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Loewner Theory for Bernstein functions I: evolution families and differential equations

One-parameter semigroups of holomorphic functions appear naturally in various applications of Complex Analysis, and in particular, in the theory of (temporally) homogeneous Markov processes. A suitable analogue of one-parameter semigroups in the inhomogeneous setting is the notion of a (reverse) evolution family. In this paper we study evolution families formed by Bernstein functions, which play the role of Laplace exponents for inhomogeneous continuous-state branching processes. In particular, we characterize all Herglotz vector fields that generate such evolution families and give a complex-analytic proof of a qualitative description equivalent to Silverstein's representation formula for the infinitesimal generators of one-parameter semigroups of Bernstein functions. We also establish several sufficient conditions for families of holomorphic self-maps, satisfying the algebraic part in the definition of an evolution family, to be absolutely continuous and hence to be described as solutions to the generalized Loewner - Kufarev differential equation. Most of these results are then applied in the sequel paper [currently in preparation] to study continuous-state branching processes.


Introduction
Particular interest to look into families of Bernstein functions from the viewpoint of Loewner Theory comes from Probability Theory. In general, complexanalytic tools are known to play an important role in the study of Markov branching processes, see e.g. [3,28,33]. A very classical example is the Galton -Watson process, which is a (temporally) homogeneous Markov chain with the one-step transition probabilities encoded in the so-called generating (holomorphic) function f : D → D := {z ∈ C : |z| < 1} as its Taylor coefficients at z = 0. Evolution of the stochastic process corresponds to a (deterministic) dynamics in a complex domain: the generating function for the n-step transition probabilities coincides with the n-th iterate of f defined as usual by In case of a homogeneous branching process with continuous time, iterates are replaced by a continuous one-parameter semigroup of holomorphic self-maps, see Section 2.1 for the definition and basic properties.
In a similar manner, inhomogeneous branching processes lead to non-autonomous holomorphic dynamical systems. In this setting the role of one-parameter semigroups is played by the so-called (reverse) evolution families, see Section 2.2.
Independently from applications to stochastic processes, one-parameter semigroups and iteration of holomorphic functions have been a subject of deep studies in Complex Analysis since early 1900s. Evolution families, in turn, constitute one of the cornerstone concepts in Loewner Theory, which originated from Loewner's seminal paper [40] of 1923 and since then has been serving as a powerful tool in the theory of conformal mappings, see e.g. [2,11] for a detailed survey.
Applications of Loewner Theory to branching processes (with discrete states) were studied in detail by Goryainov [23 -27]. It is worth mentioning that similar relationships between stochastic processes and evolution families (or one-parameter semigroups in the homogeneous case) have been recently discovered in Non-Commutative Probability, see e.g. [4,20,49,21]. This paper is motivated by yet another interesting case, in which the transition probabilities can be encoded in an evolution family formed by holomorphic self-maps of the right half-plane. Continuous-state (Markov) branching processes introduced in 1958 by Jiřina [35], see also [30], have been attracting increasing interest in the last decades, see e.g. [19] or [43] and references therein. Because of the branching property, the transition probability measures k s,t (x, ·) in such a process (X t ) are infinitely divisible and, as a result, Bernstein functions come into play via the Laplace transform: E e −ζXt X s = x = +∞ 0 e −ζy k s,t (x, dy) = exp(−xv s,t (ζ)), x, ζ 0, where v s,t : [0, +∞) → [0, +∞) is a Bernstein function, which is uniquely defined by the transition kernel k s,t and which we refer to as the Laplace exponent of the process. In turn, the Chapman -Kolmogorov equation leads to a sort of semigroup property v s,t • v t,u = v s,u for any s, t, u with 0 s t u; essentially, this means that the Laplace exponents form a reverse evolution family. For more details we refer the reader to our sequel paper [32].
While homogeneous continuous-state branching processes are well understood, see e.g. [38,Chapter 10], a systematic study of the inhomogeneous case has been launched only recently. In particular, in [19] the authors construct inhomogeneous continuous-state branching processes starting from an integral evolution equation for the Laplace exponents v s,t regarded as self-maps of (0, +∞).
A well-known remarkable fact is that every Bernstein function f ≡ 0 extends to a holomorphic self-map of the right half-plane H := {z ∈ C : Re z > 0}. Another important property is that the composition of two Bernstein functions is again a Bernstein function. In other words, the holomorphic extensions of Bernstein functions f ≡ 0 form a subsemigroup BF in the topological semigroup Hol(H, H) of all holomorphic self-maps of H endowed with the operation of composition and the usual topology of locally uniform convergence. Moreover, BF is topologically closed in Hol(H, H). See Section 2.4 for more details.
The purpose of this paper is to develop complex-analytic methods based on the modern Loewner Theory which would be useful, in particular, for studying inhomogeneous continuous-state branching processes. A part of our results are then applied in the sequel paper [32]. Other results are aimed to future investigation of branching processes and/or represent some general interest in the frames of Loewner Theory and its applications.
1.1. Organization of the paper and summary of main results. The paper is organized as follows. In the next section, Section 2, we present necessary background on one-parameter semigroups and (reverse) evolution families and on boundary fixed points of holomorphic self-maps; the last subsection is devoted to basic properties of Bernstein functions (regarded as holomorphic functions in the right-half plane H).
In Section 3 we establish our main results. We start by proving Theorems 1 and 2 in Sections 3.1 and 3.2, respectively, which concern infinitesimal description of an arbitrary topologically closed (sub)semigroup U in Hol(D, D), where D C is a simply connected domain. Theorem 1 is a general version of a result known for several special cases; it states that the infinitesimal generators associated to oneparameter semigroups in U form a topologically closed convex cone. Theorem 2 deals with the evolution families. We prove that an evolution family (w s,t ) generated by a Herglotz vector field φ is contained in U if and only if for a.e. "frozen" instant t, φ(·, t) is an infinitesimal generator of a one-parameter semigroup in U. Both theorems apply to the case of the semigroup BF ⊂ Hol(H, H) consisting of all Bernstein functions different from the identical zero.
