Totally Ordered Measured Trees and Splitting Trees with Infinite Variation

Combinatorial trees can be used to represent genealogies of asexual individuals. These individuals can be endowed with birth and death times, to obtain a so-called `chronological tree'. In this work, we are interested in the continuum analogue of chronological trees in the setting of real trees. This leads us to consider totally ordered and measured trees, abbreviated as TOM trees. First, we define an adequate space of TOM trees and prove that under some mild conditions, every compact TOM tree can be represented in a unique way by a so-called contour function, which is right-continuous, admits limits from the left and has non-negative jumps. The appropriate notion of contour function is also studied in the case of locally compact TOM trees. Then we study the splitting property of (measures on) TOM trees which extends the notion of `splitting tree' studied in \cite{MR2599603}, where during her lifetime, each individual gives birth at constant rate to independent and identically distributed copies of herself. We prove that the contour function of a TOM tree satisfying the splitting property is associated to a spectrally positive L\'evy process that is not a subordinator, both in the critical and subcritical cases of compact trees as well as in the supercritical case of locally compact trees.

Model 1. Individuals have i.i.d. lifetimes in (0, ∞] during which they give birth at times of independent Poisson point processes with the same intensity to independent copies of themselves, giving rise to exactly one offspring at each birth event. The global history of the population can be encoded by a so-called 'chronological tree' as depicted in Figure 1(D). The vertical segments represent the individuals together with their lifetimes, and the vertical axis represents time flowing upwards. Hence, the lower endpoint of each vertical segment represents the birth-time while the upper endpoint represents death-time. The ancestors of segments can be found by following the dotted lines at the bottom of each segment. Following segments downwards and dotted lines to the left gives us the ancestral lines of the tree. In [Lam10] a generalization of the preceding model is proposed and called a 'splitting tree', following the terminology introduced in [GK97]. [Lam10] proposes to encode the tree by its contour, obtained by using a total order which is increasing when descending along ancestral lines; we then traverse the tree at unit speed respecting the total order and recording the height to obtain a piecewise linear function with jumps such as the one depicted in Figure 1(A). Conversely, starting from a non-negative piecewise linear function with positive jumps and negative slopes, we can interpret it as the contour of a tree as follows. Jumps of the function correspond to individuals and the size of the jumps are interpreted as lifetimes (which we explore from the top of the jump). The bottom of each jump is joined backwards to the first vertical line (or jump) it encounters (by joining along the dotted lines) which then becomes its parent while the height at which it is joined is the birth-time. This produces a notion of genealogy between the jumps. This interpretation of a function as giving rise to a tree was introduced and explored in [LGLJ98] and interpreted via last in first out (LIFO) queues with a single server.
The process counting the number of individuals alive as time varies in Model 1 is known as (binary, homogeneous) Crump-Mode-Jagers (CMJ) process. This process is Markovian only in the case of exponentially distributed (or a.s. infinite) lifetimes. On the other hand, the numbers of individuals in each successive generation of the genealogy evolve as a Galton-Watson process. However, the real surprise of [Lam10] is that the contours of splitting trees are Markovian: they are Lévy processes in the subcritical case and otherwise the splitting tree truncated at a given time has a reflected Lévy process as its contour. For descriptions of the contour of non-binary and non-homogeneous versions of Model 1, we refer to [SS15].
The chronologies of [Lam10] are basically discrete, in the sense that their associated genealogy is a discrete (or combinatorial) tree from which the chronological tree is obtained by specifying the birth and death times of individuals. In short, we wish to extend these results to real trees as follows.
(1) To generalize the notion of chronological tree based on the formalism of real trees; these will be the TOM trees alluded to in the abstract, (2) To show that these TOM trees also give rise to a contour which characterizes the tree, (3) To introduce the splitting property for measures on TOM trees, (4) To identify the law of contours under measures with the splitting property in terms of reflected Lévy processes. Items 1 and 2 are based on modifying the formalism of structured real trees of [Duq08] to suit our present needs. On the other hand, items 3 and 4 are motivated by [Lam10], although our arguments differ. In forthcoming work, we use the above measures with the splitting property to give a construction of supercritical Lévy trees (which have been constructed as limits of Galton-Watson trees consistent under Bernoulli leaf percolation in [DW07] or by relating them to subcritical L'evy trees via Girsanov's theorem in [Del08] and [AD12]) by applying the height process of [DLG02] to the contours constructed in this paper. This gives us access to the chronology driving the genealogy encoded in supercritical Lévy trees and lets us identify the prolific individuals (those with an infinite line of descent) introduced in [BFM08]. Finally, we believe that our construction of supercritical Lévy trees will provide a snake construction of supercritical superprocesses (with spatially independent branching mechanisms, as in [LG99] for (sub)critical cases) and give an interpretation for the backbone decomposition of [BKMS11].
1.2. Statement of the results. The central notion of this work, that of a TOM tree, is based on the metric spaces called real trees. The definition of real trees mimics the concept of a combinatorial tree defined as a combinatorial graph which is connected and has no cycles.
When σ 1 σ 2 and σ 1 = σ 2 we write σ 1 ≺ σ 2 . Since a tree has no loops, there is a unique element, called the most recent common ancestor of σ 1 and σ 2 and denoted The real tree coded by a function has been introduced for continuous functions in [LG06] and for càglàd (left-continuous with right limits) functions with negative jumps in [Duq08]. We now recall its construction in the setting of càdlàg functions, that is right-continuous functions which admits limits from the left, with non-negative jumps. Given a càdlàg function f : [0, m] → [0, ∞) with non-negative jumps and such that f (m) = 0 we can define a compact real tree (τ f , d f , ρ f ) as follows. Consider the pseudo-distance d f on [0, m] given by as well as the corresponding equivalence relationship ∼ f given by We now adopt (and modify) the insight of [Duq08] which is to notice that the function f endows the real tree τ f with additional structure: a total order ≤ where two equivalence classes σ 1 , σ 2 ∈ τ f satisfy σ 1 ≤ σ 2 if and only if sup σ 1 ≤ sup σ 2 , and a measure µ equal to the push-forward of Lebesgue measure under the canonical projection, denoted p f , from [0, m] to τ f . The abstraction of this situation (i.e., not defining the tree from any given f ) lies at the heart of the notion of TOM trees.
We will be exclusively interested in locally compact TOM trees. In this case, the measure µ is actually finite when the tree is compact and hence σ-finite otherwise. As defined, TOM trees are intimately linked with the structured trees of [Duq08]. The main difference between TOM trees and the structured trees of [Duq08] is that the total order ≤ is in a sense the opposite of ours, since it is an extension of the genealogical partial order. Extending instead to the inverse genealogical partial order is important in order to obtain a coding function which is càdlàg and has non-negative jumps, since we will be interested in random TOM trees whose coding function is a Markov process. Indeed, to get a structured tree, it suffices to define a new total order ≤ st so that s ≤ st t if t ≤ s. Then the tree ((τ, d, ρ) , ≤ st , µ) is a structured tree. When coding a tree by a càdlàg function f , it suffices to consider the function t → f (m − t) to be in the setting of [Duq08]. We make the additional assumption of diffusivity of µ to obtain what we think is a slightly easier definition of a coding function for TOM trees.
With our definition, not every càdlàg function f : [0, m] → [0, ∞) with non-negative jumps and f (m) = 0 codes a compact TOM tree. We have to assume additionally that the equivalence classes f gives rise to have zero Lebesgue measure, so that the pushforward measure is non-atomic. One way to ensure this is to impose that the equivalence classes generated by f are at most countable. This assumption is satisfied with probability 1 if f is the sample path of a Lévy process with no negative jumps that is not a subordinator (cf. Proposition 3 in Section 2).
There is a simpler way to define a total order satisfying Or when the underlying tree is binary. A real tree τ is said binary when τ \ {σ} has at most 3 connected components for every σ ∈ τ . In that case, σ is called a branching point if τ \ {σ} has 3 connected components, the one containing ρ and two others. It can be easily seen that these three connected components are also trees. Suppose that for every branching point σ we are given an orientation, which declares, for the two components of τ \ {σ} which do not contain ρ, which one is the 'left' subtree rooted at σ and which one is the 'right' subtree rooted at σ. Then we can define a total order ≤ on τ as follows. For any σ 1 , σ 2 ∈ τ , σ 1 σ 2 implies σ 2 ≤ σ 1 (so that ≤ satisfies Or1). Otherwise, note that σ 1 ∧ σ 2 is a branching point and we then define Then ≤ is a total order on τ which satisfies Or.
