Green Function for Gradient Perturbation of Unimodal Lévy Processes in the Real Line

We prove that the Green function of the generator of symmetric unimodal Lévy process with the weak lower scaling order bigger than one and the Green functions of its gradient perturbations are comparable for bounded C1,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,1}$$\end{document} subsets of the real line if the drift function is from an appropriate Kato class.


Introduction
Perturbations of Markovian generators are widely studied for many years. This theory may be considered from various points of view. Such perturbations appear, e.g., in local and non-local partial differential equations [10,11,33,34], semigroup theory [4,7,8,26,30], stochastic processes [25,31,31] and potential theory [9,12,17]. One of the natural questions is how this perturbation affects the solutions of the equations related to the unperturbed operator (e.g., the transition density of the semigroup and the Green function). In this paper we are interested in the gradient perturbations and the potential theory of the perturbed operator. We briefly recall some results closely related to our research. Cranston and Zhao in [16] considered the operator + b(x)∇ in R d for d 2. They proved that the Green function and harmonic measures of Lipschitz domains are comparable with those of for the drift b from an appropriate Kato class. In [24] and [25] Jakubowski studied the α-stable Ornstein-Uhlenbeck process. He proved estimates for the first exit time from the ball and Harnack inequality for this process. In [9] Bogdan and Jakubowski proved similar results as Cranson and Zhao for α/2 + b(x)∇ in C 1,1 domains in R d , d 2.
In the recent paper [19] these results were generalized to the case of pure-jump symmetric unimodal Lévy processes possessing certain weak scaling properties. We note that in the papers [9,19] the case d = 1 was omitted. The aim of this paper is to fill this gap and prove analogous results in one-dimensional case.
We will denote by {X t } a pure-jump symmetric unimodal Lévy process on R, that is, a process with the symmetric density function p t (x) on R\{0} which is non-increasing on R + . The characteristic exponent of {X t } equals where ν is a Lévy measure, i.e., R 1 ∧ |z| 2 ν(dz) < ∞. For general information on unimodal processes, we refer the reader to [5,18,35]. A primary example of the mentioned class of processes is the symmetric α-stable Lévy process having the fractional Laplacian α/2 as a generator. Let be the generator of the process X t . We will consider a non-empty bounded open C 1,1 set D and the Green function G D (x, y) for L. Now, letG D (x, y) be the Green function forL where b is a function from the Kato class K 1 . (For definitions of C 1,1 sets, the Green function and the Kato class, we refer the reader to Sects. 2 and 4). Our main result is Theorem 1.1 Let b ∈ K 1 , and let D ⊂ R be an union of finitely many open, bounded intervals with positive distance between every two intervals. We assume that the characteristic exponent ψ ∈ WLSC(α, 0, c) ∩ WLSC(α 1 , 1, c 1 ) ∩ WUSC(α, 0, C), where α 1 > 1.
Then, there exists a constant C such that for x, y ∈ D, Here WLSC and WUSC are the classes of functions satisfying a weak lower and weak upper scaling condition, respectively (see Sect. 2 for definitions). Generally, we follow the approach of [9] and [19]; however, there are some important differences. Although the geometry of the set D is much simpler than in higher dimensions, it seems that the one-dimensional case sometimes demands more delicate arguments. One of the main difficulties is to obtain the proper estimates for derivative of the Green function. We use them to show the uniform integrability of ∂ x G D (x, y) against |b(z)|dz (see Lemma 3.11). As it was mentioned in [9], for d = 1, the available estimates are not integrable near y. We emphasize here that we make no additional assumption on the monotonicity of ν (r )/r as mentioned above. Like in the mentioned papers, our main tool is the perturbation formula. First, we use it to obtain estimates for sets D with a small radius. Since the Green function G D (x, y) is bounded, we do not use the perturbation series as in [9] and [19]. Instead, we propose a simpler iteration argument. We note also that one of our standing assumptions is α 1 > 1. It may be understood that the rank of the operator L is larger than 1. Without this assumption the drift term may have a stronger effect than L on the behavior of the Green function of theL. Any results concerning the cases α 1 would be interesting; however, for α < 1, Theorem 1.1 cannot hold in the form above (see the Introduction of [9] for more details) This paper is organized as follows. In Sect. 2 we define the process X and present its basic properties. In Sect. 3, we introduce the Green function of X and prove the estimates for its derivative and some 3G-like inequalities. In Sect. 4, we define the operatorL and the Green function of the underlying Markov process. Lastly, in Sect. 5, we prove Theorem 1.1.
When we write f (x) C ≈ g(x), we mean that there is a number 0 < C < ∞ independent of x, i.e., a constant, such that for every x, We use the convention that numbered constants denoted by capital letters do not change throughout the paper. For a symmetric function f : R → [0, ∞) we shall often write f (r ) = f (x) for any x ∈ R with |x| = r .

