Boundary Harnack principle for the absolute value of a one-dimensional subordinate Brownian motion killed at $0$

We prove the Harnack inequality and boundary Harnack principle for the absolute value of a one-dimensional recurrent subordinate Brownian motion killed upon hitting $0$, when $0$ is regular for itself and the Laplace exponent of the subordinator satisfies certain global scaling conditions. Using the conditional gauge theorem for symmetric Hunt processes we prove that the Green function of this process killed outside of some interval $(a,b)$ is comparable to the Green function of the corresponding killed subordinate Brownian motion. We also consider several properties of the compensated resolvent kernel $h$, which is harmonic for our process on $(0,1)$.


Introduction
Let X = (X t ) t≥0 be a recurrent subordinate Brownian motion on R such that 0 is regular for itself with φ : (0, ∞) → (0, ∞) the Laplace exponent of the corresponding subordinator. We assume that φ is a complete Bernstein function satisfying a certain global scaling condition (H).
The goal of this paper is to establish the Harnack inequality and boundary Harnack principle for nonnegative harmonic functions of the absolute value of process X killed upon hitting {0}, denoted by Z = (Z t ) t≥0 . In order to do so, we show that the Green function for the killed process Z (a,b) on a finite interval (a, b), a > 0, is comparable to the Green function of X (a,b) . We introduce a third process Y = (Y t ) t≥0 on (0, ∞), obtained from X (0,∞) by creation through the Feynman-Kac transform with rate equal to the killing density κ (0,∞) , see (4.1). Process Y is called the resurrected (censored) process on (0, ∞) corresponding to X. Using the conditional gauge theorems from [3] and the sharp two-sided Green function estimates (4.4) for X (a,b) obtained in [4] we first show that the Green functions of processes X (a,b) and Y (a,b) are comparable. In the second step we relate Y (a,b) and Z (a,b) through a Feynman-Kac transform by a discontinuous additive functional and apply the corresponding conditional gauge theorem.
We examine a function h : R → (0, ∞) defined by which is harmonic for Z on (0, ∞). Here u q is the q-potential density of process X, so h is sometimes called the compensated resolvent kernel. This function is often considered in relation to the local time of Lévy processes and its properties were extensively studied in [10], [12] and [5]. By expressing the Green function of Z through function h we obtain estimates of the probability that Z does not die upon exiting the interval (0, R), as well as estimates of the expected exit time of Z from the same interval in terms of h. These results can be also found in a recent paper [5], where they have been considered in a similar setting.
Using these results, as well as sharp two-sided Green function estimates for Z (a,b) obtained through the conditional gauge theorem, we arrive to the main results of this paper by applying standard methods from [7] and [8].
The paper is composed as follows. In Section 2 we recall some basic results for a one-dimensional subordinate Brownian motion and consider several properties of function h. Applying these results, in Section 3 we prove several properties of the first exit time of Z from a finite interval (0, R). In Section 4 we prove that process Z killed outside of a finite interval (a, b), 0 < a < b, can be obtained from the killed censored process Y (a,b) by a combination of a discontinuous and continuous Feynman-Kac transform and show that the Green functions for X (a,b) and Z (a,b) are comparable. Finally, in Section 5 we give the proof of the Harnack inequality and boundary Harnack principle for Z (a,b) .
where the Lévy measure satisfies the condition ∞ 0 (1 ∧ t)ν(t)dt < ∞. Note that every Bernstein function φ satisfies the condition Let W = (W t ) t≥0 be a 1-dimensional Brownian motion and S = (S t ) t≥0 a subordinator independent of W with the Laplace exponent φ, that is Define a one-dimensional subordinate Brownian motion X = (X t ) t≥0 by X t = W St . Then X is a Lévy process with the characteristic exponent and a decreasing Lévy density Furthermore, we will only consider the case when 0 is regular for itself, i.e. when By [1, Theorem II.16] there exists a bounded and continuous density u q of the q-resolvent Since the transition density p t (x) is decreasing in x it follows that u q is decreasing as well.
where τ B = inf{t > 0 : X t ∈ B} is the first exit time of X from B. If (2.3) holds also for D in place of B, we say that f is regular harmonic on D.
Here we assume that the expectation in (2.3) is finite, X ∞ = ∂, where ∂ is the so-called cemetery point and that f (∂) = 0.
The function h is symmetric and since u q is decreasing, h is increasing on [0, ∞).
Let X 0 be the process X killed at 0 and Z = (Z t ) t≥0 the absolute value of that process, Since 0 is not polar, that is P x (σ 0 < ∞) > 0 for all x ∈ R, X 0 is a proper subprocess of X. Also, if X is recurrent then by [12, Theorem 3.1] P x (σ 0 < ∞) = 1 for all x ∈ R. By [12, Theorem 1.1] h is harmonic for the process X 0 on R \ {0} and since it is symmetric, it is also harmonic for Z on (0, ∞). Let G X 0 (x, dy) and G Z (x, dy) be the Green measures for X 0 and Z respectively. Note that for every x > 0 and A ∈ B(0, ∞) and thus the Green function of Z is equal to Furthermore, the Green function G X 0 of X 0 can be represented in terms of the function h. By [1, Note that by the Chung-Fuchs type criteria for recurrence and (2.2) κ = 0 if and only if X is recurrent. It follows from (2.5) that The following lemma is implied by [5, Proposition 2.2, Proposition 2.4], which establish similar bounds for G X 0 in terms of h.
for every x, y > 0.
Proof. First we show that h is a subadditive function on R. Since h is symmetric it follows that By (2.6) and subadditivity of h for 0 < x < y we get Since h is increasing on (0, ∞), when κ = 0 it follows that Throughout this paper we will assume that the complete Bernstein function φ satisfies the following global scaling condition (H): There exist constants a 1 , a 2 > 0 and 1 2 < δ 1 ≤ δ 2 < 1 such that Note that since δ 1 > 1 2 the regularity condition (2.2) is satisfied and κ = 0, i.e. X is recurrent and 0 is regular for itself.
We will use the following estimate of h in terms of the characteristic function ψ several times in the following chapter, see also [5, Lemma 2.14].