Theorem 2 reduces the study Herglotz vector fields generating evolution families of Bernstein functions to the study of infinitesimal generators associated to one-parameter semigroups in BF. Such generators, which we refer to as Bernstein generators, appear naturally in Stochastic Processes. A representation formula for Bernstein generators was found long time ago by Silverstein [52]. Using complexanalytic tools, in particular, the theory of one-parameter semigroups, in Section 3.3 we obtain a qualitative characterization of Bernstein generators, see Theorem 3. As mentioned after the statement of the theorem, it is easily seen to be equivalent to Silverstein's representation. Moreover, as corollaries, we obtain analogous representation of Bernstein generators for the cases when the corresponding one-parameter semigroup has a boundary regular fixed point at 0 or at ∞. Section 3.4 is devoted to finding sufficient conditions for families satisfying the first two (algebraic) conditions in the definitions of reverse and usual evolution families, see Section 2.2, to be absolutely continuous, i.e. to admit associated Herglotz vector fields and hence to be described with the help of (a general version of) the Loewner -Kufarev equations (2.3) -(2.6). Theorems 4 and 5 deal with general (usual and reverse) evolution families having a boundary Denjoy -Wolff point, while Theorem 6 and its Corollary 3.17 concern specifically families of Bernstein functions and give sufficient conditions for absolute continuity in terms of the local behaviour at 0. Motivated by the latter result, in the last section, Section 3.5, we make sure that for absolutely continuous evolution families in BF, the well-known formulas for the derivatives of the first and the second order w.r.t. the initial condition at an internal fixed point can be extended to the boundary regular fixed point at 0.

Preliminaries
2.1. One-parameter semigroups of holomorphic self-maps. We start by introducing some general notation and terminology. For an arbitrary domain D ⊂ C we denote by Hol(D, C) the set of all holomorphic functions in D. We endow Hol(D, C) with the open-compact topology. It is easy to see that a sequence (f n ) converges in Hol(D, C) if and only if it converges (to the same limit function) locally uniformly in D. Furthermore, for a set E ⊂ C we denote by Hol(D, E) the topological subspace of Hol(D, C) formed by all f ∈ Hol(D, C) such that f (D) ⊂ E.
Definition 2.1. By a semigroup of holomorphic self-maps of a domain D ⊂ C we mean a set U ⊂ Hol(D, D) containing the identity map id D and such that f • g ∈ U for any f, g ∈ U. We say that such a semigroup U is topologically closed if U is (relatively) closed in Hol(D, D).
Definition 2.2. Let U be a semigroup of holomorphic self-maps of some domain D ⊂ C. A one-parameter semigroup in U is a continuous semigroup homomorphism [0, +∞) ∋ t → v t ∈ U from the semigroup [0, +∞), + with the Euclidian topology to the semigroup (U, •) endowed with the topology inherited from Hol(D, D).
Equivalently, (v t ) ⊂ U is a one-parameter semigroup if and only if it satisfies the following three conditions: Thanks to Montel's normality criterion, see e.g. [22, § II.7, Theorem 1], if C \ D contains at least two distinct points, then the continuity condition (iii) is equivalent to the pointwise convergence: v t (z) → z as t → 0 + for each z ∈ D.
Remark 2.3. It is known, see e.g. [9, §8.4], that in one complex variable, the theory of one-parameter semigroups of holomorphic self-maps is non-trivial only in the following two cases: (a) D is conformally equivalent to the unit disk D := {z : |z| < 1}, i.e. D is simply connected and different from the whole plane C; or (b) D is conformally equivalent to D * := D \ {0}. Moreover, case (b) reduces easily to case (a) because any one-parameter semigroup (φ t ) of holomorphic self-maps of D * extends holomorphically to z = 0 by setting φ t (0) = 0 for all t 0; see [9,Theorem 8.4.7 on p. 224].
Most of the theory in the literature is developed for D := D or D := H, sometimes for the strip D := {z : 0 < Im z < π}. For the sake of notational uniformity, we will keep working with an arbitrary simply connected D C, unless explicitly stated otherwise. Clearly, most of the results proved for any particular choice of D extend, more or less automatically, to the general case with the help of conformal mappings. In particular, it is known [5], see also [8], [9,Chapter 10], or [50, §3.2] that for any one-parameter semigroup (v t ) in Hol(D, D) the limit The holomorphic function φ is called the infinitesimal generator of (v t ). The one-parameter semigroup can be recovered from its infinitesimal generator; namely, for each z ∈ D the map [0, +∞) ∋ t → w := v t (z) is the unique solution to Conversely, if φ ∈ Hol(D, C) is a semicomplete vector field in D, i.e. if for any z ∈ D the initial value problem (2.2) has a unique solution w = w z (t) ∈ D defined for all t 0, then the mappings v t (z) := w z (t), z ∈ D, t 0, form a one-parameter semigroup in Hol(D, D), whose infinitesimal generator coincides with φ.
In the literature one can find two different conventions concerning the choice of the sign in formula (2.1). Here we follow the convention from [50], which is also often adopted in the literature on Probability Theory. The opposite choice of the sign is made in most of the works in Loewner Theory, e.g. in [7,10,15], as well as in many publications on one-parameter semigroups, including [9].
In what follows, the set of all infinitesimal generators of one-parameter semigroups in the domain D will be denoted by G, or by G Hol(D, D) when we need to specify the domain D explicitly. Furthermore, for a semigroup U ⊂ Hol(D, D), we will denote by G(U) the set of all φ ∈ G for which the associated one-parameter semigroup (v φ t ) is contained in U. There are two ways to extend the notion of a one-parameter semigroup to the non-autonomous (or, in probabilistic terminology, time-inhomogeneous) case. The following definitions are slight modifications of the definitions given in [7,15]. Most of the results proved in these papers can be easily extended to our setting. satisfying the following conditions: EF1: w s,s = id D for any s ∈ I; EF2: w s,u = w t,u • w s,t for any s, t, u ∈ I with s t u; EF3: for each z ∈ D there exists a non-negative function f z ∈ AC loc (I) such that for any s, t, u ∈ I with s t u.
