Potential Theory of Subordinate Brownian Motions Revisited

Transcription

1 Potential Theory of Subordinate Brownian Motions Revisited Panki Kim Renming Song and Zoran Vondraček Abstract The paper discusses and surveys some aspects of the potential theory of subordinate Brownian motion under the assumption that the Laplace exponent of the corresponding subordinator is comparable to a regularly varying function at infinity. This extends some results previously obtained under stronger conditions. AMS Mathematics Subject Classification: Primary 6J45 Secondary 6G5, 6J35, 6J75 Keywords and phrases: Subordinator, subordinate Brownian motion, potential theory, Green function, Lévy density, Harnack inequality, boundary Harnack principle Introduction An R d -valued process X = X t : t is called a Lévy process in R d if it is a right continuous process with left limits and if, for every s, t, X t+s X s is independent of {X r, r [, s]} and has the same distribution as X s X. A Lévy process is completely characterized by its Lévy exponent Φ via E[exp{i ξ, X t }] = exp{ tφξ}, t, ξ R d. The Lévy exponent Φ of a Lévy process is given by the Lévy-Khintchine formula Φξ = i l, ξ + ξ, AξT + e i ξ,x i ξ, x { x <} Πdx, ξ R d, R d where l R d, A is a nonnegative definite d d matrix, and Π is a measure on R d \ {} satisfying x Πdx <. A is called the diffusion matrix, Π the Lévy measure, and l, A, Π the generating triplet of the process. Nowadays Lévy processes are widely used in various fields, such as mathematical finance, actuarial mathematics and mathematical physics. However, general Lévy processes are not very easy to deal with. A subordinate Brownian motion in R d is a Lévy process which can be obtained by replacing the time of Brownian motion in R d by an independent subordinator i.e., an increasing Lévy process. This work was supported by Basic Science Research Program through the National Research Foundation of KoreaNRF grant funded by the Korea governmentmest-984. Supported in part by the MZOS grant

2 More precisely, let B = B t : t be a Brownian motion in R d and S = S t : t be a subordinator independent of B. The process X = X t : t defined by X t = B St is a rotationally invariant Lévy process in R d and is called a subordinate Brownian motion. The subordinator S used to define the subordinate Brownian motion X can be interpreted as operational time or intrinsic time. For this reason, subordinate Brownian motions have been used in mathematical finance and other applied fields. Subordinate Brownian motions form a very large class of Lévy processes. Nonetheless, compared with general Lévy processes, subordinate Brownian motions are much more tractable. If we take the Brownian motion B as given, then X is completely determined by the subordinator S. Hence, one can deduce properties of X from properties of the subordinator S. On the analytic level this translates to the following: Let φ denote the Laplace exponent of the subordinator S, that is, E[exp{ λs t }] = exp{ tφλ}, λ >. Then the characteristic exponent Φ of the subordinate Brownian motion X takes on the very simple form Φx = φ x our Brownian motion B runs at twice the usual speed. Therefore, properties of X should follow from properties of the Laplace exponent φ. The Laplace exponent φ of a subordinator S is a Bernstein function, hence it has a representation of the form φλ = bλ + e λt µdt, where b and µ is a measure on, satisfying, t µdt <. If µ has a completely monotone density, the function φ is called a complete Bernstein function. The purpose of this work is to study the potential theory of subordinate Brownian motion under the assumption that the Laplace exponent φ of the subordinator is a complete Bernstein function comparable to a regularly varying functions at infinity. function l slowly varying at infinity such that More precisely, we will assume that there exist α, and a φλ λ α/ lλ, λ. Here and later, for two functions f and g we write fλ gλ as λ if the quotient fλ/gλ stays bounded between two positive constants as λ. A lot of progress has been made in recent years in the study of the potential theory of subordinate Brownian motions, see, for instance [3, 4, 5, 6, 7, 3, 35] and [6, Chapter 5]. In particular, an extensive survey of results obtained before 7 is given in [6, Chapter 5]. At that time, the focus was on the potential theory of the process X in the whole of R d, the results for the killed subordinate Brownian motion in an open subset still being out of reach. In the last few years significant progress has been made in studying the potential theory of subordinate Brownian motion killed upon exiting an open subset of R d. The main results include the boundary Harnack principle and sharp Green function estimates. For the processes having a continuous component see [6] for the one-dimensional case and [, 3, 4] for multi-dimensional case. For purely discontinuous processes, the boundary Harnack principle was obtained in [5] and sharp Green function estimates were discussed in the recent preprint [7]. The main assumption in [3, 4, 5]

3 and [6, Chapter 5] is that the Laplace exponent of the subordinator is regularly varying at infinity. The results were established under different assumptions, some of which turned out to be too strong and some even redundant. Time is now ripe to put some of the recent progress under one unified setup and to give a survey of some of these results. The survey builds upon the work done in [6, Chapter 5] and [5] The setup we are going to assume is more general than all these of the previous papers, so in this sense, most of the results contained in this paper are extensions of the existing ones. In Section we first recall some basic facts about subordinators, Bernstein functions and complete Bernstein functions. Then we establish asymptotic behaviors, near the origin, of the potential density and Lévy density of subordinators. In Section 3 we establish the asymptotic behaviors, near the origin, of the Green function and the Lévy density of our subordinate Brownian motion. These results follow from the asymptotic behaviors, near the origin, of the potential density and Lévy density of the subordinator. In Section 4 we prove that the Harnack inequality and the boundary Harnack principle hold for our subordinate Brownian motions. The materials covered in this paper by no means include all that can be said about the potential theory of subordinate Brownian motions. One of the omissions is the sharp Green function estimates of killed subordinate Brownian motions in bounded C, open sets obtained in the recent preprint [7]. The present paper builds up the framework for [7] and can be regarded as a preparation for [7] in this sense. Another omission is the Dirichlet heat kernel estimates of subordinate Brownian motions in smooth open sets recently established in [8, 9,, ]. One of the reasons we do not include these recent results in this paper is that all these heat kernel estimates are for particular subordinate Brownian motions only and are not yet established in the general case. A third notable omission is the spectral theory for killed subordinate Brownian motions developed in [7, 8, 9]. Some of these results have been summarized in [34, Section.3]. A fourth notable omission is the potential theory of subordinate killed Brownian motions developed in [,, 37, 38, 4, 4]. Some of these results have been summarized in [6, Section 5.5] and [34, Chapter 3]. In this paper we concentrate on subordinate Brownian motions without diffusion components and therefore this paper does not include results from[3, 4, 6]. One of the reasons for this is that subordinate Brownian motions with diffusion components require a different treatment. We end this introduction with few words on the notations. For functions f and g we write ft gt as t + resp. t if the quotient ft/gt converges to as t + resp. t, and ft gt as t + resp. t if the quotient ft/gt stays bounded between two positive constants as t + resp. t. 3

