Fluctuation theory for L\'evy processes with completely monotone jumps

We study the Wiener-Hopf factorization for L\'evy processes $X_t$ with completely monotone jumps. Extending previous results of L.C.G. Rogers, we prove that the space-time Wiener-Hopf factors are complete Bernstein functions of both the spatial and the temporal variable. As a corollary, we prove complete monotonicity of: (a) the tail of the distribution function of the supremum of $X_t$ up to an independent exponential time; (b) the Laplace transform of the supremum of $X_t$ up to a fixed time $T$, as a function of $T$. The proof involves a detailed analysis of the holomorphic extension of the characteristic exponent $f(\xi)$ of $X_t$, including a peculiar structure of the curve along which $f(\xi)$ takes real values.


Introduction
This is the first in a series of papers, where we study a class of one-dimensional Lévy processes X t with completely monotone jumps, introduced by L.C.G. Rogers in [70].
The main objective of this article is to provide a detailed description of characteristic (Lévy-Khintchine) exponents f of these Lévy processes and their Wiener-Hopf factors κ + (τ, ξ), κ − (τ, ξ). In particular, we extend the result of [70], which asserts that κ + (τ, ξ) and κ − (τ, ξ) are complete Bernstein functions of ξ (or, equivalently, that the ladder height processes associated with X t have completely monotone jumps). Our main result states that κ + (τ, ξ) and κ − (τ, ξ) are additionally complete Bernstein functions of τ (that is, also the ladder time processes have completely monotone jumps). In fact, we prove an even stronger statement.
The notion of a Rogers function f (ξ) and its domain D f is introduced in Section 3.1. If f is a Rogers function, the notation Γ f , Z f , ζ f (r) and λ f (r) is introduced in Section 4.1, while the symbols Γ f , D + f and D − f are defined in Section 4.2.

Essentials of fluctuation theory for Lévy processes 2.1 Lévy processes
Throughout this work we assume that X t is a one-dimensional Lévy process. In other words, X t is a real-valued stochastic process with independent and stationary increments, càdlàg paths, and initial value X 0 = 0. We allow X t to be killed at a uniform rate, that is, the probability of X t being alive at time t is equal to e −ct for some c ≥ 0.

R\{0}
(1 − e iξx + iξ(1 − e −|x| ) sign x)ν(dx), (2.2) where a ≥ 0 is the Gaussian coefficient, b ∈ R is the drift, c ≥ 0 is the rate at which X t is killed and ν(dx) is the Lévy measure, a non-negative Borel measure on R \ {0} such that R\{0} min{1, x 2 }ν(dx) < ∞; ν(dx) describes the intensity of jumps of X t . If ν(dx) is absolutely continuous, we denote its density function again by the same symbol ν(x). A Lévy process X t is a compound Poisson process if the paths of X t are piece-wise constant with probability one. This is the case if and only if a = 0, ν is a finite measure and b = R\{0} (1 − e −|x| ) sign x ν(dx).

Fluctuation theory
Fluctuation theory studies the properties of the supremum and the infimum functionals of a Lévy process X t , which are defined by Fluctuation theory for Lévy processes with completely monotone jumps as well as times at which these extremal values are attained, denoted by T t = inf{s ∈ [0, t] : X s = X t }, T t = inf{s ∈ [0, t] : X s = X t }.
It is known that the sets in the definition of T t and T t almost surely contain only one element s, unless X t is a compound Poisson process.
For a detailed introduction to the fluctuation theory of Lévy processes, we refer to [3,6,18,22,49,72]. Here we limit our attention to the results that are used in this work.
Corollary 2.1. Suppose that X t is a one-dimensional Lévy process with completely monotone jumps, possibly killed at a uniform rate. Let S be an exponentially distributed random variable independent from the process X t . Then for all ξ > 0, P(X S > x), P(T S > s) and E exp(−ξX t ) (2.4) are completely monotone functions on (0, ∞) of x, s and t, respectively. If X t converges almost surely to −∞, then additionally P(X ∞ > x) and P(T ∞ > s) are completely monotone functions on (0, ∞) of x and s, respectively. Similar statements hold for X t and T t .
Proof. In the following argument we use well-known properties of complete Bernstein, Stieltjes and completely monotone functions that are discussed in Section 3.2.
Suppose that S is exponentially distributed with intensity σ. The Laplace transforms of the expressions given in (2.4) are then where τ, ξ, σ > 0.
Similarly, P(T S > s) is a completely monotone function of s, and an even shorter argument proves that E exp(−ξX t ) is a completely monotone function of t.

