Diffusions on a space of interval partitions: construction from Bertoin's ${\tt BES}_0(d)$, $d\in(0,1)$

In 1990, Bertoin constructed a measure-valued Markov process in the framework of a Bessel process of dimension between 0 and 1. In the present paper, we represent this process in a space of interval partitions. We show that this is a member of a class of interval partition diffusions introduced recently and independently by Forman, Pal, Rizzolo and Winkel using a completely different construction from spectrally positive stable L\'evy processes with index between 1 and 2 and with jumps marked by squared Bessel excursions of a corresponding dimension between $-2$ and 0.


Introduction
We define interval partitions, following Aldous [1, Section 17] and Pitman [6,Chapter 4]. We write β to denote M . We refer to the elements of an interval partition as its blocks. The Lebesgue measure of a block is called its mass or size.
In this paper we construct diffusion processes in a space of interval partitions in Bertoin's [2,3] framework of a Bessel process of dimension d ∈ (0, 1). Bertoin studied the excursions of such a Bessel process. Specifically, he first decomposed the Bessel process While the interval lengths λ y (t) − λ y (t−) of β y are (twice) the locations R(t) of atoms of µ y [0,T ] , the order of the intervals is not captured by µ y [0,T ] . Hence, this theorem is not an immediate consequence of Bertoin's corresponding results for (µ y [0,T ] , y ≥ 0) Indeed, we prove this theorem by identifying this diffusion process as an instance of a class of diffusion processes introduced in [4], where we gave a general construction of processes in a space of interval partitions based on spectrally positive Lévy processes (scaffolding) whose point process of jump heights (interpreted as lifetimes of individuals) is marked by excursions (spindles, giving "sizes" varying during the lifetime, one for each level crossed). Informally, the interval partition evolution, indexed by level, considers for each level y ≥ 0 the jumps crossing that level and records for each such jump an interval whose length is the "size" of the individual (width of the spindle) when crossing that level, ordered from left to right without leaving gaps. This construction and terminology is illustrated in Figure 1.1.
Specifically, if N = i∈I δ (s i ,f i ) is a point process of times s i ∈ [0, S] and excursions f i of excursion lengths ζ i (spindle heights), and X is a real-valued process with jumps ∆X(s i ) := X(s i ) − X(s i −) = ζ i at times s i , i ∈ I, we define the interval partition skewer(y, N, X) at level y, as follows.
The skewer of (N, X) at level y, denoted by skewer(y, N, X), is defined as This definition is meaningful when X has finitely many jumps as in Figure 1.1, and also when X has a dense set of jump times and the f i are such that M N,X is finite. In [4], we established criteria under which skewer(N, X) is a diffusion. Specifically, N is a Poisson random measure (PRM) with intensity measure Leb ⊗ ν, where ν is the Pitman-Yor excursion law [7] associated with a suitable (self-similar) [0, ∞)-valued diffusion, and X is an associated Lévy process, suitably stopped at a time S when X is zero. In this interval partition evolution, each interval length (block) evolves independently according to the [0, ∞)-valued diffusion, which we call block diffusion, while between (the infinitely many) blocks, new blocks appear at the pre-jump levels of X. The PRM of jumps is obtained by mapping the PRM of spindles onto the spindle heights. Conversely, we may view the PRM of spindles as marking the PRM of jumps by block excursions. See Section 3 for more details.
Theorem 1.4. When the block diffusion is BESQ(−2(1 − d)), a squared Bessel process of dimension −2(1−d) ∈ (−2, 0) and the scaffolding Lévy process is Stable(2−d) stopped at an inverse local time τ 0 X (v) of X at 0, the interval partition evolution associated via skewer is distributed as the diffusion in Theorem 1.2, for u = 2 d−1 v.
The remainder of this paper is organised, as follows. In Sections 2 and 3, we state the main results of [2,3] and [4,5], exhibiting the parallels. In Section 4, we make precise the connections between the two frameworks and deduce the theorems we have stated. In Section 5, we discuss some further observations.