The importance of the notion of TOM trees is that they allow the construction of a contour which codes the tree in the aforementioned sense. To formalize this, we will say that two compact TOM trees are isomorphic if there is an isometry from one to the other which respects the total order and maps the associated diffuse measures one onto the other.
Theorem 1. Let c be a compact TOM tree and let m = µ(τ ). There exists a càdlàg function f c : [0, m] → [0, ∞) with no negative jumps such that the tree coded by f c is isomorphic to c. In particular, if c andc are non-isomorphic TOM trees then f c = fc.
The function f c is called the contour process for reasons that will be clear upon its construction. Also, note that the above correspondence gives us a canonical way to explore the tree: at time t we visit the equivalence class [t] of t under f c . The mapping φ : t → [t] is called the exploration process. We can then interpret the measure µ on the compact TOM tree as a measure of time. Indeed, if we traverse the interval [0, m] from left to right at unit speed, the exploration process gives us a way to traverse the tree respecting the order ≤ (except at a countable number of exceptional points) and µ(V ) is the time it takes to explore the set V ⊂ τ .
Theorem 1 follows from Theorem 1.1 in [Duq08], where the functions f c are given a characterization (as those satisfying a certain property Min). For completeness, we give a proof of Theorem 1 in Appendix A.
In order to relate the compact and locally compact cases, we introduce the truncation of a locally compact TOM tree. Let r > 0 and c = ((τ, d, ρ) , ≤, µ) be a locally compact TOM tree.
Definition. The truncation of c at level r, is the TOM tree c r = ((τ r , d r , ρ r ) , ≤ r , µ r ) where d r is the restriction of d to τ r ×τ r , ρ r = ρ, ≤ r is the restriction of ≤ to τ r ×τ r and µ r (A) = µ(A ∩ τ r ).
It is simple to see that (τ r , d r , ρ r ) is a real tree. As noted in [DW07], the closed balls of a locally compact real tree are compact by the Hopf-Rinow theorem (cf. [Gro07] or [BBI01]); this is where the completeness assumption in our definition of real trees comes into play since, without it, [0, 1) would be a locally compact tree without compact truncations at level r ≥ 1. Hence, the truncation c r at level r of a locally compact TOM tree c is a compact TOM tree and can therefore be coded by a function f r . The functions f r are related by a time-change in Proposition 4 of Section 3.
Let C c stand for the set of equivalence classes of isomorphic compact TOM trees (cf. Remark 7.2.5 of [BBI01,p. 251] for why this class actually constitutes a set, although this is also a consequence of Theorem 1). We define X t : C c → [0, ∞) ∪ { †} as follows, where † is a so-called cemetery state. Given (a representative) c = ((τ, d, ρ) , ≤, µ) ∈ C c , let f be the contour process of c and We then endow C c with the σ-field C = σ(X s : s ≥ 0). We now turn to the TOM trees coded by Lévy processes. We consider the canonical process on two functional spaces: the usual Skorohod space D of càdlàg trajectories f : [0, ∞) → [0, ∞) (as introduced for example in [Kal02,Ch 19,p. 380]) and the associated excursion space Note that the cemetery state † is isolated and absorbing and ζ = ζ(f ) is termed the lifetime of the excursion f .
On both functional spaces D and E we abuse our previous notation and define the canonical process X by X t (f ) = f (t) as well as the σ-fields D and E on D and E respectively as σ(X t : t ≥ 0). Note that in view of the correspondence between functions and trees, if c = ((τ, d, ρ) , ≤, µ) is a compact TOM tree, we can define ζ = ζ(c) = µ(τ ). We also introduce the canonical filtration (F t , t ≥ 0) where We now consider spectrally positive (i.e., with no negative jumps) Lévy processes. The reader can consult [Ber96] (especially Chapter VII) for the adequate background. Let Ψ be the Laplace exponent of a spectrally positive Lévy process which is not a subordinator (equivalent to assuming Ψ(∞) = ∞). This is a convex function written as where κ, β ≥ 0, α ∈ R and π is the Lévy measure of Ψ, which satisfies ∞ 0 1 ∧ x 2 π(dx) < ∞. Let P x (or P Ψ x when Ψ is not clear from the context) be the law of a spectrally positive Lévy process with Laplace exponent Ψ which starts at x. When κ > 0, the Ψ-Lévy process has the same law as the Lévy process with Laplace exponent ψ + κ killed after an independent exponential time with parameter κ; we will assume that a killed Lévy process jumps to the cemetery state † upon killing. We say that X is (sub)critical if X does not drift or jump to +∞ (or equivalently Ψ (0) ≥ 0 and κ = 0) and otherwise it will be termed supercritical. Let b be the largest root of the convex function Ψ, so that b > 0 if and only if Ψ is supercritical. If X stands for the cumulative minimum process given by then the reflected process X − X is a strong Markov process (cf. [Ber96, Ch. VI]) and X is a version of its local time at zero. Given a supercritical exponent Ψ we can define an associated subcritical exponent Ψ # (λ) = Ψ(λ + b) which corresponds to conditioning a Ψ-Lévy process on reaching arbitrarily low levels, in the sense that its law equals on every F s . (Cf. Lemma 7 in [Ber96, Ch. VII] and Lemme 1 in [Ber91].) We let ν stand for the excursion measure of X − X away from zero; the measure ν may charge excursions of infinite length. Indeed, if P → denotes the law of the Lévy process after it approaches its overall minimum from the left for the last time then ν = ν # + bP → , where ν # is the excursion measure associated to Ψ # which is concentrated on excursions with finite length.
Also, let us define the law Q x on E as the law of the Lévy process killed (and sent to †) upon reaching zero under P x . Then, under ν, the canonical process is strongly Markovian with the same semigroup as Q x . Define and let X r be the process obtained by stopping X − X r when it reaches zero. (In the killed case, X − X r might be killed before X − X r reaches zero; in this case, we simply concatenate independent copies started at r until a copy reaches zero). We let ν r be the push-forward of ν by X r . We also consider the law η r of the tree coded by the canonical process under ν r . Note that the measures η r are concentrated on binary trees. Our next result shows that we can define a measure η Ψ whose image under truncation at level r equals η r and that the measure η Ψ is characterized by a self-similarity property termed the splitting property.
To define such a property, suppose that c = ((τ, d, ρ) , ≤, µ) is a TOM tree. Truncate c at height r to obtain c r and suppose that c r has a total measure greater than t. Let φ stand for the exploration process of c r . Then, on the interval [ρ, φ(t)) we can define left subtrees and right subtrees. The right subtrees are characterized by the existence of σ ∈ [ρ, φ(t)) such that The right subtree at σ is then equal to and we can collect all these right subtrees in the measure In the following definition, recall that X(c) is the contour associated to the tree c.
Definition. A measure κ on locally compact TOM trees satisfies the splitting property if for any r > 0, on defining κ r as the image of κ under truncation at height r, (1) for any t > 0, κ r (ζ > t) < ∞ and (2) on the set {ζ > t} and conditionally on X t = x, Ξ r t is a Poisson random measure on [0, x]×C c with intensity Leb(ds) ⊗ κ r−s (dc): if h : [0, ∞) ∪ { †} → [0, ∞) is measurable and vanishes at † and g : Note that no assumption is made concerning conditional independence of Ξ r t and σ(X s , s ≤ t) given X t .
Remark. As consequences of the definition, we will see that when κ is concentrated on compact TOM trees then the splitting property admits the non-truncated version and implies the integrability properties See the proof of the forthcoming Theorem 2.
It turns out that a class of measures on binary TOM trees with the splitting property are in correspondence with Lévy processes, under an additional assumption of constant sojourn, as the next result shows.
The splitting trees of Model 1 previously introduced have the property that they treat elements of the tree in an equal manner, as in the following definition.
In other words, a TOM tree has sojourn a if the measure µ equals a times Lebesgue measure (on the tree). It is interesting that the sojourn a can be 0 in the above definition even if µ is a non-trivial measure. This happens in particular for the trees coded by Brownian excursions since Brownian motion spends zero time (with respect to Lebesgue measure) at its cumulative minimum process.
