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When Ψ(ξ)= ψ(ξ2) for a complete Bernstein function ψ(ξ), the above re- sults can be significantly improved. Following the approach of [30], a (rather complicated) explicit formula for P(Mt Notation. We denote by C, C1, C2, etc. constants in theorems, and by c, c1, c2, etc. temporary constants in proofs. Any dependence of a constant on some parameters is always indicated by writing, for example, c(n, ε). We write f(x) g(x) when f(x)/g(x) 1. We use the terms increasing, decreasing, concave∼ , convex function, etc.→ in the weak sense. 2. Preliminaries. 2.1. Complete Bernstein and Stieltjes functions. A function ψ(ξ) is said to be a complete Bernstein function (CBF) if 1 ∞ ξ µ(dζ) (2.1) ψ(ξ)= c1 + c2ξ + , ξ C ( , 0), π Z0+ ξ + ζ ζ ∈ \ −∞ where c1, c2 0, and µ is a measure on (0, ) such that the integral ∞ −1 ≥−2 ˜ ∞ 0 min(ζ ,ζ )µ(dζ) is finite. A function ψ(ξ) is said to be a Stieltjes functionsR if ∞ c˜1 1 1 (2.2) ψ˜(ξ)= +c ˜2 + µ˜(dζ), ξ C ( , 0], ξ π Z0+ ξ + ζ ∈ \ −∞ for somec ˜1, c˜2 0 and some measureµ ˜ on (0, ) such that the integral ∞ −1 ≥ ∞ 0 min(1,ζ )˜µ(dζ) is finite. See [34] for a general account on complete RBernstein functions, Stieltjes functions and related notions. It is known that ψ(ξ) is a CBF if and only if ψ(ξ) is nonnegative and increasing on (0, ), holomorphic in C ( , 0], and Im ψ(ξ) > 0 when Im ξ> 0. Furthermore,∞ if ψ(ξ) is a CBF, then\ −∞ξ/ψ(ξ) is a CBF, and 1/ψ(ξ) and ψ(ξ)/ξ are Stieltjes functions The function ψ˜(ξ) given by (2.2) is the Laplace transform ofc ˜2δ0(dx)+ (˜c1 + µ˜(x)) dx ([34], Theorem 2.2). Furthermore, πc˜1δ0(dζ) +µ ˜(dζ) is the limitL of measures Im(ψ˜( ζ + iε)) dζ as ε 0+ ([34], Corollary 6.3 and Comments 6.12), so,− in a sense,− it is the boundary→ value of ψ˜. Therefore, we use a shorthand notation Im(ψ˜+( ζ)) dζ forµ ˜(dζ). Furthermore, we have − − c˜1 = limξ→0(ξψ˜(ξ)) andc ˜2 = limξ→∞ ψ˜(ξ). Following [30], we define ∞ 2 † 1 ξ log ψ(ζ ) (2.3) ψ (ξ)=exp 2 2 dζ , Re ξ> 0, π Z0 ξ + ζ 4 M. KWASNICKI,´ J. MALECKI AND M. RYZNAR for any function ψ(ξ) such that min(1,ζ−2) log ψ(ζ2) is integrable in ζ> 0. By a simple substitution, ∞ 2 2 † 1 log ψ(ξ ζ ) (2.4) ψ (ξ)=exp 2 dζ , ξ> 0. π Z0 1+ ζ By [30], Lemma 4, if ψ(ξ) is a CBF, then also ψ†(ξ) is a CBF (this was independently proved in [24], Proposition 2.4), and (2.5) ψ†(ξ)ψ†( ξ)= ψ( ξ2), ξ C R. − − ∈ \ Proposition 2.1. If ψ(ξ) is nonnegative on (0, ), and both ψ(ξ) and ξ/ψ(ξ) are increasing on (0, ), then ∞ ∞ (2.6) e−2C/π ψ(ξ2) ψ†(ξ) e2C/π ψ(ξ2), p ≤ ≤ p where 0.916 is the Catalan constant. Note that e2C/π 2. If, inC≈ addition, ψ(ξ) is regularly varying at , then ≤ ∞ (2.7) ψ†(ξ) ψ(ξ2), ξ . ∼ p →∞ An analogous statement for ξ 0 holds for