4 Subordinators. Subordinators and Bernstein functions Let S = S t : t be a subordinator, that is, an increasing Lévy process taking values in [, with S =. A subordinator S is completely characterized by its Laplace exponent φ via E[exp λs t ] = exp tφλ, λ >. The Laplace exponent φ can be written in the form cf. [, p. 7] φλ = bλ + e λt µdt. Here b, and µ is a σ-finite measure on, satisfying t µdt <. The constant b is called the drift, and µ the Lévy measure of the subordinator S. A C function φ :, [, is called a Bernstein function if n D n φ for every positive integer n. Every Bernstein function has a representation cf. [34, Theorem 3.] φλ = a + bλ + e λt µdt, where a, b and µ is a measure on, satisfying, t µdt <. a is called the killing coefficient, b the drift and µ the Lévy measure of the Bernstein function. Thus a nonnegative function φ on, is the Laplace exponent of a subordinator if and only if it is a Bernstein function with φ+ =. A Bernstein function φ is called a complete Bernstein function if the Lévy measure µ has a completely monotone density µt, i.e., n D n µ for every non-negative integer n. Here and below, by abuse of notation we will denote the Lévy density by µt. Complete Bernstein functions form a large subclass of Bernstein functions. Most of the familiar Bernstein functions are complete Bernstein functions. See [34, Chapter 5] for an extensive table of complete Bernstein functions. Here are some examples of complete Bernstein functions: i φλ = λ α/, α, ]; ii φλ = λ + m /α α/ m, α,, m ; iii φλ = λ α/ + λ β/, β < α, ]; iv φλ = λ α/ log + λ γ/, α,, γ, α; v φλ = λ α/ log + λ β/, β < α, ]. 4

5 An example of a Bernstein function which is not a complete Bernstein function is e λ. It is known cf. [34, Proposition 7.] that φ is a complete Bernstein function if and only if the function λ/φλ is a complete Bernstein function. For other properties of complete Bernstein functions we refer the readers to [34]. The following result, which will play an important role later, says that the Lévy density of a complete Bernstein function cannot decrease too fast in the following sense. Lemma. [7, Lemma.] Suppose that φ is a complete Bernstein function with Lévy density µ. Then there exists C > such that µt C µt + for every t >. Proof. Since µ is a completely monotone function, by Bernstein s theorem [34, Theorem.4], there exists a measure m on [, such that µt = [, e tx mdx. Choose r > such that [,r] e x mdx r, e x mdx. Then, for any t >, we have e tx mdx e t r e x mdx [,r] e t r Therefore, for any t >, µt + e t+x mdx e r [,r] [,r] [,r] r, e x mdx e tx mdx e r The potential measure of the subordinator S is defined by UA = E r, [, e tx mdx. e tx mdx = e r µt. {St A} dt, A [,.. Note that UA is the expected time the subordinator S spends in the set A. The Laplace transform of the measure U is given by LUλ = e λt dut = E exp λs t dt = φλ.. We call a subordinator S a complete subordinator if its Laplace exponent φ is a complete Bernstein function. The following characterization of complete subordinators is due to [4, Remark.] see also [6, Corollary 5.3]. Proposition. Let S be a subordinator with Laplace exponent φ and potential measure U. Then φ is a complete Bernstein function if and only if Udt = cδ dt + utdt for some c and completely monotone function u. 5

6 In case the constant c in the proposition above is equal to zero, we will call u the potential density of the subordinator S. An inspection of the argument, given in [6, Chapter 5] or [4] leading to the proposition above yields the following two results cf. [6, Corollary 5.4 and Corollary 5.5] or [4, Corollary.3 and Corollary.4]. Corollary.3 Suppose that S = S t : t is a subordinator whose Laplace exponent φλ = bλ + e λt µdt is a complete Bernstein function with b > or µ, =. Then the potential measure U of S has a completely monotone density u. Proof. By [6, Corollary 5.4] or [4, Corollary.3], if the drift of complete subordinator S is zero or the Lévy measure µ has infinite mass, then the constant c in Proposition. above is equal to zero so the potential measure U of S has a density u. The completeness of the density follows directly from Proposition.. Corollary.4 Let S be a complete subordinator with Laplace exponent φ. Suppose that the drift of S is zero and the Lévy measure µ has infinite mass. Then the potential measure of a subordinator with Laplace exponent ψλ := λ/φλ has a completely monotone density v given by vt = µt,. Proof. Since the drift of S is zero and the Lévy measure µ has infinite mass, by [6, Corollary 5.5] or [4, Corollary.4], we have that ψλ = a + e λt νdt where a = tµtdt, the Lévy measure ν of ψ has infinite mass and the potential measure of a possibly killed i.e., a > subordinator with Laplace exponent ψ has a density v given by vt = µt,. The completeness of the density follows from [6, Corollary 5.3], which works for killed subordinator.. Asymptotic behavior of the potential and Lévy densities From now on we will always assume that S is a complete subordinator without drift and that the Laplace exponent φ of S satisfies lim λ φλ = or equivalently, the Lévy measure of S has infinite mass. Under this assumption, the potential measure U of S has a completely monotone density u cf. Corollary.3. The main purpose of this subsection is to determine the asymptotic behaviors of u and µ near the origin. For this purpose, we will need the following result due to Zähle cf. [46, Theorem 7]. 6

7 Proposition.5 Suppose that w is a completely monotone function given by wt = where f is a nonnegative decreasing function. Then e st fs ds, fs e s ws, s >. If, furthermore, there exist δ, and a, s > such that then there exists C = C w, f, a, s, δ > such that wλt aλ δ wt, λ, t /s,.3 fs C s ws, s s. Proof. Using the assumption that f is a nonnegative decreasing function, we get that, for any r >, we have Thus In particular, we have and wt = t t s f t r r f t e s f e s f s t ds s t ds t f r t e r. twt, t >, r >. e r fs e s ws, s >, e t s w On the other hand, for r, ], we have twt = r r e s f e s f s t ds + s t ds + f t, s >, t >..4 s r e s f r t e r s t ds e r t e s t r s w ds + f e r, s t where in the last line we used.4. Now we assume.3, then we get that t w as δ wt, t /s, s < r. s 7

8 Thus, for r, ], we have, Choosing r, ] small enough so that twt a e r twt e s s δ ds + f a e r e s s δ ds, we conclude that for this choice of r, we have r f c twt, t /s t for some constant c >. Since w is decreasing, we have r r fs c s w c s ws, s rs, s r t e r. where c = c r. From this we immediately get that there exists c 3 > such that fs c 3 s ws, s s. Corollary.6 The potential density u of S satisfies Proof. Apply the first part of Proposition.5 to the function ut C 3 t φt, t >..5 wt := e st us ds = φt, We introduce now the main assumption on our Laplace exponent φ of the complete subordinator S that we will use throughout the rest of the paper. Recall that a function l :,, is slowly varying at infinity if lλt lim =, for every λ >. t lt Assumption H: There exist α, and a function l :,, which is measurable, locally bounded and slowly varying at infinity such that φλ λ α/ lλ, λ..6 8

