Rigid representations of the multiplicative coalescent with linear deletion

We introduce the multiplicative coalescent with linear deletion, a continuous-time Markov process describing the evolution of a collection of blocks. Any two blocks of sizes $x$ and $y$ merge at rate $xy$, and any block of size $x$ is deleted with rate $\lambda x$ (where $\lambda\geq 0$ is a fixed parameter). This process arises for example in connection with a variety of random-graph models which exhibit self-organised criticality. We focus on results describing states of the process in terms of collections of excursion lengths of random functions. For the case $\lambda=0$ (the coalescent without deletion) we revisit and generalise previous works by authors including Aldous, Limic, Armendariz, Uribe Bravo, and Broutin and Marckert, in which the coalescence is related to a"tilt"of a random function, which increases with time; for $\lambda>0$ we find a novel representation in which this tilt is complemented by a"shift"mechanism which produces the deletion of blocks. We describe and illustrate other representations which, like the tilt-and-shift representation, are"rigid", in the sense that the coalescent process is constructed as a projection of some process which has all of its randomness in its initial state. We explain some applications of these constructions to models including mean-field forest-fire and frozen-percolation processes.

1. Introduction 1.1. The multiplicative coalescent and MCLD(λ). The multiplicative coalescent (or briefly MC) is a continuous-time Markov process describing the evolution of a collection of blocks (components). The dynamics are as follows: for each pair of components with masses m i and m j , the pair coalesces at rate m i m j to form a single component of mass The process is simple to construct from any initial state with finitely many components. In [2, Section 1.5], Aldous uses a graphical construction to show that the process is welldefined starting from any initial state in which the sum of the squares of the masses is finite. Writing the masses in decreasing order, define the space Then Proposition 5 of [2] says that the MC is a Feller process in ( ↓ 2 , d).
We generalise the multiplicative coalescent to include deletion as well as coalescence. Let λ ≥ 0. Now we want the dynamics of the process to satisfy: (i) any two components of mass m i and m j merge with rate m i m j , (ii) any component of mass m i is deleted with rate λm i . (1.3) We call such a process a multiplicative coalescent with linear deletion, with deletion rate λ, and denote it by MCLD(λ).
Again, if the initial state has finitely many components, the process is easily seen to be well-defined. For initial states in ↓ 2 , a graphical construction is given in [30], which gives rise to a well-behaved continuous-time Markov process taking values in ↓ 2 . This process has the Feller property with respect to the distance d(·, ·) -see Theorem 1.2 of [30].
The motivation of our study of the MCLD process is twofold. Firstly, ideas involving the representation of the state of a MC process in terms of the set of excursion lengths of some random function began with Aldous [2], and have been developed in a series of works including those of Aldous and Limic [5], Armendáriz [8], Uribe Bravo [35], Broutin and Marckert [18] and Limic [23]. We find that this theory has a surprising and elegant extension to processes involving deletion, which we build in this paper.
Secondly, the MCLD arises as a scaling limit of certain discrete processes of coalescence and fragmentation or deletion, such as the mean-field forest-fire model introduced by Ráth and Tóth in [32] and studied by Crane, Freeman and Tóth in [19], and the mean-field frozen percolation process introduced by Ráth in [31]. These processes are of particular interest because of their self-organised criticality properties. These scaling limits are the subject of a future paper [24], but we discuss related properties here in Section 6.
We further discuss related literature in Section 1.4.

1.2.
Tilt representations of MC. We start by revisiting previous results giving representations of the MC in terms of excursions of random functions.
We first define Brownian motion with parabolic drift. For a function h(x), x ≥ 0, let E ↓ (h) denote the lengths of the excursions of h above its running minimum, written in non-increasing order (see Section 2 below for a formal definition). Aldous [2] showed that there is an eternal multiplicative coalescent process, whose marginal distributions can be written in terms of excursions of a Brownian motion with parabolic drift.
We call this representation a tilt representation; for each t, the function h t is obtained from h 0 by adding the linear function tx. It is easy to see that the tilt representation indeed produces a coalescent process as t increases. Also note that only adjacent excursions can coalesce under the tilt representation.
Part of the approach of Armendáriz (described in [8,Section 4]) is further elaborated in Chapter 4 of the PhD thesis [35] of Uribe Bravo. In [18,Corollary 4], Broutin and Marckert give an alternative proof of Proposition 1.3. Their proof involves considering the connected components of an Erdős-Rényi graph process on n vertices, and then taking the weak limit as n → ∞. The key idea is to define an ordering of the vertices (the Prim ordering, related to invasion percolation and to the minimal spanning tree) which is consistent with the coalescent process in the sense that at all times, the components are intervals of the Prim order (and thus only adjacent intervals can coalesce); nonetheless, for each fixed time, exploring the graph in the Prim ordering yields a random walk with the same distribution as is obtained from a standard traversal in, say, depth-first or breadth-first order.
Aldous and Limic [5] characterized the set of all eternal multiplicative coalescents, i.e. those defined for all times t ∈ (−∞, ∞). They defined a three-parameter set of random processes W κ,τ,c (x), x ≥ 0, such that if (m t , t ∈ R) is a non-constant eternal MC which is extremal (that is, its law is not the mixture of the laws of other eternal MC processes) then for each t ∈ R, the marginal distribution of m t is the same as that of the sequence E ↓ (h t ) of excursion lengths of h t (x) := W κ,τ,c (x) + tx for some (κ, τ, c). The processes W κ,τ,c have been called Lévy processes without replacement; see Section 6.1 for their definition and the precise statement of the cited result of [5].
1.3. Contributions of this paper. In this section we summarize the main results of our paper. The central results are the following: • MC admits a tilt representation from any initial state: In Theorem 2.8 we show that for any possible initial state m ∈ ↓ 2 , there exists a random function f 0 : [0, ∞) → [−∞, 0] such that if we define f t (x) = f 0 (x) + tx (1.5) then E ↓ (f t ), t ∈ R is a multiplicative coalescent process with initial state m. • MCLD admits a "tilt-and-shift" representation: Let m ∈ ↓ 2 , and define f t for all t ≥ 0 as in (1.5). Let λ > 0. In Theorem 2.13 we show that there exists a σ(f 0 )-measurable non-decreasing function Φ : [0, ∞) → [0, ∞) such that if we define then E ↓ (g t ), t ≥ 0 is a MCLD(λ) process with initial state m. Here Φ is a non-decreasing pure jump process; Φ(t) is the total weight of components deleted by time t. In addition to the "tilt" given by (1.5), we now have a "shift" since in (1.6), the function f t is applied at x + Φ(t). If a component of size a is deleted, then Φ(t) − Φ(t−) = a, and we see a shift to the left of size a at time t; that is, the graph of g t− on [a, ∞] becomes the graph of g t on [0, ∞].
Note that the tilt representation of the MC is the λ = 0 case of the tilt-and-shift representation of the MCLD. Under the tilt-and-shift representation, only adjacent excursions coalesce and only the "leftmost" excursion gets deleted. We call the above representations "rigid" because all of the randomness is contained in the initial state f 0 , and the rest of the evolution of f t (resp. g t ) is deterministic.
The function f 0 above is constructed so that it has the exponential excursion levels property: conditional on the sequence of excursion lengths, the levels of the excursions are independent, and an excursion of length m occurs at level −E where E ∼ Exp(m). One of the central observations behind the rigid representation results outlined above is that f t and g t for any t ≥ 0 inherit the exponential excursion levels property.
We prove these main results in two stages: we first consider the case of finitely many blocks in Section 3, then we extend the tilt-and-shift representation to ↓ 2 using truncation and approximation arguments in Section 5.
The proof of the finite tilt-and-shift representation result in Section 3 involves various other representations which are also of independent interest. Their constructions rely heavily on the notion of size-biased orderings, and on the realisation of such orderings using independent exponential random variables, c.f. Section 3.1. Now we outline these representations and how they relate to each other.
• Interval coalescent representation: Given an initial state m with finitely many blocks, we consider a process whose states are finite sequences of lengths (depicted as intervals arranged in some order). In the initial state, the interval lengths are the masses of blocks of m, and they are arranged in a size-biased random order. An interval I merges with the interval on its right at a rate equal to the product of the length of I with the combined length of all of the intervals to the right of I. The leftmost interval is deleted at a rate equal to λ times the total length of intervals. For an illustration, see  Figure 1: An illustration of the rates of the interval coalescent process: the lengths of the three intervals (from left to right) are b 1 , b 2 and b 3 . The leftmost block marked with an arrow represents the "cemetery". The rate at which the first interval gets deleted is λ · (b 1 + b 2 + b 3 ). The rate at which the first interval merges with the second one is b 1 · (b 2 + b 3 ). The rate at which the second and third intervals merge is b 2 · b 3 .
1 Figure 1.1. An illustration of the rates of the interval coalescent process: the lengths of the three intervals (from left to right) are b 1 , b 2 and b 3 . The leftmost block marked with a lightning represents the "cemetery". The rate at which the first interval gets deleted is . The rate at which the first interval merges with the second one is b 1 · (b 2 + b 3 ). The rate at which the second and third intervals merge is b 2 · b 3 .
• Particle representation: Given an initial state m = (m 1 , . . . , m n ) as above, we initially put a particle of mass m i at initial height −E i , where E i ∼ Exp(m i ), 1 ≤ i ≤ n are independent. Now the particles start to move up until they reach height 0 and die. The speed of a particle is equal to λ plus the total weight of particles strictly above it and strictly below 0. Note that once a particle reaches the one above it, they stick together until they die. Accordingly, we group the particles that share the same height into time-t blocks. In Section 3.3 we show that the the vector of sizes of the time-t blocks of the particle system, in decreasing order of their height, evolves like the above described interval coalescent process. See  A simulation of the particle system which realises the MCLD. Each particle moves upwards at rate equal to λ plus the mass of particles strictly above it. When a particle reaches 0 it dies and is removed from the system. Particles at the same height at time t form a time-t block. The figure shows a system with n = 20 particles and λ = 1.3, evolving on the time interval [0, 0.5]. We see 5 deletion events, and the deleted blocks have sizes 1, 1, 14, 1, 3 respectively. Compare to Figure 1.3, where λ = 0 (realising the multiplicative coalescent without deletion), and to Figure 6.1, where deleted particles reenter at independent exponential heights (realising the forest fire model).
Note that the particle representation outlined above is also "rigid". In Section 3.4 we explain how particles in the particle representation correspond to excursions of g t in the tilt-and-shift representation, completing the proof of the validity of the tilt-and-shift representation of MCLD(λ) with finitely many blocks.
In Section 6 we give some applications of our rigid representation results: • In Section 6.1 we give a sketch proof of the fact that the Lévy processes without replacement W κ,τ,c have the exponential excursion levels property, and conclude that all eternal multiplicative coalescents have a tilt representation. This gives an alternative way to approach the main result of Limic [23] (see Section 1.4). • In Section 6.2 we apply the tilt-and-shift representation to Brownian motion with parabolic drift. What we find can be non-rigorously summarized as follows: if we start from BMPD(u) then at time t ≥ 0 we see BMPD(u + t − Φ(t)). We show that the resulting MCLD(λ) process is the scaling limit of the list of component sizes in the mean field frozen percolation model of [31] started from a near-critical Erdős-Rényi graph, thus extending the result [2, Corollary 24], which is the λ = 0 case. • In Section 6.3 we give a particle representation of the mean field forest fire model of [32]. We demonstrate how this representation allows us to give a new probabilistic interpretation of the non-linear controlled PDE problem (and the associated characteristic curves) which played a central role in the theory developed in [32] and [19].
1.4.1. Rigid representations of coalescent processes. In this subsection we discuss links to previous work on rigid representations for the multiplicative coalescent, which is somewhat scattered (and in some cases unpublished). The various approaches and methods of Armendáriz [8], Uribe Bravo [35], Broutin and Marckert [18] and Limic [23] resulting in versions of the "tilt" representation are all related to ours, although these similarities are often implicit. Let us now try to sketch some of these connections. In [8] Armendáriz draws attention to the fact that Brownian motion with parabolic drift has the exponential excursion levels property which we informally described in Section 1.3. This observation had been made by Aldous in [2, equation (72)] (see also Bertoin [9]). Based on this, Armendáriz constructs a representation of the MC with finitely many blocks in [8,Section 4] -this representation appears to be equivalent to the λ = 0 case of our particle representation (see Section 3.3), although the methods are different.
In Chapter 4.2 of his PhD thesis [35], Uribe Bravo recalls and further elaborates the representation of [8, Section 4] (without proofs). In particular, he points out that (what we call) the initial heights of particles are in size-biased order and formulates the crucial property corresponding to our (3.16) about the height gaps of particles. Again, the representation outlined in [35,Chapter 4.2] is equivalent to the λ = 0 case of our particle representation; moreover the representation stated (without proof) in [35,Chapter 4.3] is equivalent to our tilt representation of MC (c.f. Section 2.1) in the case of finitely many blocks. In [35,Chapter 4.4] Uribe Bravo gives a sketch proof of Proposition 1.2 which is quite similar to our short sketch proof of Proposition 1.3 given in Section 6.1.1.
At first, the approach of Broutin and Marckert in [18] seems quite different from ours. Their construction relies on exploration processes for random graphs, defined using the Prim ordering for a graph with edge-weights; our methods make no explicit reference to the graph structure underlying the MC (or MCLD). However, one can observe that the components of the dynamic Erdős-Rényi graph process, arranged in Prim order, give the interval coalescent process (see Section 3.2) without deletion, started from an initial state with equal-sized blocks. Broutin and Marckert also extend the representation to obtain a construction of the standard augmented multiplicative coalescent of Bhamidi, Budhiraja and Wang [11].
In a paper available in interim form at [23], Limic obtains tilt representations of all eternal multiplicative coalescents. The main result can be written as follows. Let f t (x) = W τ,κ,c (x) + tx, where W τ,κ,c is a Lévy processes without replacement. Then the process E ↓ (f t ), t ∈ R is an eternal multiplicative coalescent. Limic's approach involves the generalisation of the breadth-first walks used in [2] and [5], to give a system of simultaneous breadth-first walks relating to the state of the multiplicative coalescent at different times. Limic recalls from [8,Section 4] and [35,Chapter 4.2] the definition of Uribe's diagram (which is equivalent to the λ = 0 case of our particle representation) and proves (via a calculation involving iterated integration) that this representation is indeed a valid representation of MC. In [23,Section 4] Limic explores the connection between Uribe's diagram and simultaneous breadth-first walks. We describe an alternative approach to the tilt representation of the set of eternal coalescents in Section 6.1. Note that the connection between Prim's algorithm [18] and simultaneous breadth-first walks [23] is the topic of ongoing research [34], c.f. Open question 3 of [23, Section 7].
Finally we mention that also in the case of the standard additive coalescent [6], a tilt representation can be given. Let e(x) be a standard Brownian excursion on [0, 1], and for λ > 0 define the function h λ on [0, 1] by h λ (x) = e(x) − λx. A consequence of the results of Bertoin [9] is that the process E ↓ (h e −t ), t ∈ R is a version of the standard additive coalescent. An alternative proof of this result is given by Broutin and Marckert [18].