2. Bertoin's study of Bessel processes [2,3] Consider a Bessel process , the Bessel process R satisfies an SDE that yields .
Furthermore, this singular integral is finite as t ↑ T R (0). By time reversal, this means that this integral is also well-defined under the excursion measure of the Bessel process. While the excursions can be stitched together to form a Bessel process that has 0 as a reflecting boundary, the positive values of these integrals are not summable so that the representation (2.1) fails beyond T R (0). However, (2.1) can be extended beyond T R (0) if some compensation is introduced, as follows. It is well-known that the Bessel process R has jointly continuous space-time local times on (0, ∞) 2 . To obtain a family of local times that extends continuously to [0, ∞) 2 , it is convenient to choose the level-a local time (L a (t), t ≥ 0), a > 0, of R such that the occupation density of R is (a d−1 L a (t), a > 0, t ≥ 0). By the occupation density formula and since L 0 (t) = 0 for t < T R (0), we can write for t < T R (0). Bertoin showed that defining H(t) by the right-most integral in (2.2) also for t ≥ T R (0) yields a path-continuous process H with unbounded variation, but zero quadratic variation (the finiteness of H follows from the Hölder continuity of L a (t) in a). Clearly, this process H is increasing on all excursion intervals of R away from zero, but the effect of the compensating local time at zero is that H does not increase across the zero-set of R. With this notation, we have Bertoin noted that (R, H) is a Markov process and that (0, 0) is recurrent for this Markov process. It is instructive to consider the excursions of (R, H) away from (0, 0) by plotting R(t) against "time" H(t). Since H increases during each excursion of R away from 0, on ( s , r s ), say, such a plot shows a time-changed excursion of R starting from 0 at "time" H( s ) and returning to 0 at "time" H(r s ) > H( s ).
As H does not increase across the zero-set of R, the excursions for different ( s , r s ) overlap, in general, when included in the same plot.
Since H is increasing when R is away from 0, and can only decrease across the zero-set of R, the excursions of (R, H) away from (0, 0) typically consist of many excursions of R. Specifically, each excursion of (R, H) can be decomposed into three parts: first, at the "beginning", there is an escape from (0, 0) towards the left by an accumulation of short R-excursions until, in the "middle", one R-excursion takes (R, H) across to positive H-values and, at the "end", there is a final approach back to (0, 0) from the right by an accumulation of short R-excursions.
The main objects of interest in Bertoin's work [2,3] are • the excursions away from (0, 0) of (R, H), and associated quantities, • the excursions away from 0 of R : for some T > 0. Specifically, some of the main results of [2] are the following. We use Bertoin's numbering for ease of reference.

The inverse local time
The main additional results of [3] are the following. I. 6 The semi-group of R is characterised by its Laplace transforms, for γ ≥ 0, 1} admits a continuous version in the space N ((0, ∞)) of point measures that are finite on (ε, ∞) for all ε > 0, equipped with the topology of vague convergence.
3. Skewer processes of marked Lévy processes [4,5] Let α ∈ (0, 1) and X a spectrally positive Stable(1+α)-process with Laplace exponent ψ(c) = c 1+α /2 α Γ(1+α). We call X scaffolding and proceed to decorate it. Specifically, consider the PRM i∈I δ (s i ,∆X(s i )) of its jumps. For each jump ∆X(s i ), These excursions were studied by Pitman and Yor [7], who also noted, in their Remark (5.8) on pp. 453f., that when conditioned on their length, they are BESQ(4+2α) bridges from 0 to 0. By standard marking of PRMs, N : where E is the space of (continuous) excursion paths. This is illustrated in a simplified way in Figure 1.1. The intensity measure Leb ⊗ ν of N is the Pitman-Yor excursion measure of [7], which can be described by entrance laws and a further evolution as unconditioned BESQ(−2α) processes.
