Squared Bessel processes of positive and negative dimension embedded in Brownian local times

The Ray--Knight theorems show that the local time processes of various path fragments derived from a one-dimensional Brownian motion $B$ are squared Bessel processes of dimensions $0$, $2$, and $4$. It is also known that for various singular perturbations $X= |B| + \mu \ell$ of a reflecting Brownian motion $|B|$ by a multiple $\mu$ of its local time process $\ell$ at $0$, corresponding local time processes of $X$ are squared Bessel with other real dimension parameters, both positive and negative. Here, we embed squared Bessel processes of all real dimensions directly in the local time process of $B$. This is done by decomposing the path of $B$ into its excursions above and below a family of continuous random levels determined by the Harrison--Shepp construction of skew Brownian motion as the strong solution of an SDE driven by $B$. This embedding connects to Brownian local times a framework of point processes of squared Bessel excursions of negative dimension and associated stable processes, recently introduced by Forman, Pal, Rizzolo and Winkel to set up interval partition evolutions that arise in their approach to the Aldous diffusion on a space of continuum trees.


Introduction and statement of main results
Squared Bessel processes are a family of one-dimensional diffusions on [0, ∞), defined by continuous solutions Y = (Y (x), 0 ≤ x ≤ ζ) of the stochastic differential equation where δ is a real parameter, B = (B(x), x ≥ 0) is standard Brownian motion, and ζ is the lifetime of Y , defined by It is known [24, Chapter XI], [11] that this SDE has a unique strong solution, with ζ < ∞ and Y (ζ) = 0 almost surely if δ ≤ 0, when we make the boundary state 0 absorbing by setting Y (x) = 0 for x ≥ ζ. The distribution on the path space C[0, ∞) of the process Y so defined, for each starting state y ≥ 0 and each real δ, is denoted BESQ y (δ). For each real δ, the collection of laws (BESQ y (δ), y ≥ 0) defines a Markovian diffusion process on [0, ∞), the squared Bessel process of dimension δ, denoted BESQ(δ). Several other constructions and interpretations of BESQ(δ) are known. In particular, • BESQ(δ) for δ = 1, 2, . . . is the squared norm of standard Brownian motion in R d ; • BESQ(δ) may be understood for all real δ as a continuous-state branching process, with an immigration rate δ if δ > 0, emigration rate |δ| if δ < 0, and lifetime ζ at which the population dies out. The case of immigration has been well-studied [13,26,25,24,14,18]. The literature on the case of emigration is rather sparse [21,9], but scaling limit results for discrete branching processes with emigration [27,28] with BESQ(δ) limits for δ < 0 can be obtained from [1,2,21]. While dimension δ = 0 is critical for whether the BESQ(δ) process has finite or infinite lifetime, dimension δ = 2 is well known to be critical in another respect: for a BESQ v (δ) process Y with hitting times T y := inf{x ≥ 0 : Y (x) = y}, • for δ > 2 the process is upwardly transient, with P v (T 0 < ∞) = 0 and P v (T w < ∞) = 1 for 0 < v < w, while • for δ < 2 the process is either recurrent if 0 < δ < 2, or downwardly transient if δ ≤ 0, with P v (T a < ∞) = 1 for all 0 ≤ a < v in either case. A remarkable duality between BESQ(δ) processes of dimensions δ = 2 ± 2α was pointed out in [22,Theorem (3.3) and Remark (4.2)(ii)] and [23,Section 3]: • for each real α ≥ 0 and 0 < u < v, the conditional distribution of a BESQ v (2 + 2α) process up to time T u , given the event (T u < ∞) which has probability (u/v) α , equals the unconditional distribution of a BESQ v (2 − 2α) process up to time T u . There is a similar description of BESQ u (2 + 2α) up to T v for 0 < u < v as BESQ u (2 − 2α) up to T v given T v < T 0 . This duality relation between dimensions 2 ± 2α is best known for α ∈ (0, 1). Then it relates the recurrent dimensions 2 − 2α ∈ (0, 2), for which the inverse local time process of BESQ 0 (2 − 2α) at 0 is a stable subordinator of index α, to the transient dimensions 2 + 2α ∈ (2, 4). For α = 1/2 this is the well known relation between Brownian motion on [0, ∞) with either reflection or absorption at 0, and the three-dimensional Bessel process, expressed here in terms of squared Bessel processes. But as emphasized in [23,Example (3.5)], the duality relation between dimensions 2±2α holds also for α ≥ 1, when it relates the downwardly transient BESQ(−δ) process for −δ = 2 − 2α ≤ 0 to the upwardly transient BESQ(4 + δ) process.