1.4.2.
Mean-field graph models of self-organized criticality (SOC). The mean-field frozen percolation model [31] is a dynamic random graph model where the initial number of vertices is n, two connected components of size k and l merge at rate kl n and a connected component of size k disappears at rate λ(n)k. Note that up to scaling, this is MCLD(λ). If 1/n λ(n) 1 then the evolution of the densities of small components is asymptotically described by Smoluchowski's coagulation differential equations with multiplicative kernel [31,Theorem 1.2], the solutions of which exhibit SOC [31,Theorem 1.5].
The definition of the mean-field forest fire model [32] is the same as that of the above described frozen percolation model, with one difference: instead of removing a component, we only remove its edges, with the vertices remaining as singletons. A variant of Smoluchowski's system of equations describes the asymptotic densities of small components [32,Section 1.3], leading to SOC. The proof of the well-posedness of this infinite system of differential equations involves a non-linear PDE which is a controlled variant of the Burgers equation.
In [19] Crane, Freeman and Tóth describe the time evolution of the size of the component of a fixed vertex in the above forest fire model in the n → ∞ limit. They also give a probabilistic meaning to the characteristic curves of the controlled Burgers equation in [19,Remark 3.11]. In Section 6.3.2 we give a different probabilistic meaning to these characteristic curves by comparing them to particle trajectories in the particle representation.
The model proposed by Aldous in [4,Section 5.5] is studied by Merle and Normand in [26]: two connected components of size k and l merge at rate kl n and connected components disappear if their size exceeds a threshold ω(n). In order to achieve SOC, one chooses 1 ω(n) n. Theorem 1.1 of [26] states that the densities of small components converge to the solution of Smoluchowski's coagulation equations with multiplicative kernel. Note that this is a special case of the main result of Fournier and Laurencot in [21], who study discrete models of Smoluchowski's coagulation equation with more general coagulation kernels. Theorem 1.3 of [26] identifies the Benjamini-Schramm limit of the "threshold deletion" graph model as n → ∞.
Note that by comparing [31, Theorem 1.2] and [26, Theorem 1.1] one sees that (in the SOC regime) the densities of small components converge to the same hydrodynamic limit in the model with linear deletion and the model with threshold deletion. However, if one is interested in the scaling limit of big component dynamics, the exact deletion mechanism does crucially enter the picture. We discuss scaling limits of the model with linear deletion in Section 6.2. The scaling limit of the threshold deletion model is not yet known, but in Remark 6.16 we give a particle representation of it.
In [27] Merle and Normand identify the Benjamini-Schramm limit of a different selforganized critical model of aggregation, where vertices can only connect to a fixed number of other vertices.

1.4.3.
Scaling limits of critical random graph models. In [2, Corollary 2] Aldous identifies the scaling limit of component sizes in a near-critical Erdős-Rényi graph in terms of the excursion lengths of BMPD and in [2,Corollary 24] he shows that the standard MC is the scaling limit of the evolution of component sizes of the dynamic Erdős-Rényi graph in the critical window. Let us now list some related results.
The papers [10,15,20,22,29,33] explore the universality class of graph models whose scaling limits are described by BMPD and the standard MC.
The family of eternal multiplicative coalescent processes are characterized in [5]. The scaling limits of some classes of inhomogeneous random graph models are given by Lévy processes without replacement and non-standard eternal MC processes, see [5,16,22].
The continuum scaling limit of the metric structure of critical random graphs is studied in the "BMPD" universality class in [1,12,13] and in the "Lévy processes without replacement" universality class in [14].
Martin and Yeo [25] study the Erdős-Rényi random graph within the critical window, conditioned to be acyclic; alternatively expressed, this is the uniform distribution over forests with a given number of vertices and edges. Analogously to Proposition 1.2, they obtain a scaling limit for the collection of component sizes which is described by the sequence of excursions of an appropriate reflected diffusion; here the drift of the diffusion depends on space as well as on time.
In the future paper [24] we describe the possible scaling limits that can arise from a frozen percolation process with lightning rate λn 1/3 , started from an empty graph. The possible limit objects are eternal MCLD(λ) processes. The "arrival" of the process at the critical window gives rise to a non-stationary eternal MCLD(λ) scaling limit, while the scaling limit in the "self-organized critical" regime is a stationary MCLD(λ).
1.5. Plan of the paper. In Section 2 we give the main results about tilt representations for MC and tilt-and-shift representations for MCLD, and discuss their interpretation.
In Section 3 we prove the tilt-and-shift representation result of MCLD in the case of finitely many blocks. Along the way, we introduce the interval coalescent representation and the particle representation and show how all these representations relate to each other.
In Section 4 we collect some preparatory results about random point measures and excursions that we will use in Section 5.
In Section 5 we extend our rigid representation results to any initial state m ∈ ↓ 2 by approximating m with a sequence of truncated initial states. Since the particle representation is essentially the same as the tilt-and-shift representation, our proof involves a careful analysis of the effect of the insertion of a new particle on the death times of other particles.
In Section 6 we present the applications of the theory of rigid representations that we mentioned in Section 1.3.