Recall the measures η r on binary trees defined from the (reflected) Lévy process with Laplace exponent Ψ.
Theorem 2. There exists a unique measure η Ψ on locally compact TOM trees whose truncation at level r equals η r . The measure η Ψ satisfies the splitting property, is concentrated on binary trees, has constant sojourn and assigns finite measure to non-compact TOM trees. When Ψ is (sub)critical, η Ψ is the pushforward of ν under the mapping taking functions to trees. η Ψ charges non-compact trees if and only if Ψ is supercritical. Conversely, if a non-zero measure κ on locally compact TOM trees has the splitting property, is concentrated on binary trees and there exists a ≥ 0 such that under κ the tree has sojourn a almost everywhere, then there exists a spectrally positive Lévy process with Laplace exponent Ψ such that κ = η Ψ .
The assumption that the measure κ has sojourn a is necessary to be able to code the tree by a Lévy process as the next example shows. Consider Model 1. Lambert shows in Theorem 4.1 and Remark 2 of [Lam10, p 373] that if b times the expected lifetime is less than or equal to 1 then the resulting real tree is compact with probability one. In this setting, one can use Lebesgue measure on each of the (finite number of) segments comprising the tree, thus obtaining a tree of sojourn 1, and the contour process of the compact TOM tree is a compound Poisson process minus a drift of slope 1 until it reaches zero. However, one might use an additional speed distribution S on (0, ∞) and associate to individuals iid speeds with distribution S at which their lifetimes will be traversed (or equivalently, defining the multiple of Lebesgue measure which will be used on each interval in the tree representing the lifetime). The compact TOM tree we obtain has the splitting property but does not have sojourn a and is not coded by a Lévy process (since there is no unique drift). However, there is a (random) time-change which (by giving all individuals the same non-random speed) takes the coding process into a Lévy process and this begs the question on whether there is a characterization of binary trees with the splitting property as time-changes of Lévy processes. This question is left open. Another question that is not addressed is on characterizing splitting trees which are not binary.
1.3. Organization. The paper is organized as follows. In Section 2, we study σ-finite measures on compact TOM trees and how they are related to Lévy processes through a proof of Theorem 2 in this case. In Section 3, we study the truncation operator on compact real trees and how locally compact TOM trees can be therefore constructed as direct limits of consistent families of compact TOM trees. We then give in Section 4 some constructions of the reflected Lévy processes which allow us, in Section 5 to construct σ-finite measures on locally compact TOM trees which have the splitting property and to give the proof of Theorem 2 in full generality. Finally, in Appendix A, we handle the deterministic aspects of the space of TOM trees culminating in a proof of Theorem 1.

Measures on compact TOM trees with the splitting property
In this section, we provide the characterization of measures on compact TOM trees with the splitting property stated in Theorem 2. We first show that the canonical process codes a compact real tree under the excursion measure of a (sub)critical Lévy process.
Proposition 3. Let X be a (sub)critical spectrally positive Lévy process with Laplace exponent Ψ. Let ν be the intensity measure of excursions of X above its cumulative minimum. Then, ν-almost surely, the equivalence classes have at most three elements for all s ∈ [0, ζ).
We now show that the right-hand set has ν-measure zero. By countable subadditivity, it suffices to show ν X [u 1 ,u 2 ] = X [u 2 ,u 3 ] , u 3 < ζ = 0. for ε < u 1 < · · · < u 3 . However, using the Markov property at u 1 under ν, we see that Hence, again by the Markov property, it suffices to show that P x X u 1 = X [u 1 ,u 2 ] = 0. However, However, since X is spectrally positive, then 0 is regular for (−∞, 0) (cf. Thm. 1 [Ber96, Ch. VII]) and so Lemma 1 in [PR69] (or Theorem 1 in both [Cha13] and [PUB12]) tell us that the law of X u 2 −u 1 is non-atomic, which implies that h is actually zero.
We now do an analysis on TOM trees to reduce our proof of the splitting property of η Ψ to a simple property of the excursion measures and to motivate the construction of the Lévy process out of a measure on trees with the splitting property.
Let f : [0, m] → R be a càdlàg function which codes a TOM tree c and consider s ∈ (0, m) and the class of s under ∼ f denoted σ = [s] f . We first identify the subtrees attached to the right of (σ, ρ] with the excursions of f above its cumulative minimum on [s * , m], where We have already identified R σ with the image of [s * , m] by the exploration process in Subsection A.3. Also, when discussing the tree coded by a function in Section 1, we identified the interval can be identified with the excursions of f above its cumulative minimum on the interval [s * , m]. To be more precise, say that 'σ ∈ [ρ, σ) is a branching point of R σ if R σ \ {σ} has at least three components. One of these components contains σ and a different one contains ρ. If we join all the other components toσ (of which there is only one in the case of binary trees), we obtain a compact TOM tree which can be termed the tree atσ to the right of [σ, ρ]. Let us call it cσ ,σ . The set ofσ ∈ [σ, ρ] for which cσ ,σ is not reduced toσ is at most countable. Indeed, by compactness, there can be only finitely many of them with height exceeding a given ε > 0. Let σ 1 , σ 2 , . . . be the branching points of R σ in [σ, ρ]. Notice that if (σ n i , n ∈ N) is a sequence in [σ, σ i ) converging to σ i then the real tree of c σ i ,σ , denoted τ σ i ,σ is given by Hence, c η i ,σ can be coded by However, if x i = d(η i , ρ), we also have the representation by our previous identification of the elements of [σ, ρ], which shows that c η i ,σ can be coded by an excursion of f above its cumulative minimum.
Under the additional assumptions that c is binary with sojourn a and that s = µ(L σ ), we can reconstruct f (s + ·) from f (s), the sequence (x i ) and each of the functions f i . Indeed, for any l ∈ [0, f (s)], let σ l be the point of [σ, ρ] at distance l from σ and consider S l to be the measure of R σ ∩ L σ l . Then because of constant sojourn, say a, is the Lebesgue measure of the interval of definition of f i , so that S can be constructed from the aforementioned quantities. Also, Now let Λ be the right-continuous inverse of S. By considering the dense set of measures of left sets (cf. Lemma 18), we see that We therefore obtain the representation Proof of Theorem 2. Let us start with the proof that the tree coded by the excursion measure of a Lévy process has the splitting property. Since we have just identified right subtrees along [ρ, σ) with excursions of the contour above its cumulative minimum, it suffices to prove that, for any t > 0 and on the set {ζ > t}, the excursions of X ·+t above its cumulative minimum form a Poisson point process. Let us recall that, under ν, on the set {X t ∈ (0, ∞)} and conditionally on X t = x, the path X ·+t has law Q x . Hence, the point process with atoms at starting levels of the excursions and the excursions themselves is a Poisson point process on [0, x] × E with intensity Leb ⊗ ν. We conclude the splitting property of η Ψ .
Consider now a non-zero measure κ on compact TOM trees with the splitting property and let ν be the push-forward of κ by the contour map.
We start by proving some consequences of the splitting property of κ. First, we prove that, in the compact case, the splitting property can be recast as Indeed, note that if c is a compact TOM tree, then its height sup σ∈τ d(ρ, σ) is finite. Hence, for any r greater than the height, c coincides with its truncation c r at level r. Therefore, for any measurable subset A of the set of compact TOM trees, we have that We deduce that κ r converges to κ in the following sense: Hence, for any g vanishing on trees with small height: The identity between the extremes can then be generalized to non-negative and measurable g. We now prove the integrability conditions Since κ is non-zero, there exists t > 0 such that κ(ζ > t) > 0. Note that in Ξ ∞ t , there can only be a finite number of atoms (s i , c i ) such that c i has measure > ε for any ε > 0. By the splitting property, applied with h = 1 [0,∞) and g(s, c) = 1 ζ(c)>ε , we get: We conclude that κ(ζ > ε) < ∞ for any ε > 0. We now choose g(s, c) = ζ(c). Next, under κ, the right of φ(t) has finite measure almost everywhere, so that, using the fact that κ has constant sojourn, say equal to a: We conclude that 1 ∧ ζ(c) κ(dc) < ∞.