ψ(ξ) regularly varying at 0. In particular, (2.6) holds for→ any CBF. Likewise, (2.7) holds for any reg- ularly varying CBF. A result similar to (2.6) was obtained independently in [25], Proposi- tion 3.7, while (2.7) for CBFs was derived in [22], Proposition 2.2. Proof. By the assumptions, we have (2.8) ψ(ξ2) min(1,ζ2) ψ(ξ2ζ2) ψ(ξ2) max(1,ζ2), ξ,ζ> 0. ≤ ≤ It follows that ∞ 2 2 † 1 log ψ(ξ ζ ) ψ (ξ)=exp 2 dζ π Z0 1+ ζ 1 ∞ log ζ2 ψ(ξ2)exp dζ = e2C/π ψ(ξ2). π 1+ ζ2 ≤ p Z1 p The lower bound is obtained in a similar manner. The second statement of the proposition is proved in a very similar manner to Lemma 15 in [30]. Define an auxiliary function h(ξ,ζ)= ψ(ξ2ζ2)/ψ(ξ2). By (2.8) we have log h(ξ,ζ) 2 log ζ , ξ,ζ > 0. Since ψ is regularly vary- | |≤ | | 2α ing at infinity, for some α, limξ→∞ h(ξ,ζ)= ζ for each ζ> 0. Hence, by dominated convergence, ∞ log h(ξ,ζ) ∞ log ζ2α lim 2 dζ = 2 dζ = 0. ξ→∞ Z0 1+ ζ Z0 1+ ζ SUPREMA OF LEVY´ PROCESSES 5 It follows that ∞ 2 2 log ψ(ξ ζ ) π 2 lim 2 dζ log ψ(ξ ) = 0, ξ→∞Z0 1+ ζ − 2 † 2 and so finally limξ→∞ ψ (ξ)/ ψ(ξ ) = 1, as desired. Regular variation at 0 is proved in a similar way. p As in [30], for differentiable functions ψ(ξ) with positive derivative, we define 1 ξ/λ2 (2.9) ψ (ξ)= − , λ> 0,ξ C ( , 0). λ 1 ψ(ξ)/ψ(λ2) ∈ \ −∞ − 2 2 2 ′ 2 This definition is extended continuously by ψλ(λ )= ψ(λ )/(λ ψ (λ )). Note † † that if ψ(0) = 0, then ψλ(0) = 1. For simplicity, we denote ψλ(ξ) = (ψλ) (ξ). By [30], Lemma 2, if ψ(ξ) is a CBF, then ψλ(ξ) is a CBF for any λ> 0. 2.2. Estimates for the Laplace transform. This short section contains some rather standard estimates for the inverse Laplace transform. Proposition 2.2. Let a> 0, c 1. If f is nonnegative and f(x) cf(a) max(1, x/a) (x> 0), then for any≥ ξ> 0, ≤ ξ f(ξ) f(a) L . ≥ c(1 + (aξ)−1e−aξ) Proof. We have a ∞ ξ f(ξ)= ξe−ξxf(x) dx + ξe−ξxf(x) dx L Z0 Za a cf(a) ∞ cf(a) ξe−ξx dx + ξxe−ξx dx ≤ Z0 a Za cf(a) = cf(a)(1 e−aξ)+ (1 + aξ)e−aξ = cf(a)(1 + (aξ)−1e−aξ), − aξ as desired. Proposition 2.3. If f is nonnegative and increasing, then for a, ξ > 0, f(a) eaξξ f(ξ). ≤ L Proof. As before, a ∞ ξ f(ξ)= ξe−ξxf(x) dx + ξe−ξxf(x) dx L Z0 Za ∞ f(a) ξe−ξx dx = f(a)e−aξ, ≥ Za as claimed. 6 M. KWASNICKI,´ J. MALECKI AND M. RYZNAR Proposition 2.4. If f is nonnegative and decreasing, then for a, ξ > 0, ξ f(ξ) f(a) L . ≤ 1 e−aξ − Proof. Again, a ∞ ξ f(ξ)= ξe−ξxf(x) dx + ξe−ξxf(x) dx L Z0 Za a f(a) ξe−ξx dx = f(a)(1 e−aξ), ≥ Z0 − as claimed. 