Main results
We begin by introducing some notation. Let We will use the topology of coordinate-wise convergence on ↓ ∞ . For m ∈ ↓ 0 with n non-zero entries, we will sometimes ignore the infinite trailing string of zeros and regard m as an element of R n >0 . For m, m ∈ ↓ 2 we define the distance d(m, m ) by (1.2). The metric space ↓ 2 , d(·, ·) is complete and separable.
Next, we introduce some definitions relating to excursions. (i) g(x) > g(l) for all x < l.
We say that r − l is the length of the excursion and g(l) is the level of the excursion. We say that the excursion is strict if g(x) > g(l) for any x ∈ (l, r).
From now on, we say simply "excursion" to mean excursion above the minimum, and we say that l is a "minimum" if l is the left endpoint of an excursion of g.
Definition 2.2. Suppose that for any ε > 0, g has only finitely many excursions with length greater than ε. Then let E ↓ (g) ∈ ↓ ∞ be the sequence of the lengths of the excursions of g, arranged in non-increasing order.
We writeḡ for the function defined bȳ (2.1) Note that if g is a lower semi-continuous càdlàg function, then the excursions of g andḡ have the same lengths and levels; in particular, 2.1. Tilt representation of multiplicative coalescent. The aim of this section is to formulate the "rigid" representation of the MC process from any initial condition in Theorem 2.8.
Note that f µ is non-increasing, lower semi-continuous and càdlàg.
However, as long as m ∈ ↓ 2 , the mass distribution is locally finite: in Lemma 4.1 we will show that almost surely µ(A) < ∞ for every bounded set A ⊆ (−∞, 0]. (2.6) Remark 2.6. Let f 0 be defined by Definition 2.5.
(i) An alternative characterization of the function f 0 is as follows: f 0 is the nonincreasing càdlàg function such that the interval I j on which it takes the value −E j has length m j , moreover the Lebesgue measure of the complement of ∪ ∞ j=1 I j is zero. (ii) If m = (m 1 , . . . , m n ) ∈ ↓ 0 and E σ 1 < · · · < E σn is the increasing rearrangement of E i , 1 ≤ i ≤ n, then an equivalent way to write the function f 0 is For an illustration of (2.7), see We say that the function f t is a "tilt" of f 0 .
In Lemma 4.4 we will show that, with probability 1, for all t the function f t satisfies the criteria of Definition 2.2.
The process E ↓ (f t ), t ≥ 0 has the law of the multiplicative coalescent started from m.
We will prove Theorem 2.8 for m ∈ ↓ 0 in Section 3, and extend this result to m ∈ ↓ 2 in Section 5.
We say that Theorem 2.8 gives a "rigid" representation of the MC process, because all of the randomness is contained in the initial state of the representation and the rest of the dynamics is rigid, i.e., deterministic.

2.2.
Tilt-and-shift representation of MCLD(λ). Similarly to the rigid representation of the MC in terms of the excursion lengths of (f t (·)) in Theorem 2.8, we will give a rigid representation of the MCLD(λ) in terms of the excursion lengths of another function (g t (·)) in Theorem 2.13 below. We begin with the case of finitely many components.
is defined by (2.6) (or, equivalently, (2.7)). We will now define g t (x) for t, x ≥ 0 such that for all x ≥ 0, the function t → g t (x) is càdlàg. The time evolution of g t (·) consists of two parts: (1) Tilt: If g t− (0) < 0 then we let d dt g t (x) = λ + x. (2) Shift: If g t− (0) = 0, then we let g t (x) = g t− (x + x * (t)), where is the length of the first excursion of g t− (·) (see Definition 2.1).
Let us define ν to be the measure on [0, ∞) given by where x * (t) > 0 is the size of the shift to the left at time t (see (2.9)); and if no shift occurred at time t, then we let x * (t) = 0. Let us also define Proposition 2.10. Let m ∈ ↓ 0 , λ > 0 and let f 0 be defined by Definition 2.5. If we define g t (·) by Definition 2.9, then the process E ↓ (g t ), t ≥ 0 has the law of the MCLD(λ) process m t , t ≥ 0 started from m 0 = m.
We will prove Proposition 2.10 in Section 3.
Remark 2.11. In the MCLD(λ) interpretation, Φ(t) corresponds to the total amount of mass deleted up to time t.
x * (t) Figure 1: An illustration of Definition ??. Dashed lines are used to denote the running minimum. The second graph g t− is obtained from the first graph g t ′ using the tilt operation. Observe that some excursions of g t ′ got merged with the tilt operation. The third graph g t is obtained from the second graph by shifting it to the left by x * (t), i.e., the length of the first excursion of the second graph. If we denote by y the level of the first excursion of g t ′ then t = t ′ + |y|/λ is the time instant when this excursion reaches level zero and thus gets shifted.
1 Figure 2.2. An illustration of Definition 2.9. Dashed lines are used to denote the running minimum. The second graph g t− is obtained from the first graph g t using the tilt operation. Observe that some excursions of g t got merged with the tilt operation. The third graph g t is obtained from the second graph by shifting it to the left by x * (t), i.e., the length of the first excursion of the second graph. If we denote by y the level of the first excursion of g t then t = t + |y|/λ is the time instant when this excursion reaches level zero and thus gets shifted.
From Definition 2.9 it follows that we have Remark 2.12. Extending the dynamics of g t for initial states in m ∈ ↓ 2 will amount to finding Φ(·) corresponding to g 0 := f 0 , c.f. Definition 2.5. We will then define g t using the formula (2.12): the question is how to define Φ(·) in a way that will appropriately extend Definition 2.9 from m ∈ ↓ 0 to m ∈ ↓ 2 . The technical issue that we have to overcome is that for a typical t ≥ 0 our functions g t− (·) does not have a "first excursion" (c.f. (2.9)) if m ∈ ↓ 2 \ ↓ 1 . For example, a Brownian motion with parabolic drift does not have a first excursion, yet in Section 6.2 we will apply our tilt-and-shift representation to BMPD(u).
We are ready to state the main result of the paper.
. We will prove Theorem 2.13 in Section 5.
Remark 2.14. (i) Put another way, Theorem 2.13(iii) says that for any fixed t, the random variable Φ(t) := ν[0, t] is a stopping time with respect to the filtration (F + x ) x≥0 . We will make good use of this together with the strong Markov property of Brownian motion in Section 6.2.
(iii) We construct the measure ν that appears in Theorem 2.13 by extending our earlier construction given in Definition 2.9 from ↓ 0 to ↓ 2 in the sense that we obtain ν as the weak limit as n → ∞ of the measures ν (n) corresponding to initial conditions m (n) truncated at index n, see Lemma 5.8 and Corollary 5.9. Lemmas 4.4,5.3). We conjecture that for any such function g 0 , there is a unique measure ν such that the function g t (x) controlled by Φ(t) = ν[0, t] according to (2.12) satisfies The function g 0 is constant on each of its excursions. Suppose that h 0 is another càdlàg and lower semi-continuous function such that g 0 =h 0 , that is, the excursions of h 0 have the same lengths and levels as those of g 0 . Define, analogously to (2.12), so that h t is constructed using the same tilt-and-shift procedure as g t (and using the same control function Φ). Then it is straightforward to see thath t =ḡ t , and so, as at (2.2), E ↓ (g t ) and E ↓ (h t ) are the same process. We will make use of this observation in Section 6.2 when we consider the tilt-and-shift construction started from Brownian motion with a parabolic drift.

Rigid representations: finite state space
In this section we restrict the MCLD(λ) process to the space ↓ 0 of states with only finitely many blocks. The ultimate goal of this section is to prove Proposition 2.10.
In Section 3.1 we recall the notion of a size-biased rearrangement of m ∈ ↓ 0 and how this notion can be extended to ↓ 2 using an appropriate family of independent exponential random variables.
In Section 3.2 we define the interval coalescent with linear deletion (or briefly ICLD(λ)) and show that an ICLD(λ) with a size-biased initial state gives a representation of the MCLD(λ).
In Section 3.3 we define our particle representation and show that if the initial heights of particles form an appropriate family of independent exponential random variables, then we obtain a representation of of the ICLD(λ) with a size-biased initial state.
In Section 3.4 we show that a copy of the particle system is embedded in the tilt-andshift representation introduced in Definition 2.9.
3.1. Construction of size-biased sequences using independent exponential random variables. Now we recall some useful definitions from [2, Section 3.3].
The proof of the next claim follows from the memoryless property of exponential random variables and we omit it. Claim 3.3. The law of ≺ introduced in Definition 3.2 is size-biased (with respect to the sizes m i ), in the sense that for any n ∈ N the restriction of ≺ to [n] is size-biased with respect to (m 1 , m 2 , . . . , m n ), c.f. Definition 3.1.