Let Ξ = i δ (x i ,f i ) be a Poisson random measure on [0, ∞) × E with intensity Leb ⊗ ν. Define the process Y by the following procedure inspired by Itô's synthesis theorem (cf. [Itô72]): Let in analogy with (1). Heuristically, Y is the contour process of a tree obtained by grafting trees with law ν to the right of the vertical interval (−∞, 0]. We claim that Y is a (sub)critical spectrally positive Lévy process. If so, let Ψ be its Laplace exponent. Since the compact TOM tree coded by the excursion measure of Y above its cumulative minimum process has law κ by construction, and if Y is a (sub)critical Lévy process this law should equal, by definition, η Ψ , we see that κ = η Ψ . Let us prove that Y is a (sub)critical spectrally positive Lévy process. By construction, Y has no positive jumps. Also, note that the running infimum of Y is −Λ, which goes to −∞. Hence, if Y is proved to be a Lévy process, then it is spectrally positive and is critical or subcritical. It remains to see that Y has independent and stationary increments.
Let us note that the process Y x obtained by killing Y upon reaching −x has the following construction in terms of Poisson random measures: LetQ x be the law of x + Y x . By translation invariance of Lebesgue measure and independence properties of Poisson random measures we see that the two processes are independent and have respective lawsQ x andQ y . Also, note that if we concatenate a process with the same law as Y to an independent process with lawQ x when it gets killed, then we get a process with law Y . For any t ≥ 0, let Notice that the interval [g t , d t ] can be reduced to a point but that when it isn't, it is an interval of constancy for Λ. Also, note that g t = d t if and only if S is continuous at Λ t . Let Ξ t be the restriction of Ξ to [0, Λ t ] × E and define the random measure Ξ t characterized by where A and B are Borel subsets of [0, ∞) and of the excursion space E respectively. Let G l and G l be the σ-fields generated by the restriction of Ξ to [0, l] × E and [l, ∞) × E. Then G l is independent of G l . It is simple to see that Λ t is a stopping time for (G l ) for any t ≥ 0 and using the independence of G l and G l , the translation invariance of Lebesgue measure and discretizing Λ t by Λ t 2 n /2 n , one shows the independence of Ξ t and Ξ t and that Ξ t has the same law as Ξ by mimicking the proof of the strong Markov property for Feller processes. Define Using the analysis of the excursions straddling a given time of [RY99, Ch. XIII §3, p. 488], we see that defining the σ-field F Y gt as σ(Y s∧gt : s ≥ 0), then the law of the excursion of Y straddling t given F Y gt equals 1 s<t ν t−s + 1 s=t δ † on the set g t = s. The above discussion gives us access to 3 different parts of the path of Y : before g t , between g t and d t (conditionally on F Y gt ) and after d t (which, if we shift it to start it at zero, is independent of the pre-d t part and has the same law as Y since it is constructed from Ξ t in the same manner as Y is constructed from Ξ). We now study the measure ν to make a further description of the parts of the path between g t and t and between t and d t .
Consider the shift operators θ t on canonical space which take f to the function s → f (t + s). Thanks to the splitting property of κ, we see that conditionally on X s and under ν, the post-s process has lawQ Xs . Formally the splitting property translates into for nonnegative measurable h and F defined on [0, ∞) and E respectively (with g vanishing at †).
We now prove an analogous result underQ x : It is obtained from (2) as follows. Fix s > 0. Using the Poissonian description ofQ x , we see that under this measure, the trajectory splits into 3 independent subpaths: 1) the subpath X ·∧gs before g s , 2) the subpath X gs+·∧ds − X gs− between g s and d s , and 3) the subpath X ds+· after d s . We now work conditionally on X s = y, X s = z, g s = s ; note that X s− = z also. After d s , the process has lawQ y . Between g s and d s , the process has law ν s−s . Using (2), we see that under ν s−s , the post s − g s part of the trajectory has lawQ z−y . By concatenation, we see that underQ x and on the set X s = y, X s = z, g s = s , the post-s part of the trajectory has lawQ Xs , which implies (3). We deduce thatQ x is Markovian and that, therefore ν also is. Finally, we finish the proof that Y is a Lévy process. Let t > 0 and note that the process Y has 3 independent subpaths by our analysis of its Poissonian construction: before g t , between g t and d t (shifted by −Λ t to end at zero) and after d t (shifted by −Λ t to start at zero). Also, the law of the (shifted) process between g t and d t is ν t−gt and by the Markov property (under ν) we see that the conditional law of the process between t and d t (minus Λ t , to start at zero) given the pre-t process isQ Xt−Lt . By concatenating the post d t -part, which has law the law of Y (when shifted to start at zero) we see that the law of Y t+· − Y t given σ(Y s : s ≤ t) is the law of Y . Hence, Y is a Lévy process.

Locally compact TOM trees
We have already noted in Section 1 that truncations of locally compact real trees are compact and defined the truncation of a locally compact TOM tree. In this short section, we explore the relationship between the contour processes of two truncations of a same locally compact TOM tree. We then use this information to see how to build locally compact TOM trees out of sequences of consistent compact TOM trees.
The truncation at level r of a locally compact TOM tree c is the compact TOM tree c r = ((τ r , d| τ r , ρ) , ≤ | τr , µ| τ r ) .
We now study the relationship of the contour processes of two different truncations of the same tree.
Proposition 4. Let c be a compact TOM tree. The contour processes f and f r of c and c r are related by a time-change as follows: let C r (f ) be the right-continuous inverse of the continuous and non-decreasing function A r = A r (f ) given by Then Proof. As verified at the end of the proof of Proposition 20, we have the equality µ = Leb • φ −1 , where φ is the exploration process of c. Hence Let σ ∈ τ r and define s = µ r (L σ ) and t = µ(L σ ) so that As we just computed, s = µ r (L σ ) = A r t . However, by taking σ n ∈ [ρ, σ) satisfying σ ≤ σ n+1 ≤ σ n and setting t n = µ(L σn ), we see that t n decreases to t and f (t n ) < r, so that A r increases in any right neighborhood of t and so t = C r s . Hence f r = f • C r on the set {µ r (L σ ) : σ ∈ τ r }, which by Lemma 18 is dense. We conclude that We now use the preceding proposition to identify the space of locally compact TOM trees with the direct limit of compact TOM trees. Indeed, note that if c is a locally compact TOM tree and c n is its truncation at height n, then the truncation of c n+1 at height n equals c n . Conversely, let (r n , n ∈ N) be a sequence of non-negative real numbers increasing to ∞ and (c n , n ∈ N) be a sequence of compact TOM trees.
Definition. The sequence (c n , n ∈ N) is said to be consistent (at levels (r n , n ∈ N)) if the truncation of c n+1 at height r n is isomorphic to c n .
Proposition 5. If (c n , n ∈ N) is a consistent sequence of compact TOM trees at levels r 1 < r 2 < · · · then there exists a unique locally compact TOM tree c such that the truncation of c at level r n is isomorphic to c n .
Proof. Let us construct a locally compact TOM tree from a consistent sequence (c n , n ∈ N) at levels r 1 < r 2 < · · · ; suppose that c n = ((τ n , d n , ρ n ) , ≤ n , µ n ), let ψ n be an isomorphism between c n and the truncation of c n+1 at level r n (which is actually a mapping between τ n and τ n+1 preserving distances, root, total order and which maps the measure µ n into the restriction of µ n+1 to τ rn n+1 ). We then define ψ n m for m ≤ n as the composition ψ n−1 • · · · • ψ m . We begin by specifying the tree part τ of c. Let τ be the direct limit of (τ n , n ∈ N) with respect to the mappings (ψ n ). In other words, τ has elements of the type (l, σ) where σ ∈ τ l and we identify (l, σ) and (m,σ) if there exists n ≥ l, m such that ψ n l (σ) = ψ n m (σ). We then deduce that the preceeding equality holds for any n ≥ l, m.
The root of τ is taken as the equivalence class of (1, ρ 1 ). The triple (τ, d, ρ) is locally compact rooted real tree. Indeed, note that if (l, σ) and (m,σ) are two representatives of elements of τ , we can embed them in τ n for any n ≥ m, l and since τ n is a tree, we can obtain an isometry of an interval which starts at ψ n l (σ) and ends at ψ n m (σ). Lack of loops can be proved by noting that an isometry of an interval into τ must remain bounded, so that it can be transformed into an isometry of an interval into τ n for some n. Local compactness and completeness can also be proved by a similar argument.
