasing on (0, ∞) then for xy ≥ 0 we have Proof By subadditivity of K we have Hence

Proposition 2.7
Let K be non-decreasing on [0, ∞). For R > 0 and 0 < |x| < R, If κ = 0, then additionally Proof Let 0 < x < R. By Proposition 2.4, Lemma 2.5 and subadditivity of K, On the other hand, by Lemma 2.1 and subadditivity of K, which combined with the first bound above provide the first estimate. Moreover, if κ = 0 then We also consider the renewal function V of the (properly normalized) ascending ladderheight process of X t . The ladder-height process is a subordinator with the Laplace exponent Silverstein studied V and its derivative V as g and ψ in [36, (1.8) and Theorem 2]. The Laplace transform of V is The definition of V is rather implicit and properties of V are delicate. In particular the decay properties of V are not yet fully understood. For a detailed discussion of V we refer the reader to [5] and [36].
It is known that V is absolutely continuous and harmonic on (0, ∞) for X t . Also V is a positive harmonic function for X t on (0, ∞), hence V is actually (strictly) increasing. For the so-called complete subordinate Brownian motions [35] V is monotone, in fact completely monotone, cf. [30,Proposition 4.5]. This property was crucial for the development in [15,27], but in general it fails in the present setting cf. [5,Remark 9]. One of the important features of the function V is the fact that the Green function of (0, ∞) can be written as This follows from [2,Theorem VI.20].
Below, in Lemmas 2.8-2.11 we collect useful facts which are true for general symmetric Lévy processes, which are not compound Poisson.

Lemma 2.11 ([22], Proposition 3.7)
There is an absolute constant C 4 such that for any x ∈ (0, R), In fact we may take For a continuous non-decreasing function φ : The function φ −1 is non-decreasing and càglàd (left continuous with right-hand side limits).
Below we often consider the (unbounded) characteristic exponent ψ of a symmetric Lévy process with infinite Lévy measure and its maximal function ψ * , and denote This short notation is motivated by the following equality: It is rather natural to assume (relative) power-type behaviour for the characteristic exponent ψ of X. To this end we consider ψ as a function on (0, ∞). We say that ψ satisfies the global weak lower scaling condition (WLSC) if there are numbers α > 0 (called the index of the lower scaling) and γ ∈ (0, 1], such that In short we write ψ ∈ WLSC(α, γ ) or ψ ∈ WLSC. The global weak upper scaling condition (WUSC) means that there are numbers β < 2 (called the index of the upper scaling) and ρ∈ [1, ∞) such that We call α, γ , β, ρ the scaling characteristics of ψ or simply the scalings. In most of our results we assume only the lower scaling condition.
We are thus led to the behavior of ψ −1 .
The following technical lemma is the main tool in estimating the tail function of T 0 via its Laplace transform. Recall thatK(x) = 1
To obtain the upper bound we apply Lemma 2.12 with f (r) .
where the last step follows from Lemma 2.14.

Hitting Times of Points
In this section we examine the tail function P x (T 0 > t) under various assumptions on ψ or K. Under the monotonicity of K we find the lower and upper bounds of the tail function. On the other hand comparability of ψ and ψ * is another source of the estimates via approximate inversion of the Laplace transform. We also derive asymptotics of P x (T 0 > t) as t → ∞ by applying Tauberian theorems.

Proposition 3.1 We have for any
Proof Observe that for the Laplace transform of P x (T 0 > ·) we have It follows from Eq. 13 that in general case, while under the assumption ψ(x) ≥ aψ * (|x|),

By [7, Lemma 5] we have
in general case, and in the other considered case which ends the proof.
Proof Let R, t > 0. We have Let |x| < R. By Chebyshev's inequality and Proposition 2.6 we obtain while by Lemma 2.7, Setting RK(R) = t we obtain the conclusion.

Lemma 3.3 Let K be non-decreasing on
Let R = 1/ψ −1 (1/t). Then for 0 < x < R, by Proposition 2.7 and the strong Markov property The proof is completed.
The assumption about monotonicity of K can be removed if we assume the lower scaling condition of ψ.
where c depends only on the scalings.
From the above lower and upper bounds we derive two corollaries providing two sided sharp estimates.