3.2.
A size-biased interval representation of MCLD(λ). We introduce a related process b t , t ≥ 0 in which only neighbouring blocks are allowed to merge, and only the leftmost block is allowed to be deleted.
Definition 3.5. With some abuse of notation, we denote by 0 = n≥0 R n >0 the space of finite sequences with positive entries.
Definition 3.6 (Interval coalescent with linear deletion, ICLD(λ)). The state space of the continuous-time Markov process (b t ) is 0 . The dynamics consist of coalescence and deletion: then the rate of the transition from b to b is then the rate of this transition is All other rates of the ICLD(λ) are 0.
We are now ready to state the main result of Section 3.2.
We will prove Theorem 3.7 in Section 3.2.1.
Remark 3.8. (i) To the best of our knowledge, even the λ = 0 case of Theorem 3.7 is novel, and it gives an interval coalescent representation of the multiplicative coalescent process (on the state space ↓ 0 ).
(ii) Theorem 3.7 generalizes in a natural way to any initial condition m ∈ ↓ 1 . For initial conditions with infinite total mass, the situation is more complicated since under the natural extension of the concept of size-biased order (see Definition 3.2) there is no smallest element of the order, and any two elements are separated by infinitely many other elements in the order, c.f. Remark 3.4(i).
Lemma 3.9. Let b ∈ 0 and m = R ↓ (b). Then the total jump rate of the MCLD(λ) out of the state m is the same as the total jump rate of the ICLD(λ) out of the state b.
Proof. The total jump rate in the MCLD(λ) is λ m i + i<j m i m j . Since m is a reordering of b, we obtain the same total rate for the ICLD(λ) by adding the deletion rate in (3.4) to the sum over 1 ≤ k < n of the coalescence rate in (3.3). In particular, in (2) it is the case that the conditional distribution of b given R ↓ (b ) is that of a size-biased reordering of R ↓ (b ), so the same is also true in (1).
Before proving Lemma 3.10, we use it to prove Theorem 3.7.
Proof of Theorem 3.7. Assume that the initial state has n non-zero blocks. Then the process b t will make n jumps before being absorbed in the all-0 state. Let τ 0 = 0 and let τ 1 , . . . , τ n be the jump times of the process, so that We claim that for 0 ≤ k ≤ n, the following properties hold: . . , m τ k ) has the distribution of the first k steps of the jump-chain of the MCLD(λ). P2(k): Given m τ 1 , . . . , m τ k , the distribution of b τ k is that of a random size-biased reordering of m τ k .
As soon as we prove these properties, the proof of Theorem 3.7 follows using Lemma 3.9 and the memoryless property.
We now prove P1(k) and P2(k) by induction on k. The case k = 0 is immediate from the definition of b 0 in the statement of Theorem 3.7.
For the induction step, suppose that P1(k) and P2(k) hold for a given k with 0 ≤ k < n. Now we condition on m τ 1 , . . . , m τ k and apply Lemma 3.10 with the choices From P2(k), we know that given m τ 1 , . . . , m τ k , the distribution of b τ k is a random sizebiased reordering of m τ k ; then, as in (1) of Lemma 3.10, we obtain b τ k+1 by taking a step of the ICLD(λ) chain from b τ k .
But also, comparing with (2) of Lemma 3.10, the distribution we obtain of m τ k+1 = R ↓ (b τ k+1 ) is the same as we would have obtained by taking a step of the MCLD(λ) chain from m τ k . So indeed we can extend P1(k) to give P1(k + 1) also.
This completes the induction step. In this way we obtain that P1(n) and P2(n) hold, and this completes the proof of Theorem 3.7.
Proof of Lemma 3.10. To avoid awkward notation, we will write the proof under the extra assumption that m has all its block sizes distinct, and so does m for each m that can be obtained from m by a coalescence step. The general statement can then be easily obtained by continuity.
Under the above stated extra assumption we can write the transition rates of the MCLD(λ) chain as follows. If m is obtained from m by merging the two blocks of size m I and m J , then the rate of the transition from m to m is If m is obtained from m by deleting the block of size m I , then the rate of the transition from m to m is (3.6) Let π m denote the probability distribution on 0 which arises from the size-biased reordering of m. Again if m has distinct block sizes, we can write (3.7) Now, to prove Lemma 3.10 it is enough to demonstrate the following claim: (Rescaled by the total jump rate, which by Lemma 3.9 are the same for the ICLD(λ) and the MCLD(λ), the left side represents the probability of obtaining a given value b using procedure (1) in Lemma 3.10, while the right side represents the same for procedure (2).) To prove (3.8), there are two cases that we have to handle, corresponding to coalescence and deletion.
We first treat the case of coalescence, that is we assume that the state R ↓ (b ) = m arises from m by merging some blocks with sizes m I and m J . Then b has an interval of size m I + m J , say b k = m I + m J . There are exactly two reorderings b of m for which Now let us rewrite the two sides of (3.8). The left side is Using (3.5) and (3.7), the right side can be rewritten as These are the same, so (3.8) holds in the case of coalescence.
We now turn to the case of deletion; that is, we assume that the state R ↓ (b ) = m arises from m by deleting a block of size m I . There is one rearrangement b of m for which (3.9) So (3.8) holds in this case also. This completes the proof of Lemma 3.10.
3.3. Particle representation. The representation of the MCLD(λ) in Section 3.2 moved some of the randomness of the process into the choice of an initial condition (using a sizebiased reordering). Thereafter the possible transitions of the process were restricted (only neighbouring blocks were allowed to merge, and only the first block could be deleted).
In this section we take this to an extreme by giving a natural construction of the process in which all the randomness is in the initial condition; the evolution of the process thereafter is entirely deterministic, but nonetheless the process projects to the MCLD(λ). Such processes might be called "rigid". Definition 3.11 (Particle system). Let m = (m 1 , . . . , m n ) ∈ ↓ 0 . Let E 1 , . . . , E n be independent with E i ∼ Exp(m i ). For i = 1, . . . , n, let Y i (0) = −E i be the initial height of a particle i with mass m i . The heights of particles evolve over time; we describe the joint evolution of the heights Y 1 (t), . . . , Y n (t) using a system of ordinary differential equations.
Analogously to the definition (2.5), we define (3.10) The system of differential equations governing We say that the particle i "dies" at time t i , where t i is defined by A "time-t block" consists of the union of all the particles that share the same (strictly negative) height at time t.
In words, particles start at negative locations and move up. If a particle reaches zero then it stops there and dies. Before it dies, the speed of a particle is equal to λ plus the total weight of particles strictly above it and strictly below zero. Observe that if two particles ever reach the same height, then they stay together forever.
Recall the definition of the ICLD(λ) process from Definition 3.6. Proposition 3.12. Let b t ∈ 0 be the vector of sizes of the time-t blocks of the particle system, in decreasing order of their height. Then the process b t , t ≥ 0 has the law of ICLD(λ), started from an initial state b 0 .
Before proving Proposition 3.12, let us state the following corollary.
Corollary 3.13. The process m t = R ↓ (b t ) has the law of MCLD(λ) started from the state m.
Proof of Corollary 3.13. It follows from Definition 3.11 and Claim 3.3 that b 0 is a sizebiased reordering of m. Now the statement of Corollary 3.13 follows from Theorem 3.7 and Proposition 3.12.
Before we prove Proposition 3.12, let us introduce a useful notation.
and all these gaps are independent.
Proof of Proposition 3.12. Suppose there are n(t) time-t blocks. Let be the vector of their heights in decreasing order. Then µ t , defined at (3.10), is also given by is the decreasing rearrangement of the initial heights of the particles, and Y (i) (0) is the initial height of a particle of mass (b 0 ) i .
From here onwards, let us condition throughout By repeatedly applying the memoryless property of the exponential distribution, we see that where Exp ≺ (·) was introduced in Definition 3.14.
Let us consider the time τ 1 at which the first jump of the process (b t ) occurs. This jump may be either a deletion (if a block reaches height 0 and so dies) or it may be a coalescence (if two blocks reach the same height). Between times 0 and τ 1 , the kth highest Let us therefore define These variables are all independent. Then τ 1 = min 0≤k≤n−1 t (k) , and we have Further, let κ = argmin 0≤k≤n−1 t (k) . This argmin is uniquely defined with probability 1.
If κ = 0, then the event at time τ 1 is a deletion of the highest block, while if κ = k ≥ 1 then the event at time τ 1 is a coalescence of the kth and (k + 1)st highest blocks. Then κ and τ 1 are independent; the probability that κ = k is proportional to λ n i=1 b i for k = 0, and to b k Comparing the previous paragraph with the rates of the ICLD(λ) at (3.3) and (3.4), we see that the distribution of the new state b = b τ 1 , together with the time τ 1 at which it occurs, is the same as for the ICLD(λ) process.
From here on, let us condition further on τ 1 and b τ 1 = b as well as on b 0 = b. Again applying the memoryless property, the conditional distribution of the remaining height gaps just before τ 1 , excluding the one which reaches 0 at that time, is unchanged from what it was at time 0.
The heights and masses of blocks at time τ 1 are given by and for i > κ. (3.20) We therefore obtain that and all these gaps are independent. Thus we have shown that So we can repeat the argument above starting from time τ 1 , until the time τ 2 of the next jump, obtaining that b t continues to evolve as an ICLD(λ) process. Continuing recursively until the time of the nth jump (when the last death occurs and b t becomes identically zero), we obtain the desired result. The proof of Proposition 3.12 is complete.
We will now prove a corollary of the results and methods developed in Sections 3.2 and 3.3. This corollary will only be used in Section 6.
We consider the particle system introduced in Definition 3.11. Let b t ∈ 0 be the vector of sizes of the time-t blocks of the particle system, in decreasing order of their height and let m t = R ↓ (b t ). Let τ 0 = 0 and denote by τ k , k ≥ 1 the kth jump time of the process (b t ). Recall the notion of the point measure µ t from (3.10) (or from (3.15)) and notion of Exp(m) from Definition 2.4.
Before we prove Corollary (3.15), let us state the key claim needed for the proof. The proof of this claim follows from the memoryless property and we omit it.
). Proof of Corollary 3.15. Given m τ 1 , . . . , m τ k , the distribution of b τ k is that of a random size-biased reordering of m τ k (c.f. P2(k) in the proof of Theorem 3.7 and the proof of Corollary 3.13).
In the proof of Proposition 3.12 we have seen that given b τ 1 , . . . , b τ k , the conditional joint distribution of the heights is Combining the above observations with Claim 3.16 we obtain that the statement of Corollary 3.15(i) indeed holds. Now Corollary 3.15(ii) follows from the combination of part (i), the observation that between the jumps of the process (b t ) the height gaps between particles decrease at constant speed and the memoryless property of the height gap distribution (c.f. Definition 3.14).

3.4.
Tilt-and-shift representation. In this section we connect the particle system introduced in Definition 3.11 to the "tilt-and-shift" dynamics presented in Definition 2.9 and prove Proposition 2.10. Let m = (m 1 , . . . , m n ) ∈ ↓ 0 . Let E i ∼ Exp(m i ) independently for i = 1, . . . , n. Consider the particle system of Section 3.3 in which a particle of size m i starts at height Y i (0) = −E i for each i = 1, . . . , n.
Define the function g 0 ≡ f 0 as at (2.6) or (2.7), and then g t , t > 0 using the tilt-and-shift procedure of Definition 2.9.
Proposition 3.18. Let b t ∈ 0 be the vector of sizes of the time-t blocks of the particle system in decreasing order of height. Then (b t , t ≥ 0) = (E(g t ), t ≥ 0) with probability 1. (3.21) Then from Proposition 3.12, Corollary 3.13 and Theorem 3.7 we can immediately deduce the following result: Corollary 3.19. (E(g t ), t ≥ 0) has the law of ICLD(λ) started from an initial state which is a size-biased reordering of m. Hence E ↓ (g t ), t ≥ 0 has the law of MCLD(λ) started from m and thus Proposition 2.10 holds.
Proof of Proposition 3.18. We will show how to find a copy of the particle system embedded within the tilt-and-shift process. Namely, the excursions of g t correspond to the time-t blocks in the particle system. Specifically, an excursion [x, x ) of g t corresponds to a time-t block whose size is the length x − x of the excursion, and whose height is the level g t (x) of the excursion.
The idea is that the "tilt" part of the construction produces an upward drift of the excursion levels which corresponds to the upward movement of the blocks in the particle system; this drift causes neighbouring excursions to merge, corresponding to the merging of blocks in the particle system when they reach the same height. Meanwhile the "shift" mechanism which removes the leftmost excursion when its level reaches 0 corresponds to the death of a block when it reaches height 0 in the particle system.
Recall that Y (1) (t), . . . , Y (n(t)) (t) are the heights of the time-t blocks of the particle system in decreasing order, and b t is the vector of the sizes of those blocks (in decreasing order of height).
Let us denote b 0 = (b 1 , . . . , b n ). Write also Then the excursions of g 0 are the intervals and the levels of these excursions are Recall the time τ 1 , the time of the first merging or deletion in the particle system, defined in the proof of Proposition 3.12, which we can also write as (3.25) Since each block moves upwards at a rate equal to the sum of the sizes of blocks above it plus λ, we have that for 1 ≤ k ≤ n and t ∈ [0, τ 1 ), But also if we definẽ then by the definition of the tilt mechanism in Definition 2.9, we have that for 1 ≤ k ≤ n and t ∈ [0,τ 1 ), Since the derivatives on the right-hand sides of (3.26) and (3.28) are the same, and the values at t = 0 are also the same by (3.24), we have that τ 1 =τ 1 and that Y (k) (t) = g t (x k−1 ), k = 1, . . . , n, t ∈ [0, τ 1 ). (3.29) By (3.27), also g t (x k−1 ) > g t (x k ) for all such k and t, and so for all t ∈ [0, τ 1 ), the excursions of g t are again given by (3.23). So indeed we find that throughtout [0, τ 1 ) the block heights and the excursion levels continue to correspond, and the block sizes and excursion lengths do not change.
Now we look at what happens at time τ 1 , which is the first time that the particle system has a merge or deletion event. Recall from the proof of Proposition 3.12 the value κ which describes which kind of event happens at time τ 1 .
• If 1 ≤ κ ≤ n − 1 then the event is a merge of the blocks with sizes b κ and b κ+1 . In that case we have that Y κ (τ 1 −) = Y κ+1 (τ 1 −) and so also (by (3.29)) we have g τ 1 − (x k−1 ) = g τ 1 − (x k ). Hence at time τ 1 also the excursions [x κ−1 , x κ ) and [x κ , x κ+1 ) merge into a single excursion which is [x κ−1 , x κ+1 ). • If instead κ = 0 then Y (1) (τ 1 −) = 0 and g τ 1 − (0) = 0. Then time τ 1 is the death time of the block of size b 1 ; also, following (2.9), at time τ 1 there is a shift of size Then we obtain g τ 1 (x) = g τ 1 − (x + b 1 ). In both of these cases the heights and masses of the particle system at time τ 1 are given by (3.19) and (3.20). If we now define then we obtain that the excursions of g τ 1 are the intervals and the levels of these excursions are From here, proceeding from (3.31) and (3.32) just as we did from (3.23) and (3.24), we can repeat the argument above from time τ 1 until the time τ 2 of the next jump of E(g t ).
Continuing recursively until the time of the nth jump (when the last death occurs), we obtain that the excursion lengths and block sizes continue to correspond, as required for Proposition 3.18.
Remark 3.20. Observe that written symbolically, we have shown that whereḡ t is the minimum process of g t as defined at (2.1); from the expression (3.15) for µ t , and the definition of f µ in Definition 2.3, we have that f µt is the non-increasing piecewise constant càdlàg function taking value Y (i) (t) on an interval of length (b t ) i for i = 1, . . . , n(t), and otherwise takes the value −∞.
In this formulation the statement of Proposition 3.18 can be seen to follow immediately, since b t = E(f µt ) and E(ḡ t ) = E(g t ).
Corollary 3.21. Recalling the definition of ν from (2.10) and t i from (3.12), we have (3.34) Remark 3.22. We observed in Remark 3.8 that the ICLD(λ) process naturally generalizes to any initial condition m ∈ ↓ 1 . Similarly, the definitions and results of Sections 3.3 and 3.4 also extend to m ∈ ↓ 1 . As a corollary, Proposition 2.10 generalizes to m ∈ ↓ 1 . For initial conditions with infinite total mass we cannot naively extend Definition 2.9, as explained in Remark 2.12. The extension of the tilt-and-shift representation to m ∈ ↓ 2 \ ↓ 1 will be carried out in Section 5.