A total order ≤ on τ can be defined by stating that (l, σ) ≤ (m,σ) if and only if ψ n l (σ) ≤ n ψ n m (σ) for some n ≥ l, m. It is easy to see that ≤ satisfies Or1 and Or2. Finally, we place the measure µ on τ . We abuse our notation to say that µ n is the push-forward of the measure µ n by the mapping that sends σ to the equivalence class of (n, σ). Note that (µ n ) increases and so we can define the measure µ by It is simple to see that µ satisfies Mes1 and Mes2.
We end the proof by noting that c = ((τ, d, ρ) , ≤, µ) is a locally compact TOM tree whose truncation at height r n is isomorphic to c n .
Let (f n ) be a sequence of càdlàg functions where f n is defined on [0, m n ], f (0) = 0 and such that f n codes a tree c n .
Definition. We say that the sequence (f n ) is consistent under time change if there exists a sequence of heights r 1 , r 2 , . . . increasing to infinity and such that From Propositions 4 and 5 we see that (the equivalence class under isomorphism of) a locally compact TOM tree can be identified with a sequence of functions consistent under time change (and such that their induced equivalence classes have zero Lebesgue measure). Let us then define the function X r t defined on TOM trees and with values in R ∪ { †} by stating that X r t (c) is the value at time t of the contour process of the truncation c r of c at level r (if the measure of c r is greater than t; otherwise we define it as †). We can then use the σ-field F = σ(X r t , t, r ≥ 0) when discussing measurability issues for functions defined on (or with values on) locally compact TOM trees.
Corollary 6. Let P r be a sequence of measures on Skorohod space such that the push-forward of P r under the time change at level r ≤ r gives P r . Suppose that under P r , the canonical process is non-negative, approaches zero at death time and that the equivalence classes [s] X induced by the canonical process have zero Lebesgue measure. Then there exists a unique measure P on locally compact TOM trees such that, under P the contour process of the truncation at level r has law P r .

Consistent families of reflected Lévy processes
The objective of this section is to give a further study of a family of stochastic processes consistent under time-change: the reflected Lévy processes introduced in Section 1. This will allow us, through Corollary 6, to construct random locally compact TOM trees. In this section, we give two additional constructions of reflected Lévy processes. Both constructions make sense even in the case of positive killing coefficient. The first construction is based on time-changes and has a pathwise time-change consistency which is useful in constructing locally compact TOM trees. This enables us to construct the measure η Ψ in the statement of Theorem 2 in the non-compact case. The second construction is through Poisson random measures. It is a consequence of the time-change construction and is fundamental to our proof of Theorem 2 in the locally compact case. 4.1. Reflected Lévy processes through time-change. We place ourselves on the canonical space of càdlàg trajectories from [0, ∞) to R together with the canonical process X = (X s , s ≥ 0) and the canonical filtration (F t , t ≥ 0). Let P x be the law of a spectrally positive Lévy process (spLp) with Laplace exponent Ψ started at x. We will suppose throughout that X is not a subordinator under P 0 .
Let us consider a reflecting threshold r and define the cumulative maximum process from r, X r given by We will write X when r is clear from the context, as in the next definition. Recall that a spectrally positive Lévy process with Laplace exponent ψ started at x ≤ r and reflected below r is a stochastic process having the law P r x of X − X under P x . Under P r x , the canonical process is a strong Markov process as proved in Proposition 1 of [Ber96, Ch. VI]. We will also make use of the Lévy process reflected below r and killed upon reaching zero and denote its law by Q r x ; recall that killing preserves the strong Markov property. The laws Q r x were introduced in Definition 4.5.4.1 of [Lam08] since they coincide with the law of the (jump-chronological) contour process J r of a splitting tree truncated at height r associated to a finite-variation spLp with Laplace exponent Ψ whose progenitor has lifetime x. The relationship between the contour processes at different heights (cf. Proposition 4) suggests the following corresponding property of the laws Q r x : let Let C r be the right-continuous inverse of A r .
Proposition 7. If x ≤ r 1 < r 2 then the law of X • C r 1 under P r 2 x is P r 1 x . By stopping at the first hitting time of zero, we obtain: [Lam10] proves Corollary 8 when the underlying Lévy process is of finite variation. We are interested in extending the consistency under time-change in order to construct random locally compact TOM trees out of the trajectories of more general Lévy processes.
In the Brownian case (when Ψ(λ) = λ 2 /2), the equality in law between X • C 0 under P r 2 x and P r 1 x is reduced to the equality in law between Brownian motion reflected below its supremum and Brownian motion time changed by C 0 to remain negative. This can be proved using Tanaka's formula, Knight's Theorem and Skorokhod's reflection lemma. These tools are not available for general spectrally positive Lévy processes. However, this already tells us that some local time argument should be involved in the proof. Indeed, our argument is based on excursion theory for Markov processes, in which local time plays a fundamental role.
Proposition 7 is a generalization of the results for Brownian motion or finite variation Lévy processes. The only caveat is to see that the equality between time-change and reflection holds as a consequence of the assumed lack of negative jumps. For example, if P 0 is the law of the difference of two independent stable subordinators of index α ∈ (0, 1), then under P r 1 x , the barrier r 1 is touched. However, since points are polar (see for example [Ber96, p. 63]) then X • C r 1 does not touch the barrier r 1 almost surely under P r 2 x so that no equality in law can be valid in this case. Proof of Proposition 7. Recall that the time-changed processes X • C r 1 under P r 2 x are also strongly Markovian, as shown for example in [Dyn65] or [Sha88].
The argument is based on excursion theory, as introduced in [Ber96, Ch. 4]. Indeed, we prove that the (regenerative) sets of times at which the two (strong Markov) processes visit r 1 , being the images of subordinators as in [Ber99, Ch. 2], have the same drift (zero in this case), and that the excursions of both processes below r 1 admit the same stochastic descriptions. To see that the drift is equal to zero, by Proposition 1.9 of [Ber99], it suffices to see that the sets of times at which the two processes visit r 1 have zero Lebesgue measure almost surely.
The law of X under P r 1 x until hitting r 1 is equal to the law of X under P x until it surpasses r 1 . After X hits r 1 under P r 1 x , the process has law P r 1 r 1 , which is the push-forward of P 0 0 under f → r 1 + f . Similarly, X equals X • C r 1 until X surpasses r 1 , and under P r 2 x , these processes have the law of X under P x until it surpasses r 1 . After X • C r 1 hits r 1 , it has the law under P r 2 x of X • C r 1 under P r 2 r 1 , which by spatial homogeneity is the image of the law of X • C 0 under P r 2 0 . Hence, it suffices to focus on the case 0 = x = r 1 < r 2 ; set r 2 = r to simplify notation.
Formula 9.2.9 of [Don07, p. 98] implies that the Laplace exponent of the upward ladder time process of X under P 0 is equal to α/Φ(α) where Φ is the right-continuous inverse of Ψ and since P 0 is not the law of a subordinator, then Φ(α) → ∞ as α → ∞ implying that the drift of the upward ladder time process, which is equal to lim α→∞ 1/Φ(α), is zero. Hence, the set of times X is at its supremum under P 0 has zero Lebesgue measure almost surely and so ∞ 0 1 Xs=0 ds = 0 almost surely under P 0 0 . Let us obtain a corresponding statement for X •C 0 under P r 0 . Since the time-change just removes parts of the trajectory, we are reduced to proving that X spends zero time at 0 under P r 0 which is equivalent to proving that X − X r spends zero time at zero under P 0 . Note however that X spends zero time at 0 under P 0 , even if X is compound Poisson minus a drift since the drift is forced to be nonzero having excluded subordinators. Hence, until the trajectory of X − X r reaches r, it spends zero time at zero under P 0 . After this process reaches r, its law is that of r + X − X under P 0 . Note that Hence, we now show that if T is a standard exponential time independent of X then X T − X T is almost surely not −r. By the Pečerskiȋ-Rogozin formula (cf. Th. 5 in [Ber96, Ch. VI] or Equation (5) in [PUB12]), we know that X T − X T is a negative infinitely divisible random variable with zero drift and Lévy measure equal to Recall that since X is not a subordinator under P 0 then 0 is regular for (−∞, 0), as verified in Since ν is an infinite measure the law of X T − X T has no atoms under P 0 (cf. Theorem 27.4 in [Sat99]). Since the laws P r 0 and P 0 0 are strongly Markovian, the sets t ≥ 0 : X • C 0 t = 0 and {t ≥ 0 : X t = 0} are regenerative under P r 0 and P 0 0 ; we have just proved that in both cases their corresponding inverse local time at zero has zero drift.