Corollary 3.5
If X is unimodal then for x ∈ R and t > 0, The comparability constant is absolute.
Proof If X is unimodal then K is increasing and ψ ≥ π −2 ψ * (see [7,Proposition 2]), hence the upper bound follows from Proposition 3.1, while the lower bound is a consequence of Lemma 3.3.
Proof By Lemma 2.14 we have K(x) ≈ 1 |x|ψ(1/x) and the conclusion follows immediately from Proposition 3.1 and Proposition 3.4.

Example 1
Let ψ(x) = |x| + x 2 . The corresponding process, which is the sum of the Cauchy process and independent Brownian motion, is obviously unimodal. Then, by Lemma 2.14, and by Corollary 3.5,
Remark 1 There are recent results obtained by Juszczyszyn and Kwaśnicki [25] where not only the behaviour of the tail function P x (T 0 > t) was described but also its derivatives. Their assumptions on the process were much more restrictive than ours. They assumed complete monotonicity of the Lévy density and some additional property of the first two derivatives of the symbol of the process. The results of our paper regarding the tail functions are more general, however our methods do not allow us to treat the derivatives. The processes from Example 1 and 2 do not satisfy the assumptions of [25]. In Example 1 the symbol ψ fails the requirements of [25], while the Lévy measures in Example 2 are singular. Now we turn to asymptotics of the tail function when t → ∞ not only in the case of hitting {0} but for hitting arbitrary compact set as well.
If ψ is regularly varying at 0 with index 1 then there is a function L(u) slowly varying at 0 such that Since K is continuous and B is compact, by the dominated convergence theorem and Lemma 2.1, lim λ→0 λu λ (0)L(P x (T B > ·))(λ) = K(x) − E x K(X T B ).

Remark 2
We again compare [25] with our results with regard to asymptotics of the tail function of T 0 . For example the case ψ(x) = |x| + |x| 2 is not covered in [25]. In Example 2 we provided sharp estimates of P x (T 0 > t) and Corollary 3.8 exhibits the asymptotics at infinity. Note that ψ −1 (u) = u u+ 1 4 + 1 2 ∼ = u, u → 0. Next, Then, by the second part of Corollary 3.8, The next example illustrates that the decay of the tail function of T 0 can be very slow. Note that the intensity of small jumps of the process below is larger than the corresponding intensity of the Cauchy process while it is smaller than the corresponding intensity for any symmetric α-stable process, α > 1. Therefore the considered process is in some sense between the Cauchy process and any symmetric α-stable processes, α > 1. Note that the Cauchy process hits points with 0 probability, while α-stable processes, α > 1, hit points with probability 1.

Behaviour of Harmonic Functions
This section prepares some tools used in the sequel for estimating the tail function for the hitting time of an interval. On the other hand the results are interesting on their own. In the first subsection we prove the global Harnack inequality under global weak scaling assumption for ψ, while in the second we provide some boundary type estimates for harmonic functions.
We start with a lemma which shows a very useful property of the compensated kernel.

Harnack inequality
We say that the global Harnack inequality holds if there is a constant C H such that for every R > 0 and any non-negative harmonic function on (−R, R) we have Here we prove that ψ ∈ WLSC(α, γ ), α > 1 is a sufficient condition. The Harnack inequality will be very important in the next subsection to find the boundary behaviour of certain harmonic functions.
, with comparability constant dependent on the scalings, then with c dependent on the scalings. Next, we can use WLSC property for K with index α − 1 to choose δ < 1/2 (dependent only on the scalings) small enough, such that Then Let R 0 = δR, with δ chosen in the preceding proposition.
Next, we use the standard chain argument to get where C H = C H (c 1 , a).

Boundary behaviour
In this subsection we prove certain estimates of non-negative functions which are harmonic on (0, R), 0 < R ≤ ∞. We show that under appropriate assumptions the function V (x) provides the right order of decay at the boundary at 0 for harmonic functions we consider. The obtained results are then used in Section 5 to estimate the tail function of the hitting time of an interval. In our development the following Property (H) of the derivative of V is crucial. Recall that V is the renewal function defined on page 8. Below, in Remark 3, we discus the situations when it holds. We also mention that we do not know any example of a symmetric Lévy process with an unbounded symbol for which the property is not satisfied.
We say that X satisfies (H) if there is a constant H ≥ 1 such that for any 0