Baxter-Donsker formulae
The expressions for κ + (τ, ξ) and κ − (τ, ξ) discussed below are similar to those obtained by Baxter and Donsker in [4]. We derive them from (2.5), although in fact the article [4] predated the works of Pecherski-Rogozin and Fristedt. They correspond directly to the meaning of the term Wiener-Hopf factorisation in analysis, and they can also be proved by solving a Riemann-Hilbert problem for log(τ + f (ξ)) with a fixed τ > 0. Suppose that τ 1 , τ 2 > 0 and Re ξ 1 , Re ξ 2 > 0. By (2.5), we have By combining the above results, we obtain a variant of the Baxter-Donsker formula: This can be simplified in the following way: by (2.5) and dominated convergence theorem, Using the first of the above identities and (2.8), we easily find that Indeed, the integral of (ξ 1 + iz) −1 − (ξ 2 + iz) −1 over R is 0, so we may replace τ 2 + f (z) in (2.8) by 1 + f (z)/τ 2 . Passing to the limit as τ 2 → ∞ and applying dominated convergence theorem, we obtain (2.10) with τ = τ 1 . We remark that if f (ξ)/ξ is integrable in the neighbourhood of 0, then we may pass to the limit as ξ 2 → 0 + in (2.10) to get the expression found in [4]; however, the more general form (2.10) is more convenient for our needs.
The corresponding expression for κ − (τ, ξ) reads In a similar manner, using the identity and by setting τ 1 = τ and considering the limit as τ 2 → ∞, we obtain we omit the details and refer to the proof of Proposition 5.3 for an analogous argument. These are the main expressions that we will work with.

Main idea of the proof
Our goal is to mimic the argument used in [60], which can be summarised as follows.
Suppose that X t is symmetric, so that f (ξ) is real-valued for ξ ∈ R, and assume additionally that f (ξ) is an increasing and differentiable function of ξ on (0, ∞). In this case, integrating by parts in (2.10), we obtain It is well-known that the righthand side defines a complete Bernstein function of τ if 0 ≤ ξ 1 ≤ ξ 2 , and the essential part of Theorem 1.1 follows in the symmetric case.
The above approach does not work for asymmetric processes, because then f (ξ) takes complex values. For this reason, we restrict our attention to processes with completely monotone jumps, discussed in detail in the next section. In this case f (ξ) has a holomorphic extension to C \ iR, and there is a unique line Γ f along which f (ξ) takes real values. Our strategy is to deform the contour of integration in (2.10) to Γ f and only then integrate by parts. Implementation of the above plan requires a detailed study of the class of characteristic exponents of Lévy processes with completely monotone jumps: the Rogers functions. Definitions and basic properties of Rogers functions are studied in the next section.

Lévy processes with completely monotone jumps and Rogers functions
Some of the properties discussed below are not used in the proof of Theorem 1.1. However, we gather them here to facilitate referencing in forthcoming works.