Lemma 2.3
There exists a constant c 1 > 1 such that for all x > 0 .
On the other hand, denoting the Fourier transform operator by F we have .
From the previous lemma and (H) it follows that h also satisfies a global scaling condition, i.e. there exist constants d 1 , d 2 > 0 such that This condition implies the following lower bound for the Green function G Z (0,R) of the killed process Z (0,R) , R > 0, which is obtained similarly as in [5,Lemma 4.2]. We omit the proof.
3 Properties of the exit time of Z from a finite interval The following probability estimate that Z does not die upon exiting (0, R) was also obtained in [5, Proposition 2.7].
Lemma 3.1 For every R > 0 and x ∈ (0, R) Proof. First we prove the right inequality. For ε > 0 by harmonicity of h on (0, ∞), Since h is continuous and h(0) = 0 by the dominated convergence theorem and quasi-left continuity of Z it follows that h is regular harmonic for Z on (0, R), Since h is increasing it follows that For the other inequality, by continuity and harmonicity of the Green function G Z (·, 2R) on (ε, R) and Lemma 2.2, it follows that The following estimate for the tail distribution function of the lifetime of Z was proven in [5, Corollary 3.5.]. Under additional assumptions it is also possible to obtain estimates of the derivatives of the tail distribution with respect to the time component. For more detail see [6].
Lemma 3.2 If there exist a 1 > 0 and δ 1 ∈ (0, 1] such that φ(λt) ≥ a 1 λ δ 1 φ(t) hold for all λ ≥ 1 and t > 0, then there exists a constant c 2 = c 2 (n, φ) such that Using this result we can easily derive the following estimates for the expected exit time of Z from interval (0, R) in terms of the function h.
, for x small enough.
Proof. (i) By Lemma 2.2 (ii) For the other inequality note that for all t > 0 Note that by (H) there exists a constantsc 2 > 0 such that for all t ≤ 1 so there exists t 0 = t 0 (φ, R) ∈ (0, 1) such that f R (t) > 0 for all t < t 0 . Therefore, .