It was shown by Shiga and Watanabe [26] that the distribution of BESQ y (δ) for all real y ≥ 0 and δ ≥ 0 is uniquely determined by the prescription that BESQ y (1) is the distribution of ( √ y + B) 2 , and the following additivity property: for y, y ≥ 0 and δ, δ ≥ 0, and two independent processes Y and Y , if Y is a BESQ y (δ) and Y is a BESQ y (δ ) then Y + Y is a BESQ y+y (δ + δ ). (3) The distribution of BESQ y (−δ) for all y > 0 and δ > 0 is determined in turn by the duality between dimensions −δ and 4 + δ. Pitman and Yor [23] used the additivity property to construct a BESQ y (δ) process Y (δ) y for y, δ ≥ 0 as a sum of points in a C[0, ∞)-valued Poisson point process, whose intensity measure involves the local time profile induced by Itô's law of Brownian excursions. The C[0, ∞)-valued process (Y (δ) y , y ≥ 0, δ ≥ 0) then has stationary independent increments in both y ≥ 0 and δ ≥ 0. This construction, and the duality between dimensions 0 and 4, explained the multiple appearances of BESQ(δ) processes and their bridges for δ = 0, 2 and 4 in the Ray-Knight descriptions of Brownian local time processes.
This model of Brownian local times and BESQ processes, driven by a Poisson point process of local time pulses from Brownian excursions, led to a number of further developments. In particular, as recalled later in Lemmas 5 and 6, if a reflecting Brownian motion |B| is perturbed by adding a multiple µ of its local time process at 0, to form X := |B| + µ , where µ might be of either sign, then the resulting perturbed Brownian motion X has a local time process from which it is possible, by varying µ, and sampling at suitable random times, to construct BESQ(δ) processes for all real δ. The more recent notion of a Poisson loop soup [16] greatly generalizes this construction of local time fields from one-dimensional Brownian motion to one-dimensional diffusions [19] and much more general Markov processes.
Despite these constructions of BESQ(δ) for all real δ in the local time processes of perturbed Brownian motions, and the general importance of additivity properties in the construction of local time fields [20] [16], it is known [24, Exercise XI.(1.33)] and [11, top of p.332] that the additivity property (3) of BESQ processes fails without the assumption that both δ ≥ 0 and δ ≥ 0. Our starting point here is a weaker form of additivity of BESQ processes, involving both positive and negative dimensions: Proposition 1. For arbitrary real δ, δ and y, y ≥ 0, let Y , Y and Y 1 be three independent processes, with • Y a BESQ y (δ) with lifetime ζ; • Y a BESQ y (δ ) with lifetime ζ ; • Y 1 a BESQ 1 (δ + δ ). Let T be a stopping time relative to the filtration generated by the pair of processes (Y, Y ), with T ≤ ζ ∧ ζ , and let Z be the process Then Z is a BESQ y+y (δ + δ ).