Definition of Rogers functions
Recall that a function ν(x) on (0, ∞) is said to be completely monotone if we have (−1) n ν (n) (x) ≥ 0 for all x > 0 and n = 0, 1, 2, . . . By Bernstein's theorem, ν(x) is completely monotone on (0, ∞) if and only if it is the Laplace transform of a non-negative Borel measure on [0, ∞), known as the Bernstein measure of ν(x). The following class of Lévy processes appears to have been first studied by Rogers in [70]. Definition 3.1. A Lévy process X t has completely monotone jumps if the Lévy measure ν(dx) of X t is absolutely continuous with respect to the Lebesgue measure, and there is a density function ν(x) such that ν(x) and ν(−x) are completely monotone functions of x on (0, ∞).
We propose the name Rogers functions for the class of characteristic exponents of Lévy processes with completely monotone jumps, possibly killed at a uniform rate. Among numerous equivalent characterisations of this class, we take the following one as the definition. The following theorem provides four equivalent definitions of a Rogers function. Its proof is a mixture of standard arguments from the theory of complete Bernstein and Stieltjes functions (as in [73] or [54]) and the argument given in [70] (see formulae (14)-(18) therein). (a) f (ξ) extends to a Rogers function; (b) f (ξ) is the characteristic exponent of a Lévy process with completely monotone jumps, possibly killed at a uniform rate; for all ξ ∈ R, where a ≥ 0, b ∈ R, c ≥ 0 and µ(ds) is a Borel measure on R \ {0} such that R\{0} |s| −3 min{1, s 2 }µ(ds) < ∞; (d) either f (ξ) = 0 for all ξ ∈ R or for all ξ ∈ R, where c > 0 and ϕ(s) is a Borel function on R with values in [0, π]. In particular, formula (3.2) defines a holomorphic function on the set where ess supp ϕ denotes the essential support of ϕ. We call D f the domain of f , and we keep the notation D f throughout the article. Note that if s ∈ R \ ess supp ϕ, then f (−is) is a positive real number, and also formula (3.1) extends to D f .
We always identify the function f (defined originally on R, or even on (0, ∞)) and its holomorphic extension given by (3.2) to the set D f . We remark that this is the maximal holomorphic extension of f which takes values in C \ (−∞, 0]. However, f may extend to a holomorphic function in an even larger set. For example, if f (ξ) = ξ 2 , then ϕ(s) = π for almost all s ∈ R and D f = C \ iR, despite the fact that f extends to an entire function.
(b) In Theorem 3.3(b), f (ξ) has the representation (2.2), where ν(dx) has a density function ν(x) with respect to the Lebesgue measure, and ν(x) and ν(−x) are completely monotone functions of x on (0, ∞). (e) In Theorem 3.3(c) we may equivalently write, for a fixed r > 0, for the same a, c and µ(ds), and someb ∈ R.
(f) The constants a, b, c in Theorem 3.3(c) andb above are given by The measure µ(ds) satisfies with the vague limit of measures in the right-hand side.
(g) If f (ξ) is not identically zero, then, for almost every s ∈ R, −ϕ(s) sign s in Theorem 3.3(d) is the non-tangential limit of Arg f (ξ) at ξ = −is. In particular, for almost all s ∈ R.
(h) Formula (3.1) will be referred to as Stieltjes representation of f (ξ), while (3.2) will be called the exponential representation of f (ξ).
Conversely, if f (ξ) is a Rogers function and f (ξ) is not identically equal to zero, then Arg f (ξ) is a bounded harmonic function in H, which takes values in [−π, π]. By Poisson's representation theorem, with ξ = x + iy and x > 0, y ∈ R, we have for a Borel functionφ(s) on R with values in [−π, π]. As in the previous part of the proof, it follows that for some b ∈ R, which is equivalent to (3.2) with c = e b and ϕ(s) = −φ(s) sign s. Furthermore, , we conclude thatφ(s) ∈ [−π, 0] for s > 0 andφ(s) ∈ [0, π] for s > 0. Equivalence of conditions (a) and (d) follows, and the proof is complete.
Suppose that f (ξ) is a Rogers function. Then Re(f (ξ)/ξ) is non-negative and harmonic in H, and hence it is either everywhere positive or identically equal to 0. In the former case, f (ξ) is said to be non-degenerate; otherwise, f (ξ) = −ibξ for some b ∈ R, and f (ξ) is said to be degenerate. In particular, either f (ξ) = 0 for all ξ ∈ H, in which case we say that f (ξ) is non-zero, or f (ξ) is identically zero in H. A non-zero Rogers function corresponds to a non-constant Lévy process (with completely monotone jumps), while a non-degenerate Rogers function is the characteristic exponent of a non-deterministic Lévy process.
We introduce two additional classes of Rogers functions. A Rogers function is said to be bounded if it is a bounded function on (0, ∞); note that a bounded Rogers function typically fails to be a bounded function on H. Furthermore, a Rogers function Bounded and symmetric Rogers functions correspond to compound Poisson processes and symmetric Lévy processes, respectively.
Noteworthy, if the measure µ in Theorem 3.3(b) is purely atomic, with atoms forming a discrete subset of R, then f is meromorphic, and it is the characteristic exponent of a meromorphic Lévy process, studied in detail in [47].