Multiplying both sides by V (u) and integrating over [0, δ] we obtain
which completes the proof.
According to [2,Theorem VI.20] we have for any non-negative function f on [0, ∞), Applying this to the indicator of the interval

Lemma 4.7 Suppose that (H) holds. Let F(z) be non-negative subadditive on R and
The constant c is absolute.
Proof By the Ikeda-Watanabe formula, By Lemma 4.6, Hence,
Note that by Eq. 18, V (2)V (4) ≤ 6 0 G (0,∞) (2, y)dy,and F (w − y) ≤ F (w) + F * (6) if 0 < y < 6 and w < −6 which imply Next, by Lemma 2.9, Combining all the estimates obtained above and using subaddativity of F * and V we conclude that there is an absolute constant c such that which ends the proof.

Lemma 4.8
Let ψ ∈ WLSC(α, γ ), α > 1 and let F be a non-negative harmonic function on (0, 2R), R > 0. Suppose that r > 0 is such that V (R) ≥ 2V (r)/C 4 , where C 4 is the constant from Lemma 2.11. Then for 0 < x < r, where C H is the constant from the Harnack inequality (17), which depends only on the scalings.
Since F is harmonic then using the Harnack inequality (Theorem 4.5) we have for every r ≤ x, y ≤ R such that |x − y| < r, F (x) ≥ C H F (y). By the chaining argument we have for any By Lemma 2.11, Note that by Eq. 19, quasi left-continuity of X and harmonicity of F ,

Hitting Times of Intervals
Throughout this section B R = [−R, R], R > 0. The goal is to find sharp estimates for the tail function of T B R and we start with the case R = 1. Once this is done we use the scaling argument to treat any R > 0. The proposition below provides an effective tool for the upper bound.

Proposition 5.1 Suppose that the condition (H) holds. If
The constant c is absolute.
Proof By Remark 3 we find a constant H dependent only on the scalings such that the property (H) holds. Therefore, by applying Propositions 5.1 and 3.1 together with Lemma 2.14 we end the proof.
K(1/ψ −1 (1/t)) ∧ 1 the WLSC assumption is merely to assure the property (H). However there are many examples for which V is non-increasing and then this property holds automatically with the constant H = 1. For example if X is a special subordinate Brownian motion satisfying (1), then the estimate from the preceding corollary holds with an absolute constant. In particular ψ(x) = |x|+|x| 2 defines a special subordinate Brownian motion and it does not have the lower scaling property with index α > 1.

Proposition 5.3 Let
The constant C 5 depends only on the scalings.
For x ≥ x * ≥ 2 and t ≤ 1 ψ * (1/x) we apply subaddativity of V and Lemma 2.8 to get Next, applying Lemma 2.10 to arrive at Therefore we have proved that for x ≥ x * and any t > 0 we have where c 5 depends on the scalings. In particular taking t = 2/b we obtain where the last inequality follows from scaling property for K and ψ −1 (see Lemma 2.14 and Eq. 10). The constant c 6 depends on the scalings.
Proof We may and do assume that 1 ≤ x < x * . By the strong Markov property we have for any z ≥ 1, and Using Lemma 4.8 we estimate the harmonic function F (z) = E z P X τ (1,∞) (T B 1 > 1), with the constant c 1 dependent only on the scalings. From Lemma 2.10 and subaddativity of V we infer that P x (τ (1,∞) (1,∞) > 1), with c 2 dependent only on the scalings. Hence, K(1/ψ −1 (1)) ∧ 1 , which completes the proof. Now we are ready to state and prove the main result of this section.
The comparability constants depend only on the scaling characteristics.
Proof If t ≤ 1/ψ * (1/R) the estimates hold by [5,Remark 6] and Lemma 2.10. Let t > 0, R > 0 be fixed. We consider a space and time rescaled process Y s = X ts /R, s ≥ 0. Let K t R , etc. be objects corresponding to the process Y . Then ψ t R (x) = tψ(x/R), Let T Y B 1 be the hitting time of B 1 by the process Y . Observe that ψ t R (x) has exactly the same scaling property (with the same scaling characteristics) as ψ(x). Let t > 1/ψ * (1/R) or equivalently 1 > 1/(ψ t R ) * (1). We now apply Corollary 5.2, Proposition 5.3 and Lemma 5.4 to get where the comparability constants depend only on the scalings of ψ.
Proof If the second moment is finite, Hence we obtain the conclusion applying Theorem 5.5.