Preparatory results about µ 0 and excursions
In this section we collect preliminary results that we will later use in Section 5 when we extend the rigid representation result of Section 3 from ↓ 0 to ↓ 2 . In Section 4. Proof. For any 0 ≤ a < b, , and this is already enough to give (i). For (ii), we have and also .  Proof. It is enough to prove that for all pairs of rational numbers 0 < a < b we have µ[−b, −a] > 0 with probability 1. This follows from the second Borel-Cantelli lemma and the fact that  (iii) For any ε > 0, g has only finitely many excursions above its minimum with length greater than or equal to ε. (iv) Let x max = sup{ x : x > −∞ } ≤ ∞. The Lebesgue measure of the set of points in (0, x max ) which are not contained in some excursion above the minimum is 0.
If g ∈ G, then E ↓ (g) (see Definition 2.2) is well-defined.
Lemma 4.4. Suppose f 0 is defined from m ∈ ↓ 2 by Definition 2.5. Define f t by (2.8). Then with probability 1, we have f t ∈ G for all t ≥ 0.
Proof. Properties (i), (ii) in Definition 4.3 can be deduced for f t directly from the definitions (2.6) and (2.8). Property (iv) for f t follows from the fact that (iv) holds for f 0 (see Remark 2.6(i)) and the observation that every excursion of f 0 is contained in an excursion of f t .
It remains to justify property (iii). The function f 0 is non-increasing, and Lemma 4.1(ii) tells us that the length of the interval on which f 0 takes values in [y, y + 1] tends to 0 as y → −∞. Hence for every ε there exists a K ε such that f 0 ( , so all excursions intersecting (K ε , ∞) must have length less than ε, as desired.
Lemma 4.5. Given some f 0 ∈ G define f t by (2.8) and assume that f t ∈ G for all t ≥ 0.
Since we use the topology of coordinatewise convergence on ↓ ∞ , it is enough to show that the function t → m i (t) is càdlàg for all i.
Consider 0 ≤ t < t. Since f t is obtained from f t by adding on an increasing function, any minimum of f t is also a minimum of f t , and any excursion of f t is a sub-interval of an excursion of f t .
Also, there is some x ∈ (r, r + ε/2) with f t (x) < f t (l). Hence if δ is sufficiently small, then f t+δ has a minimum in [r, r + ε/2].
So for any ε, we can find δ such that the length of the excursion of f t+δ which includes (l, r) is at most r − l + ε. Now we will argue that for any ε > 0 there exists small enough δ such that the length m 1 (t + δ) is at most m 1 (t) + ε.
Fix any T > t and consider δ ∈ (0, T − t). Since the excursions of f t+δ are contained in the excursions of f T , any excursion of f t+δ of length more than m 1 (t) + ε must be contained in an excursion of f T whose length also exceeds that. There are only finitely many such excursions of f T since f T ∈ G. Let U be the union of those excursions, which has finite total length, say L. Now let us look at all the excursions of f t contained in U. There are at most countably many. We can take a finite number of them whose total length is at least L − ε. Each of them has length no more than m 1 (t). From the property above, if we choose δ small enough, then at time t + δ, none of them is contained in an excursion of length more than m 1 (t) + ε. But also, since the remaining length of U outside this set is only ε, then also no other point in U is contained in an excursion of length more than m 1 (t) + ε.
In similar fashion we can also obtain that k i=1 m i (t+δ) ≤ ε+ k i=1 m i (t) for sufficiently small δ, for any k. But note that k i=1 m i (t) is non-decreasing in t. So for each k, k i=1 m i (t) is right-continuous with left limits, and hence the same is true for m i (t) for each i. Corollary 4.6. Given some f 0 ∈ G define f t by (2.8) and assume that f t ∈ G for all t ≥ 0. Then the set of times t such that f t has a non-strict excursion (c.f. Definition 2.1) is countable.
Proof. By Lemma 4.5 the function t → E ↓ (f t ) is càdlàg, therefore it has countably many jumps, c.f. [17, Section 12, Lemma 1]. Thus we only need to show that if f t 0 has a non-strict excursion for some t 0 > 0 then t → E ↓ (f t ) jumps at t 0 . If [l, r) is a non-strict excursion of f t 0 , then f (l) = f (x) for some x ∈ (l, r). Now if t < t 0 and [l , r ) is an excursion of f t , then x is not in the interior of [l , r ). This implies that for any t < t 0 the function f t has at least two disjoint excursions contained in [l, r) that are separated by x, and these excursions merge at time t 0 , thus if k 0 is the smallest index k for which m k (t 0 ) < r − l then the non-decreasing function t → k 0 −1 i=1 m i (t) jumps at time t 0 . Definition 4.7. A family of good functions f (i) ∈ G, i ∈ I is said to be uniformly good if for any ε there exists K ε ∈ R such that for any i ∈ I the excursions of f (i) intersecting [K ε , ∞) are all shorter than ε.
Lemma 4.8. Let f ∈ G be continuous and assume that all of the excursions of f are strict (c.f. Definition 2.1). Let f (n) ∈ G, n ∈ N be a sequence of (not necessarily continuous) functions that converge to f uniformly on bounded intervals. Let us also assume that the family consisting of f and f (n) , n ∈ N is uniformly good. Then E ↓ (f (n) ) → E ↓ (f ) as n → ∞ in the product topology on ↓ ∞ .
Proof. Suppose (l, r) is an excursion of f . For any given γ > 0 (with γ < l), there is a δ > 0 such that the following properties hold: Here (i) holds since f has a minimum at l and, being continuous, must achieve its bounds on [0, l − γ]; (ii) holds since the excursion is strict, and (iii) holds since by the definition of excursions, there must be points arbitrarily close to the right of r which take value lower than f (l). Now suppose n is large enough that |f (n) (x) − f (x)| < δ/2 for all x ∈ [0, r + γ]. Then we obtain the following properties: ( Then f (n) must have an excursion which starts somewhere in [l − γ, l + γ] and ends somewhere in [r − γ, r + γ]. Now let ε > 0 and choose K ε such that the excursions of f and f (n) , n ∈ N intersecting [K ε , ∞) are all shorter than ε. Now by Definition 4.3(iv) there exists a finite collection of excursions (l j , r j ) of f , whose union covers all of [0, K ε + ε] except for a set of total length less than ε/2. Let k be the total number of these excursions. Apply the above argument to all of these excursions with γ = ε/4k. Then if n is sufficiently large, we have that for each of these excursions of f , there is a corresponding excursion of f (n) whose length is within ε/2k; the remaining length in [0, K ε + ε] amounts to no more than ε; and we know that outside [0, K ε + ε], all excursions (either of f (n) or f ) have length less than or equal to ε.
It follows that for any i > 0, the ith largest excursion of f (n) and the ith largest excursion of f differ by at most ε. Hence indeed E ↓ (f (n) ) converges componentwise to E ↓ (f ), as desired.

Extension of rigid representation to
In this section we will extend the rigid representation results of Section 3 to any initial condition m ∈ ↓ 2 . As we have discussed in Remark 3.22, extension from ↓ 0 to ↓ 1 is automatic, so in this section we will assume that m ∈ ↓ 2 \ ↓ 1 . In Section 5.1 we prove that the MC admits a tilt representation, i.e., Theorem 2.8. In Section 5.2 we prove that the MCLD admits a tilt-and-shift representation, i.e., Theorem 2.13.