By the scaling property of squared Bessel processes, for each fixed z > 0 the scaled process (zY 1 (w/z), w ≥ 0) is a BESQ z (δ + δ ). So (4) sets Z := Y + Y on [0, T ], and makes Z evolve as a BESQ(δ + δ ) on [T, ∞). This proposition and its proof are a straightforward generalization of the case with δ = −1, δ = 0 and T = ζ ∧ ζ , which was established as [9, Lemma 25]. The proof is by consideration of the SDE (1), as in the proof of the additivity property (3) in [24, Theorem XI (1.2)]. If both δ, δ ≥ 0, the conclusion Proposition 1 holds even without the assumption T ≤ ζ ∧ ζ , by combining the simpler additivity property (3) with the strong Markov property of BESQ(δ + δ ). We are particularly interested in the instance of Proposition 1 with y = 0, y = v, δ = −δ < 0 and T = ζ , which may be paraphrased as follows: The subtlety here is that we create dependence between Y and Y by specifying that Y only follows BESQ 0 (δ) independently of Y until time ζ (v), when Y hits zero, and then Y continues as needed for the additivity to hold. In [8], the authors encountered the case δ = 1 of Corollary 2 in a more elaborate context which we review in Section 4.
Let L = (L(x, t), x ∈ R, t ≥ 0) be the jointly continuous space-time local time process of Brownian motion B = (B(t), t ≥ 0) and let τ (v) = inf{t ≥ 0 : L(0, t) > v} be the inverse local time of B at 0. According to one of the Ray-Knight theorems, the process (L(x, τ (v)), x ≥ 0) is a BESQ v (0). This raises the following question: Can we find the pair (Y, Y ) of Corollary 2 embedded in the local times of B?
The following theorem provides a positive answer to this question. See Figure 1 for an illustration of the embedding.
Theorem 3. For each δ > 0, there is an increasing family of stopping times S δ (x), x ≥ 0 such that the following two families of random variables are independent: • Y (δ) 0  Thinking of BESQ(δ) as a branching processes with immigration or emigration, according to the sign of δ, Theorem 3 provides a frontier S δ varying with x, across which the emigration of BESQ(−δ) is the immigration of BESQ(δ).
The rest of this article is organized as follows: Section 2 presents the proofs of Theorem 3 and Corollary 4. In Section 3, we explore the implications of Proposition 1 by checking the laws of some marginals and functionals. We conclude in Section 4 by pointing out some related developments.

Proofs of Theorem 3 and Corollary 4
Our proof of Theorem 3 exploits the following known variants of the Ray-Knight theorems for perturbed Brownian motions R ± µ := |B| ± µ , where is the local time of B at 0 normalized so that |B| − d = B, and we assume µ > 0.
Lemma 5 (Théorème 2 of Le Gall and Yor [15], Theorems 3.3-3.4 of [32]). The spacetime local time process Lemma 6 (Theorem 3.3 of Carmona, Petit and Yor [5]). For each fixed v ≥ 0 the local time process 1]. Consider the excursions away from level 0 of reflected Brownian motion. Independently multiply each excursion by −1 with probability 1 2 (1 − γ). The resulting process X γ = (X γ (t), t ≥ 0) is known as skew Brownian motion. See [17] for a recent survey of constructions of this process, including the construction of X γ by Harrison and Shepp [12] as the unique strong solution to the equation where B is Brownian motion and γ is the local time process at 0 of X γ , that is Lemma 7. For γ ∈ (0, 1), let X γ be the skew Brownian motion driven by B as in (5). Let I + := (0, ∞) and I − := (−∞, 0), and consider time changes Proof. Denote by n ex the excursion intensity measure of reflecting Brownian motion relative to increments of . See e.g. [24,Chapter XII]. Relative to increments of the local time process γ of X γ , with γ d = , the absolute values of excursions of X γ away from 0 into I ± form independent Poisson point processes with intensity measures 1 2 (1 − ±γ) n ex (dω)ds. Note that γ (t) splits naturally into the contributions from these positive and negative excursions: By Knight's theorem [24, Theorem V.(1.9)], the time changes κ ± γ give rise to two independent reflecting Brownian motions X ± γ = X γ • κ ± γ . This argument is detailed in [24, page 242] for the case γ = 0, and extends easily to general |γ| < 1. See also the discussion after [17,Proposition 11].