Definition 2.6. An (absolutely continuous) reverse evolution family in a simply connected domain D C over the interval I is a two-parameter family (v s,t ) (s,t)∈∆(I) ⊂ Hol(D, D) satisfying the following conditions: REF1: v s,s = id D for any s ∈ I; REF2: v s,u = v s,t • v t,u for any s, t, u ∈ I with s t u; REF3: for each z ∈ D there exists a non-negative function f z ∈ AC loc (I) such that for any s, t, u ∈ I with s t u.
If (w s,t ) satisfies Definition 2.5 with condition EF3 replaced by the requirement that the map ∆(I) ∋ (s, t) → w s,t ∈ Hol(D, D) is continuous, then it is called a topological evolution family. In a similar way, topological reverse evolution families can be defined.
Remark 2.7. Clearly, if (v t ) ⊂ Hol(D, D) is a one-parameter semigroup, then (w s,t ) 0 s t defined by w s,t := v t−s for all (s, t) ∈ ∆ is an absolutely continuous evolution family and, at the same time, an absolutely continuous reverse evolution family.
Remark 2.8. A family (v s,t ) (s,t)∈∆(I) ⊂ Hol(D, D) is a reverse evolution family if and only if for any S ∈ I the formula w s,t := v S−t,S−s , 0 s t S, defines an evolution family (w s,t ) over [0, S]. The same is true with "reverse evolution family" and "evolution family" interchanged. For topological (reverse) evolution families, these assertions follow directly from the definitions, while in the absolutely continuous case, the proof is less trivial, see [15,Proposition 4.3].
An observation similar to Remark 2.3 is valid for (reverse) evolution families, see [12], which explains why assuming that D C is simply connected is of no loss of generality. At the same time, notwithstanding Remark 2.8, both Definitions 2.5 and 2.6 seem to be useful in our context. Reverse evolution families, as mentioned in Introduction, spring up in Probability Theory, e.g. in connection with branching processes. On the other hand, usual evolution families are more natural from the Dynamics point of view. In particular, absolutely continuous evolution families, similarly to one-parameter semigroups, are semiflows of certain holomorphic vector fields in D, but in this case the vector fields are non-autonomous, i.e. they may depend explicitly on time. In order to state this result in a precise form, we need the following definition.
By L 1 loc (I) we denote the class of all measurable (real-or complex-valued) functions which are integrable on each compact subinterval of I. Definition 2.9. A function φ : D × I → C is said to be a Herglotz vector field in a simply connected domain D C if it satisfies the following three conditions: HVF1: for any t 0, φ(·, t) ∈ G Hol(D, D) ; HVF2: for any z ∈ D, φ(z, ·) is measurable on I; HVF3: for any compact set K ⊂ D, there is a non-negative function M K ∈ L 1 loc (I) such that max z∈K |φ(z, t)| M K (t) a.e. t ∈ I. , t /f ′ (ζ), ζ ∈ D, t ∈ I, is a Herglotz vector field in D.
As usual we identify two Herglotz vector fields φ, ψ : D × I → C if φ(·, t) = ψ(·, t) for a.e. t ∈ I. Some of the main results of [7] can be now summarized and stated in our notation as follows. For s ∈ I, let I s := I ∩ [s, +∞).
Theorem A. Let D C be a simply connected domain and (w s,t ) an absolutely continuous evolution family in D over the interval I. Then there exists a unique Herglotz vector field φ : D × I → C such that for any z ∈ D and any s ∈ [0, T ), the map I s ∋ t → w(t) := w s,t (z) ∈ D is a solution to the initial value problem Conversely, let φ : D × I → C be a Herglotz vector field. Then for any z ∈ D and any s ∈ I the initial value problem (2.3) has a unique solution t → w = w(t; z, s) defined on some interval containing the set I s . Setting w s,t (z) := w(t; z, s) for all z ∈ D and all (s, t) ∈ ∆(I) one obtains an absolutely continuous evolution family (w s,t ) over the interval I.
According to Theorem A, there is a one-to-one correspondence between absolutely continuous evolution families and Herglotz vector fields. An evolution family and the vector field, corresponding to this evolution family in the sense of Theorem A, are usually said to be associated to each other. An analogous one-to-one correspondence exists between absolutely continuous reverse evolution families and Herglotz vector fields, see [15,Theorem 4.2 (i)]. In this respect, if (v s,t ) (s,t)∈∆(I) is an absolutely continuous reverse evolution family and φ is the Herglotz vector field associated to (v s,t ), then for each z ∈ D and each t ∈ is the unique solution to the initial value problem Remark 2.11. Although the correspondences between a Herglotz vector field φ on the one side and the associated absolutely continuous (usual and reverse) evolution families (w s,t ) and (v s,t ) on the other side are expressed rather explicitly, there does not seem to exist any simple way to relate the families (w s,t ) and (v s,t ) directly, i.e. without involving the Herglotz vector field φ.
Remark 2.12. Clearly, if (v s,t ) is an absolutely continuous reverse evolution family over the interval I, then for each fixed s ∈ [0, T ), the mappingsṽ s ′ ,t ′ := v s+s ′ ,s+t ′ also form an absolutely continuous reverse evolution family over a suitable interval. Therefore, combining [15, Theorems 4.1, 4.2 (ii) and 1.11], one can easily conclude that there is an equivalent way to express the one-to-one correspondence between Herglotz vector fields and absolutely continuous reverse evolution families introduced above. Namely, as a function of the parameter t, any absolutely continuous reverse evolution family (v s,t ) satisfies, in the sense of [15, Definition 2.1], the PDE where φ is the Herglotz vector field associated with (v s,t ). Conversely, given any Herglotz vector field φ : D×I → C, for each s ∈ [0, T ) the initial value problem (2.5) has a unique solution (z, t) → v s,t (z), which is defined for all (z, t) ∈ D × I s , and moreover, (v s,t ) (s,t)∈∆(I) is exactly the reverse evolution family associated with φ.