3. Suprema of general L´evy processes. We briefly recall the basic notions of the fluctuation theory for L´evy processes. Let Lt be the local time of the −1 process Xt reflected at its supremum Mt, and denote by Ls the right- continuous inverse of Lt, the ascending ladder-time process for Xt. This −1 −1 is a (possibly killed) subordinator, and Hs = X(Ls )= M(Ls ) is another (possibly killed) subordinator, called the ascending ladder-height process. The Laplace exponent of the ascending ladder process, that is, the (possibly −1 killed) bivariate subordinator (Ls ,Hs) (s (Note that in [5], a weak inequality M x is used in the definition of V z(x).) t ≤ Hence, for a symmetric process Xt which is not a compound Poisson process, we have ∞ z −zt V (x) (3.3) e P(Mt Theorem 3.1. Let Xt be a L´evy process, Mt = sup0≤s≤t Xs and let κ(z,ξ) be the bivariate Laplace exponent of its ascending ladder process. Suppose that ∞ κ(z, 0) (3.4) K(s)= 2 dz < , s> 0 Zs z ∞ and that κ(z, 0)/z is unbounded (near 0). For t,x> 0, we have min(C1,C2(κ, t)κ(1/t, 0)V (x)) P(Mt Let σ = inf t 1/z : X = M = L−1(L ); σ is a stopping time. Since z { ≥ t t} 1/z z the support of the measure dLt is contained in the set t : Xt = Mt of zeros of the reflected process, we have { } ∞ ∞ E 1 E 1 [0,x)(Mt) dLt = [0,x)(Mt) dLt; M1/z Next, observe that Mσz = Xσz , so that Mt Mσz = sup (Xσz +s Xσz ), t σz. − s≤t−σz − ≥ Hence, ∞ E 1 [0,x)(Mt) dLt Z1/z ∞ E 1 [0,x) sup (Xσz +s Xσz ) dLt; M1/z Fix ε (0, 1) (later we choose ε = 1/4). Note that the function κ(z, 0)/z is continuous,∈ decreasing and unbounded. Hence, it maps the interval (0, 1/t] onto the interval [tκ(1/t, 0), ). Furthermore, κ(z, 0) is increasing, so that ∞ 2 K(z) κ(z, 0)/z. In particular, e K(1/t) > K(1/t) tκ(1/t, 0). It fol- ≥ ε(e−1) ≥ lows that we can choose z = z(t) < 1/t such that κ(z, 0) e2 = K(1/t). z ε(e 1) − Setting k = zt< 1, the above equality can be rewritten as e2 zK(z/k) (3.8) = ε. e 1 κ(z, 0) − Suppose now that V (x)κ(z, 0) ε(e 1)/e. Then, by the upper bound P ≤ P − of (3.5), we have (M1/z x) = 1 (M1/z Corollary 3.2. Let Xt be a L´evy process, Mt = sup0≤s≤t Xs and let κ(z,ξ) be the bivariate Laplace exponent of its ascending ladder process. If 10 M. KWASNICKI,´ J. MALECKI AND M. RYZNAR ∞ −2 κ(z, 0) satisfies condition (3.9) with 0 <̺< 1 and the integral 1 κ(z, 0)z dz is finite, then R (3.12) C(κ) min(1, κ(1/t, 0)V (x)) P(M Proof. We begin with the first part of the statement. By condition (3.9), z ̺ κ(z, 0) c (κ) κ(s, 0), s z 1. ≤ 1 s ≤ ≤ In particular, κ(s, 0)/s is unbounded. Furthermore, using also finiteness of ∞ −2 the integral 1 κ(z, 0)z dz, we obtain R κ(s, 0) (3.13) K(s) c (κ) , s 1. ≤ 2 s ≤ This implies that the assumptions of Theorem 3.1 are satisfied. Let t 1 and define z = z(t) (0, 1/t) as in Theorem 3.1. By condi- tion (3.9≥) we have ∈ κ(1/t, 0) c (κ) 3 . κ(z, 0) ≤ (zt)̺ By definition of z and (3.13) (with s = 1/t), we have 1 4e2 K(1/t) 4e2c (κ)c (κ) t = 2 3 , z e 1 κ(z, 0) ≤ e 1 (tz)̺ − − which gives zt c (κ). Hence, the