Preliminaries
In what follows, R denotes the Euclidean space of real numbers and dy stands for the Lebesgue measure on R. Without further mention we will only consider Borel sets, measures and functions in R. As usual, we write a ∧ b = min(a, b) and a ∨ b = max(a, b). We let B(x, r ) = {y ∈ R : |x − y| < r }. For the arbitrary set A ⊂ R, the distance to the boundary of A will be denoted by To simplify the notation, while referring to the set D, we will omit the superscript, i.e., Definition 1 Let θ ∈ [0, ∞) and φ be a nonnegative nonzero function on (0, ∞). We say that φ satisfies the weak lower scaling condition (at infinity) if there are numbers α > 0 and c ∈ (0, 1] such that In short, we say that φ satisfies WLSC(α, θ, c) and write φ ∈ WLSC(α, θ, c). If φ ∈ WLSC(α, 0, c), then we say that φ satisfies the global weak lower scaling condition.
Throughout the paper, X t will be a pure-jump symmetric unimodal Lévy process on R. The Lévy measure ν of X t is symmetric and non-increasing, so it admits a density ν, i.e., ν(dx) = ν(|x|)dx. Hence, the characteristic exponent ψ of X t is symmetric as well.

Remark 1
The threshold (0, 1) in scaling of V in (2.7) may be replaced by any bounded interval at the expense of a constant 2C 1 /c 1 (see [5,Section 3]), i.e., for any R > 1, there is a constant c such that We define We note that M(·) is decreasing and lim r →0 + M(r ) = ∞. To simplify the notation on how the constants depend on the parameters, we put σ = (α, C, α, c) and σ = (σ, α 1 , c 1 ).

Definition 2
We say that a function b : R → R belongs to the Kato class K 1 if r 2 for small r > 0, in this paper we will use condition (2.15) in the form We consider the time-homogeneous transition probabilities By Kolmogorov's and Dynkin-Kinney's theorems the transition probability P t defines in the usual way Markov probability measures {P x , x ∈ R} on the space of the rightcontinuous and left-limited functions ω : [0, ∞) → R. We let E x be the corresponding expectations. We will denote by X = {X t } t 0 the canonical process on , X t (ω) = ω(t). Hence, For any open set D, we define the first exit time of the process X t from D, Now, by the usual Hunt's formula, we define the transition density of the process killed when leaving D [2,6,15]: We briefly recall some well-known properties of p D (see [6]). The function p D satisfies the Chapman-Kolmogorov equations Furthermore, p D is jointly continuous when t = 0, and we have In particular,

Green Function of L
We define the Green function of X t for D, and the Green operator From now on, every time we mention the Green function, it should be understood as the Green function of D, and then G = G D .

Definition 3 We say that a function
We say that a function f is a regular L−harmonic (or simply regular harmonic) Definition 4 corresponds to the definition of multidimensional C 1,1 set with localization radius r 0 . In what follows, we assume that D is a C 1,1 set with diam(D) < ∞ and localization radius r 0 = r 0 (D).
Some constants will depend on the ratio diam(D)/r 0 called the distortion of the set D.

and Theorem 4.5])
where the comparability constant c 1 depends on the scaling characteristics in (2.6) and (2.7) and the distortion of D. Now, by the same calculation as in the proof of [13, Theorem 7.3 (iii) and Corollary 7.4], we obtain where the comparability constant c 2 depends on the scaling characteristics in (2.6) and (2.7), the distortion of D and 1 ∨ diam(D). Let us consider x, y ∈ D such that Without loss of generality we may and do assume δ x δ y . Then, . By monotonicity and subadditivity of V we obtain that As a consequence of (2.6), we obtain Again, by (2.6), we get where We only need to show that 2|x − y| 2 δ x δ y . Without loss of generality we can and do assume δ x δ y . By monotonicity of V , |x − y| δ x . Hence, δ y |x − y| + δ x 2|x − y|, which completes the proof.