Complete Bernstein and Stieltjes functions
Recall In this section we recall some standard properties of these classes of functions. Theorem 3.5 (see [73,Chapter 6]). Let f (ξ) be a non-negative function on (0, ∞). The following conditions are equivalent: (a) f (ξ) extends to a complete Bernstein function; (b) f (ξ) is the characteristic (Laplace) exponent of a non-negative Lévy process with completely monotone jumps, possibly killed at a uniform rate; that is,  (a) f (ξ) extends to a Stieltjes function; (b) f (ξ) is, up to addition by a non-negative constant, the Laplace transform of a completely monotone function on (0, ∞); that is, for all ξ > 0, where c ≥ 0 and ν(x) is a completely monotone function on (0, ∞), locally integrable near 0; for all ξ > 0, where b, c ≥ 0 and µ(ds) is a non-negative Borel measure on (0, ∞) such that (0,∞) min{1, s −1 }µ(ds) < ∞; Remark 3.7. An analogue of Remark 3.4 applies to Theorems 3.5 and 3.6. In particular, as it was the case with Rogers functions, we always identify a complete Bernstein function f (ξ) with its holomorphic extension to D f = C \ (− ess supp ϕ), given by the exponential representation (3.10).

Basic properties of Rogers functions
The following three results are direct consequences of the definition of a Rogers function, and we omit their proofs.
13. If f (ξ) is a Rogers function and g(ξ) is a complete Bernstein function, then g(f (ξ)) is a Rogers function. Proposition 3.14. If f (ξ) is a Rogers function, then the limit f (0 + ) = lim ξ→0 + f (ξ) exists. More precisely, if f (ξ) has Stieltjes representation (3.1), then f (0 + ) = c, and if f (ξ) has the exponential representation (3.2), then where we understand that exp(∞) = ∞ Proof. With the notation of (3.1), we have by the dominated convergence theorem. Existence of f (∞ − ) follows now by the above argument applied to the Rogers function g(ξ) With the notation of (3.2), for ξ > 0 we have in a similar way Formula (3.14) is obtained by passing to the limit as ξ → 0 + and using monotone convergence theorem. Formula (3.15) is obtained in a similar way by considering the limit as ξ → ∞.
One easily checks that a Rogers function f (ξ) is bounded if and only if Proof. By (3.16), |s| .
Letμ(dt) be the push-forward of µ(ds) by the substitution s = −1/t. We obtain By (3.16), this proves our claim.
Fluctuation theory for Lévy processes with completely monotone jumps Remark 3.16. With the notation of the Stieltjes representation (3.9) in Theorem 3.3, a sequence f n (ξ) of Rogers functions converges locally uniformly in H to a finite limit f (ξ) if and only if the corresponding coefficients b n converge to b, and the corresponding measures |s| −3 min{1, s 2 }µ n (ds) + 1 π c n δ 0 (ds) + 1 π a n δ ∞ (ds) converge to |s| −3 min{1, s 2 }µ(ds) + 1 π cδ 0 (ds) + 1 π aδ ∞ (ds) vaguely on R ∪ {∞}, the one-point compactification of R. In this case the limit f (ξ) is clearly again a Rogers function. This property is an immediate consequence of the corresponding result for Nevanlinna-Pick functions f n (ξ)/ξ; see [2, Section 2]. Alternatively, it can be deduced from a similar characterisation of convergence of positive harmonic functions Re(f n (ξ)/ξ).
If a seqence of Rogers functions f n converges point-wise in H, then it automatically converges locally uniformly (and thus the limit is again a Rogers function). We claim that it is in fact sufficient to assume point-wise convergence on an arbitrary infinite set S ⊆ H with an accumulation point in H. Indeed, in this case the sequence f n (ξ) can only have one partial limit f (ξ) in the topology of locally uniform convergence on H, and as it is discussed in [2], convergence of f n (ξ) at any single point already asserts boundedness of the corresponding sequence of measures |s| −3 min{1, s 2 }µ n (ds) + 1 π c n δ 0 (ds) + 1 π a n δ ∞ (ds). Therefore, every subsequence of f n (ξ) necessarily contains a further subsequence which converges locally uniformly to f (ξ), which is equivalent to our claim.
Another equivalent condition for convergence of a sequence f n (ξ) of Rogers functions is given in terms of the exponential representation (3.10) in Theorem 3.3: the corresponding coefficients c n must converge to c, and the corresponding measures ϕ n (s)ds must converge to ϕ(s)ds vaguely on R. This again follows from a similar property of Nevanlinna-Pick functions f n (ξ)/ξ proved in [2, Section 2], or from an analogous result for bounded harmonic functions Arg f n (ξ).
For further discussion, we refer to [2]; see also [73] and the references therein.