9 Remark.7 The precise interpretation of.6 will be as follows: There exists a positive constant c > such that c φλ λ α/ c for all λ [,. lλ The choice of the interval [, is, of course, arbitrary. Any interval [a, would do, but with a different constant. This follows from the assumption that l is locally bounded. Moreover, by choosing a > large enough, we could dispense with the local boundedness assumption. Indeed, by [3, Lemma.3.], every slowly varying function at infinity is locally bounded on [a, for a large enough. Although the choice of interval [, is arbitrary, it will have as a consequence the fact that all relations of the type ft gt as t respectively t + following from.6 will be interpreted as c ft/gt c for t respectively < t. The assumption.6 is a very weak assumption on the asymptotic behavior of φ at infinity. All the examples in i, iii and v, we need to take α < above Lemma. satisfy this assumption. In fact they satisfy the following stronger assumption φλ = λ α/ lλ,.7 where l is a function slowly varying at infinity. By inspecting the table in [34, Chapter 5], one can come up with a lot more examples of complete Bernstein functions satisfying this stronger assumption. In the next example we construct a complete Bernstein function satisfying.6, but not the stronger.7. Example.8 Suppose that α,. Let F be a function on [, defined by F x = on x < and F x = n, n /α x < n/α, n =,,.... Then clearly F is non-decreasing and x α/ F x x α/ for all x. This implies that for all t >, t α/ lim inf x F tx F x lim sup x F tx F x tα/. If F were regularly varying, the above inequality would imply that the index was α/, hence the limit of F tx/f x as x would be equal to ct α/ for some positive constant c. But this does not happen because of the following. Take t = /α and a subsequence x n = n/α. Then tx n = n+/α and therefore F tx n /F x n = n+ / n+ = which should be equal to ct α/ = c /α α/ = c, implying c =. On the other hand, take any t, /α and the same subsequence x n = n/α. Then tx n [ n/α, n+/α implying 9

10 F tx n = F x n. Thus the quotient F tx n /F x n = which should be equal to ct α = t α for all t, /a. Clearly this is impossible, so F is not regularly varying. This also shows that F x is not to any cx α/, as x. Let σ be the measure corresponding to the nondecreasing function F in the sense that σdt = F dt: σ := n δ n/α. n= Since, + t σdt <, σ is a Stieltjes measure. Let gλ :=, λ + t σdt = n λ + n/α be the corresponding Stieltjes function. It follows from [3, Theorem.7.4] or [45, Lemma 6.] that g is not regularly varying at infinity. Moreover, since F x x α/, x, it follows from [45, Lemma 6.3] that gλ λ α/, λ. Therefore, the function fλ := /gλ is a complete Bernstein function which is not regularly varying at infinity, but satisfies fλ λ α/, λ. Now we are going to establish the asymptotic behaviors of u and µ under the assumption H. First we claim that under the assumption.6, there exist δ, and a, s > such that n= φλt aλ δ φt, λ, t /s..8 Indeed, by Potter s theorem cf. [3, Theorem.5.6], for < ɛ < α/ there exists t such that lt t ɛ λt ɛ lλt max, = λ ɛ, λ, t t. λt t Hence, φλt c λt α/ lλt = c t α/ α/ lλt ltλ lt c 3φtλ α/ ɛ, λ, t t. By taking δ := α/ ɛ,, a = c 3, and s = /t we arrive at.8. Theorem.9 Let S be a complete subordinator with Laplace exponent φ satisfying H. Then the potential density u of S satisfies ut t φt tα/ lt, t +..9 Proof. Put then by.8 we have wt := e st us ds = φt, wλt a λ δ wt, λ, t /s.

11 Applying the second part of Proposition.5 we see that there is a constant c > such that ut ct wt, for small t >. Combining this inequality with.5 we arrive at.9. Theorem. Let S be a complete subordinator with Laplace exponent φ satisfying H. Then the Lévy density µ of S satisfies µt t φt t α/ lt, t +.. Proof. Since φ is a complete Bernstein function, the function ψλ := λ/φλ is also a complete Bernstein function and satisfies ψλ λ α/ lλ, λ, where α, and l are the same as in.6. It follows from Corollary.4 that the potential measure of a subordinator with Laplace exponent ψ has a complete monotone density v given by Applying Theorem.9 to ψ and v we get vt = µt, = t µsds. µt, = vt t ψt = φt, t.. By using the elementary inequality e cy c e y valid for all c and all y >, we get that φcλ cφλ for all c and all λ >. Hence φs = φs φs for all s >. Therefore, by., for all s, / for some constants c, c >. Since we have for all t,, for some constant c 3 >. vs c φs c φs c vs vt/ vt/ vt = Using.8 we get that for every λ t t/ µs ds t/µt, µt t vt/ c t vt c 3 t φt, φs = φλλs aλ δ φλs, s s λ.

12 It follows from. that there exists a constant c 4 > such that c 4 φs vs c 4 φs, s <. Fix λ := /δ c 4 a /δ. Then for s s /λ, by our choice of λ. Further, vλs c 4 φλs c 4 a λ δ φs c 4a λ δ vs vs λ sµs This implies that for all small t µt λs s µt dt = vs vλs vs vs = vs. λ t vt = c 5 t vt c 6 t φt for some constants c 5, c 6 >. The proof is now complete. 3 Subordinate Brownian motion 3. Definitions and technical lemma Let B = B t, P x be a Brownian motion in R d with transition density pt, x, y = pt, y x given by pt, x = 4πt d/ exp x, t >, x, y R d. 4t The semigroup P t : t of B is defined by P t fx = E x [fb t ] = R d pt, x, yfy dy, where f is a nonnegative Borel function on R d. Recall that if d 3, the Green function G x, y = G x y, x, y R d, of B is well defined and is equal to G x = pt, x dt = Γd/ 4π d/ x d+. Let S = S t : t be a complete subordinator independent of B, with Laplace exponent φλ, Lévy measure µ and potential measure U. In the rest of the paper, we will always assume that S satisfies H. Hence lim λ φλ =, and thus S has a completely monotone potential density u. We define a new process X = X t : t by X t := BS t. Then X is a Lévy process with characteristic exponent Φx = φ x see e.g.[33, pp.97 98] called a subordinate Brownian motion. The semigroup Q t : t of the process X is given by Q t fx = E x [fx t ] = E x [fbs t ] = P s fx PS t ds. The semigroup Q t has a density qt, x, y = qt, x y given by qt, x = ps, x PS t ds.