According to [15,Theorem 1.11], the inverse mappings v −1 s,t also can be recovered by solving an ODE driven by the Herglotz vector field φ associated with (v s,t ).
Namely, for each s ∈ [0, T ) and each z ∈ D, the map t → w(t) := v −1 s,t (z) is the unique solution to the initial value problem Note that in this case solutions do not have to be defined for all t ∈ I s . In fact, the set of all z ∈ D for which the solution to (2.6) exists up to (and including) a given instant t coincides with v s,t (D).
Remark 2.13. Equations (2.3) -(2.6) are essentially equivalent to each other and are known as generalized Loewner -Kufarev differential equations; see, e.g. [11]. A good account on the classical Loewner -Kufarev equations can be found in [ All other fixed points are located on ∂D and should be understood in the sense of angular limits. Namely, σ ∈ ∂D is said to be a boundary fixed point for w ∈ Hol(D, D) if the angular limit ∠ lim z→σ w(z) exists and coincides with σ. It is known that the angular derivative Definition 2.14. A boundary fixed point σ ∈ ∂D of a holomorphic self-map w : D → D is said to be regular (BRFP for short) if w ′ (σ) = ∞.
Using the Cayley map D ∋ z → H(z) := (1 + z)/(1 −z) ∈ H, the above definitions can be extended to the holomorphic self-maps of the half-plane H. In fact, for σ ∈ ∂H different from ∞ this leads to literally the same definitions. In case σ = ∞, the angular derivative w ′ (∞) should be understood in the sense of Carathéodory as defined in the following version of Julia's Lemma.
locally uniformly in D. The point τ is called the Denjoy -Wolff point of w or the DW-point for short. The limit behaviour of holomorphic self-maps with interior fixed points is similar. Namely, if τ ∈ D is a fixed point of w ∈ Hol(D, D) and if w is not an automorphism of D, then (2.7) holds. In this case we also refer to τ as the H}, is a non-trivial one-parameter semigroup, then v t 's different from id D share the same BRFPs and in particular, the same DW-point. According to a well-known result of Berkson and Porta [5], see also [9, Theorem 10.1.10] for another approach, in case D = D the infinitesimal generator φ of a one-parameter semigroup (v t ) can be written as where τ is the DW-point of (v t ) and p is a suitable (uniquely defined) holomorphic function in D with Re p 0. Conversely, if φ ∈ Hol(D) is given by (2.8) with some τ ∈ D and some p ∈ Hol(D, H), then φ is the infinitesimal generator of a oneparameter semigroup with the DW-point at τ .
For the non-autonomous case the following extension of [17,Theorem 1] holds.
is finite for a.e. t 0, and λ ∈ L 1 loc (I). If the above conditions are satisfied, then for all (s, t) ∈ ∆(I), 2.4. Bernstein functions. We start with the following classical definition.
Definition 2.20. By a Bernstein function we mean a non-negative function f of class C ∞ in (0, +∞) such that (−1) n+1 f (n) (x) 0 for all x > 0 and all n ∈ N.
In the following theorem we collect some important basic facts, which can be found together with proofs in literature on Bernstein functions, in particular, in the monograph [48,Chapter 3].
Theorem D. The following statements hold.
(B) Every Bernstein function f has a (unique) holomorphic extension f * : H → C with Re f * 0. Moreover, f * extends continuously to i R and representation (2.11) still holds if f is replaced by f * and x > 0 by an arbitrary z ∈ H.
(C) For the coefficients α and β in representation (2.11) we have (D) If f ≡ 0 and g are Bernstein functions, then g•f is also a Bernstein function and trivially (g (E) Bernstein functions form a convex cone; namely, for any two Bernstein functions f 1 , f 2 and any constant a, b 0, af 1 +bf 2 is again a Bernstein function.
(F) If a sequence of Bernstein functions converges pointwise in (0, +∞) to a function f , then f is a Bernstein function.
In what follows we will not make any notational distinction between a Bernstein function f and its holomorphic extension f * . The class of all v ∈ Hol(H, C) such that v| (0,+∞) is a Bernstein function and not identically zero will be denoted by BF. According to assertions (B), (D) and (F) in the above theorem, BF is a topologically closed semigroup in Hol(H, H). Taking into account the uniqueness of holomorphic extension, we will refer to elements of BF as Bernstein functions as well.
In what follows we will need another definition, closely related to the notion of a Bernstein function.
Similarly, a function ψ : I → R is said to be absolutely monotone in I if it is of class C ∞ and ψ (k) (x) 0 for all x ∈ I and all k = 0, 1, 2, . . . Another simple observation is that according to Leibniz's rule for higher order derivatives, the product of two absolutely monotone functions is again absolutely monotone.
Remark 2.23. In a similar way, it follows immediately from Faà di Bruno's formula for the derivatives of a composite function that if g : I → J, where I and J are two open intervals in R, is such that g ′ is absolutely monotone in I, and if ψ is absolutely monotone in J, then the composition ψ • g is absolutely monotone in I.  Hol(D, D), the cone G(U) formed of all infinitesimal generators of one-parameter semigroups contained in U has been explicitly characterized, see e.g. [28]. In each case we are aware of, the cone G(U) turns out to be convex. However, we have not been able to find in the literature any general results in this respect. By this reason, below we give a proof of the following basic theorem. Theorem 1. Let D C be a simply connected domain and U a topologically closed semigroup of holomorphic self-maps of D. Then the set G(U) formed by all infinitesimal generators of one-parameter semigroups contained in U is a topologically closed (real) convex cone in Hol(D, C).
Since by Theorem D, BF is a topologically closed semigroup in Hol(H, H), the above theorem directly implies the following Proof of Theorem 1. The fact that G(U) is a cone is trivial. Indeed, if φ ∈ G(U) generates a one-parameter semigroup (v φ t ), then for any α 0 the function αφ is the infinitesimal generator of the one-parameter semigroup (ṽ t ) formed by the mappings v t := v φ αt . To show that G(U) is topologically closed, consider an arbitrary sequence (φ n ) ⊂ G(U) converging locally uniformly in D to some φ ∈ Hol(D, C). By [9, Theorem 10.5.6, p. 300], φ is also an infinitesimal generator in D and, for each t 0, we have v n t → v t in Hol(D, D) as n → +∞, where (v n t ) and (v t ) stand for the one-parameter semigroups generated by φ n and φ, respectively. By the hypothesis, (v n t ) ⊂ U and U is closed in Hol(D, D). Therefore, (v t ) ⊂ U and hence φ ∈ G(U). This means that G(U) is a closed subset of Hol(D, C).