In this framework of skew Brownian motion, we now give a more explicit statement of the local times decomposition claimed in Theorem 3.
Theorem 8. Let δ > 0 and γ := 1/(1 + δ). Let X γ be skew Brownian motion driven by B as in (5), with local time γ at zero. Let S δ (x) = inf{t ≥ 0 : γ γ (t) > x}, x ≥ 0. Then the following two families of random variables are independent For each v ≥ 0, the random level ζ (v) := inf{x ≥ 0 : S δ (x) > τ (v)} is almost surely finite, and coincides with the absorption time of Y v . Conditionally given ζ (v) = a, Note that, with γ = 1/(1+δ), we have B(S δ (x)) = X γ (S δ (x))+γ γ (S δ (x)) = 0+x = x, since S δ (x) is an inverse local time of X γ . Hence, S δ is a right inverse of B. Right inverses of Lévy processes were studied by Evans [6], also [30], to construct stationary local time processes. The main focus has been on the minimal right inverse, which for B is the first passage process. Theorem 8 involves a family of non-minimal right inverses.
Proof of Theorem 8. The definition of S δ (x) is such that the increasing path in Figure  1 is a multiple of the local time of the skew Brownian motion X γ . We write (5) as B(t) = X γ (t) + γ γ (t). The meaning of this expression is that the positive excursions of X γ are found in B as excursions above γ γ (t), while the negative excursions of X γ are found in B as excursions below γ γ (t). Recall that W + γ and W − γ comprise excursions of B above and below γ γ , respectively. The theorem identifies the distributions of these local times by application of Lemmas 5 and 6, as will now be detailed.
In the setting of Lemma 7, we can apply Lemma 6 to W − γ to see that for all v > 0, the process W − γ up to the inverse τ − γ (v) = inf{s ≥ 0 : L − γ (0, s) > v} of the local time (L − γ (0, t), t ≥ 0) of W − γ at zero has two independent local time processes i.e. 2 − 2/µ − γ = 1 − 1/γ = −δ < 0, since γ = 1/(1 + δ) ∈ (0, 1). Similarly, Lemma 5 with a = ∞, yields that W + γ has ultimate local time process i.e. 2/µ + γ = −1 + 1/γ = δ. Let us rewrite the results of the last two paragraphs in terms of the local times L = (L(x, t), x ∈ R, t ≥ 0) of B. Recall that S δ (x) = inf{t ≥ 0 : γ γ (t) > x}, x ≥ 0, where γ = 1/(1 + δ), and also set S δ (x) := 0 for x < 0. Note that where for each t ≥ 0 and in each of the two statements, at least one of the inequalities is an equality. Let Then the occupation measure U + γ of W + γ can be related to the occupation measure U of B by the usual change of variables u = κ + γ (r), separately on each excursion interval of X γ of a positive excursion to give By the occupation density formula for U in its general form for time-varying integrands, we obtain Hence (L(y, κ + γ (s) ∧ S δ (y)), y ∈ R, s ≥ 0) is a local time for W + γ that is right-continuous in s and in y. In particular, we deduce from the continuity of y → L + γ (y, ∞) that L + γ (y, ∞) = L(y, S δ (y)) for all y ≥ 0 almost surely. Similarly, we use the joint continuity of local times of W − γ of [5, Theorem 3.2] to obtain that L(y, κ − γ (s)∧S δ (y))−L(y, S δ (y)) = L − γ (y, s) almost surely. In particular, we have ). To complete the proof, we consider the random level ζ (v) = inf{x ≥ 0 : L(x, τ (v)) = L(x, S δ (x)∧τ (v))}. Since L and S δ are both increasing and B(τ By the proof of Theorem 8, we have (Υ top , Υ mid ) independent of Υ rest , and we have Υ top independent of (Υ mid , Υ rest ). Hence, Υ mid is independent of Υ top . By additivity, Υ mid

Some checks on Proposition 1