We now describe the excursions below 0. Let E − be the set of negative excursions. That is, the set of functions f : [0, ∞) → (−∞, 0] ∪ { †} for which there exists ζ = ζ(f ) > 0, termed the lifetime, such that f is strictly negative and càdlàg with no negative jumps on (0, ζ) and equal to † after ζ; hence, † is again interpreted as a cemetery state. The set E − is equipped with the natural filtration of the canonical process. Since we have proved that the canonical process X under P r 1 x (resp. X • C r 1 ) under P r 2 x ) spends zero time at r 1 , excursion theory tells us that X under P r 1 x has the same law as the stochastic process Z given by where the random measure i δ (Λ i ,E i ) is a Poisson random measure on [0, ∞) × E − with intensity Leb ⊗ ν for some σ-finite measure ν = ν 0 (resp. ν = ν r ) on E − and The measure ν 0 (resp. ν r ) is characterized, up to a multiplicative constant which does not affect the law of Z, as follows. Let T ε denote the hitting time of −ε by the canonical process. Since functions in E − start at zero and leave 0 immediately, any measure on E − is determined by its values on ∪ ε>0 σ(X t+Tε : t ≥ 0). LetP r 0 be the law of X • C 0 under P r 0 . Let O = {t ≥ 0 : X t < 0} .
Since X has non-negative jumps under P 0 0 andP r 0 then O is almost surely open under both measures. Its connected components are termed the excursion intervals of X below 0. If (a, b) is a connected component of O then the non-negativity of jumps forces X a = 0, and so the path e given by e s = X a+s if 0 ≤ s < b − a and e s = † if s ≥ b − a is an element of E − . The depth of the excursion e is defined as min s<b−a e s . For any ε > 0, there is a well-defined sequence of successive excursions of depths falling below −ε which are iid with common laws ν 0 ε (respectively ν r ε ) under P 0 0 (respectively underP r 0 ); this follows from the strong Markov property under P 0 0 andP r 0 . Then, the intensity measure ν 0 is the unique σ-finite measure (up to a multiplicative constant that does not affect the law of Z) such that ν 0 is finite on the set of excursions falling below −ε and ν 0 ε equals ν 0 conditioned on falling below −ε. Indeed, the measure ν 0 can be defined by .
We construct ν r in the same manner, based on the probability measures ν r ε giving us the law of successive excursions of X • C 0 below zero of depth below −ε under P r 0 . Hence, to prove that ν 0 = ν r , it suffices to see that ν 0 ε = ν r ε . Let us describe the left-hand side: under P 0 , consider the process X • θ Tε until it becomes positive, after which we send it to ∞; this law equals the image P −ε by killing upon becoming positive by lack of positive jumps and this is exactly the measure ν 0 ε • θ −1 Tε . However, the law ν r ε • θ Tε admits the same description. Indeed, it suffices to note that if we define the stopping time T 0 ε = inf t ≥ 0 : X • C 0 t = −ε for the process X •C 0 and note that the process X(C 0 T 0 ε +· ) is equal to X Tε+t until the latter becomes positive. Hence ν 0 = ν r and so P 0 0 equals the law of X • C 0 under P r 0 . Note that the proof remains valid even in the subcritical case where there are excursions of infinite length.
As mentioned in [Lam08], Corollary 8 and Kolmogorov's consistency theorem imply the existence of a doubly indexed process X r t such that X r has law P r x for every r ≥ x and which is consistent under time-change in the following pathwise sense: X r = X r • C r ,r for r < r where C r ,r is the inverse of A r ,r and A r ,r t = t 0 1 X r s ≤r ds.
We now give a representation of this doubly indexed process by time-changing independent copies of a Lévy process. Let X 1 , X 2 , . . . be independent copies of X and define: T 1 = inf t ≥ 0 : x + X 1 t = 0 , T n = inf {t ≥ 0 : r + X n t = 0} for n ≥ 2 as well as Let C n be the right continuous inverse of A n for any n ≥ 1. Consider the process which, by construction, is consistent under time-change.
Proposition 9. The process Y r has law Q r x . Proof. Note that if X is (sub)critical, then C 1 ∞ = ∞ since T < ∞; hence, we only need X 1 in the definition of Y r . But then, Y r coincides with X • C r under Q r x , so that the result is valid in this case.
It remains to consider the supercritical case. Also, before Y r hits r it has the same law as the Lévy process until it exceeds r, so that it suffices to consider the case x = r.
IfỸ r admits the same construction as Y r but without killing upon reaching zero (that is, we useÃ n t = t 0 1 r+X n s ≤r ds instead of A n ), we now prove thatỸ r has law P r r , which proves the stated result.
The processỸ is strongly Markovian and, using the proof of Proposition 7, one sees that t 0 1 Y r s =r ds = 0 almost surely.
Also, for each ε > 0, the successive excursions of Y r reaching r − ε are also the excursions of some X i (below r and reaching r − ε). After they reach r − ε their law is the image of P r−ε by killing upon reaching (r, ∞). Therefore, the intensity of excursions of Y r below r is ν 0 . By the arguments of the proof of Proposition 7, we see that Y r has law P r r . 4.2. Poissonian construction of reflected Lévy processes. We now present a Poissonian construction of reflected Lévy processes which will be useful in establishing Theorem 2 in the locally compact case.
Let Ψ be the Laplace exponent of a spectrally positive Lévy process which is not a subordinator and let Φ be its right-continuous inverse. Let d be the drift of Φ, so that λ As before, we let P r x be the law of a Ψ-Lévy process reflected under r > 0 and started at x ≤ r. Suppose for the moment that Ψ is subcritical or critical (that is, Ψ (0+) ≥ 0). Then we can perform the following Poissonian construction of P r x : let ν be the excursion measure of X − X t under P 0 and define ν r as the push-forward of ν by the truncation operator above r (which sends f to f • C r (f )). Let be a Poisson random measure on [0, ∞) × E with intensity Leb(ds) ⊗ ν r−x+s (df ) and define Proposition 10. For a (sub)critical Laplace exponent Ψ, the law of W is P r x . Proof. It suffices to consider the case x = 0 because the push-forward of P r−x 0 by the operator which adds x to a given function is P r x . Recall that under P 0 , −X is the local time of X − X at zero and its right-continuous inverse, say S, is a subordinator with Laplace exponent Φ, as shown in [Ber96,Ch. VII]. Also, the point process of excursions of X − X above zero under P, say is a Poisson point process on [0, ∞) × E with intensity Leb(ds) ⊗ ν(df ). Finally, since excursion intervals become jumps of S and S is a subordinator with drift d, X and S can be recovered from Ξ and d by means of the formulae Since the law of X • C r under P 0 is P r 0 , we only need to see that the excursion process of X • C r has the same law as Ξ r and that X • C r can be written in terms of its point process of excursions as in (5).
An excursion of X • C r above its cumulative minimum, when the latter equals s is an excursion of X − X reflected below level r + s by time-change. Hence the point process of excursions of X • C r above its cumulative minimum is which is a Poisson point process with intensity Leb(ds) ⊗ ν r+s (df ).
Also, the cumulative minimum process of X•C r is obtained from X by erasing the time projection of excursions above the reflection threshold, which means erasing parts of jumps of S. Hence, the process S r = (−X • C r ) −1 is given by Finally, X • C r minus its cumulative minimum can be obtained by concatenating excursions, so that indeed X • C r can be reconstructed from its excursion process as in (5).