Extension of MC tilt representation to ↓
2 . The aim of this section is to prove Theorem 2.8.
f 0 is defined from m ∈ ↓ 2 by Definition 2.5. f t is defined by (2.8). Definition 5.1. By Lemma 4.4 and Corollary 4.6 the (random) set of times t such that f t has a non-strict excursion (c.f. Definition 2.1) is countable. Hence for all but countably many t, the probability that all excursions of f t are strict is equal to 1. Let T denote this (deterministic) set of "good" times t.
From Lemma 4.4 and Lemma 4.5 it follows that t → E ↓ (f t ) is a càdlàg process with respect to the product topology on ↓ ∞ . The graphical representation of the multiplicative coalescent m t constructed in [2, Section 1.5] is also a càdlàg process with respect to the topology of the d(·, ·)-metric defined in (1.2) (see [30,Lemma 2.8]), thus it is càdlàg with respect to the weaker product topology on ↓ ∞ . Hence, since T is dense, if we can show that for any finite collection t 1 , . . . , t r ∈ T , we have then indeed the law of E ↓ (f t ) is that of the MC. For each n, let m (n) be given by For each n, m (n) ∈ ↓ 0 , and m (n) → m in ↓ 2 as n → ∞. We couple the processes starting from m (n) , n ≥ 1, by using the same height variables where m (with respect to the topology of ↓ 2 and hence also coordinatewise).
It remains to show (5.5). We will achieve this by checking that the conditions of Lemma 4.8 almost surely hold if t ∈ T . We may assume that m ∈ ↓ 2 \ ↓ 1 , as discussed in the first paragraph of Section 5. Proof. Recalling Definitions 2.3 and 2.5 we see that Proof. Since f By considering the interval on which the function f 0 takes the same value Y k , we have For all n ≥ n 0 we have δ ≤ j>n : E j <S m j . This goes to 0 as n → ∞ by Lemma 4.1 and dominated convergence. Then by the uniform continuity of f 0 on bounded intervals, we have that f Finally, we can insert the properties derived in Definition 5.1 and Lemmas 5.2, 5.3 into Lemma 4.8 to obtain (5.5). This completes the proof of Theorem 2.8.

Extension of MCLD tilt-and-shift representation to ↓
2 . The aim of this section is to prove Theorem 2.13.
In Section 5.2.1 we give quantitative bounds on the effect of the insertion of a new particle in a finite particle system on the death times of other particles.
In Section 5.2.2 we construct the death times of each particle in the infinite particle system associated to the initial state m ∈ ↓ 2 by inserting particles one by one and showing that death times converge.
In Section 5.2.3 we collect some technical lemmas that allow us to deduce the convergence of the ordered sequence of excursion lengths from the approximation results of Section 5.2.2.
In Section 5.2.4 we extend the MCLD tilt-and-shift representation from ↓ 1 to ↓ 2 , i.e., we prove Theorem 2.13. 5.2.1. Perturbation of the particle system. Recall the particle system introduced in Section 3.3. The main result of Section 5.2.1 is Lemma 5.4, which quantifies the effect of the insertion of a new particle on the death times of other particles. This perturbation result will play an important role when we extend our rigid representation of MCLD(λ) from ↓ 0 to ↓ 2 in Section 5.2.2 using truncation and approximation. Note that Lemma 5.4 is a deterministic result, i.e., it holds for any initial configuration of particles. Let us initialize a new particle system Y 0 (t), . . . , Y n (t) by letting Y i (0) = Y i (0) for any i = 1, . . . , n and by adding a new particle with initial height Y 0 (0) and mass m.
Let us then define Y 0 (t), . . . , Y n (t), µ t and t 0 , . . . , t n analogously to Definition 3.11. Then we have The rest of Section 5.2.1 is devoted to the proof of Lemma 5.4.
Without loss of generality we may assume Y 1 (0), Y 2 (0), . . . Y n (0) are all distinct, because if Y i (0) = Y j (0) for some i = j then we can replace our particle system by another one with fewer particles in which these two particles are merged.
With a slight abuse of notation, for the rest of Section 5.2.1 we will relabel our particles so that we have Y 1 (0) > Y 2 (0) > · · · > Y n (0) and denote by m i the weight and t i the death time of the particle with initial location Y i (0). Thus we have The next lemma gives a recursive formula for t 1 , . . . , t n .
Lemma 5.5. Let us formally define t 0 = 0. We have Proof. First observe that we have We will prove (5.7) by considering two cases separately.
First case: If t i > t i−1 then where in ( * ) we used (5.8) and the fact that therefore (5.7) holds.
Second case: Now we assume t i = t i−1 . First note that and therefore (5.7) holds.
Recall the notion introduced in the statement of Lemma 5.4. Denote by By (3.11) the speed of Y i (t) only depends on the locations of particles strictly above it, so we have Y i (t) ≡ Y i (t) for any 1 ≤ i < i * , thus we have t i = t i for any 1 ≤ i < i * , and (5.6) trivially follows for these particles.
For any i * ≤ i ≤ n we will think about t i = t i ( m) as a function of the variable m ≥ 0, which represents the weight of the inserted particle. With this definition we have t i (0) = t i . Let us define the Lipschitz constant L i by In order to prove (5.6), we only need to show We begin with the following claim. The proof is trivial and we omit it. Denote by t 0 ( m) the death time of particle Y 0 (t) in the particle system Y 0 (t), Y 1 (t), . . . , Y n (t). Note that we have t 0 ( m) = t 0 (0), i.e., the death time of the inserted particle does not depend on its weight, so we will omit dependence of t 0 on m.
Lemma 5.7. For any i * ≤ i ≤ n we have Proof. For i * ≤ i ≤ n, the recursion (5.7) for the new particle system reads (5.14) We will prove (5.13) by induction on i. We begin with i = i * . Since t i * −1 ( m) = t i * −1 and thus L i * −1 = 0, we obtain from Claim 5.6 and (5.14) that L i * ≤ t 0 /λ, thus (5.13) holds if i = i * .
As for the induction step, we obtain from Claim 5.6 and (5.14) that Now (5.13) holds for i by (5.15) and the induction hypothesis (i.e., the assumption that (5.13) holds for i − 1).
From the recursive inequalities (5.13) one readily deduces by induction on i the following explicit bound: Now we are ready to prove (5.12) for any i * ≤ i ≤ n: The proof of (5.12) and Lemma 5.4 is complete.

Truncation and approximation.
The main result of Section 5.2.2 is Lemma 5.8 which extends the "shift" operator from ↓ 0 to ↓ 2 using truncations and approximation, c.f. Remarks 2.12 and 2.14(iii).
Given some m ∈ ↓ 2 , let us generate E 1 , E 2 , . . . as in (2.4). Define the truncation m (n) by (5.2). We define g (n) 0 (·) ≡ f (n) 0 (·) using E 1 , E 2 , . . . , E n by (2.6). Note that we still denote by f t (·) the function constructed from the un-truncated m by (2.6) and (2.8), and also that we use the same sequence of random variables E 1 , E 2 , . . . to obtain a coupling of g (n) 0 (·), n ∈ N. We define g (n) t (·) from g (n) 0 (·) using Definition 2.9. This gives rise to the measure ν (n) by (2.10) and the function Φ (n) by (2.11). By (2.12) we have Our next result states that P-almost surely ν (n) vaguely converges to some ν as n → ∞. Proof of Lemma 5.8. We will use the particle representation (see Section 3.3) Let us assume that the function h : [0, ∞) → R for which we want to show (5.18) is supported on [0, T ].
Recalling the definition . Then by Lemma 4.1 there exists a P-almost surely finite random variable K 0 such that which implies that the time of death t (n) i of particle i (see (3.34)) satisfies Our aim is to show that the sequence ∞ 0 h(t) dν (n) (t), n ∈ N is Cauchy. In order to show this we let n ≤ m and bound In order to bound the first term on the right-hand side of (5.23) we observe that if Note that Lemma 4.1 implies that with probability 1 we have If we combine this with (5.23), (5.24) and the fact that h(·) is uniformly continuous, we can conclude that ∞ 0 h(t) dν (n) (t), n ∈ N is a Cauchy sequence for any h ∈ C 0 (R), from which it follows that the exists ν for which (5.18) holds.
Lemma 5.10. If ν is the random measure obtained in Lemma 5.8, then for every t ≥ 0 there exists y ∈ (−∞, 0] such that and ν (n) → ν vaguely, we can conclude that ν is also an atomic measure with From this (5.25) readily follows.