Estimates of Rogers functions
The following simple estimate of a Rogers function follows from the Stieltjes representation (3.4). A more refined result given in Proposition 3.18 uses the exponential representation (3.2).
On the other hand, |s| .
The lower bound for |f (ξ)| is a consequence of the upper bound for the Rogers function Suppose that f (ξ) is a non-zero Rogers function with exponential representation (3.2). The derivative of the integrand in (3.2) with respect to ξ is equal to iϕ(s) sign s/(ξ + is) 2 and it is bounded by an absolutely integrable function of s whenever ξ is restricted to a compact subset of D f . It follows that the expression for log f (ξ) can be differentiated under the integral sign. Thus, for ξ ∈ H. We will need the following improvement of the above estimate. (3.20) and, for some c > 0, More precisely, c is the constant in the exponential representation (3. 2) of f (ξ), and with the notation of (3.2), we have Proof. Since f (−ξ) = f (ξ), with no loss of generality we may assume that Re ξ ≥ 0. Let ξ = re iα , where r = |ξ| > 0 and α = Arg ξ ∈ [− π 2 , π 2 ]. Then |ξ + is| 2 = r 2 + 2rs sin α + s 2 and Im ξ ξ + is = Im(|ξ| 2 − iξs) |ξ + is| 2 = − rs cos α r 2 + 2rs sin α + s 2 .
Using the above identity, the exponential representation (3.2) and the identity Arg f (ξ) = Im log f (ξ), we obtain Arg f (re iα ) = r cos α π ∞ −∞ −s r 2 + 2rs sin α + s 2 ϕ(s) |s| ds. (3.23) It follows that the two integrals in the right-hand side are equal. Our goal is to estimate their sum: by (3.18), we have Since the integrals over (0, ∞) and (−∞, 0) are equal, it suffices to estimate one of them.
Suppose that α ≥ 0. Since 0 ≤ ϕ(s) ≤ π for all s ∈ R, we have A similar argument, involving an estimate of the integral over s ∈ (−∞, 0), leads to the same bound for α < 0, and the proof of (3.20) is complete.
Formula (3.21) is a direct consequence of the exponential representation (3.2) and (3.22). Denote the integral in the left-hand side of (3.22) by I. Observe that Clearly, |ξ − i sign s| ≤ 1 + |ξ|. By Cauchy-Schwarz inequality, we find that Since 0 ≤ ϕ(s) ≤ π for all s ∈ R, the latter integral does not exceed 2π. The former one is bounded by π 2 /|ξ| by the first part of the proof, and (3.22) follows.