Recall that the skewer of Definition 1.3 extracts from N = i∈I δ (s i ,f i ) all level-y spindle masses f i (y − X(s i )), where y ∈ (X(s i −), X(s i )), i ∈ I, and builds the interval partition that has these as interval lengths in the order given by the s i , i ∈ I. The set I H of all interval partitions can be equipped with a distance d H that applies the Hausdorff metric to the set of points not covered by the intervals.
Since X is spectrally positive, its (càdlàg) excursions away from 0 (or any other level y) start negative, jump across zero and end positive. Applied to (N, X), the skewer at level y extracts one block from each excursion of X away from y. In [4], we denote the PRM of excursions of X away from y by G y and enhance the excursion theory of X to include N: each excursion e [ ,r] := (−y+X| [ ,r] ( +s), s ∈ [0, r− ]) of X has its jumps marked by spindles. We denote by F y the associated random measure whose points are pairs of e [ ,r] and the restriction N| [ ,r]×E shifted to [0, r − ] × E. In each excursion with spindle marks, the central spindle crossing 0 can be viewed as the "middle" of three parts, separating the spindles of the "beginning" where X is negative from the spindles of the "end" where X is positive.
We refer to excursions of (X, N) as bi-clades, to the negative part of such an excursion including the central spindle up to level 0 as an anti-clade, and to the remainder as a clade. To start an interval-partition-valued process from any interval partition β, we consider clades starting from Leb(V ), V ∈ β, as follows. For each interval We stitch together all excursions X V in the left-to-right order of V ∈ β to form a scaffolding X β , similarly build N β from N V , V ∈ β, and consider skewer(N β , X β ).

Some of the main objects of interest are
• the pair (X, N) of the Stable(1 + α) scaffolding X and the PRM N of spindles, • the random point measures F y , y ≥ 0, of bi-clades of (X, N), • the type-1 evolution (β y , y ≥ 0) := skewer(N β , X β ), extracting intervals from the spindles in jumps of X β crossing level y, for any initial interval partition β. • the total mass process ( β y , y ≥ 0). Some of the main results of [4] are the following, in the numbering of [4]. x of the spindle (R, f R ) in N at the time R when the scaffolding X crosses the line X = 0, the clade part (post-R) and the time-reversed anti-clade part (pre-R) are independent and distributed as clades starting from x. 5.5 The skewer processes of (N, X) stopped at stopping times including τ 0 X (u) and S X (−u) for u ≥ 0 are type-1 evolutions.
In [5], we further prove the following.

Construction of (X, N) from (R, H) and vice versa
Let τ 0 R (s) = inf{L 0 (t) > s}, s ≥ 0, be the inverse local time of R at 0, and K the PRM of excursions of R away from 0. For each excursion interval (τ 0 R (s−), τ 0 R (s)) = ( , r) of R, we decompose the Bessel excursion (R( + t), 3), and we define the associated occupation density local time process λ s := (λ y s , y ≥ 0) of (H( + t) − H( ), 0 ≤ t ≤ r − ). Let us work out the constant c for which stopping (X, N) at τ 0 X (cu) yields the same initial distribution for the skewer process as the stopped scaffolding-andspindles pair constructed from (R, H) stopped at τ (0,0) R,H (u). We do this using the parts of [2, Lemma 3.3] and [5, Proposition 3.2] that we recalled in Sections 2 and 3 here. Specifically, the statistics of excursions of (R, H) of H-infima directly transfer to H • τ 0 R -infima that correspond to X-infima, which, by bi-clade reversibility (see 4.11 above) or the mid-bi-clade Markov property (see 4.15 above) have the same rates as X-suprema in a bi-clade. But the rates of H-infima and X-suprema differ by the constant c = 2 α = 2 1−d , hence X needs to run longer than H • τ 0 R , by a factor of c, to achieve the same number of excursions exceeding any given level y.
Finally, we note that the skewer process associated with ( ) is a diffusion by [4,Theorem 1.4], again as recalled in Section 3 here.