5.2.3.
Tilt-and-shift continuity lemmas. We will prove Theorem 2.13(i) in Section 5.2.4 by replicating the argument given in Section 5.1. In Section 5.2.3 we collect some ingredients of this proof.
Lemma 5.11. If m ∈ ↓ 2 \ ↓ 1 then the function t → E ↓ (g t ) is càdlàg with respect to the product topology on ↓ ∞ . Proof. Let us fix t ≥ 0 and define the auxiliary functions If m, m ∈ ↓ ∞ , we say that m m if i j=1 m j ≤ i j=1 m j for any i ∈ N. We are going to show As soon as we show (5.28)-(5.31), we immediately obtain i.e., the right-continuity of t → E ↓ (g t ) with respect to the ↓ ∞ topology. The proof of the existence of left limits is similar and we omit it.
(5.28) follows from the fact that g t+∆t is obtained from g * t+∆t by adding an increasing function (see (5.26)), thus the collection of excursions of g t+∆t are obtained by merging some excursions of g * t+∆t . (5.29) follows from the fact that g t+∆t is obtained from g * * t+∆t by a shift to the left plus an addition of a constant (see (5.27)), thus the excursions of g t+∆t are obtained by deleting/splitting some excursions of g * t+∆t .
Corollary 5.12. If m ∈ ↓ 2 \ ↓ 1 then the set of times t such that g t has a non-strict excursion (c.f. Definition 2.1) is countable.
Thus we only need to show that if g t 0 has a non-strict excursion for some t 0 > 0 and if ν({t 0 }) = 0 then t → E ↓ (g t ) jumps at t 0 .
The rest of the proof is identical to that of Corollary 4.6, with the additional observation that the jump at time t 0 created by the merger of excursions coming from the "tilt" operation cannot be cancelled by the deletion of excursions coming from the "shift" operation, since t → Φ(t) is continuous from the left at time t 0 .
Claim 5.13. Let m ∈ ↓ 2 \ ↓ 1 and define g t (·) using (2.12) and g Definition 5.14. By Corollary 5.12 the (random) set of times t such that either g t has a non-strict excursion or ν({t}) > 0 is countable. Hence for all but countably many t, the probability that ν({t}) = 0 and all excursions of g t are strict is equal to 1. Let T * denote this (deterministic) set of "good" times t.
Proof. First note that the functions g t (·) and g (n) t (·), n ∈ N are uniformly good (c.f. Definition 4.7): this follows from Lemma 5.2 and the fact that g t is a "shifted" version of f t : and similarly, g Proof of Theorem 2.13(i). Given m ∈ ↓ 2 \ ↓ 1 , we constructed two stochastic processes: • the graphical construction in [30, Section 3] of the MCLD(λ) process m t , t ≥ 0, • the process E ↓ (g t ), t ≥ 0, defined by (2.12), where the initial state g 0 (·) := f 0 (·) is defined by (2.6) and the control function Φ(·) is defined by (5.19).
We now want to show that these two ↓ ∞ -valued processes have the same law. Both processes are càdlàg with respect to the product topology on ↓ ∞ by [30, Proposition 1.1] and Lemma 5.11.
Hence, since the set T * introduced in Definition 5.14 is dense, if we can show that for any finite collection t 1 , . . . , t r ∈ T * , we have (5.34) then indeed Theorem 2.13(i) will follow. By Proposition 2.10 we have where m (with respect to the topology of ↓ 2 and hence also coordinatewise). Putting together (5.35), Lemma 5.15 and (5.36) we obtain (5.34). The proof of Theorem 2.13(i) is complete.
Proof of Theorem 2.13(ii). It is enough to show that for any K > 0 and ε > 0 we almost surely have −ε ≤ g t (0) ≤ 0 for any 0 ≤ t ≤ K. Let us fix K, ε > 0. Recall from (5.20)- > K for any i ≤ n. By Lemma 4.2 there exists an almost surely finite n 0 such that µ (n 0 ) 0 [y − ε, y] > 0 for any Y ≤ y ≤ 0, thus for any n ≥ n 0 and any In words: the gaps between consecutive particles initially located in [Y, 0] are smaller than or equal to ε. By Definition 3.11, these gaps can only decrease with time, thus for any n ≥ n 0 and t ≤ K there is a particle in [−ε, 0], i.e., we have −ε ≤ g (n) t (0) → g t (0) as n → ∞ for all except countably many values of t ∈ [0, K] by Claim 5.13, moreover g t (0) is a càdlàg function of t by (2.12), therefore −ε ≤ g t (0) ≤ 0 holds for every 0 ≤ t ≤ K.
Proof of Theorem 2.13(iii). Let us assume that m = (m 1 , m 2 , . . . ) ∈ ↓ 2 \ ↓ 1 . We recursively define The restriction of the measure µ 0 to (y, 0) can be determined by looking at F x := σ (g 0 (x ), 0 ≤ x ≤ x), moreover for any k ≥ 1 the restriction of µ 0) is also determined by F x , simply by removing all atoms with a weight strictly smaller than m k−1 .
As we have already discussed in the proof of Lemma 5.10, ν is an atomic measure with is F x -measurable, because the value of the death time t (n k −1) i can be determined by (3.11) if we know the initial particle configuration strictly above Y i (0) and the value of Y i (0) (we do not need to know the mass m i of particle i).
Therefore lim k→∞ t (n k ) i = t i is also G x -measurable. If we define t y = sup{ t i : Y i ∈ [y , 0) }, then y → t y is continuous, t y is F x -measurable; moreover by (5.37) we have This completes the proof of Theorem 2.13(iii).
6. Applications 6.1. Eternal multiplicative coalescents. In Section 6.1 we restrict to the case λ = 0, i.e. the multiplicative coalescent (with no deletion). We showed in Theorem 2.8 that a multiplicative coalescent m t , t ≥ 0 started from any initial condition m 0 in ↓ 2 has a "tilt" representation, as This leaves the question of eternal coalescents, i.e. those defined for all times t ∈ (−∞, ∞).
Recall from Definition 1.1 the notion of Brownian motion with parabolic drift, or briefly BMPD(u), where u ∈ R is the "tilt parameter".
If h ∼ BMPD(u), denote by M(u) the law of E ↓ (h). (6.1) In [2, Corollary 24] Aldous showed that there exists an eternal version of the MC, the standard multiplicative coalescent, such that the marginal distribution of the coalescent at any given time t ∈ R is given by M(t). Armendáriz [8], and then also Broutin-Marckert [18], showed that if h 0 ∼ BMPD(u), and h t (x) = h 0 (x) + tx for all t ∈ R, then the process E ↓ (h t ), t ∈ R is in fact the standard MC. We will provide an alternative proof of this result in Corollary 6.6 below.
Aldous and Limic [5] described the set of all eternal multiplicative coalescents. They showed that the marginal distributions of any of these coalescents can be given by the set of excursion lengths of a suitable stochastic process. To state their results we need to recall some more notation from [5].
Definition 6.1. Given κ ≥ 0, τ ∈ R and c ∈ ↓ 3 , define for x ≥ 0, where W is a standard Brownian motion, and where, for each i, E i ∼ Exp(c i ), independently of each other and of W .
If c = 0 and κ = 1, then W κ,τ,c ∼ BMPD(τ ). For general c, the processes defined in (6.2) have been called Lévy processes without replacement; each jump of size c i occurs at rate c i , but once such a jump has occurred it does not happen again. Define also the parameter space We can phrase the results of [5, Theorems 2 and 3] as follows: Our results can be applied to show that all these eternal coalescents also have a "tilt" representation. First, we make a definition: Definition 6.3. Let W (x), x ≥ 0 be a random function. We say that W has the exponential excursion levels property if the following holds: conditional on the sequence of excursion lengths E ↓ (W ) = (m 1 , m 2 , . . . ), the levels of the excursions of W are independent, with the excursion of length m i occurring at level −E i where E i ∼ Exp(m i ). Now we can state the key property that we need: Claim 6.4. Let (κ, τ, c) ∈ I. Then W κ,τ,c has the exponential excursion levels property.
Remark 6.5. Equation (72) of [2] states that BMPD(t) has the exponential excursion levels property for any t ∈ R. From this fact the statement of Claim 6.4 follows for W κ,τ,0 for any κ > 0 and τ ∈ R by scaling. For a short sketch proof of this fact, see the c = 0 case of Section 6.1.1 below.
Among other things, Claim 6.4 implies that the excursions of W κ,τ,c occur in size-biased order, c.f. Claim 3.3. From Claim 6.4, we can deduce the following result: Corollary 6.6. Let (κ, τ, c) ∈ I. Then the process E ↓ (W κ,τ +t,c ), t ∈ R is an eternal multiplicative coalescent.
We describe below the straightforward way in which Corollary 6.6 follows from Claim 6.4. We then give a sketch of the proof of Claim 6.4 in Section 6.1.1. The proof is not difficult but a detailed account would occupy many more pages, and require some more technical extensions of earlier results (in particular the material in Section 4.2) to the setting of functions with positive jumps. Corollary 6.6 is also the subject of current work by Vlada Limic which is available as an interim version at [23]. Limic gives a proof of this result based on the breadth-first walk construction used by Aldous [2] and by Aldous and Limic [5], and also using new ideas from the thesis by Uribe Bravo [35]. Even without giving the full proof, we feel that abstracting the property in Claim 6.4 gives valuable insight in complement to the different approach of [23].
Condition on E ↓ (W κ,τ c ) = m. Define f 0 =h 0 . Then Claim 6.4 tells us precisely that the distribution of f 0 is the same as the one defined at (2.6).
Define f t by f t (x) = f 0 (x) + tx as at (2.8). Then by the λ = 0 case of Remark 2.14(v) But Theorem 2.8 says that E ↓ (f t ), t ≥ 0 is a MC. Hence the same is true of E ↓ (h t ), t ≥ 0. That is, E ↓ (W κ,τ +t,c ), t ≥ 0 is a MC for all τ . But by considering values of τ tending to −∞, and applying Kolmogorov extension, the statement of Corollary 6.6 follows.
6.1.1. Sketch of proof of Claim 6.4. One necessary tool is an extension of Lemma 4.8, in two directions. First, we need not just that the collection of excursion lengths converges, but also that their levels converge; this is straightforward. Second, we need to cover the case where the limit f is allowed to have positive jumps (since this is true of the functions W κ,τ,c whenever c is not identically zero); this introduces a few technicalities (but doesn't require new ideas). Now the idea is to take an appropriate sequence of initial conditions m (n) ∈ ↓ 0 , and times t n . Define f (n) 0 based on m (n) according to (2.7), and define f (n) tn as in (2.8). Then we want to show convergence, in an appropriate sense, of f (n) tn to W κ,τ,c ; then, to use the extension of Lemma 4.8 to deduce that the lengths and levels of the excurions of f (n) tn converge in distribution to those of W κ,τ,c . Finally, observe that for any n, the function f (n) tn satisfies the exponential excursion lengths property (this is a translation of Corollary 3.15(ii) from the particle system context into the tilt representation context using the identity (3.33)). This property is then inherited by the limit W κ,τ,c of the sequence f (n) tn and Claim 6.4 follows.
The main part of the work here is showing the convergence of f (n) tn to W κ,τ,c in such a way that we can deduce the convergence of the lengths and levels of the excursions.
Case 1: c = 0 and κ = 1. The distribution of W 1,0,0 is simply BMPD(0). The fact that BMPD(u) satisfies the exponential excursion heights property was already discussed in Remark 6.5. Alternatively, let the initial condition m (n) consist of n blocks each of size n −2/3 , and let t n = n 1/3 . The function f (n) 0 has n excursions each of length n −2/3 . The gaps between the levels of these excursions are given by independent exponential random variables; for 0 ≤ k < n, let F k be the (k + 1)st such gap, with rate n 1/3 n−k n . Then the increments of f (n) n 1/3 on the intervals [(kn −2/3 , (k + 1)n −2/3 ] are independent over 1 ≤ k < n; the kth such increment is given by n −1/3 − F k . Let us denote We have In this way we obtain a functional limit theorem; the distribution of f (n) n 1/3 converges (in the sense of uniform convergence on finite intervals) to BMPD(0).
(By this method we get an alternative, self-contained proof of the results of [8,18] claiming that the tilt procedure applied to the BMPD family gives the (eternal) standard MC.) Case 2: κ = 0. Now the process in (6.2) is given just by the compensated jumps according to the vector c. In keeping with (6.3), we now need c ∈ ↓ 3 \ ↓ 2 . Here, for an appropriate initial condition take t n = c 2 1 +· · ·+c 2 n and m (n) = t −1 n (c 1 , c 2 , . . . , c n , 0, 0, . . . ). We can couple the sequence of initial functions f (n) 0 , n ∈ N with W 0,0,c in the following way. For k ≤ n, let f Consider what happens on this interval in f (n) tn . Because of the tilt, the function increases by t n t −1 n c k = c k over the course of the interval. The length t −1 n c k goes to 0 as n → ∞. By Remark 2.6, the horizontal location of the start of the interval is n j=1 t −1 n c j 1(E j < E k ); (6.4) conditional on E k , this has mean converging to E k and variance converging to 0, and (using simple martingale calculations) can be shown to converge almost surely as n → ∞ to E k . Hence in the limit process this interval produces a jump of size c k occurring at time E k , matching the term on the right of (6.2).
The vertical location of the start of the interval is −t n E k in the function f (n) 0 ; applying the tilt corresponding to the horizontal distance in (6.4), its vertical location in f Comparing with (6.2), this converges to W 0,0,c (E k −) as n → ∞, so in the limit process the jump indeed appears at the correct height.
In the limit process, these jumps are dense. In this case we do not have uniform convergence of f (n) tn to W 0,0,c , but by considering suitable time-changes one obtains that f (n) tn has the same excursions as a function h (tn) which converges to W 0,0,c in the Skorohod topology, and this is enough to give convergence of the lengths and levels of excursions as required.
Case 3: c = 0 and κ = 1. Now the process W 1,0,c has both a Brownian part and positive jumps. Here a suitable sequence of initial conditions is given by taking m (n) to consist of blocks of sizes n −1/3 (c 1 , c 2 , . . . , c k(n) ) along with n blocks of size n −2/3 , where k(n) is chosen such that (This is the same regime used in the proof of Lemma 8 of [5]). Similarly to the two previous cases, choose t n = m (n) −1 2 . The ideas of the two previous cases can be combined to give the desired result for W 1,0,c also. 6.2. Tilt-and-shift of BMPD. Recall the definition of BMPD(u) from Definition 1.1. We show that applying the tilt-and-shift procedure starting from an initial state h 0 which is a BMPD results in a MCLD process. Furthermore, the function h t remains in the class of BMPD processes (with a random parameter). Proposition 6.7. Let u ∈ R, and let h 0 ∼ BMPD(u). Let g 0 =h 0 . Let Φ(t), t ≥ 0 and g t , t ≥ 0 be given by the tilt-and-shift procedure in Theorem 2.13, and let h t be given by (2.13).
Remark 6.8. Proposition 6.7(ii) gives a MCLD process whose marginal distributions are all given by mixtures of distributions M(u) (c.f. Definition (6.1)). In [24] we find eternal processes with this property, which may be stationary or non-stationary. We show that such processes arise naturally as scaling limits of discrete models such as frozen percolation processes (see Definition 6.9 below) or forest fire processes (see Section 6.3).
In Section 6.2.1 below we observe a simple case of such a scaling limit.
6.2.1. Scaling limit of frozen percolation started from a critical Erdős-Rényi graph. First we recall the notion of mean-field frozen percolation process from [31] (using slightly different notation).
Definition 6.9 (FP(n, λ(n))). We start with a graph F (n) 0 on n vertices. Between each pair of unconnected vertices an edge appears with rate 1/n; also, every connected component of size k is deleted with rate λ(n) · k. (When a component is deleted, its vertices as well as its edges are removed from the graph.) Let F   t , arranged in decreasing order. Then M (n) (t), t ≥ 0 is a Markov process -let us call it here the frozen percolation component process on n vertices with lightning rate λ(n), or briefly FP(n, λ(n)).
In order to achieve self-organized criticality, one chooses n −1 λ(n) The next result gives a scaling limit for FP(n, λ(n)), in a setting where λ(n) n −1/3 and the initial state has the distribution of an Erdős-Rényi random graph at some point within the "critical window". Then as n → ∞ the sequence of processes m (n) (t), t ≥ 0 converge in law to the MCLD(λ) process E ↓ (h t ), t ≥ 0 given by Proposition 6.7.
Then the statement of Proposition 6.10 follows by applying Proposition 6.7 together with the Feller property of MCLD(λ) (see [30,Theorem 1.2]). 6.3. Forest fire model. This section contains a particle representation of the mean-field forest fire model of [32]; see Section 6.3.1. The representation is an adaptation of the one in Section 3.3, and we will briefly explain in Section 6.3.2 how it sheds some new light on a certain controlled non-linear PDE problem (see (6.17) below) which played a central role in the theory developed in [32] and [19].
In [32] Ráth and Tóth modify the dynamical Erdős-Rényi model to obtain the mean-field forest fire model: Definition 6.11 (FF(n, λ(n))). We start with a graph on n vertices. Between each pair of unconnected vertices an edge appears with rate 1/n; moreover each connected component of size k "burns" with rate λ(n) · k, i.e., the edges of the component are deleted. The total number of vertices remains n.
Denote by C n (i, t) the connected component of vertex i at time t. We define the empirical component size densities by , v n (t) = (v n k (t)) n k=1 . (6.10) With the above definitions v n (t), t ≥ 0 is a Markov process, let us call it here the forest fire component size density Markov process on n vertices with lightning rate λ(n), or briefly FF(n, λ(n)).