To complete the proof it remains to show that φ := φ 1 + φ 2 ∈ G(U) provided that φ 1 , φ 2 ∈ G(U). By [9, Corollary 10.2.7, p. 287], φ is an infinitesimal generator in D. Denote by (v 1 t ), (v 2 t ), and (v t ) the one-parameter semigroups generated by φ 1 , φ 2 , and φ, respectively. The Trotter product formula for one-parameter semigroups, see e.g. [9, Corollary 10.6.2, p. 304] or [46,Corollary 4], states that for each t 0, where the limit exists in the topology of Hol(D, D). Since by the hypothesis, U is a closed subset of Hol(D, D) and all finite compositions of its elements again belong to U, it follows that (v t ) ⊂ U, i.e. φ ∈ G(U).

3.2.
Evolution families in topologically closed semigroups. A natural question arises in connection with the correspondence discussed at the end of Section 2.2: how are the properties of the elements in a (reverse) evolution family related to those of the associated Herglotz vector field? For a particular property related to the boundary behaviour at a given point (namely, for having a boundary regular fixed point) the answer was given as Theorem C. The main new result of this section is a general statement of a similar kind.
Theorem 2. Let D C be a simply connected domain and U a topologically closed semigroup of holomorphic self-maps of D. Let (w s,t ) be a (reverse) evolution family in D with associated Herglotz vector field φ. Then the following two conditions are equivalent: (i) w s,t ∈ U for all (s, t) ∈ ∆(I); (ii) φ(·, t) ∈ G(U) for a.e. t ∈ I.
The assumption that U is topologically closed plays an important role in the above theorem. Compare, e.g., with Theorem C in Section 2.3, which concerns holomorphic self-maps with a given BRFP. They form a semigroup, but it is not a closed subset in Hol(D, D). As a result, an additional condition has to be imposed on a Herglotz vector field in order to ensure that the corresponding evolution family is contained in the semigroup in question.
Nevertheless, since by Theorem D, Bernstein functions different from the identical zero form a topologically closed semigroup in Hol(H, H), Theorem 2 implies directly the following corollary. The proof of (ii) ⇒ (i) in Theorem 2 makes use of the following rather simple observation, which however we have not met explicitly stated elsewhere.
Now the conclusion of the proposition follows easily from (3.1) and (3.2).
Proof of Theorem 2. First of all, recall that according to Remark 2.8, if (v s,t ) is reverse evolution family over I with associated Herglotz vector φ, then for any S ∈ I, the family (w s,t ) 0 s t S formed by the functions w s,t := v S−t,S−s is an evolution family over [0, S]. It is easy to see that [0, S] ∋ t → φ(·, S − t) is the Herglotz vector field associated with (w s,t ). It is, consequently, sufficient to prove the theorem for the case of evolution families.
Proof of (i) ⇒ (ii). By [14, Theorem 3.6], for each evolution family (w s,t ) in D, there exists a null-set N ⊂ I such that for any s ∈ [0, T ) and any t 0 ∈ I s \ N the limit lim Is∋t→t 0 w s,t 0 − w s,t t − t 0 exists in Hol(D, C) and coincides with z → φ w s,t 0 (z), t 0 . In particular, if s ∈ [0, T ) \ N, then taking t 0 := s we see that where the limit is attained locally uniformly in D. By the hypothesis, U is a topologically closed semigroup. Therefore, if (w s,t ) ⊂ U, then thanks to (3.3) we also have (v s t ) ⊂ U. Thus, assertion (i) implies that φ(·, s) ∈ G(U) for a.e. s ∈ [0, T ), i.e. that (ii) holds.
The variable change θ = g(t) relates the trajectories of the Herglotz vector field ψ to those of φ. Using this fact, it is easy to see that the evolution family (ω η,θ ) associated with ψ is given by ω η,θ = w f (η),f (θ) for all (η, θ) ∈ ∆(J). Therefore, it is sufficient to show that (ω η,θ ) ⊂ U.
Consider the sequence (ψ n ) of functions on D × J defined by where η(θ, n) := 2 −n ⌊2 n θ⌋ and ⌊·⌋ stands for the integer part. Denote by G 0 the set of all ϕ ∈ G(U) such that sup K |ϕ| C K for every compact set K ⊂ D. Recall that by Theorem 1, G(U) is convex and topologically closed. Therefore, G 0 is a compact convex set in the locally convex Hausdorff space Hol(D, C). By (3.5) the integrand in (3.6) belongs to G 0 for a.e. t 0, and by [7, Lemma 4.7] the map θ → ψ(·, θ) ∈ Hol(D, C) is measurable. Therefore, for any fixed n ∈ N and θ ∈ J, the integral in (3.6) can be understood as a weak integral of η → ψ(·, η), and moreover, its value belongs to 2 −n G 0 , see e.g. [36, Theorem 1 in §18. 1.3]. Hence ψ n (·, θ) ∈ G 0 ⊂ G(U) for all θ ∈ J. In its turn, the latter implies that ψ n 's are Herglotz vector fields in D.
Note that the elements of the evolution families (ω n η,θ ) associated with the Herglotz vector fields ψ n are finite compositions of mappings from one-parameter semigroups whose infinitesimal generators belong to G 0 . Therefore, (ω n η,θ ) ⊂ U and it only remains to show that for each (η, θ) ∈ ∆(J) fixed, ω n η,θ → ω η,θ in Hol(D, C) as n → +∞. Since self-maps of D form a normal family, it is sufficient to prove that ω n η,θ → ω η,θ pointwise in D.