13 We will always assume that the subordinate Brownian motion X is transient. According to the criterion of Chung-Fuchs type see [3] or [33, pp. 5 53], X is transient if and only if for some small r >, x <r Φx dx <. Since Φx = φ x, it follows that X is transient if and only if + λ d/ φλ dλ <. 3. This is always true if d 3, and, depending on the subordinator, may be true for d = or d =. In the case d, if there exists γ [, d/ such that then 3. holds. lim inf λ φλ λ γ >, 3. For x R d and A Borel subset of R d, the potential measure of X is given by Gx, A = E x {Xt A}dt = = P s A us ds = A Q t A x dt = ps, x, y us ds dy, P s A xps t ds dt where the second line follows from.. If A is bounded, then by the transience of X, Gx, A < for every x R d. Let Gx, y denote the density of the potential measure Gx,. Clearly, Gx, y = Gy x where where Gx = pt, x Udt = The Lévy measure Π of X is given by see e.g. [33, pp ] ΠA = pt, x µdt dx = Jx dx, A R d, A Jx := pt, x µdt = A pt, xut dt. 3.3 pt, xµtdt 3.4 is the Lévy density of X. Define the function j :,, by jr := 4π d/ t d/ exp r µdt, 4t r >, 3.5 and note that by 3.4, Jx = j x, x R d \ {}. Since x pt, x is continuous and radially decreasing, we conclude that both G and J are continuous on R d \ {} and radially decreasing. The following technical lemma will play a key role in establishing the asymptotic behaviors of the Green function G and the jumping function J of the subordinate Brownian motion X in the next subsection. 3

14 Lemma 3. Suppose that w :,, is a decreasing function, l :,, a measurable, locally bounded function which is slowly varying at, and β [, ], β > d/. If d = or d =, we additionally assume that there exist constants c > and γ < d/ such that wt ct γ, t. 3.6 Let a If then Ix := Ix wt 4πt d/ e x 4t wt dt. t β l/t, t, 3.7 x d+β l w x, x. x d x b If then Ix wt t β l/t, t, 3.8 Γd/ + β 4 β π d/ x d+β l x, x. Proof. a Let us first note that the assumptions of the lemma guarantee that Ix < for every x. Now, let ξ /4 to be chosen later. By a change of variable we get 4πt d/ e x 4t wt dt ξ x x = x d+ 4π d/ t d/ e t x w dt + x d+ t d/ e t w dt 4t ξ x 4t =: 4π d/ x d+ I x + x d+ I x. 3.9 We first consider I x for the case d = or d =. It follows from the assumptions that there exists a positive constant c such that ws c s γ for all s /4ξ. Thus I x It follows that ξ x x t d/ e t γ ξ x c dt c x γ t d/ γ dt = c 3 x d. 4t lim x x d+ I x x d +β l x =. 3. In the case d 3, we proceed similarly, using the bound ws w/4ξ for s /4ξ. 4

15 Now we consider I x: x d+ x I x = x d t d/ e t w dt ξ x 4t 4 β x = x d+β l t d/ +β e t x β w l x ξ x 4t 4t Using the assumption 3.7, we can see that there is a constant c > such that c w for all t and x satisfying x /4t /4ξ. x x β l 4t 4t 4t x < c, 4t l x x l 4t dt. x Now choose a δ, d/ + β note that by assumption, d/ + β >. By Potter s theorem cf. [3, Theorem.5.6 i], there exists ρ = ρδ such that l x / x δ / x δ l 4t 4t/ x x 4t/ x = 4t δ 4t δ c t δ t δ 3. whenever x > ρ and 4t x for > ρ and 4t x x we have that Let > ρ. By reversing the roles of / x and 4t/ x we also get that l x l 4t c tδ t δ 3. x > ρ. Now we define ξ := ρ 4 so that for all x with x 4ξ c c t d/ +β e t t δ t δ t d/ +β e t w c 3 := c c x x β l 4t 4t 4t l x x l 4t x and t > ξ x c c t d/ +β e t t δ t δ. 3.3 t d/ +β e t t δ t δ dt <, c 4 := c c t d/ +β e t t δ t δ dt <. The integrals are finite because of assumption d/ + β δ >. It follows from 3.3 that c 3 lim inf x lim sup x x t d/ +β e t x β 4t w l 4t 4t x x t d/ +β e t x β 4t w l 4t 4t x 5 l x l 4t x ξ x, tdt l x l 4t x ξ x, tdt c 4.

16 This means that x d+ I x x d β+ l x x = 4 β t d/ +β e t x β w l ξ x 4t 4t Combining 3. and 3.4 we have proved the first part of the lemma. 4t l x x l 4t dt. 3.4 x b The proof is almost the same with a small difference at the very end. Since l is slowly varying at, we have that This implies that lim x td/ +β e t w x x β l 4t 4t l x lim x l 4t =. x 4t l x x l 4t ξ x, t = t d/ +β e t, t. x By the right-hand side inequality in 3.3, we can apply the dominated convergence theorem to conclude that lim x x d+ I x x d β+ l x = lim x 4β = 4 β Γd/ + β. x t d/ +β e t x β w l 4t 4t 4t l x x l 4t ξ x, t dt x Together with 3.9 and 3. this proves the second part of the lemma. 3. Asymptotic behavior of the Green function and Lévy density The goal of this subsection is to establish the asymptotic behavior of the Green function Gx and Lévy density Jx of the subordinate process Y under certain assumptions on the Laplace exponent φ of the subordinator S. We start with the Green function. Theorem 3. Suppose that the Laplace exponent φ is a complete Bernstein function satisfying the assumption H and that α, d. In the case d, we further assume 3.. Then Gx x d φ x x d α l x, x. Proof. It follows from Theorem.9 that the potential density u of S satisfies ut t φt tα/ lt, t +. 6

17 Using.5 and 3. we conclude that in case d there exists c > such that ut ct γ, t. We can now apply Lemma 3. with wt = ut, β = α/ to obtain the required asymptotic behavior. Remark 3.3 i Since α is always assumed to be in,, the assumption α, d in the theorem above makes a difference only in the case d =. ii In case d 3, the conclusion of the theorem above is proved in [46, Theorem ii iii] under weaker assumptions. The statement of [46, Theorem ii] in case d is incorrect and the proof has an error. The asymptotic behavior near the origin of Jx is contained in the following result. Theorem 3.4 Suppose that the Laplace exponent φ is a complete Bernstein function satisfying the assumption H. Then Jx φ x x d l x, x. x d+α Proof. It follows from Theorem. that the potential density u of S satisfies µt t φt t α/ lt, t +. Since µt is decreasing and integrable at infinity, there exists c > such that µt ct, t. We can now apply Lemma 3. with wt = µt, β = + α/ and γ = to obtain the required asymptotic behavior. Proposition 3.5 Suppose that the Laplace exponent φ is a complete Bernstein function satisfying the assumption H. Then the following assertions hold. a For any K >, there exists C 4 = C 4 K > such that jr C 4 jr, r, K. 3.5 b There exists C 5 > such that jr C 5 jr +, r >