Spine of a Rogers function
The curve (or, more generally, the system of curves) along which a Rogers function takes positive real values plays a key role in our development.  (c) The spine Γ f is the union of pairwise disjoint simple real-analytic curves, which begin and end at the imaginary axis or at infinity. Furthermore, Γ f has parameteri- (d) For every r > 0, the spine Γ f restricted to the annular region r ≤ |ξ| ≤ 2r is a system of rectifiable curves of total length at most Cr, where one can take C = 300.
Furthermore, if ζ f (r) = re iϑ(r) for r ∈ Z f , then |r(rϑ (r)) | ≤ 9((rϑ (r)) 2 + 1) cos ϑ(r)    We denote the extension of λ f (r) described in the above result by the same symbol. The notation Γ f , Z f , ζ f (r) and λ f (r) is kept throughout the paper. Whenever there is only one Rogers function involved, we omit the subscript f (also in D f ), except in statements of results.
Spines of a sample of Rogers functions are shown in Figure 1.
Let Z be the set of those r > 0 for which ϑ(r) ∈ (− π 2 , π 2 ). By part (b), the spine of f (ξ) satisfies (4.2), that is, Γ = {ζ(r) : r ∈ Z}. Since Γ is the nodal line of the harmonic function Im f (ξ), it is a union of (at most countably many) simple real-analytic curves.
These curves necessarily begin and end at the imaginary axis or converge to infinity, and part (b) asserts that they do not intersect each other. This completes the proof of part (c).
The proof of item (d) is rather long and technical, and it is deferred to Section 7.
We need the following observation. Suppose that g(ξ) is a holomorphic function in the unit disk D and Im g(ξ) is a bounded, positive function on D. Then, by Poisson's representation theorem, Im g(ξ) has a non-tangential limit h(ξ) for almost every ξ ∈ ∂D, and Im g(ξ) is given by the Poisson integral of h. Therefore, − Re g(ξ) is the conjugate Poisson integral of h. It follows that Re g(ξ) extends to a continuous function on (Cl D) \ ess supp h. Furthermore, if this extension is denoted by the same symbol, then on every interval (α 1 , α 2 ) such h(e iα ) = 0 for almost all α ∈ (α 1 , α 2 ), the function Re g(e iα ) is continuous and has positive derivative.
Consider a connected component U of the set {ξ ∈ H : Im f (ξ) > 0}. From Theorem 4.2 it follows that U is simply connected, and the boundary of U is a Jordan curve on the Riemann sphere C ∪ {∞}, which consists of the curve {ζ(r) : r ∈ (r 1 , r 2 )} and the interval [ir 1 , ir 2 ] for some (r 1 , r 2 ) ⊆ (0, ∞). By Carathéodory's theorem, the Riemann map Φ between U and D extends to a homeomorphism of the boundaries of these domains (as subsets of the Riemann sphere). We apply the property discussed in the previous paragraph to g(ξ) = log f (Φ −1 (ξ)).
We already know that the limit of Im log f (ξ) = Arg f (ξ) is equal to zero everywhere on {ζ(r) : r ∈ (r 1 , r 2 )}: this is obvious at ζ(r) for r ∈ (r 1 , r 2 ) ∩ Z, and at ζ(r) = −ir for r ∈ (r 1 , r 2 ) \ Z it is a consequence of the inequality 0 ≤ Arg f (ξ) ≤ π 2 + Arg ξ for ξ ∈ U . It follows that Re log f (ξ) extends from U to a continous function on U ∪ {ζ(r) : r ∈ (r 1 , r 2 )}. Furthermore, if this extension is denoted again by the same symbol, then Re log f (ζ(r)) is strictly increasing in r ∈ (r 1 , r 2 ), because Φ(ζ(r)) follows an arc of ∂D in a counterclockwise direction. We conclude that λ(r) extends to a strictly increasing, continuous function on (r 1 , r 2 ).
A similar argument applies to every connected component U of the set {ξ ∈ H : Im f (ξ) < 0}. In this case − Im f (ξ) > 0 for ξ ∈ U , and Φ(ζ(r)) follows an arc of ∂D in a clockwise fashion, so again λ(r) extends to a strictly increasing continuous function on the appropriate interval (r 1 , r 2 ).
Both cases are very similar, so we discuss only the former one. We have then log λ(0 + ) = log c + 1 π lim The desired equality log λ(0 + ) = log f (0 + ) follows by an application of the monotone convergence theorem for the integral over r ∈ (0, ∞) and the dominated convergence theorem for the integral over r ∈ (−∞, −ε]. The proof of the other identity, λ(∞ − ) = f (∞ − ), is very similar, and we omit the details.

Symmetrised spine of a Rogers function
If f (ξ) is a Rogers function, then we denote by Γ f the union of Γ f , all endpoints of Γ f , and the mirror image −Γ f of Γ f with respect to the imaginary axis. We also extend the definition of ζ f (r) to all r ∈ R so that ζ f (0) = 0 and ζ f (−r) = −ζ f (r). The orientation of −Γ f is chosen in such a way that ζ f (r), r ∈ −Z, is its parameterisation. Thus, Γ f consists of at most one unbounded simple curve and at most countably many simple closed curves, and any two of them can only touch on the imaginary axis.
The system of curves Γ f naturally divides the complex plane into two open sets, D + f and D − f : the set D + f is on the left, and D − f is on the right when traveling along Γ f . More precisely, if ζ f (r) = re iϑ(r) with ϑ(r) ∈ [− π 2 , π 2 ] for r > 0 and ϑ(0) = 0, then where Int denotes the interior of a set. We remark that D + f ∩ (C \ iR) is the set of those ξ ∈ C \ iR for which Im f (ξ) > 0,

Wiener-Hopf factorisation theorem
The proof that the Wiener-Hopf factors of a Rogers function are complete Bernstein functions was essentially given in [70], where it is shown that f is a Rogers function if and only if f (ξ) + 1 = f + (−iξ)f − (iξ) for some complete Bernstein functions f + (ξ) and f − (ξ). The following statement is a minor modification. For completeness, we provide a simplified version of the proof from [70].
for some non-zero complete Bernstein functions f + (ξ), f − (ξ) and for all ξ ∈ H, or, equivalently, ξ ∈ D f . The factors f + (ξ) and f − (ξ) are defined uniquely, up to multiplication by a positive constant.