6.3.1.
Particle representation of the forest fire model. In Definition 6.14 and Proposition 6.15 below we are going to give a novel particle representation of FF(n, λ(n)) by slightly modifying Definition 3.11. Note that this correspondence is one-to-one.
In plain words, v n and m n correspond to each other if there is a graph G on n vertices such that v n and m n both arise from G. Definition 6.13. If µ n is a finite point measure on R − such that µ n (R − ) = 1 and the masses of the atoms of n µ n are integers then we define v( µ n ) to be the element of V n corresponding to the element of M n which consists of the ordered list of masses of the atoms of n µ n . Now we define the particle representation of the FF(n, λ(n)) model. Definition 6.14. Given v n (0) = (v n k (0)) n k=1 ∈ V n and the corresponding m n = m n j N j=1 ∈ M n , we define the initial heights of the particles Y i (t), 1 ≤ i ≤ n by letting Y i (0) = −E j , where E j ∼ Exp(m n j ), 1 ≤ j ≤ N are independent and vertex i initially belongs to component j in the forest fire model. We define If Y i (t − ) < 0 then we let d dt Y i (t) = λ(n) + µ n t ( Y i (t), 0), (6.13) and if Y i (t − ) = 0 then we say that vertex i burns and we let − Y i (t) have Exp(1) distribution, independently from everything else.
In words, a clustered family of particles with mass 1/n start at negative locations, move up and merge with other particle clusters, but if a time-t block of particles with total mass k/n reaches 0, then this block burns and gets replaced by k particles of mass 1/n with i.i.d. locations with negative Exp(1) distribution. Proposition 6.15. (i) For any n ∈ N + and any initial state v n (0) ∈ V n , the process v( µ n t ), t ≥ 0 is a FF(n, λ(n)) process with initial state v n (0) (see Definitions 6.11, 6.14 and 6.13 for the definitions of FF(n, λ(n)), µ n t and v( µ n ), respectively).
(ii) For any t ≥ 0, the conditional distribution of n µ n t given the σ-algebra σ(m n s , 0 ≤ s ≤ t) is Exp(m n t ) (c.f. Definition 2.4), where m n t is the M n -valued random variable corresponding to the V n -valued random variable v( µ n t ) (c.f. Definition 6.12).
See Figure 6.1 for a simulation of the particle system realising the forest fire model.
Proof of Proposition 6.15. Recalling Definition 2.4 we observe that by Definition 6.14 we have n µ n 0 ∼ Exp(m n ), where m n ∈ M n corresponds to v n (0) ∈ V n . Denote by τ the first burning time of the particle system Y i (t), 1 ≤ i ≤ n. Denote by Y i (t) := Y i (nt) and µ t := n µ n nt = n i=1 δ Y i (t) so that d dt Y i (t) (6.13) = nλ(n) + µ t (Y i (t), 0), 0 ≤ t < τ n , thus the evolution of the particle system Y i (t), 1 ≤ i ≤ n satisfies Definition 3.11 (with λ = nλ(n)) up to time τ /n, including the time-τ /n block that burns. Likewise, if v n (t), t ≥ 0 is a FF(n, λ(n)) process, then the M n -valued process corresponding to the V n -valued process v n (nt), 0 ≤ t ≤ τ /n satisfies the definition of a MCLD(nλ(n)) process, including the time of the first deletion event and the component that gets deleted. Therefore by Corollary 3.13 the statement of Proposition 6.15(i) holds for v( µ n t ), 0 ≤ t ≤ τ (including time τ , since in both Definitions 6.11 and 6.14 we add k singletons of mass 1/n if a block of size k is deleted).
Next we observe that the conditional distribution of n µ n τ given the σ-algebra σ(m n s , 0 ≤ s ≤ τ ) is Exp(m n τ ). Indeed, this follows from Corollary 3.15(i) and the fact that we insert k particles of mass 1/n with i.i.d. −Exp(1) distribution if a block of k particles burn at time τ , which is exactly what we have to do to maintain the property required by Definition 2.4.
Therefore we can inductively repeat this argument using Corollary 3.15(i) again and again to show that Proposition 6.15(i) holds for v( µ n t ), 0 ≤ t ≤ τ i , where τ i is the i'th burning time. The proof of Proposition 6.15(ii) similarly follows from Corollary 3.15(ii). Remark 6.16. Let us recall a closely related dynamic random graph model of selforganized criticality, studied by Fournier and Laurencot in [21] and by Merle and Normand in [26]. One starts with a graph on n vertices, the coagulation mechanism is the same as in Definition 6.9 and Definition 6.11 (between each pair of unconnected vertices an edge appears with rate 1/n), but the deletion mechanism is different: connected components are deleted forever when their size exceeds a threshold ω(n). In order to achieve self-organized criticality, one chooses 1 ω(n) n. Let us remark that this model also admits a particle representation: the initial weights and heights of particles should be the same as in Definition 6.14, and the particle dynamics between deletion times should be given by (6.13). The deletion mechanism is obvious: if the weight of a time-t block exceeds the threshold ω(n), we remove that time-t block. The proof of the validity of this particle representation can be carried out analogously to the proof of Proposition 6.15. This is a "rigid" representation, i.e., all of the randomness is encoded in the initial state. However, in contrast to the case of the frozen percolation model (i.e., MCLD), the above described threshold-deletion model does not admit a clean tilt-and-shift representation, since particles from the middle can be deleted (as opposed to the case of linear deletion, where only the top particle can be deleted). 6.3.2. The controlled Burgers equation. In this section we use the particle representation of the FF(n, λ(n)) process to give a new interpretation of a certain controlled non-linear PDE problem (see (6.17) below) which played a central role in the theory developed in [32] and [19]. This new interpretation will be presented in Remark 6.17.
One investigates the FF(n, λ(n)) when 1 n λ(n) 1 as n → ∞. This is called the self-organized critical regime of the lightning rate λ(n). We assume that v n k (0) → v k (0) for all k ∈ N as n → ∞, where k k 3 v k (0) < +∞.
Remark 6.17. Let us assume that µ n t converges weakly in probability to some measure µ t as n → ∞. Denote by V (t, y) = µ t (y, 0), y ≤ 0.
Moreover, if 1 n then µ n t (y, 0) ≈ V (t, y), thus by comparing (6.19) and (6.13) we see that the trajectories − Y i (s), s ≥ 0 of particles can be viewed as discrete approximations of characteristic curves.