18 Proof. For simplicity we redefine in this proof the function j by dropping the factor 4π d/ from its definition. This does not effect 3.5 and 3.6. It follows from Lemma. and Theorem. that a For any K >, there exists c = c K > such that µr c µr, r, K. 3.7 b There exists c > such that µr c µr +, r >. 3.8 Let < r < K. We have Now, jr = t d/ exp r /tµt dt t d/ exp r /tµt dt + K/ = I + I. K t d/ exp r /tµt dt I = I = K/ K/ K t d/ exp r µt dt = t d/ exp r t K/ t d/ exp r 4t exp 3r µt dt e 3K/ K t d/ exp r µt dt = 4 d/+ t K/ c 4 d/+ K/ s d/ exp r µs ds. 4s 4t exp 3r µt dt 4t K/ t d/ exp r µt dt, 4t s d/ exp r µ4s ds 4s Combining the three displays above we get that jr c 3 jr for all r, K. To prove 3.6 we first note that for all t and all r it holds that r + r t t. This implies that r + exp e /4 exp r, 4t 4t for all r >, t >

19 Now we have jr + = t d/ exp 8 t d/ exp = I 3 + I 4. r + µt dt 4t r + µt dt + 4t For I 3 note that r + 4r for all r >. Thus I 3 = 8 t d/ exp = 4 d/+ r + µt dt 4t 8 s d/ exp r µ4s ds c 4s 3 t d/ exp t d/ exp r /tµt dt 4 d/+ r + µt dt 4t s d/ exp r µs ds, 4s I 4 = t d/ r + exp µt dt t d/ exp{ /4} exp r µt dt 3 4t 3 4t = e /4 s d/ exp r µs + ds c 4s e /4 s d/ exp r µs ds. 4s Combining the three displays above we get that jr + c 4 jr for all r >. 3.3 Some results on subordinate Brownian motion in R In this subsection we assume d =. We will consider subordinate Brownian motions in R. Let B = B t : t be a Brownian motion in R, independent of S, with [ ] E e iθbt B = e tθ, θ R, t >. The subordinate Brownian motion X = X t : t in R defined by X t = B St is a symmetric Lévy process with the characteristic exponent Φθ = φθ for all θ R. In the first part of this subsection, up to Corollary 3.8, we do not need to assume that φ satisfies the assumption H. Let X t := sup{ X s : s t} and let L = L t : t be a local time of X X at. L is also called a local time of the process X reflected at the supremum. Then the right continuous inverse L t of L is a subordinator and is called the ladder time process of X. The process X L t is also a subordinator and is called the ladder height process of X. For the basic properties of the ladder time and ladder height processes, we refer our readers to [, Chapter 6]. Let χ be the Laplace exponent of the ladder height process of X. It follows from [, Corollary 9.7] that logφλθ logφλ θ χλ = exp π + θ dθ = exp π + θ dθ, λ >. 3. The next result, first proved independently in [7] and [8], tells us that χ is also complete Bernstein function. The proof presented below is taken from [7]. 9

20 Proposition 3.6 Suppose φ, the Laplace exponent of the subordinator S, is a complete Bernstein function. Then the Laplace exponent χ of the ladder height process of the subordinate Brownian motion X t = B St is also a complete Bernstein function. Proof. It follows from Theorem [34, Theorem 6.] that φ has the following representation: t log φλ = γ + + t ηtdt, 3. λ + t where η in a function such that ηt for all t >. By 3. and 3., we have log χλ = γ + t π + t dθ λ θ ηtdt + t + θ. By using ηt, we have ηt t + t λ θ + t + θ + t + θ + t λ θ + t + λ θ + t + λ t λ θ + t λ θ t λ θ + t. Since λ θ + t dθ = t λ θ t + dθ = t we can use Fubini s theorem to get log χλ = γ + t + t tλ + t = γ + t + t + t s = γ + + s ηs ds. λ + s t λ γ + dγ = π λ t, ηtdt 3. ηtdt + + t tλ + ηtdt t Applying [34, Theorem 6.] we get that χ is a complete Bernstein function. The potential measure of the ladder height process of X is denoted by V and its density by v. We will also use V to denote the renewal function of X: V t := V, t = t vs ds. The following result is first proved in [7]. Proposition 3.7 χ is related to φ by the following relation e π/ φλ χλ e π/ φλ, for all λ >.

21 Proof. According to 3., we have log χλ = γ + t + t tλ + t ηtdt. Together with representation 3. we get that for all λ > log χλ log φλ = t + t t tλ + t + t λ ηt dt + t This implies that λ t + λ λ + t tλ + t dt = π/ log χλ log φλ π/, for all λ >, λ λ + t t dt = π. i.e., e π/ χλφλ / e π/, for all λ >. Combining the above two propositions with Corollary.3, we obtain Corollary 3.8 Suppose φ, the Laplace exponent of the subordinator S, is a complete Bernstein function satisfying lim λ φλ =. Then the potential measure of the ladder height process of the subordinate Brownian motion X t = B St has a completely monotone density v. In particular, v and the renewal function V are C functions. In the remainder of this paper we will always assume that φ satisfies the assumption H. We will not explicitly mention this assumption anymore. Since φλ λ α/ lλ as λ, Lemma 3.7 implies that χλ λ α/ lλ /, t. 3.3 It follows from 3.3 that lim λ χλ/λ =, hence the ladder height process does not have a drift. Recall that V t = V, t = t vsds is the renewal function of the ladder height process of X. In light of 3.3, we have, as a consequence of Theorem.9, the following result. Proposition 3.9 As t, we have and V t φt / vt t φt / t α/ lt / tα/ lt /.

22 Remark 3. It follows immediately from the proposition above that there exists a positive constant c > such that V t cv t for all t,. It follows from 3.3 above and [9, Lemma 7.] that the process X does not creep upwards. Since X is symmetric, we know that X also does not creep downwards. Thus if, for any a R, we define τ a = inf{t > : X t < a}, σ a = inf{t > : X t a}, then we have P x τ a = σ a =, x > a. 3.4 Let G, x, y be the Green function of X in,. Then we have the following result. Proposition 3. For any x, y > we have { x G, x, y = vzvy + z xdz, x y, vzvy + z xdz, x > y. x x y Proof. Let X, be the process obtained by killing X upon exiting from,. By using 3.4 above and [, Theorem, p. 76] we get that for any nonnegative function on f on,, [ ] x E x fx, t dt = k vzfx + z yvydzdy, 3.5 where k is the constant depending on the normalization of the local time of the process X reflected at its supremum. We choose k =. Then [ ] E x dt = = = x x fx, t vz fw x x w x vy vyfx + y zdydz = vzvw + z xdzdw + vzfx + y zdzdy x x vz fw x z x vw + z xfwdwdz vzvw + z xdzdw. 3.6 On the other hand, E x [ = x ] fx, t dt = G, x, wfw dw + G, x, wfw dw x G, x, wfw dw. 3.7 By comparing 3.6 and 3.7 we arrive at our desired conclusion. For any r >, let G,r be the Green function of X in, r. Then we have the following result.