constant C in Theorem 3.1 satisfies ≥ 4 2 C2 = zt/(2e) c4(κ)/(2e). This ends the proof of the first part. The second≥ part can be justified in a similar way, since condition (3.10) implies that κ(s, 0) K(s) c (κ) , s 1. ≤ 5 s ≥ Moreover, for t< 1 and z = z(t) selected according to Theorem 3.1 we have z(1) z(t) < 1/t. Applying (3.10) [with z∗ = z(1)], we obtain ≤ κ(1/t, 0) c (κ) 1 6 , z . κ(z, 0) ≤ (zt)̺ ≤ t Finally, the last statement is a direct consequence of the previous ones. Remark 3.3. Due to Potter’s theorem ([8], Theorem 1.5.6) condition (3.9) is implied by regular variation of κ(z, 0) at zero with index 0 < ̺∗ < 1. Like- SUPREMA OF LEVY´ PROCESSES 11 wise, condition (3.10) is implied by regular variation of κ(z, 0) at with index 0 < ̺∗ < 1. ∞ In the second part of the above corollary the assumption limz→0 z/κ(z, 0) = 0 can be removed at the expence that the lower bound holds for t t , ≤ 0 where t0 = t0(κ) is sufficiently small. This is due to the fact that since limtց0 K(1/t)=0, z = z(t) in Theorem 3.1 is well defined for t small enough. By the results of [5], Theorem VI.14 and [6], the regular variation of order ̺ (0, 1) of κ(z, 0) at 0 or at is equivalent to the existence of the limit ∈ ∞+ of P(Xt 0) as t or t 0 , respectively. Hence, Corollary 3.2 implies the following≥ result.→∞ → Corollary 3.4. Let Xt be a L´evy process and Mt = sup0≤s≤t Xs. If lim P(Xt 0) (0, 1) and lim supP(Xt 0) < 1, t→∞ ≥ ∈ t→0+ ≥ then (3.12) holds for x> 0 and t 1. If ≥ lim P(Xt 0) (0, 1) and lim sup P(Xt 0) < 1, t→0+ ≥ ∈ t→∞ ≥ then (3.12) is true for x> 0 and t 1. Finally, if ≤ lim P(Xt 0) (0, 1) and lim P(Xt 0) (0, 1), t→∞ ≥ ∈ t→0+ ≥ ∈ then (3.12) holds for every x> 0 and t> 0. Proof. We only need to verify that κ(z, 0)/z2 is integrable at infinity, and that limz→0+ (z/κ(z, 0)) = 0. In each of the cases, there is ε> 0 such that P(Xt 0) 1 ε for all t> 0. Therefore, by (3.1) and the Frullani integral, κ(≥z, 0) ≤ z1−−ε for z 1, and κ(z, 0) z1−ε when 0 Remark 3.5. The uniform estimates of Corollary 3.4 complement the existing results from [17] about the asymptotic behavior of P(Mt Theorem 4.1. Suppose that Xt is a symmetric L´evy process with L´evy– Khintchin exponent Ψ(ξ). Suppose that Ψ(ξ) is increasing in ξ> 0. If Mt = 12 M. KWASNICKI,´ J. MALECKI AND M. RYZNAR sup0≤s≤t Xs, then 1 ∞ ξΨ′(λ) Ee−ξMt = 2 2 π Z0 (λ + ξ ) Ψ(λ) (4.1) p ∞ 2 2 1 ξ log(λ ζ )/(Ψ(λ) Ψ(ζ)) −tΨ(λ) exp − 2 2 − dζ e dλ. × π Z0 ξ + ζ Since P(Mt Corollary 4.2. With the notation and assumptions of Theorem 4.1, ∞ −ξx e P(τx >t) dx Z0 Ee−ξMt = ξ (4.2) 1 ∞ Ψ′(λ) = 2 2 π Z0 (λ + ξ ) Ψ(λ) p ∞ 2 2 1 ξ log((λ ζ )/(Ψ(λ) Ψ(ζ))) −tΨ(λ) exp − 2 2 − dζ e dλ. × π Z0 ξ + ζ Proof of Theorem 4.1. Let ψ(ξ)=Ψ(√ξ) for ξ > 0. For any z C ( , 