We claim that for a.e. θ ∈ J, ψ n (·, θ) → ψ(·, θ) in Hol(D, C) as n → +∞. Indeed, consider a sequence (z k ) ⊂ D having an accumulation point in D. By compactness of G 0 , it is only necessary to prove that for each k ∈ N and for a.e. θ ∈ J, ψ n (z k , θ) → ψ(z k , θ) as n → +∞. The latter holds for any Lebesgue point θ, i.e. for any point θ ∈ J at which the derivative of F k (θ) := θ 0 ψ(z k , η)dη exists and equals ψ(z k , θ). Since the Lebesgue points have full measure, our claim is proved.
Fix now a compact set K ⊂ D. As we have just proved, for a.e. θ ∈ J. Since ψ n (·, θ) and ψ(·, θ) belong to G 0 for all n ∈ N and a.e. θ ∈ J, we have 0 δ K n (θ) 2C K for a.e. θ ∈ J. Therefore, thanks to Lebesgue's Dominated Convergence Theorem, for each compact interval Y ⊂ J and each function h : for all z ∈ K. The fact that ω n η,θ → ω η,θ as n → +∞ follows now from general results on ODEs, see e.g. [

Bernstein generators.
By a Bernstein generator we mean an infinitesimal generator of a one-parameter semigroup whose elements are Bernstein functions. In the notation introduced in Preliminaries, the class of all Bernstein generators is exactly G(BF). It is worth mentioning that in the literature on stochastic processes, see e.g. [38, Sect. 12.1], Bernstein generators appear in connection to continuous-state branching processes; in this probabilistic context, they are referred to as branching mechanisms.  If φ is given by (3.7), then clearly φ ∈ Hol(H, C) and satisfies conditions (i) and (ii) in Theorem 3. Moreover, (3.7) implies that φ ′′ is the Laplace transform of a non-negative Borel measure µ on [0, +∞) given by  λe −λ π(dλ).
Remark 3.5. With the help of the monotone convergence theorem, it is easy to see that in the above notation, In this section we will present a complex-analytic proof of Theorem 3 independent from representation (3.7). On the other hand, with the help of the above computation, (3.7) can be adapted for the case of a BRFP at z := 0. Corollary 3.6. A function φ : H → C is a Bernstein generator associated with a one-parameter semigroup (v φ t ) having a BRFP at σ := 0 if and only if φ admits the representation where c ∈ R, b 0 and π is a non-negative Borel measure on (0, +∞) such that Moreover, if the above necessary and sufficient condition is satisfied, then In particular, σ = 0 is the Denjoy -Wolff point of (v φ t ) if and only if φ admits representation (3.9) with c 0. Note that the Bernstein generator φ in Corollary 3.6 admits both representations (3.7) and (3.9). As we will see from the proof, the coefficient b and the measure π in the two representations are the same.
As a representation of a branching mechanism φ, formula (3.9) appeared in [ In a similar way, the case of a BRFP at ∞ can be treated.
Moreover, if the above necessary and sufficient condition is satisfied, then and (v φ t ) ′ (∞) = e −c t for all t 0. In particular, σ = ∞ is the Denjoy -Wolff point of (v φ t ) if and only if φ admits representation (3.12) with c 0. The representation (3.12) with c 0 has a probabilistic interpretation that the corresponding branching process is exactly a process with non-decreasing sample paths because it can be expressed as a randomly time-changed subordinator (with killing) according to the Lamperti transformation (see e.g. [38,Theorem 12.2]).
Remark 3.8. According to Corollary 3.6, a function φ given by representation (3.9) with c 0 is an infinitesimal generator of a one-parameter semigroup with the DWpoint τ = 0. It is therefore interesting to compare (3.9) with the Berkson -Porta formula (2.10a): φ(ζ) = ζ 2 P (ζ), ζ ∈ H, where P is the holomorphic function with non-negative real part given by with the Borel measure µ defined by (3.8). It is worth mentioning that in the case of arbitrary c ∈ R, the above representation of φ corresponds to the extension of the Berkson -Porta formula due to Bracci, Contreras and Díaz-Madrigal [6, Theorem 1.3].

Auxiliary statements and proofs.
Proof. Let f : H → C be a Bernstein function. Then Re f 0 in H and hence, by Remark 2.19, φ := −f is an infinitesimal generator in H. Clearly, we may suppose that φ ≡ 0. By Theorem B, the limit f ′ (∞) := ∠ lim z→∞ f (z)/z exists and belongs to [0, +∞).
Note that we may restrict ourselves to the case f ′ (∞) > 0. To cover the case f ′ (∞) = 0, it would be sufficient to consider the Bernstein functions f + δ id H , δ > 0, and use the fact that by Corollary 3.1, G(BF) is topologically closed in Hol(H, C).
For any ε > 0, Euler's approximations of the one-parameter semigroup (v φ t ) generated by φ, which are defined recurrently by v ε 0 = id, v ε n := v ε n−1 + εf • v ε n−1 , n ∈ N, are Bernstein functions, because BF is a convex cone closed w.r.t. compositions and containing id H , see Theorem D. Taking into account that by Theorem D (F), BF is topologically closed in Hol (H, H), we only need to show that for any t > 0, v t/n n → v φ t locally uniformly in H as n → +∞. Notice that v ε n is the n-th iterate of ψ ε (z) := z + εf (z). Recall that f ′ (∞) > 0. Therefore, ψ ε is a hyperbolic self-map of H with the DW-point at ∞. Therefore, see e.g. [53,Chapter 6, §46], for any ε > 0 the limit g ε (z) := lim n→+∞ ψ •n ε (z)/|ψ •n ε (1)| exists for all z ∈ H; and moreover, g ε ∈ Hol(H, H) is a non-constant solution to the functional equation where log stands for the branch of the logarithm in H satisfying log 1 = 0. For each ε > 0, the function h ε solves Abel's functional equation Recall that g ε (H) ⊂ H. Hence h ε (H) lies in the horizontal strip of width If h εn → h locally uniformly in H for some sequence ε n → 0 + as n → +∞, then the integral in the above equality tends along the sequence (ε n ) to h ′ (z)f (z). Taking into account that h ε (1) = 0 for all ε > 0, we therefore conclude that The function h is the Koenigs function of (v φ t ), see e.g. [9, Theorem 10.1.4, p. 274]. In particular, it satisfies Abel's equation h • v φ t = h + t for all t 0. On the other hand, if for some fixed t 0, and some sequence (n k ) ⊂ N tending to +∞, we have v Re z for all z ∈ H. It follows that either v * ∈ Hol(H, H) or v * ≡ ∞. The latter alternative is impossible because, as a holomorphic self-map of H, ψ t/n k is non-expansive w.r.t. the hyperbolic distance ρ in H, and hence for any z ∈ H, we have n k ρ(z, z + tf (z)/n k ) → t|f (z)| 2 Re z as k → +∞. Thanks to (3.13) with ε := t/n k , we have h t/n k • v t/n k n k = h t/n k + t. Passing to the limit as k → +∞, we obtain the desired relation h • v * = h + t.