23 Proposition 3. For all r > and all x, r r G,r x, y dy V xv r. In particular, for any R >, there exists C 6 = C 6 R > such that for all r, R and all x, r, r Proof. For any x, r, we have r x x = = G,r x, ydy C 6 φr φx / G,r x, ydy x x y x vz r G, x, ydy vzvy + z xdzdy + x z vy + z xdydz + r x x x vz r α/ lr / x α/ lx /. vzvy + z xdzdy r x vy + z xdydz V r V x. Now the desired conclusion follows easily from Proposition 3.9. As a consequence of the result above, we immediately get the following. Corollary 3.3 For all r > and all x, r r G,r x, y dy V r V x V r x. In particular, for any R >, there exists C 7 = C 7 R > such that for all x, r, and r, R, r G,r x, ydy C 7 φr / φx / φr x / r α/ lr / x α/ lx / r x α/ lr x / Proof. The first inequality is a consequence of the identity r G,r x, ydy = r G,r r x, ydy which is true by symmetry of the process X. The second one now follows exactly as in the proof of Proposition 3... Remark 3.4 With self-explanatory notation, an immediate consequence of the above corollary is the following estimate r r G r,r x, y dy V r V r + x V r x

24 4 Harnack inequality and Boundary Harnack principle From now on we will always assume that X is a subordinate Brownian motion in R d. Recall that H is the standing assumptions on the Laplace exponent φ. We additionally assume that α, d and in the case d, we further assume 3.. The goal of this section is to show that the Harnack inequality and the boundary Harnack principle hold for X. The infinitesimal generator L of the corresponding semigroup is given by Lfx = fx + y fx y fx{ y } Jydy 4. R d for f C b Rd. Moreover, for every f C b Rd fx t fx t LfX s ds is a P x -martingale for every x R d. We recall the Lévy system formula for X which describes the jumps of the process X: for any non-negative measurable function f on R + R d R d with fs, y, y = for all y R d, any stopping time T with respect to the filtration of X and any x R d, E x [ T ] fs, X s, X s = E x fs, X s, yjx s, ydy ds. 4. s T R d See, for example, [5, Proof of Lemma 4.7] and [6, Appendix A]. 4. Harnack inequality It follows from Theorem 3.4 and the -version of [3, Propositions.5.8 and.5.] that and D}. r r r s d+ jsds lr r α φr, r 4.3 s d jsds lr r α φr, r. 4.4 For any open set D, we use τ D to denote the first exit time from D, i.e., τ D = inf{t > : X t / Lemma 4. There exists a constant C 8 > such that for every r, and every t >, P x sup X s X > r C 8 φr t. s t 4

25 Proof. It suffices to prove the lemma for x =. Let f C b Rd, f, f =, and fy = for all y. Let c = sup y j,k / y j y k fy. Then fz + y fz y fz c y. For r,, let f r y = fy/r. Then the following estimate is valid: f r z + y f r z y f r z { y r} c y r { y r} + { y r} y c { y r} r + { y r}. By using 4.3 and 4.4, we get the following estimate: Lf r z f r z + y f r z y f r z y r Jydy R d y c { y r} + { y r} Jydy R d c 3 φr, where the constant c 3 is independent of r. Further, by the martingale property, implying the estimate r τb,r t E f r Xτ B,r t f r = E Lf r X s ds E f r Xτ B,r t c 3 φr t. If X exits B, r before time t, then f r Xτ B,r t =, so the left hand side is larger than P τ B,r t. Lemma 4. For every r,, and every x R d, where C 8 is the constant from Lemma 4.. Proof. Let z Bx, r/. Then inf E [ ] z τbx,r z Bx,r/ 4C 8 φr/, P z τ Bx,r t P z τ Bz,r/ t C 8 φr/ t. Therefore, E z [ τbx,r ] t Pz τ Bx,r t t C 8 φr/ t. Choose t = /C 8 φr/ so that C 8 φr/ t = /. Then [ ] E z τbx,r C 8 φr/ = 4C 8 φr/. 5

26 Lemma 4.3 There exists a constant C 9 > such that for every r, and every x R d, [ ] C 9 sup E z τbx,r z Bx,r φr. Proof. This lemma follows immediately from Theorem 3. and the -version of [3, Proposition.5.8]. An improved version will be given in Proposition 4.9 later on. Here we present a different proof that does not need the transience of X. Let r,, and let x R d. Using the Lévy system formula 4., we get P z Xτ Bx,r x > r = G Bx,r z, y j u y du dy, c Bx,r Bx,r where G Bx,r denotes the Green function of the process X in Bx, r. Now we estimate the inner integral. Let y Bx, r, u Bx, r c. If u Bx,, then u y u x, while for u / Bx, we use u y u x +. Then j u y du = c Bx,r = Bx,r c Bx, Bx,r c Bx, Bx,r c Bx, Bx,r c c j u x du, j u y du + j u y du Bx,r c Bx, c j u x du + j u x + du Bx,r c Bx, c c j u x du + c j u x du Bx,r c Bx, c where in the next to last line we used 3.5 and 3.6. Now, It follows from 4.4 that G Bx,r z, y dy c j u x du Bx,r Bx,r c [ ] = E z τbx,r c c v d jv dv r = c φr [ ] E z τbx,r which implies the lemma. Lemma 4.4 There exists a constant C > such that for every r,, every x R d, and any A Bx, r A P y TA < τ Bx,3r C, for all y Bx, r. Bx, r 6

27 Proof. Without loss of generality assume that P y T A < τ Bx,3r < /4. Set τ = τ Bx,3r. By Lemma 4., P y τ t P y τ By,r t c φr t. Choose t = /4c φr, so that P y τ t /4. Further, if z Bx, 3r and u A Bx, r, then u z 4r. Since j is decreasing, j u z j4r. Thus, P y T A < τ E y s T A τ t {Xs X s,x s A} TA τ t = E y TA τ t E y A A j u X s du ds j4r du ds = j4r A E y [T A τ t ], where in the second line we used properties of the Lévy system. Next, The last two displays give that E y [T A τ t ] E y [t ; T A τ t ] = t P y T A τ t t [ P y T A < τ P y τ < t ] t = 8c φr. P y T A < τ j4r A 8c φr = A j4r 8c φr. The claim now follows immediately from.6 and Theorem 3.4. Lemma 4.5 There exist positive constant C and C, such that if r,, x R d, z Bx, r, and H is a bounded nonnegative function with support in Bx, r c, then E z HXτ Bx,r C E z [τ Bx,r ] Hyj y x dy, and E z HXτ Bx,r C E z [τ Bx,r ] Hyj y x dy. Proof. Let y Bx, r and u Bx, r c. If u Bx, we use the estimates u x u y u x, 4.5 while if u / Bx, we use u x u y u x