Baxter-Donsker formulae
Recall that every Rogers function f is automatically extended from H to C \ iR in such a way that f (−ξ) = f (ξ) for ξ ∈ C \ iR. This extension is again given by the Stieltjes representation (3.1) and, for a non-zero Rogers function f , by the exponential representation (3.2).
We begin with a Baxter-Donsker-type expression, similar to the one found in [4]. A simpler proof of this result can be given, which uses Cauchy's integral formula. However, we choose a more technical argument involving Fubini's theorem, in order to illustrate the key idea of the proof of Theorem 5.5.
On the other hand, let I denote the logarithm of the left-hand side of (5.3). Using (3.2), we find that The integral of (z − ξ 1 ) −1 − (z − ξ 2 ) −1 over z ∈ R is absolutely convergent and equal to zero by Cauchy's integral formula; we omit the details. It follows that The integrand in the right-hand side is an absolutely integrable function: (1 + |z|) |z|

1
(1 + |s|) |s| (5.5) for some positive number C(ξ 1 , ξ 2 ). Therefore, by Fubini's theorem, The inner integral can be evaluated explicitly by the residue theorem: the integrand is a meromorphic function in the lower complex half-plane, which decays faster than |z| −1 as |z| → ∞. If s > 0, it has one pole, located at z = −is, with residue for s < 0, there are no poles. We conclude that and the first part of (5.3) follows. The second one is proved in a very similar way.
The proof of the third part requires some modifications. Suppose that Im ξ 1 > 0 and Im ξ 2 < 0. In this case the logarithm of the left-hand side of (5.3) is again given by (5.4), but the integral of (z − ξ 1 ) −1 − (z − ξ 2 ) −1 over z ∈ R is absolutely convergent and equal to 2πi rather than 0; again we omit the details. It follows that As before, we may use Fubini's theorem, and we obtain Again, the inner integral can be evaluated explicitly by the residue theorem. If s > 0, the integrand is a meromorphic function in the upper complex half-plane, which decays faster than |z| −1 as |z| → ∞. It has a single pole, located at z = ξ 1 , with residue ξ 1 /(ξ 1 + is) − 1/(1 + |s|). On the other hand, if s < 0, the integrand is a meromorphic function in the lower complex half-plane, which decays faster than |z| −1 as |z| → ∞. It has a single pole, located at z = ξ 2 , with residue −ξ 2 /(ξ 2 + is) + 1/(1 + |s|). Therefore, By the definition (5.2) of Wiener-Hopf factors f + (−iξ 1 ), f − (iξ 2 ), we conclude that I = log f + (−iξ 1 ) + log f − (iξ 2 ), and the third part of formula (5.3) follows.