As a Koenigs function of a one-parameter semigroup, h is univalent in
Since v t/n n ∈ Hol(H, H) for any n ∈ N, t 0, and since Hol (H, H) is a normal family, the latter means that for each t 0, v t/n n → v φ t in Hol(H, H) as n → +∞. This completes the proof.
Proof of Theorem 3. Suppose first that φ is a Bernstein generator. As usual, denote by (v φ t ) the one-parameter semigroup in BF associated with φ. Since φ(z) = lim t→0 + z − v φ t (z) /t and the limit is attained locally uniformly in H, it follows that φ is real-valued on (0, +∞) and that for all x ∈ (0, +∞) and all n 2, because (−1) n−1 v φ t (n) (x) 0 for all t > 0 and all n ∈ N.
Since φ ′′ (x) 0 for all x ∈ (0, +∞), the limit φ(0) := lim R∋x→0 + φ(x) exists and belongs to (−∞, +∞]. Considering the equation dw/dt + φ(w(t)) = 0 with the initial condition w(0) = x in a right neighbourhood of the origin on the real line, we see that the solution leaves H at finite time if φ(0) > 0, which would contradict the fact that φ is an infinitesimal generator in H.
Using (3.15), we get: Since by Corollary 3.1, G(BF) is closed in Hol(H, C), it follows that ϕ is a Bernstein generator and hence the proof is complete.
With a similar technique, for all x > 0 we further obtain Extending this formula holomorphically to H and taking into account the first equality in (3.16), we arrive at the desired representation (3.9).
Conversely, bearing in mind Bernstein's theorem on completely monotone functions, see e.g. [48, Theorem 1.4 on p. 3], one can easily see that if φ is given by (3.9) with some c ∈ R, b 0, and a Borel measure π on (0, +∞) subject to the integrability condition (3.10), then φ is a holomorphic function in H satisfying conditions (i) -(iii) in Theorem 3 and (3.16), with φ ′ (0) = c. Therefore, φ is a Bernstein generator and the associated one-parameter semigroup (v φ t ) has a BRFP at 0. Moreover, with the different sign in the definition of the infinitesimal generator taken into account, from [17,Theorem 1]  We omit the rest of the proof since it is very similar to that of Corollary 3.6.  Suppose that (w s,t ) (s,t)∈∆(I) ⊂ U τ satisfies conditions EF1 and EF2 in Definition 2.5. If there exits z 0 ∈ D such that the map I ∋ t → w 0,t (z 0 ) belongs to AC loc (I), then (w s,t ) is an absolutely continuous evolution family.
The family ( w s,t ) defined by w s,t := L t • w s,t • L −1 s for all (s, t) ∈ ∆(I), where L t (z) := z − ib(t) /a(t), satisfies conditions EF1 and EF2 in Definition 2.5. Moreover, its elements share the same DW-point at 1 ∈ H and a boundary regular fixed point at ∞. By the Chain Rule for angular derivatives, see e.g. [17,Lemma 2], elements of a topological evolution family are univalent [12,Proposition 2.4], we have v 0,t (0, τ ) ⊂ (0, τ ) for all t ∈ I. Hence the monotonicity of f follows from the fact that the holomorphic maps v s,t do not increase the hyperbolic distance in H. For z ∈ {z 0 , τ }, the map t →ṽ 0,t (z) belongs AC loc (J). Again by univalence of elements of a topological evolution family,ṽ 0,t (z 0 ) =ṽ 0,t (τ ) for all t ∈ J . By [14,Proposition 3.7], it follows that (ṽ s,t ) is an absolutely continuous evolution family.
Suppose now that (v s,t ) is a topological reverse evolution family. If I is of the form [0, T ], 0 < T < +∞, then applying the above argument to the topological evolution family (v s,t ) (s,t)∈∆(I) defined byv s,t := v T −t,T −s , we directly arrive to the desired conclusion.
To cover the case I = [0, T ), 0 < T +∞, the argument has to be slightly modified. As above, we suppose that τ = ∞ or τ ∈ (0, +∞) and fix some z 0 ∈ (0, τ ). Next we choose an increasing sequence (T n ) ⊂ (0, T ) converging to T and set f (s) := s + +∞ n=1 ′ z 0 − v s,Tn (z 0 ) a n 1 [0,Tn] (s), a n := 2 n max where the prime in the summation symbol means that, in order to cover the possibility of v 0,Tn = id H for some n ∈ N, we omit all terms for which a n = 0.
Let u := f −1 : J → I, where J := f (I). Fix some n ∈ N and notice that where f n is a strictly increasing function on I. Since s → z 0 − v s,Tn (z 0 ) is non-decreasing, the function f n • u is Lipschitz continuous on f ([0, T n ]). This fact allows us to show, arguing to as above, that v Tn−u(t),Tn−u(s) is an absolutely continuous reverse evolution family on [0, T n ]. Since n ∈ N is arbitrary in this argument, we may conclude that (v u(s),u(t) ) is an absolutely continuous evolution family, as desired. Now we prove an analogue of Theorem 4 for reverse evolution families.