28 Let B Bx, r c. Then using the Lévy system we get ] τbx,r E z [ B X τbx,r = E z B j u X s du ds. By use of 3.5, 3.6, 4.5, and 4.6, the inner integral is estimated as follows: j u X s du = j u X s du + j u X s du B B Bx, B Bx, c j u x du + j u x du B Bx, B Bx, c cj u x du + cj u x du B Bx, B Bx, c = c j u x du Therefore B ] E z [ B X τbx,r τbx,r E z c = c E z τ Bx,r B j u x du B uj u x du. Using linearity we get the above inequality when B is replaced by a simple function. Approximating H by simple functions and taking limits we have the first inequality in the statement of the lemma. The second inequality is proved in the same way. Definition 4.6 Let D be an open subset of R d. A function u defined on R d is said to be harmonic in D with respect to X if E x [ ux τb ] < and ux = E x [ux τb ], x B, for every open set B whose closure is a compact subset of D; regular harmonic in D with respect to X if it is harmonic in D with respect to X and for each x D, ux = E x [ux τd ]. Now we give the proof of Harnack inequality. The proof below is basically the proof given in [39] which is an adaptation of the proof given in []. However, the proof below corrects some typos in the proof given in [39]. Theorem 4.7 There exists C 3 > such that, for any r, /4, x R d, and any function u which is nonnegative on R d and harmonic with respect to X in Bx, 6r, we have ux C 3 uy, for all x, y Bx, r. 8

29 Proof. Without loss of generality we may assume that u is strictly positive in Bx, 6r. Indeed, if ux = for some x Bx, 6r, then by harmonicity = ux = E x [ux τb ] For x B = Bx, ɛ Bx, 6r. This and the fact that the Levy measure of X is supported on all of R d and has a density imply that u = a.e. with respect to Lebesgue measure. Moreover, by the harmonicity, for every y Bx, 6r, uy = E y [ux τb ] = where B = By, δ Bx, 6r. Therefore, if ux = for some x, then u is identically zero in Bx, 6r and there is nothing to prove. We first assume u is bounded on R d. Using the harmonicity of u and Lemma 4.4, one can show that u is bounded from below on Bx, r by a positive number. To see this, let ɛ > be such that F = {x Bx, 3r \ Bx, r : ux > ɛ} has positive Lebesgue measure. Take a compact subset K of F so that it has positive Lebesgue measure. Then by Lemma 4.4, for x Bx, r, we have ux = E x [uxt K τ Bx,3r {TK τ Bx,3r< } ] > c ɛ K Bx, 3r, for some c >. By taking a constant multiple of u we may assume that inf Bx,r u = /. Choose z Bx, r such that uz. We want to show that u is bounded above in Bx, r by a positive constant independent of u and r, /4. We will establish this by contradiction: If there exists a point x Bx, r with hu = K where K is too large, we can obtain a sequence of points in Bx, r along which u is unbounded. Using Lemmas 4., 4.3 and 4.5, one can see that there exists c > such that if x R d, s, and H is nonnegative bounded function with support in Bx, s c, then for any y, z Bx, s/, E z HXτ Bx,s c E y HXτ Bx,s. 4.7 By Lemma 4.4, there exists c > such that if A Bx, 4r then A P y TA < τ Bx,6r c Bx, 4r, y Bx, 8r. 4.8 Again by Lemma 4.4, there exists c 3 > such that if x R d, s, and F Bx, s/3 with F / Bx, s/3 /3, then Let P x TF < τ Bx,s c η = c 3 3, ζ = 3 c η. 4. Now suppose there exists x Bx, r with ux = K for K > K := Bx, c ζ chosen so that Bx, s 3 = Bx, 4r c ζk Note that this implies d c ζ. Let s be <. 4. /d s = rk /d < r. 4. c ζ 9

30 Let us write B s for Bx, s, τ s for τ Bx,s, and similarly for B s and τ s. Let A be a compact subset of A = {y Bx, s : uy ζk}. 3 It is well know that ux t is right continuous in [, τ Bx,6r. Bx, s 3 Bx, r, we can apply 4.8 to get Hence uz E z [uxt A τ Bx,6r {TA <τ Bx,6r}] ζkp z T A < τ Bx,6r A c ζk Bx, 4r. A Bx, s 3 Bx, 4r c ζk Bx, s 3 =. Since z Bx, r and A This implies that A / Bx, s/3 /. Let F be a compact subset of Bx, s/3 \ A such that Let H = u B c s. We claim that F Bx, s E x [uxτ s ; Xτ s / B s ] ηk. If not, E x HXτ s > ηk, and by 4.7, for all y Bx, s/3, we have uy = E y uxτ s E y [uxτ s ; Xτ s / B s ] E xhxτ s c ηk ζk, c contradicting 4.3 and the definition of A. Let M = sup Bs u. We then have or equivalently K = ux = E x [uxτ s T F ] = E x [uxt F ; T F < τ s ] + E x [uxτ s ; τ s < T F, Xτ s B s ] +E x [uxτ s ; τ s < T F, Xτ s / B s ] ζkp x T F < τ s + MP x τ s < T F + ηk = ζkp x T F < τ s + M P x T F < τ s + ηk, M K η ζ P x T F < τ s + ζ. Using 4.9 and 4. we see that there exists β > such that M K + β. Therefore there exists x Bx, s with ux K + β. 3

31 Now suppose there exists x Bx, r with ux = K > K. Define s in terms of K analogously to 4.. Using the above argument with x replacing x and x replacing x, there exists x Bx, s with ux = K + βk. We continue and obtain s and then x 3, K 3, s 3, etc. Note that x i+ Bx i, s i and K i + β i K. In view of 4., /d x i+ x i r + 4 r K /d /d i r + 4 K /d + β i /d c i= ζ c i= ζ i= /d r K /d /d + 4 K /d + β i/d = c 4 rk /d c ζ c ζ where c 4 := 5 c ζ /d i= + β i/d. So if K > c d 4 K then we have a sequence x, x,... contained in Bx, r with ux i + β i K, a contradiction to u being bounded. Therefore we can not take K larger than c d 4 K, and thus sup y Bx,r uy c d 4 K, which is what we set out to prove. In the case that u is unbounded, one can follow the simple limit argument in the proof of [39, Theorem.4] to finish the proof. By using the standard chain argument one can derive the following form of Harnack inequality. Corollary 4.8 For every a,, there exists C 4 = C 4 a > such that for every r, /4, x R d, and any function u which is nonnegative on R d and harmonic with respect to X in Bx, r, we have i= ux C 4 uy, for all x, y Bx, ar. 4. Some estimates for the Poisson kernel Recall that for any open set D in R d, τ D is the first exit time of X from D. We will use G D x, y to denote the Green function of X in D. Using the continuity and the radial decreasing property of G, we can easily check that G D is continuous in D D \ {x, x : x D}. We will frequently use the well-known fact that G D, y is harmonic in D \ {y}, and regular harmonic in D \ By, ε for every ε >. Using the Lévy system for X, we know that for every bounded open subset D, every f and all x D, E x [fx τd ; X τd X τd ] = D c D G D x, zjz ydzfydy. 4.4 For notational convenience, we define K D x, y := G D x, zjz ydz, x, y D D c. 4.5 D Thus 4.4 can be simply written as E x [fx τd ; X τd X τd ] = Dx, yfydy, D c 4.6 3