Remark 5
If X is a special subordinate Brownian motion satisfying (1), then the upper bound from Theorem 5.5 is true without the assumption ψ ∈ WLSC(α, γ ), α > 1. This follows from the fact that property (H) holds for such processes (see Remark 3). In particular ψ(x) = |x| + |x| 2 defines a special subordinate Brownian and it does not have a lower scaling property with α > 1 but we have and Here the constant c is independent of R, By inspecting the proof of Proposition 5.3 it is clear that we can prove a lower bound but the constant c R will be dependent on R (to choose x * as in Proposition 5.3 one can use unboundedness of K instead of the scaling property).

Heat Kernel Estimates
This section is devoted to finding sharp estimates of the heat kernel of the process X killed after hitting an interval. We apply the previous results on hitting times and the estimates of the heat kernel of the free process obtained in [7] under the assumption of unimodality of X and both lower and upper scaling property of ψ. At the end of the section we suggest a certain extension of the main result, which allows to treat symmetric processes which are not unimodal. We denote D R = (−R, −1) ∪ (1, R), R > 1.
where the constant C 6 depends on the scalings.
Proof Let x > 1. By Proposition 2.4, Then by the strong Markov property we have (1,R) ).
Proof By subaddativity of V and K it is enough to consider 1 < x ≤ (R ∨ 3)/2. By Eq. 9, and then Proposition 6.1, The proof is completed by observing that K(R) ≈ V 2 (R)/R with the comparability constant dependent only on the scalings, which folows from Lemma 2.14.
The constant C 9 depends only on the scalings.
Proof We may assume that |x| < y. We find the estimates in the case of fixed t = 2 or t = 3 and r = 1, keeping all arising constants dependent only on the scalings. Then applying the scaling argument we will be able to extend the estimates for the whole range of times and any r > 0. We also assume that t = 1 > V 2 (1). The case 1 ≤ V 2 (1) can be deduced from a general bound for the killed semigroup obtained in [8, Corollary 2.4, Theorem 3.3 and the beginning of Section 5]. In what follows all comparabilities hold with comparability constants which are either depend only on the scalings or they are absolute. The same remark applies to all constants appearing in the proof. As mentioned above throughout the proof we fix D = (−∞, −1) ∪ (1, ∞).
We start with the upper bound. First, we prove that there is a constant c 0 such that To this end we consider two cases.
To get a general bound for any t, r > 0 we consider Y s = 1 r X st , s ≥ 0. For such a process (fixing t and r) its characteristic exponent is ψ r t (u) = tψ(u/r) so it has the same scaling characteristics as ψ(u). Let p Y , p D,Y denote the transition densities for the free and killed process Y , respectively. We have p 2t (x − y) = p Y 2 ((x − y)/r) and p rD 2t (x, y) = p D,Y 2 (x/r, y/r). Moreover P y/r (τ Y D > 2) = P y (τ rD > 2t). Hence applying (25) to Y we obtain p rD (2t, x, y) = p D,Y 2 (x/r, y/r) ≤ c 8 P y/r (τ Y D > 2)P x/r (τ Y D > 2)p Y 2 ((x − y)/r) = c 8 P y (τ rD > 2t)P y (τ rD > 2t)p 2t (x − y).
Remark 7 Let X be a pure jump process. The assumption of unimodality of X can be removed by assuming certain estimates of the symmetric Lévy density of X. Suppose that where f : [0, ∞) → [0, ∞) is non-decreasing and f ∈ WLSC(α, γ 1 ) ∩ WUSC(β, ρ 1 ), 1 < α < β < 2. Then according to [7,Proposition 28] the characteristic exponent of X, ψ ≈ f , so ψ ∈ WUSC(β, ρ) ∩ WUSC(β, ρ). Moreover, by the result of Chen, Kim and Kumagai [13] we have We also note that the renewal function of the ladder process is V X (x) ≈ Hence we conclude that Lemma 6.5 holds in this case. Moreover the density p t (x) is almost non-increasing for x ∈ (0, ∞) that is there is symmetric q t (x) non-increasing on (0, ∞) and a constant c ≥ 1 such that Therefore we can repeat, with necesarry slight modifications, all the steps from the proof of Theorem 6.7 and obtain its conclusion in this case. The details are left to interested readers.