0] and ξ> 0, we define [see (1.1) and (2.3)] ∈ \ −∞ 1 ∞ ξ log(z + Ψ(ζ)) √ ϕ(ξ, z)= z exp 2 2 dζ −π Z0 ξ + ζ 1 ∞ ξ log(1 + ψ(ζ2)/z) = exp 2 2 dζ . −π Z0 ξ + ζ For any ξ> 0, the function ϕ(ξ, z) is positive and increasing in z (0, ). As z 0 or z , ϕ(ξ, z) converges to 0 and 1, respectively. Furthermore,∈ ∞ if Im→z> 0, then→∞ arg(1 + ψ(ζ2)/z) ( π, 0) for all ζ> 0, and therefore ∈ − 1 ∞ ξ arg(1 + ψ(ζ2)/z) arg ϕ(ξ, z)= 2 2 dζ (0, π/2). −π Z0 ξ + ζ ∈ Hence, for any ξ> 0, ϕ(ξ, z) [and even (ϕ(ξ, z))2] is a complete Bernstein function of z. Note that the continuous boundary limit ϕ+(ξ, z) exists for − z> 0: if z = ψ(λ2), or λ = ψ−1(z), then ∞p − 2 2 + 1 ξ log (1 ψ(ζ )/ψ(λ )) ϕ (ξ, z)=exp −2 2 dζ − −π Z0 ξ + ζ SUPREMA OF LEVY´ PROCESSES 13 1 ∞ ξ log 1 ψ(ζ2)/ψ(λ2) ∞ ξ = exp | −2 2 | dζ + i 2 2 dζ −π Z0 ξ + ζ Zλ ξ + ζ ∞ 2 2 2 1 ξ(log ψλ(ζ ) log 1 ζ /λ ) ξ = exp −2 2| − | dζ + i arctan ; π Z0 ξ + ζ λ see (2.9) for the notation. Here log− denotes the boundary limit on ( , 0) approached from below, log−( ζ)= iπ/2 + log ζ for ζ> 0. The function−∞ log 1 ζ2/λ2 is harmonic in the− upper− half-plane Im ζ> 0, so that | − | 1 ∞ ξ log 1 ζ2/λ2 1 ξ2 |2 − 2 | dζ = log 1+ 2 . π Z0 ξ + ζ 2 λ Furthermore, exp(i arctan(ξ/λ)) = (λ + iξ)/ λ2 + ξ2. Therefore, with z = ψ(λ2), p ∞ 2 + λ(λ + iξ) 1 ξ log ψλ(ζ ) ϕ (ξ, z)= 2 2 exp 2 2 dζ − λ + ξ π Z0 ξ + ζ (4.3) λ(λ + iξ)ψ† (ξ) = λ ; λ2 + ξ2 see (2.3) for the notation. Note that if ψ(ξ) is bounded on (0, ) and z + ∞ ≥ supξ>0 ψ(ξ), then ϕ (ξ, z) is real. By (1.1), ϕ(ξ, z)/z is− the double Laplace transform of the distribution of Mt. But for all ξ> 0, ϕ(ξ, z)/z is a Stieltjes function of z. Therefore, by (2.2), ϕ(ξ, z) 1 ∞ ϕ+(ξ, ζ) 1 = Im − dζ z π Z0 ζ z + ζ ∞ + 2 1 ′ 2 ϕ (ξ, ψ(λ )) 1 = 2λψ (λ )Im − 2 2 dλ π Z0 ψ(λ ) z + ψ(λ ) ∞ ′ 2 † 2 λψ (λ ) λξψλ(ξ) 1 = 2 2 2 2 dλ. π Z0 ψ(λ ) λ + ξ z + ψ(λ ) Note that the second equality holds true also when ψ(ξ) is bounded. Since 2 ∞ −tψ(λ2 ) −zt 1/(z + ψ(λ )) = 0 e e dt, we have R ∞ ∞ ′ 2 † ϕ(ξ, z) 2 λψ (λ ) λξψλ(ξ) −tψ(λ2 ) −zt = 2 2 2 e dλ e dt. z Z0 π Z0 ψ(λ ) λ + ξ The theorem follows by the uniqueness of the Laplace transform. Let V (x)= V 0(x) be the renewal function for the ascending ladder-height process Hs corresponding to Xt; see Section 3 for the definition. When Xt satisfies the absolute continuity condition [e.g., if 1/(1 + Ψ(ξ)) is integrable 14 M. KWASNICKI,´ J. MALECKI AND M. RYZNAR in ξ], then V (x) is the (unique up to a multiplicative constant) increasing harmonic function for X on (0, ), and V ′(x) is the decreasing harmonic t ∞ function for Xt on (0, ); cf. [35]. It is known ([5], formula (VI.6)) that for ξ> 0, ∞ 1 V (ξ)= , L ξκ(0,ξ) Moreover, if Xt is not a compound Poisson process, then by [14], Corol- lary 