32 revealing K D x, y as a density of the exit distribution of X from D. The function K D x, y is called the Poisson kernel of X. Using the continuity of G D and J, one can easily check that K D is continuous on D D c. [44]. The following proposition is an improvement of Lemma 4.3. The idea of the proof comes from Proposition 4.9 For all r > and all x R d, E x [τ Bx,r] V rv r x x, x Bx, r. In particular, for any R >, r, R and x R d, E x [τ Bx,r] C 7 φr φr x x / where C 7 = C 7 R is the constant form Proposition 3.3. r α/ r x x α/ lr / lr x x /, x Bx, r, Proof. Without loss of generality, we may assume that x =. For x, put Z t = Xt x x. Then Z t is a Lévy process on R with Ee iθzt = Ee iθ x x Xt = e tφ θ x x = e tφθ θ R. Thus Z t is of the type of one-dimensional subordinate Brownian motion studied in Section 3.3. It is easy to see that, if X t B, r, then Z t < r, hence E x [τ B,r ] E x [ τ], where τ = inf{t > : Z t r}. Now the desired conclusion follows easily from Proposition 3.3 more precisely, from 3.8. As a consequence of Lemma 4., Proposition 4.9 and 4.5, we get the following result. Proposition 4. There exist C 5, C 6 > such that for every r, and x R d, K Bx,rx, y C 5 j y x r φr φr x x / j y x r r α/ lr / r x x α/ lr x x /, 4.7 for all x, y Bx, r Bx, r c and j y x K Bx,rx, y C 6 φr/ j y x lr r α 4.8 for all y Bx, r c. 3

33 Proof. Without loss of generality, we assume x =. For z B, r and r < y < y r y z z y z + y r + y y, and for z B, r and y B, c, y r y z z y z + y r + y y +. Thus by the monotonicity of j, 3.5 and 3.6, there exists a constant c > such that cj y j z y j y r, z, y B, r B, r c. Applying the above inequality, Lemma 4. and Proposition 4.9 to 4.5, we have proved the proposition. Proposition 4. For every a,, r, /4, x R d and x, x Bx, ar, K Bx,rx, y C 4 K Bx,rx, y, y Bx, r c, where C 4 = C 4 a is the constant from Corollary 4.8. Proof. Let a,, r, /4 and x R d be fixed. For every Borel set A Bx, r c, the function x P x X A is harmonic in Bx τbx,r, r. By Corollary 4.8 and 4.6, we have for all x, x Bx, ar, K Bx,rx, y dy = P x X A τbx,r A C 4 P x X A = τbx K,r Bx,rx, y dy. This implies that K Bx,rx, y C 4 K Bx,rx, y for a.e. y Bx, r c, and hence by continuity of K Bx,rx, for every y Bx, r c. The next inequalities will be used several times in the remainder of this paper. Lemma 4. There exists C > such that s α/ ls / C r α/, < s < r 4, 4.9 lr / s α/ r α/ C, < s < r 4, 4. ls / lr / s α/ ls / C r α/ lr /, < s < r 4, 4. A 33

34 r r ls / lr / s +α/ ds C r α/, < r 4, 4. ls / lr / s α/ ds C r α/, < r 4, 4.3 and r r r ls s +α ds C lr r α, < r 4, 4.4 ls s α ds C lr, < r 4, 4.5 rα s α ls ds C rα lr, < r Proof. The first three inequalities follow easily from [3, Theorem.5.3], while the last five from the -version of [3,.5.]. Proposition 4.3 For every a,, there exists C 7 = C 7 a > such that for every r, and x R d, K Bx,rx, y C 7 Proof. By Proposition 4., r α/ d l y x r / lr / y x r α/, x Bx, ar, y {r < x y r}. K Bx,rx, y c r d Bx,ar K Bx,rw, ydw for some constant c = c a >. Thus from Proposition 4.9, 4.7 and Remark 3. we have that K Bx,rx, y c r d G Bx,rw, zjz ydzdw Bx,r Bx,r = c r d E z [τ Bx,r]Jz ydz Bx,r c r α/ r z x α/ r d lr / lr z x Jz ydz / Bx,r for some constant c = c a >. Now applying Theorem 3.4, we get K Bx,rx, y c 3r α/ d lr / Bx,r r z x α/ l z y lr z x / z y d+α dz for some constant c 3 = c 3 a >. Since r z x y z 3r 3, from 4.9 we see that r z x α/ lr z x / c y z α/ 4 l y z / 34

35 for some constant c 4 >. Thus we have K Bx,rx, y c 5r α/ d lr / c 5r α/ d lr / c 6r α/ d lr / Bx,r l z y / z y d+α/ dz By, y x rc l z y / z y d+α/ dz ls / ds y x r s +α/ for some constants c 5 = c 5 a > and c 6 = c 6 a >. Using 4. in the above equation, we conclude that K Bx,rx, y for some constant c 7 = c 7 a >. c 7r α/ d l y x r / lr / y x r α/ Remark 4.4 Note that the right-hand side of the estimate can be replaced by 4.3 Boundary Harnack principle V r r d V y x r. The proof of the boundary Harnack principle is basically the proof given in [5], which is adapted from [4, 4]. The following result is a generalization of [4, Lemma 3.3]. Lemma 4.5 There exists a positive constant C 9 > such that for any r, and any open set D with D B, r we have P x X τd B, r c C 9 r α lr D G D x, ydy, x D B, r/. Proof. We will use Cc R d to denote the space of infinitely differentiable functions with compact supports. Recall that L is the L -generator of X in 4. and that Gx, y and G D x, y are the Green functions of X in R d and D respectively. We have L Gx, y = δ x y in the weak sense. Since G D x, y = Gx, y E x [GX τd, y], we have, by the symmetry of L, for any x D and any nonnegative φ Cc R d, G D x, ylφydy = G D x, ylφydy D R d = Gx, ylφydy E x [GX τd, y]lφydy R d R d = Gx, ylφydy Gz, ylφydyp x X τd dz R d D c R d = φx + φzp x X τd dz = φx + E x [φx τd ]. D c 35