When Ψ is supercritical, let b > 0 be the largest root of the convex function Ψ and Ψ # (λ) = Ψ(λ + b). Recall that Ψ # is the Laplace exponent of the (subcritical) spectrally positive Lévy process with Laplace exponent Ψ conditioned to reach arbitrarily low levels in the sense that its law equals lim on every F s . (Cf. Lemma 7 in [Ber96, Ch. VII] and Lemme 1 in [Ber91].) Let us use the notation P x and P # x to distinguish both Lévy processes. Let H a denote the hitting time of −a, defined by H a = inf {t ≥ 0 : X t ≤ −a}. Recall that conditionally on H a < ∞, the laws of X stopped at H a coincide under P and P # . We deduce that, conditionally on H a < ∞, the point process of excursions of X above its cumulative minimum that start at levels deeper than −a under P is a Poisson point process with the same intensity as under P # (let us call it ν # ). Furthermore, −X ∞ under P has an exponential law with rate b and the post minimum process is independent of the pre-minimum process (cf. Theorem 25 in [Don07, Ch. 8, p.85]). The law of the former had been denoted P → . With these preliminaries, we can give a construction of the excursion process of X above its cumulative minimum X under P and P r . Let Ξ = n δ (sn,fn) be a Poisson point process with intensity ν # + bP → , interpreting the law P → as that of an excursion of infinite length. Construct the processes Proposition 11. If Ψ is supercritical, the law of W is P x .
An analogous Poissonian construction is valid in the reflected case. To construct the excursion measure, let V r stand for the operator that removes (by time change) the trajectory above level r and let ν r be the law of the concatenation of P → • (V r ) −1 followed by Q r r (which is the law of the Lévy process started at r, reflected below r, and killed (or sent to †) upon reaching zero. Consider also the push-forward ν #,r of ν # by truncation at level r. Finally, let Ξ r = n δ (sn,fn) be a Poisson point process with intensity Leb(ds) ⊗ ν #,r−x+s + bν r−x+s . As before, consider Proposition 12. If Ψ is supercritical, the law of W r is P r x . Proof sketch. Again, it suffices to consider the case x = 0. The arguments are very close to those of Proposition 10. The differences stem from the fact that in the supercritical case, under P 0 , there are two types of excursions: those corresponding to the associated subcritical Lévy process with exponent Ψ # , and the unique excursion of infinite length (which appears at rate b for the cumulative minimum process). Appealing to the time-change construction of P r 0 in the proof of Proposition 9 we also see the two type of excursions. The first type of excursions come from the associated subcritical process, which are then affected by a time-change to remain below level r − x + s if the cumulative minimum takes the value −s. On the other hand, the excursion of infinite length under P 0 will also be affected by a time-change to remain below r − x + s, except that after death time, we must concatenate a process with law P r r which is killed upon reaching −s. These second types of excursions arrive at rate b.

Measures on locally compact TOM trees with the splitting property
In this section, we will establish the relationship between Lévy processes and measures on locally compact TOM trees stated in Theorem 2. Part of the proof of this theorem is similar to the proof in the compact case of Theorem 2 which we will follow closely, highlighting the differences.
We first use excursion theory for Lévy processes as in the compact case to construct the measure η Ψ of Theorem 2 in the non-compact case and to prove that it has the splitting property.
Proof of Theorem 2 in the non-compact case: construction of η Ψ and proof of its splitting property. Let Ψ be a supercritical Laplace exponent. Recall the definition of the truncated excursion measure ν r associated to Ψ given in Section 1 before the definition of the splitting property.
From Proposition 3, the local absolute continuity of P Ψ and P # , and the fact that Y r is a concatenation of time-changed trajectories of P Ψ 0 we see that the canonical process codes a compact TOM tree ν r -almost surely. Let η r be the image measure of ν r under the mapping sending functions to TOM trees. Thanks to Corollary 8, we see that the measures (ν r , r ≥ 0) are consistent under time-change. Hence, Corollary 6 implies the existence of a measure η Ψ on locally compact TOM trees such that the image of η Ψ by truncation at level r equals η r . By the Markov property under ν r , we see that on the set ζ > t and conditionally on X t = x, the post-t process has law Q r x . By the Poissonian construction of P r x of Proposition 12, we see that the point process of excursions above the cumulative minimum of the post-t process is a Poisson random measure with intensity Leb(ds) ⊗ ν r−x+s (df ). We conclude that η Ψ has the splitting property.
We now consider a measure η on locally compact TOM trees which is not concentrated on compact ones. Our aim is to construct the Laplace exponent of a supercritical spectrally positive Lévy process such that η = η Ψ . In the compact case, we could construct the corresponding Lévy process directly, while in the locally compact case, we will construct first the Lévy process reflected at a given level r and obtain the Lévy process in the limit as r → ∞.
Proof of Theorem 2 in the non-compact case, Construction of a Laplace exponent such that η = η Ψ . Let η r be the push-forward of η by truncation at level r and let n r the law of the contour process under η r . Let be a Poisson random measure on [0, ∞) × E with intensity Leb(ds) ⊗ η r+s (the definition of the intensity was simpler in the compact case since truncation was not necessary). If a is the sojourn of η, define S l = al + We claim that Y r is a reflected Lévy process. To prove it, however, we need to let r → ∞ to see that we get a limit which is a Lévy process. However, truncation allows us to use the splitting property in a way that is parallel to the compact case. Note that the measure η r satisfies an r-dependent version of the splitting property: under η r and conditionally on X t = x, Ξ t is a Poisson random measure on [0, x] × C c with intensity Leb(ds) ⊗ η r−s (df ). This is an important difference with the compact case. For example, if we denote by Y r,x the process obtained by killing Y r when it reaches level −x, then (with the same proof as in the compact case) we see that the two processes are independent and have the same laws as Y r,x and Y r+x,y respectively. Denote the law of x + Y r−x,x by Q r x . Just as in the compact case, use of the splitting property implies that both n r and Q r x are Markovian: n r (g 1 (X s 1 ) · · · g n (X sn ) F • θ sn ) = n r g 1 (X s 1 ) · · · g n (X sn ) Q r Xs n (F ) and Q r x (g 1 (X s 1 ) · · · g n (X sn ) F • θ sn ) = Q r x g 1 (X s 1 ) · · · g n (X sn ) Q r Xs n (F ) . Let P r x be the law of x + Y r−x (defined for x ≤ r). The above Markov property for Q r x implies that P r x is also Markovian. Using Proposition 4 we see that if r < r then the image of n r under truncation at level r equals n r . Hence, the image of P r x under truncation at level r equals P r x . If H + r equals the hitting time of [r, ∞) by the canonical process we then see that for any A ∈ F t P r x A, H + r > t = P r x A, H + r > t which implies, through a projective limit theorem such as Theorem 3.
This law is then easily seen to be subMarkovian. To see that P 0 is the law of a Lévy process, note that P r x (F (y + X)) = P r−y x+y (F (X)) . If F is F t -measurable and bounded and vanishes if H + r ≤ t, we can take limits as r → ∞ to obtain E x (F (y + X)) = E x+y (F (X)) which implies that the family of laws (P x , x ∈ R) is spatially homogeneous and so P 0 is the law of a (possibly killed) Lévy process. The Lévy process has to be spectrally positive since contour processes have only positive jumps. Let Ψ be its Laplace exponent. We now assert that P r x has the law of a Lévy process with Laplace exponent Ψ reflected below r, following the proof of Proposition 7. Indeed, recall that under P r x the canonical process is Markov. Also, if we stop the canonical process when it surpasses r, this stopped process has the same law under P r x and under P x by construction. Finally, note that under P x , the level set at r {t ≥ 0 : X t = r} has Lebesgue measure zero and by truncating at r > r and time changing, we see that the same holds under P r x . Hence, P r x coincides with the reflection of P x at level r. We conclude that η = η Ψ . Note that Ψ is then supercritical since otherwise η Ψ would be concentrated on compact TOM trees and η was assumed to charge non-compact trees.
Appendix A. Preliminaries on TOM trees and a Proof of Theorem 1 The objective of this section is to collect some technical results on TOM trees in Subsection A.1 which will enable us to construct a coding function for them in Subsection A.2 and to prove Theorem 1.
A.1. Preliminary results. We start with an analysis of the greatest common ancestor operator denoted ∧ in Section 1 (page 3). Recall that σ 1 ∧ σ 2 is characterized by Since on the interval [σ 1 , σ 2 ] the distance d coincides with the usual distance on an interval, for every σ ∈ [σ 1 , σ 2 ] we have: As usual, the open ball of radius ε centered at σ ∈ τ is denoted B ε (σ).
We then note that ifσ 1 ,σ 2 ∈ B ε (σ) then, by Equation (7), We tacitly assumed the sets involved in Mes1 of the definition of a TOM tree are measurable. They are actually closed as we now show. For any σ ∈ τ , define the left, strict left, right, and strict right of σ as follows: Lemma 14. For any σ ∈ τ , the strict left of σ is open.