Green function estimates for Z (a,b)
Let X (a,b) and Z (a,b) be processes X and Z killed outside of interval (a, b), 0 < a < b. In this section we obtain sharp bounds on the Green function G Z (a,b) by comparing it to the Green function of X (a,b) .
Let Y be the process obtained from X (0,∞) through the Feynman-Kac transform with respect to the positive continuous additive functional A κ with potential for every bounded Borel function f on (0, ∞). We call Y the resurrected (censored) process on (0, ∞) corresponding to X, see [2] for a study of the censored process corresponding to a symmetric α-stable Lévy process, α ∈ (0, 2).
From the representation of Beurling-Deny and LeJan, the jumping measure associated with the Dirichlet form (E Z , F Z ) corresponding to the process Z has a density equal to i(x, y) = j(|x − y|) + j(|x + y|).
The Dirichlet forms corresponding to the processes X (a,b) , Y (a,b) and Z (a,b) are therefore equal to where the killing densities κ 1 , κ 2 and κ 3 are of the form i(x, y)dy.
Note that Y (a,b) can be obtained from X (a,b) by creation through the Feynman-Kac transform at rate κ (0,∞) . Therefore, by [3,Lemma 3.4] we can relate the Green functions of processes X (a,b) and Here ζ X (a,b) = inf{t > 0 : X t ∈ (a, b)} is the lifetime of X (a,b) and P y x denotes the probability measure of the G X (a,b) (·, y)-conditioned process starting from x, i.e. the process with transition probability Next we recall the definition of the Kato class S ∞ from [3].
for all measurable sets B ⊂ K such that λ(B) < δ.
Also, (H) implies that Φ −1 satisfies the following scaling condition: there exists a constant c 5 > 1 such that for all 0 < r ≤ R < ∞ .
Next, we associate the Green functions for processes Y (a,b) and Z (a,b) . Since where F (x, y) = j(|x+y|) j(|x−y|) and q = κ 3 − κ 2 , Z (a,b) can be obtained from Y (a,b) through the Feynman-Kac transform driven by a discontinuous additive functional (4.7) By [3, Lemma 3.9] the ratio of Green functions G Z (a,b) (x, y) and G Y (a,b) (x, y) is equal to the gauge function u 2 (x, y) = E y x e A q+F (ζ Y (a,b) ) and ζ Y (a,b) = inf{t > 0 : Y t ∈ (a, b)} is the lifetime of Y (a,b) and P y x is the probability measure of the G Y (a,b) (·, y)-conditioned process starting from x, see (4.3). We recall the definition of the Kato class A ∞ .
where B ⊂ K is a measurable set such that B E F (x, y)J(x, dy) dx < δ.
By [3, Theorem 3.8] the conditional gauge function u 2 is bounded between two positive constants when q ∈ S ∞ (Y (a,b) ) and F ∈ A ∞ (Y (a,b) ). This is shown by using (4.4) similarly as in Theorem 4.2, so we omit the proof.
Theorem 4.4 Let the assumptions from Theorem 4.2 hold and A q+F be the discontinuous additive functional for Y (a,b) from (4.7). Then q ∈ S ∞ (Y (a,b) ) and F ∈ A ∞ (Y (a,b) ) and consequently the Green functions of the processes Y (a,b) and Z (a,b) are comparable.

Boundary Harnack principle for Z
The exit distribution of Z τ (a,b) starting from x is equal to where K Z (a,b) is the Poisson kernel of Z (a,b) given by Recall that the process Z can exit the interval (a, b) only by jumping out, since by [11,Theorem 1] for all x ∈ (a 1 , a 2 ) ⊂ R. Using the results from the previous sections we can similarly as in [7,Section 4] prove the Harnack inequality and boundary Harnack principle for nonnegative harmonic functions of process Z (a,b) .

Theorem 5.2 Boundary Harnack principle
Let R > 0. There exists a constant c 7 = c 7 (R, φ) > 0 such that for all r ∈ (0, R), and every nonnegative function u which is harmonic for Z in (0, 3r) and continuously vanishes at 0 it holds that where λ 1 is the constant from Lemma 2.4.
By the last two displays, (2.1) and Lemma 2.3 we get the required inequality, i.e.