A similar argument to work out c can be based on the values of R when crossing H = 0 and the mass of the central spindle of N when crossing X = 0. Note, however, that these also differ by a factor of 2, by (4.1). Proposition 4.1 makes precise the sense in which the framework of a single Bessel process R ∼ BES 0 (d) of [2,3], via (R, H), yields the scaffolding-and-spindles framework (N, X) of [4,5]. The main step in the proof is time-changing the excursions of R away from 0 to form BESQ(−2(1 − d)) spindles. Let us invert this time-change and construct R from the spindles of N. To this end, recall our notation ν for the Pitman-Yor excursion measure of BESQ(−2α) of Section 3. Then i∈I δ (s i ,e i ) has the same distribution as the Itô excursion process K of R ∼ BES 0 (d), up to a linear time change, with d = 1 − α. In particular, the e i can be stitched together in the order of the s i , i ∈ I, to yield a process R ∼ BES 0 (1 − α).
Proof. This follows from Proposition 4.1. Specifically, mapping f i to e i is elementary since all f i are continuous with compact support a.s.. In present notation, we can write (4.1) as This is a.s. well-defined for all e i , i ∈ I, so the time-changes relating f i and e i are bijective, and hence the associated PRMs are bijectively related by standard mapping of PRMs. In particular, we deduce the claimed distributional identities up to a linear time change. The construction of Markov processes from excursions has been well-studied [10]. Note that a linear time change of the PRM has no effect on the BES 0 (1 − α)-excursions themselves. Specifically, we define τ (s) = i∈I : s i ≤s ζ i , s ≥ 0, and R(τ (s i −) + t) = e i (t), 0 ≤ t ≤ ζ i , also setting R(t) = 0 for t ∈ i∈I [τ (s i −), τ (s i )] and obtain R ∼ BES 0 (1−α), and this is the same process as if we replace s i by as i , i ∈ I, throughout, a > 0. The process τ is an inverse local time of R at 0, and replacing s i by as i corresponds to a different choice of local time.
Corollary 4.4. For (X, N) as in Section 3 and notation R as in Proposition 4.3, with τ (s) = i∈I : s i ≤s ζ i , s ≥ 0, define H on the range of τ as H(τ (s)) := X(s), s ≥ 0, and outside the range of τ as Then the pair (R, H) has the same distribution as (R, H) of Section 2.
Proof. Since H is determined by R via (2.2) and R d = R, it suffices to show that H relates to R in the same way. Indeed, we have H • τ = X, by construction, and X(s) is the compensated limit of its jumps ∆X( where (a d−1 L a i (∞), a ≥ 0) is the continuous version of the total occupation density local time of e i at level a. This entails that the right-most equality of (2.2) holds for t = τ (s), when (R, H) is replaced by (R, H). Since these limits exist almost surely uniformly for s in compact intervals, they also hold at t = τ (s i −), i ∈ I. Beyond the range of τ , we have, for each i ∈ I, Hence, we obtain and this completes the proof.

5.
Further consequences of the connection between [2,3] and [4,5] In the light of the results of Section 4, the results of [2,3] and [4,5] are closely related. Indeed, many results of [2,3] can now be deduced from [4,5], and the approach of [2,3] could be refined to handle the additional order structure needed for the interval partitions of [4,5].   [4] or [5], and from [2,3] that are analogues of each other.
One may note, however, that these results differ in detail, not just because (β y , y ≥ 0) and ( [4,Proposition 3.2] find stable inverse local times of different indices, but fundamentally play the same role, since they provide the time parameterisations for the PRMs of excursions of (R, H) and of bi-clades, respectively.
But as (N 0 , X 0 ) has been constructed from (R, H) as Proposition 4.1 did for the proof of Theorems 1.2 and 1.4, we read from (4.2) the coupling so the distribution of µ y [0,T H (−1/ρ)] follows from the distribution of the ranked sequence of interval lengths of the pseudo-stationary β y , which are PD(α, 0) scaled by independent Q(y), as required.