Estimates of the Poisson Kernel
If D is C 1,1 , it is known that the harmonic measure of D has a density and we call it the Poisson kernel. By the Ikeda-Watanabe formula [23] it is equal to First, we give the estimates of the Poisson kernel in the special case D = B(0, R).
If |z| 2R, by (2.5) and [20, Proposition 3.5], we get which implies the claim of the lemma.
where c 2 = c 2 (C 3 , C 4 ). If dist(z, B) = δ z , the lower bound follows by (3.9). Suppose dist(z, B) > δ z , and letB be a connected component ofD such that dist(z,B) = δ z . Note that r 0 dist(x,B) |x − z| + dist(z,B) 2|x − z|. Therefore, by Lemma 3.1, Now, (2.10) and (2.5) imply Hence, we obtain the lower bound in this case. Next, we will prove the upper bound for the second integral. Let λ = δ z ∧ diam(D) and D 1 =D ∩ {y : δ y λ} and D 2 =D ∩ {y : δ y > λ}. By weak scaling conditions, we obtain where c 5 = c 5 (C 3 , C) and the constant c 6 depends only on the scaling characteristics. This completes the proof.

Estimates of @ x G(x, y)
Below, we will prove various estimates of ∂ x G(x, y) according to the range of variables x and y. We summarize these results in Theorem 3.10. First, we will need the following auxiliary lemma.

Lemma 3.4 Let x ∈ D. There is a constant C
Proof Note that for r < R, by (2.8), we have with c 2 depending on R. Let B 1 = B(x, δ x /2). By symmetry of M, (2.5) and (3.10), we have Note that for z / ∈ B 1 , we have δ z 3|x − z| and δ x 2|x − z|. Hence, by (2.6), Proposition 3.5 There is a constant C 7 = C 7 (σ , diam(D)/r 0 , 1 ∨ diam(D)) such that Proof Since X t is translation invariant, we may and do assume that 0 / ∈ D. Let x, y ∈ D and x = y. It is known (see [21,Lemma 2 (3.12) Hence, by symmetry, By Lemma 2.1 and the dominated convergence theorem, Again, by Lemma 2.1 and (2.14), for |x − z| |x − y|/2, we have
By Lemma 3.1 and Proposition 3.5, we get a weaker but also useful estimate. (3.14) Proof Let 0 < h < δ y /2. Then, by Corollary 3.6 and monotonicity of M,
Since f ∈ K 1 and (2.15) holds, we obtain the assertion of the lemma.
Let us fix y ∈ A. For z ∈ B, we define P 1 (y, z) = P A (y, z)1 B(x,ε/2) (z) and P 2 (y, z) = P A (y, z) − P 1 (y, z). Since P 1 is bounded, we have P 1 (y, ·) ∈ K 1 and by Lemma 3.7, Since ∂ x G(x, z) is finite on the support of P 2 (y, ·), by the mean value theorem and the dominated convergence theorem, we get These imply which completes the proof. Lemma 3.9 Let x, y ∈ D and δ x < 2|x − y|. Then, there exists a constant C 9 = C 9 (σ , diam(D)/r 0 , 1 ∨ diam(D)) such that Proof Let B ⊂ D be any interval such that B ⊂ D, and put A = B c ∩ D. For any x ∈ B and y ∈ D such that x = y, by Propositions 3.5 and 3.8 and harmonicity of G, Therefore, it remains to estimate the integral

By Proposition 3.3 and Lemmas 3.1 and 3.4, we get
Since constants c 1 , . . . , c 4 depend on D only via constants C 2 , C 4 and C 6 , the proof is completed.

Theorem 3.10
There is a constant C 10 = C 10 (σ , diam(D)/r 0 , 1 ∨ diam(D)) such that Proof First, assume |x − y| > δ x /2. By Corollary 3.6, Lemma 3.9 and monotonicity of K , we have Now, we show the existence of a constant c such that when |x − y| δ x /2. In this case δ x ≈ δ y and therefore, by Lemma 3.1, Since α 1 > 1, by (2.7), we obtain that s → V 2 (s)/s is almost increasing (bounded from below by an increasing function). Hence, we get the claim.
We end this section with the proof of the uniform integrability of ∂ z G(z, y).