Contour deformation in Baxter-Donsker formulae
The contour of integration in the expression given in Proposition 5.3 need not be R: it can be deformed to a more general one. For our purposes it is important to replace it by the symmetrised spine Γ f . In this case it is in fact easier to repeat the proof of Proposition 5.3 rather than deform the contour of integration. Before we state the result, however, we need a technical lemma.
Lemma 5.4. Suppose that f (ξ) is a non-degenerate Rogers function and ξ 1 , Proof. For simplicity, we omit the subscript f . The integral is absolutely convergent by Theorem 4.2(d) and the fact that the integrand is bounded on Γ by C(ξ 1 , ξ 2 )(1 + |z|) −2 for some C(ξ 1 , ξ 2 ) that depends continuously on ξ 1 , ξ 2 ∈ D + ∪ D − . By the dominated convergence theorem, the integral is a continuous function of ξ 1 , ξ 2 ∈ D + ∪ D − , and therefore it is sufficient to prove the result when ξ 1 , ξ 2 do not lie on the imaginary axis, that is, ξ 1 , ξ 2 ∈ (D + ∪ D − ) \ iR. Denote the left-hand side of (5.6) by I. We claim that (in the last equality we used the fact that ξ 1 , ξ 2 ∈ D + ).
As it was observed in the proof of Theorem 4.3, for every s < 0 such that −is / ∈ D + we have ζ(−s) = −is, and consequently ϕ(s) = 0 for almost all s < 0 such that −is / ∈ D + .
On the other hand, if s < 0 and −is ∈ D + , then we have already found that the inner integral in (5.9) is zero. Therefore, In a similar way, ϕ(s) = 0 for almost all s > 0 such that −is / ∈ D − . On the other hand, if −is ∈ D − , then we already evaluated the inner integral. We conclude that Proof. Substituting z = ζ f (r) in (5.8), we obtain In order to prove formula (5.10), it suffices to observe that Our final result in this section is obtained from the above corollary by integration by parts.
Proof. As usual, we drop subscript f from the notation. The assertion clearly holds true for degenerate Rogers functions, so we only consider the case when f (ξ) is nondegenerate. Suppose that ξ 1 , ξ 2 ∈ (0, i∞) ∩ (D + ∪ D − ) and Im ξ 1 < Im ξ 2 . Our starting point is the integral in (5.8), and we will deduce (5.11) by integration by parts.
Note that dh(r) = dλ(r)/λ(r), so that the integral in the left-hand side of (5.18) coincides with the one in (5.11). Since h(r) is increasing and g(r) is non-negative, the integral, if convergent, is automatically absolutely convergent.
We now evaluate the integral in the right-hand side of (5.17). Recall that g(r) may have two jump discontinuities: at r = |ξ 1 |, of size π, if ξ 1 ∈ D − ; and at r = |ξ 2 |, of size −π, if ξ 2 ∈ D − . Otherwise, g(r) is absolutely continuous, with derivative g (r) given by (5.15) for r ∈ Z and equal to zero almost everywhere in (0, ∞) \ Z. It follows that The integral in the right-hand side is identical to the one in Corollary 5.6; in particular it is absolutely convergent. Furthermore, if ξ 1 ∈ D − , then ζ(|ξ 1 |) = ξ 1 , and so h( Let us summarise what we have found so far: the integral in (5.11) is convergent, and by combining (5.18), (5.19) and Corollary 5.6 we obtain , in each case the righthand side is equal to log(f + (−iξ 1 )/f + (−iξ 2 )), and consequently (5.11) is proved when ξ 1 , ξ 2 ∈ (0, i∞)∩(D + ∪D − ) and Im ξ 1 < Im ξ 2 . The case Im ξ 1 > Im ξ 2 follows by symmetry. Finally, extension to the case when ξ 1 or ξ 2 is in (0, i∞) \ (D + ∪ D − ) follows by continuity: , and both sides of (5.11) are continuous functions of ξ 1 , ξ 2 ∈ (0, i∞). Indeed, continuity of the left-hand side is obvious, while for the right-hand side continuity is a consequence of dominated convergence theorem; we omit the details.

Space-time Wiener-Hopf factorisation
In this section we return to our original problem and prove Theorem 1.1. Recall that we consider a non-constant Lévy process X t with completely monotone jumps, its characteristic exponent f (ξ), and the Wiener-Hopf factors κ + (τ, ξ) and κ − (τ, ξ). By Theorem 3.3, f (ξ) extends to a non-zero Rogers function.
Before we proceed with the proof of Theorem 1.1, we clarify one aspect of the statement of the theorem. If X t is a compound Poisson process, then, according to our definitions, the expressions κ + (τ, ξ 1 )κ − (τ, ξ 2 ) and κ • (τ )κ + (τ, ξ 1 )κ − (τ, ξ 2 ) are different. Below we prove that both of them define a complete Bernstein function of τ .
Proof of Theorem 1.1.

Local rectifiability of the spine
This section contains the proof of Theorem 4.2(d). Our strategy is as follows. We first prove (in Lemma 7.1) inequality (4.4), which can be thought of as an upper bound for the curvature of the spine Γ of f (ξ). Next, we use this bound to prove (in Lemma 7.2) that Arg ζ(r) cannot oscillate too rapidly away from the imaginary axis. To prove that Arg ζ(r) does not oscillate between ± π 2 too quickly, we show (in Lemma 7.3) that the zeroes of Arg ζ(r) are separated when the derivative of Arg ζ(r) is large. All these auxiliary results are used to prove a variant of Theorem 4.2(d) in Lemma 7.4.
Throughout this section, we use the notation Γ f , ζ f (r) and Z f introduced in Theorem 4.2, and for simplicity we omit the subscript f . We use logarithmic polar coordinates ξ = e R+iα rather than the usual polar coordinates ξ = re iα ; the two are clearly related by the relation r = e R . We writeZ = log Z, so that R ∈Z if and only if r = e R ∈ Z. We also define Θ(R) = ϑ(e R ) = Arg ζ(e R ), so that ζ(e R ) = e R+iΘ(R) .
We begin with the proof of an equivalent form of formula (4.4).
Taking into account two connected components ofZ 2 which may intersect the boundary of [R 0 , R 0 + h], we conclude that The desired results follows by combining the three estimates (7.3), (7.4) and (7.5) and the inequality h + 3π + (45πh + π) ≤ 140.