9.7, ∞ 1 ξ log Ψ(ζ) † κ(0,ξ)=exp 2 2 dζ = ψ (ξ), π Z0 ξ + ζ where Ψ(ξ)= ψ(ξ2); see (2.3) for the notation. Clearly, we have V ′(ξ)= ξ V (ξ) = 1/ψ†(ξ); here V ′ is the distributional derivative of V onL [0, ). L ∞ We remark that when Xt is a compound Poisson process, then, also by [14], Corollary 9.7, ∞ −t † 1 1 e (4.4) κ(0,ξ)= cψ (ξ) with c = exp − P(Xt = 0) dt . −2 Z0 t For simplicity, we state the next three results only for the case when Xt is not a compound Poisson process. However, extensions for compound Poisson processes are straightforward due to (4.4). As an immediate consequence of Proposition 2.1 and Karamata’s Taube- rian theorem ([8], Theorem 1.7.1), we obtain the following result, which in the case of complete Bernstein functions was derived in Proposition 2.7 of [22]. Proposition 4.3. Let Ψ(ξ) be the L´evy–Khintchin exponent of a sym- metric L´evy process Xt, which is not a compound Poisson process, and sup- pose that both Ψ(ξ) and ξ2/Ψ(ξ) are increasing in ξ> 0. If Ψ(ξ) is regu- larly varying at , then V is regularly varying at 0 and Γ(1 + α)V (x) 1/ Ψ(1/x) as x∞ 0. Similarly, if Ψ(ξ) is regularly varying at 0, then∼ → Γ(1p + α)V (x) 1/ Ψ(1/x) as x . ∼ →∞ p Another consequence of Proposition 2.1 is a uniform estimate of the re- newal function; see also Proposition 3.9 of [25]. Theorem 4.4. Let Ψ(ξ) be the L´evy–Khintchin exponent of a symmetric L´evy process Xt, which is not a compound Poisson process, and suppose that both Ψ(ξ) and ξ2/Ψ(ξ) are increasing in ξ> 0. Then 1 1 1 V (x) 5 , x> 0. 5 Ψ(1/x) ≤ ≤ Ψ(1/x) p p Proof. Let ψ(ξ) =Ψ(√ξ) for ξ > 0. By Proposition 2.1, we obtain e−2C/π/ ξ2ψ(ξ2) V (ξ) e2C/π/ ξ2ψ(ξ2), ξ> 0. Since V is increasing, ≤L ≤ p p SUPREMA OF LEVY´ PROCESSES 15 Proposition 2.3 gives e V (1/x) e1+2C/π 5 V (x) L . ≤ x ≤ ψ(1/x2) ≤ ψ(1/x2) Furthermore, using subadditivity andp monotonicityp of V (see [5], Section III.1), for x = ka + r (k 0, r [0, a)) we obtain V (x) kV (a)+ V (r) (k + 1)V (a). It follows that≥ V (∈x) 2V (a) max(1, x/a) for≤ all a, x > 0, and≤ so, by Proposition 2.2, ≤ V (1/x) 1 1 V (x) L , ≥ 2x(1 + e−1) ≥ 2(1 + e−1)e2C/π ψ(1/x2) ≥ 5 ψ(1/x2) as desired. p p We remark that when V is a concave function on (0, ) (e.g., when ψ is a complete Bernstein function, see below), then clearly V (∞x) max(1, x/a)V (a), so that the lower bound in Theorem 4.4 holds with constant≤ 2/5 instead of 1/5. If ψ(ξ) is a complete Bernstein function [CBF, see (2.1)], then ψ†(ξ) and ξ/ψ†(ξ) are CBFs, and hence 1/ψ†(ξ) is a Stieltjes function; see (2.2). There- fore, V ′(x) is a completely monotone function on (0, ), and V (x) is a Bernstein function; see [34] for the relation between completely∞ monotone, Bernstein, complete Bernstein and Stieltjes