Lemma 3.11
The function ∂ z G(z, y) is uniformly in y integrable against |b(z)|dz.
Proof It is enough to show that Let N > 0 and r N = inf{r > 0 : M(r ) N /C 8 } ∧ r 0 . Note that lim r →0 M(r ) = ∞; hence, r N → 0 as N → ∞. Fix y ∈ R, and take N such that r N r 0 . By (3.14), By Corollary 3.6 and monotonicity of M(·), we have where D = n k=1 (a 2k−1 , a 2k ). By (2.15), lim N →∞ K r N = 0, which completes the proof.

3G Inequalities
Now, we apply the estimates of the Green function and its derivative to obtain the following 3G-type inequalities.

Proposition 3.12
There is a constant C 11 = C 11 (σ , 1 ∨ diam(D)) such that Proof For x, y ∈ D, we define .

Green Function ofL
Following [8] and [27] we recursively define, for t > 0 and x, y ∈ R, p 0 (t, x, y) = p(t, x, y), where c T → 1 if T → 0, see [8,Theorem 2]. By Chapman-Kolmogorov equation (see [27]), there is C 14 > 0 such that We letP,Ẽ be the Markov distributions and expectations defined by the transition densityp on the canonical path space. By Hunt formula, Except for the symmetry,p D has analogous properties as p D , i.e., the Chapman-Kolmogorov equation holds R dp We denote byG D (x, y) the Green function ofL = L + b∂ on D, As for G, from now on, every time we will mention the Green functionG, it should be understood as the Green function ofL on D, and thenG =G D . For x ∈ D c or y ∈ D c , we observe bothp D (t, x, y) = 0 andG(x, y) = 0. Indeed, since D is a C 1,1 set, for any x ∈ D c (or y ∈ D c ), we can find some interval for every t < 1. This implies c <P x (τ D = 0) and by Blumenthal's zero-one law ( [14]), we haveP x (τ D = 0) = 1. Thus, by (4.4), we get the assertion. By (4.2), we have Thus, the intensity of jumps of the canonical process X t underP x is the same as under P x . Accordingly, we obtain the following description.
We define the Poisson kernel of D forL, By (4.5), (4.7) and (4.6), we havẽ For the case of A ⊂ ∂ D, we refer the reader to (5.8) below.

Lemma 4.2G(x, y) is continuous and
Proof In the same way as in [9,Lemma 7], we get that there are positive constants c and C such thatp is well defined, integrable and bounded on R. Hence, following [19, Theorem 3.1], we obtain the following perturbation formula (for the proof see [19]). (4.10)

Proof of Theorem 1.1
First, we will prove the comparability of G andG for small sets D from the C 1,1 class. For this purpose, we could consider the perturbed series forG as presented in [19]. We could define by induction the functions G n and show the convergence and estimates of the seriesG However, sinceG is bounded, we present a simpler proof of the following lemma (compare [19,Lemma 3.11]).
We note that the comparison constants in the proof above will improve to 1 if diam(D) → 0 and the distortion of D is bounded. By (4.8), where C 16 = C 16 (σ , b, λ, diam(D)) and diam(D) < ε from Lemma 5.1.
Following [9, Proof of Lemma 14], we obtain that the boundary of our general C 1,1 open set D is not hit at the first exit, i.e., x ∈ D. (5.8) Hence, in the context of Lemma 5.1, theP x distribution of X τ D is absolutely continuous with respect to the Lebesgue measure and has the density functioñ provided x ∈ D. This follows from (4.8) and (5.8).
The definition ofL-harmonicity is analogous to that of L-harmonicity.

Definition 5
We say that a function f :

We say that a function f is regularL-harmonic on an open bounded set
Following [9] and [19], we get the following Harnack inequality.  with C 18 = C 18 (σ, b, q, r 0 (D)).
Now, we have all the tools necessary to prove the main result of our paper. Since in the proof we follow the idea from [9], we only give its basic steps (for details see [9, Proof of Theorem 1]).
In the proof of lower bounds, we consider two cases: x close to y and x far away from y.
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