Theorem 5. Let D ∈ {D, H} and τ ∈ ∂D. Let (v s,t ) (s,t)∈∆(I) ⊂ U τ be a topological reverse evolution family. If there exists z 0 ∈ D such that the map I ∋ t → v 0,t (z 0 ) belongs to AC loc (I), then the reverse evolution family (v s,t ) is absolutely continuous.
Remark 3.12. In contrast to Theorem 4, the hypothesis in the above theorem contains the requirement that (v s,t ) is continuous in (s, t). This assumption is essential for the proof we give below. However, we do not know whether Theorem 5 remains valid if we assume that (v s,t ) satisfies REF1 and REF2 from Definition 2.6 without requiring a priori any regularity in s and t.
To prove Theorem 5 we need the following two lemmas.
The constant A K depends on K and z 0 , but not on f .
Proof. By Julia's Lemma for H, see Theorem B, either g(z) := f (z) − z is constant, or g ∈ Hol(H, H). In the former case, (3.19) holds with A K := 1. In the latter case, it is enough to apply Harnack's inequality to the function g in the same manner as in the proof of Proposition 3.4.
Let K be a compact subset of Q. Then there exists ε > 0 such that K ′ := (s,t,ζ)∈K B ε (s, t, ζ) is also a compact subset of Q. Again by Carathéodory's Kernel Convergence Theorem, the map Q ∋ (s, t, ζ) → v −1 s,t (ζ) is continuous. Therefore, it is sufficient to establish (3.20) under the additional condition that |ζ 2 − ζ 1 | ε. To this end notice that, by continuity, the function Q ∋ (s, t, ζ) Proof of Theorem 5. First of all, obviously, we may assume that D = H and τ = ∞. Further, fix some S ∈ I. According to the definition of a reverse evolution family, it is sufficient to show that for any z ∈ H there exists a constant M S,z > 0 such that for any s, t 1 , t 2 satisfying 0 s t 1 t 2 S, we have Apply Lemma 3.14 with Clearly, K S is compact. Moreover, v 0,t 2 (H) ⊂ v 0,t 1 (H) for any t 2 t 1 , t 1 , t 2 ∈ I, because v 0,t 2 = v 0,t 1 •v t 1 ,t 2 ; hence, K S ⊂ Q. In view of condition REF2 in Definition 2.6, inequality (3.20) for ζ j := v 0,t j (z 0 ), j = 1, 2, s := 0, and t := t 1 , yields Now fix an arbitrary z ∈ H. Using Lemma 3.13 and the above inequality, we obtain Since the hyperbolic and Euclidean distances are equivalent on the compact set {v s,t (z) : (s, t) ∈ ∆([0, S])} and since the holomorphic mappings v s,t 1 : H → H, s ∈ [0, t 1 ], do not increase the hyperbolic distance, from inequality (3.21) it follows that for a suitable constant M S,z > 0 and for any s, t 1 , t 2 satisfying 0 s t 1 t 2 S, This completes the proof.
Next we turn to families of holomorphic self-maps of H formed by (extensions) of Bernstein functions. First we establish a (quantitative) rigidity type result. In contrast to arbitrary holomorphic self-maps, for which the Burns -Krantz rigidity property involves angular derivatives up to the third order, see e.g. [51] and [18], for functions in BF, existence of finite limits of the first two derivatives at z = 0 is sufficient. Recall that by definition of a Bernstein function, if f ∈ BF then f ′ as well as all higher order derivatives of f are real-valued and monotonic on (0, +∞). Hence there exist (finite or infinite) limits f (n) (0) := lim R∋x→0 + f (n) (x), n ∈ N.
Moreover, there exists a non-negative finite limit f (0) := lim R∋x→0 + f (x).  Proof. Let f ∈ BF. Then for all z ∈ H, Thanks to the triangle inequality, this immediately implies (3.22). z − 0 exist, are finite and equal to f ′ (0) and f ′′ (0), respectively. Furthermore, thanks to the monotonicity of f ′ and f ′′ on (0, +∞), a sort of converse is also true: for f ∈ BF, existence of finite angular derivatives of the first and second order at 0 implies that f ′ (0) and f ′′ (0) are finite. In particular, the two ways to define f ′ (0) and f ′′ (0), as angular derivatives and via limits as R ∋ x → 0 + , agree. Now we will use the rigidity property proved above in order to give sufficient conditions for a (reverse) evolution family of Bernstein functions to be absolutely continuous.
Before proving the theorem, we obtain a corollary stating that analogous assertion holds for reverse evolution families.
Fix some neighbourhood U of ζ 0 contained in H along with its boundary. By (3.27), for all t < t 0 close enough to t 0 , the function f t (z) := w t,t 0 (z) − ζ 0 has two distinct zeros in U, namely, z = w 0,t (z j ), j = 1, 2. At the same time, f t (z) → z − ζ 0 uniformly in U as [s 0 , t 0 ] ∋ t → t 0 . The limit function has only one simple zero in U, which contradicts Hurwitz's Theorem. Therefore, J = I, i.e. w 0,t (z 1 ) = w 0,t (z 2 ) for all t ∈ I.
Arguing as above we see that for any S ∈ I, there exists a compact set K S ⊂ H such that w 0,t (z j ) ∈ K S , j = 1, 2, for any t ∈ [0, S]. Hence in view of (iii), (3.26), and Proposition 3.15, the functions I ∋ t → w 0,t (z j ), j = 1, 2, are of class AC loc (I). Thus, the conclusion of the theorem follows from [14, Proposition 3.7].
3.5. The second angular derivative of evolution families in BF. An elementary result from the theory of ODE is that for the solution w(t) = w s,t (z) to an initial value problem of the form If now φ is Herglotz vector field in H and the associated evolution family (w s,t ) has a BRFP at σ = 0, then formula (3.28), where w ′ s,t (0) and φ ′ (0, t) are to be understood as angular derivatives, is justified by Theorem C.
In view of Theorem 6, it is worth extending (3.29) to evolution families of Bernstein functions.