We now give a simple sufficient condition for a sequence to converge.
Lemma 17. The set L generates the Borel subsets of τ .
Proof. Note that B δ (σ) = {σ : d(σ, σ) ≤ δ} and denote the latter set by C. If suffices to show that C ∈ A (for every σ ∈ τ and δ > 0). Since τ \ C is open, it is the union of a countable set of open components, say C 1 , C 2 , . . .. Note that real trees are locally path connected (one way to see this is because closed balls of a real tree are real trees themselves and another one is to prove directly that if σ 1 , σ 2 ∈ B δ (σ) then [σ 1 , σ 2 ] ⊂ B δ (σ)). Hence the notions of path-wise connectedness and connectedness coincide for real trees. We assert that if σ 1 , σ 2 ∈ C i and σ 1 ≤σ ≤ σ 2 thenσ ∈ C i . Indeed, note first that [σ 1 , σ 2 ] ⊂ C i : this follows since [σ 1 , σ 2 ] ⊂ τ \C (or σ 1 and σ 2 would belong to different path-connected components of τ \ C) and then since [σ 1 , σ 2 ] is connected and has non-empty intersection with C i , it follows that C i ∪ [σ 1 , σ 2 ] is connected and by definition of connected component, If I i ∈ L i but S i ∈ L i , consider a decreasing sequence σ n such that µ L σ n → I i and note that in this case Remaining cases are handled similarly, which shows that which terminates the proof.
Hence, any measure on the Borel sets of τ is determined by its values on L .
A.2. The contour process of a compact TOM tree. The reason for introducing compact TOM trees is that they provide us with a simple way to code their elements by real numbers. This is formally done through exploration process and the height function now defined. First, define ψ : τ → [0, µ(τ )] by ψ(σ) = µ(L σ ) , where we recall that L σ is the left of σ. Note that ψ is strictly increasing thanks to Mes.
Remark. If a tree does not have a first element, then 0 ∈ D; an actual example can be seen in Figure 2.
First, notice that for any σ ∈ G t we have the inclusion L σ ⊂ G t , so that G t is necessarily of the form L σ or L σ \ {σ}, which yields µ(G t ) = s t . Now, by definition of i t , there is some ≤-decreasing sequence (σ n ) of elements of D t such that µ(σ n ) ↓ i t . Since (σ n ) is decreasing, the sequence L σn is also decreasing; let L denote its limit. If there were two elements in L \ G t , say σ 1 < σ 2 , we would have t ≤ ψ(σ 1 ) < ψ(σ 2 ) ≤ i t by Mes1. Now this contradicts the definition of i t since L \ G t ⊂ D t , so L \ G t contains at most one element. By Mes2, this shows that µ(L) = µ(G t ). Now recall that µ(G t ) = s t , so that which shows that i t = s t = t.
We can now define the exploration process φ : [0, µ(τ )] → τ by means of notice that ψ −1 (s) is ≤-decreasing if s decreases. Therefore, if s n ↓ t along D, then ψ −1 (s n ) converges thanks to Lemma 16. To see that the limit does not depend on the subsequence, notice that if s n ands n both decrease to t along D, they can be intertwined into a decreasing sequence which contains them both as subsequences.
We now prove that φ is right-continuous. Let t < µ(τ ) and t n ↓ t with t n > t. Then there exists t n ∈ D such that t < t n <t n ,t n − t n → 0 and d φ(t n ) , ψ −1 t n → 0.
By monotonicity of ψ −1 on D, we see that A similar argument shows that φ has left limits: note first that by monotonicity the limit lim s→t− s∈D ψ −1 (s) exists. If t n ↑ t with t n < t, there existt n ∈ D such that t n <t n < t,t n − t n → 0 and d φ(t n ) , ψ −1 t n → 0.
By monotonicity we see that lim Uniqueness of the extension follows from density of D.
We finally prove that Let us then consider t 1 < t < µ L φ(t 1 ) .
We now define the contour process f : Proposition 20. The contour process f is càdlàg with no negative jumps and the compact TOM tree coded by f is isomorphic to c.
In the second case, for any σ ∈ L φ(t) , we have the bound Let us now analyze the tree coded by f . Recall that if we define the pseudo-distance then the tree coded by f is the quotient space of [0, µ(τ )] under the equivalence relation ∼ f given by t 1 ∼ f t 2 if d f (t 1 , t 2 ) = 0. We first prove that φ is constant on the equivalence classes of ∼ f . Indeed, consider t 1 < t 2 such that φ(t 1 ) = φ(t 2 ) and let us prove that t 1 ∼ f t 2 . We have 3 cases: φ(t 1 ) ≺ φ(t 2 ): Then f (t 1 ) < f (t 2 ) and so t 1 ∼ f t 2 . φ(t 2 ) ≺ φ(t 1 ): Then f (t 2 ) < f (t 1 ) and so t 1 ∼ f t 2 .
We can finally turn to the proof of Theorem 1.
Proof of Theorem 1. Existence of a coding function for the tree follows from Proposition 20.
However, if two trees c 1 and c 2 have the same coding function f , then Proposition 20 also implies that c 1 and c 2 are isomorphic to c f , so that they are isomorphic to each other.
A.3. Elementary operations on compact TOM trees. Consider a compact TOM tree c = ((τ, d, ρ) , ≤, µ). The objective of this section is to consider two operations on c which allow one to construct another TOM tree and we explore the relationships between the contour processes. Consider any element σ of τ .
Definition. We define the subtree to the right of σ, denoted c ≥σ , equal to R σ with the same root as c, and with the distance, order and measure obtained by restricting those of c to R σ .
Proposition 21. For any compact TOM tree c and any σ ∈ τ , c ≥σ and c σ are compact TOM trees.
The  We have the explicit formula Proof. Note that ρ ∈ R σ . We have seen in Lemma 14 that R σ is closed. If σ ≤ σ 1 ≤ σ 2 then [σ 1 , σ 2 ] ⊂ R σ , so that R σ is a closed and pathwise connected subset of a compact real tree. Hence R σ , with the induced distance is a compact real tree. The total order, when restricted to R σ , satisfies properties Or1 and Or2. Restriction to R σ does not alter Mes1 or Mes2. This implies that c ≥σ is a compact TOM tree. To find its contour process f ≥σ , note that if σ ∈ R σ then µ(L σ ∩ R σ ) = µ(L σ ) − µ(L σ ) = µ(L σ ) − s * .
It remains to prove that if φ(s) = σ (or in other words, s is a representative of σ when considering c as the tree coded by f ) then µ(L σ ) = inf {t ≥ s : f (t) < f (s)} .
A.4. Topological remarks about the space of compact TOM trees. Given that compact TOM trees can be identified with a subset of càdlàg functions, thanks to Theorem 1, we see that TOM trees constitute a set (without the need of passing to isometry classes). Also, we can define the distance between compact TOM trees c 1 and c 2 in terms of the distance of their contours f 1 and f 2 as follows. Suppose that the supports of f i is [0, m i ] and that m 1 < m 2 , say. and λ runs over all strictly increasing continuous functions of [0, m 2 ] into itself. By uniqueness of the contour in Theorem 1, we see that d is a metric on compact TOM trees. If c n = ((τ n , d n , ρ n ) , ≤ n , µ n ) is a sequence of compact TOM trees converging to c under the metric d, then, a slight generalization of Proposition 2.10 of [ADH14] (to cover càdlàg functions as done in [Lam15]) shows us that the rooted measured metric spaces ((τ n , d n , ρ n ) , µ n ) converge in the Gromov-Hausdorff-Prokhorov topology associated to the metric d GHP defined as follows: d GHP (((τ 1 , d 1 , ρ 1 ) , µ 1 ) , ((τ n , d n , ρ n ) , µ n )) = inf φ 1 ,φ 2 ,τ d τ H (φ 1 (τ 1 ) , φ 2 (τ 2 )) + d τ (φ 1 (ρ 1 ) , φ 2 (ρ 2 )) + d τ P µ 1 • φ −1 1 , µ 2 • φ −1 2 where the infimum runs over all isometric embeddings φ i from τ i into a common metric space τ , d τ H denotes the Hausdorff distance between subsets of τ , and d τ P stands for the Prokhorov distance between finite measures on τ .