functions. More precisely, we have the following result. Proposition 4.5. Let Ψ(ξ) be the L´evy–Khintchin exponent of a sym- metric L´evy process Xt, which is not a compound Poisson process, and sup- pose that Ψ(ξ)= ψ(ξ2) for a complete Bernstein function ψ. Then V is a Bernstein function, and ∞ † 1 1 ψ (ξ) −xξ (4.5) V (x)= bx + Im + 2 (1 e ) dξ, x> 0, π Z + −ψ ( ξ ) ξ − 0 − ∞ ′ 1 1 † −xξ (4.6) V (x)= b + Im + 2 ψ (ξ)e dξ, x> 0, π Z + −ψ ( ξ ) 0 − 2 where b = limξ→0+ (ξ/ ψ(ξ )). p As explained after formula (2.2), the expression Im( 1/ψ+( ξ2)) dξ in (4.5) and (4.6) should be understood in the distributional sense,− as a− weak limit of measures Im( 1/ψ( ξ2 + iε)) dξ on (0, ) as ε 0+. The measure Im( 1/ ψ+( ξ)) dξ has− an atom− of mass πb at∞ 0, and this→ atom is not included− in the− integrals from 0+ to in (4.5) and (4.6). ∞ Proof. Since 1/ψ†(ξ) is a Stieltjes function, it has the form (2.2), ∞ ′ 1 b 1 1 C V (ξ)= † = a + + µ˜(dζ), ξ ( , 0], L ψ (ξ) ξ π Z0+ ξ + ζ ∈ \ −∞ 16 M. KWASNICKI,´ J. MALECKI AND M. RYZNAR where, using (2.5), 1 ψ†(ξ) µ˜(dξ)= Im dξ = Im dξ − (ψ†)+( ξ) − ψ+( ξ2) − − and 1 ξ a = lim , b = lim . ξ→∞ ψ†(ξ) ξ→0+ ψ†(ξ) Using Proposition 2.1, we can express a and b in terms of ψ. Since ψ is unbounded, also ψ† is unbounded [by (2.6)], and so in fact a =0. In a similar way, if ξ/ψ(ξ) converges to 0 as ξ 0+, then (2.6) gives ξ/ψ†(ξ) 0, so that b = 0. When the limit of ξ/ψ→(ξ) is positive [since ξ/ψ(ξ) is→ a CBF, the limit always exists], then ψ is regularly varying at 0, and so b = 2 limξ→0+ (ξ/ ψ(ξ )), as desired. By the uniqueness of the Laplace transform, p 1 ∞ V ′(x)= b + e−xξµ˜(dξ), x> 0. π Z0+ The result follows by integration in x. Note that for a compound Poisson process, we have a> 0, so there is an extra positive constant in (4.5). As a combination of Theorem 3.1 and Theorem 4.4, we obtain the follow- ing result. Theorem 4.6. Let Ψ(ξ) be the L´evy–Khintchin exponent of a symmetric 2 L´evy process Xt. Suppose that both Ψ(ξ) and ξ /Ψ(ξ) are increasing in ξ> 0. 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Malecki Institute of Mathematics Institute of Mathematics and Computer Science and Computer Science Wroclaw University of Technology Wroclaw University of Technology ul. Wybrzeze˙ Wyspianskiego´ 27 ul. Wybrzeze˙ Wyspianskiego´ 27 50-370 Wroclaw 50-370 Wroclaw Poland Poland and and Institute of Mathematics LAREMA Polish Academy of Sciences Universite´ d’Angers ul. Sniadeckich´ 8 2 Bd Lavoisier 00-976 Warszawa 49045 Angers cedex 1 Poland France E-mail: [email protected] E-mail: [email protected] M. Ryznar Institute of Mathematics and Computer Science Wroclaw University of Technology ul. Wybrzeze˙ Wyspianskiego´ 27 50-370 Wroclaw Poland E-mail: [email protected]