To simplify presentation, for any real δ and v ≥ 0 we will denote by BESQ v (x), x ≥ 0) a process with law BESQ v (δ). For r ≥ 0 let γ(r) denote a gamma variable, with γ(0) = 0 and Fix To show this, first recall some known facts: Here (a) is a consequence of the additivity property (3), while (b) details for −δ = 2− 2α ≤ 0 the entrance law for BESQ(2−2α) killed at T 0 , with α > 0, which was identified by [23, (3.2) and (3.5)]. The case of (b) for δ = 0 and µ = 1/(2b) is also an easy consequence of the Ray-Knight description of Brownian local times L(x, (1) . Applying this instance of (b), we find that BESQ (0) γ(1)/µ (x) has Laplace transform (in λ) On the other hand, we obtain the Laplace transform of Y (x) + Y (x) by conditioning Y (x) and Y (x) on ζ = inf{x ≥ 0 : Y (x) = 0}. Specifically, now using (b) for Y and (a) for Y , we find By the additivity property of BESQ(0) and then proceeding as for (8), we have The change of variables m = xu, dm = xdu allows the integral in (9) to be expressed as Writing a = 2λx, b = 2µx and q = 1 + δ/2, this integral is of the form where the integral is evaluated as F (1) − F (0) for the indefinite integral The identification of (8) and (9) is now elementary using (10). Consider next the distribution of ∞ 0 (Y + Y )(x)dx in the setting of Corollary 2. By the corollary, this is the distribution of the corresponding integral of a BESQ v (0), which according to the Ray-Knight theorem for local times of B at time τ (v) is that of The equivalent equality of Laplace transforms at 1 2 λ 2 reads This formula too can be checked from the construction of Y and Y by conditioning on ζ . For the BESQ 0 (δ) process Y on [0, m] continued as BESQ Y (m) (0), we have by Lemma 5 applied with a = m and µ = 2/δ, since these substitutions make with a/µ = 1 2 δm. On the other hand, given ζ = m, an application of the formula of [23, Proposition (5.10)] yields the first passage bridge functional To complete the calculation of the left side of (11) we must integrate the product of expressions in (12) and (13) with respect to the distribution of ζ , the absorption time of BESQ v (−δ), which is the distribution of v/(2γ(1 + δ/2)) with density is the gamma(r) density at t as in (7). So at the level of the total integral functional, the identification of the distribution of Y +Y as BESQ v (0) implies the identity Make the change of variables x = λm, dm = dx/λ, then set t = λv/2, δ = 2p, to see that this evaluation shows that Corollary 2 has the following consequence.
Corollary 10. For all t > 0 and p ≥ 0, The simplest case of the Corollary is for p = 0. Then it is some variation of Knight's analysis of the joint distribution of τ (v) and M (τ (v)) := max{|B(s)|, 0 ≤ s ≤ τ (v)}. See Section 11.3 of [31], especially formula (11.3.1). For p = 0 or p = 1, the integrals are easily evaluated using the elementary indefinite integrals According to Mathematica, there are similar expressions for p = 2, 3, . . ., but they get more complicated and their general structure is not readily apparent. This kind of argument can be extended to a full proof of Proposition 1 by the method of computating the Laplace functional for suitable measures ρ on (0, ∞), and showing that it equals the known Laplace functional of BESQ y+y (δ − δ ), found in [23] when δ − δ ≥ 0, for enough such ρ. Let us briefly sketch this here for the (slightly easier) case when δ + δ ≥ 0 and y = 0, y = v. Without loss of generality, δ > 0, as otherwise the statement follows from full additivity. The case y > 0 is then also straightforward, while δ − δ < 0 follow similarly. We claim that for any function f : [0, ∞) → [0, ∞) that is Lebesgue integrable on [0, z] for all z ≥ 0, the Laplace functional (14) reduces to the known Laplace functional of BESQ v (δ − δ ) when ρ(dx) = f (x)dx. By [24,Theorem XI.(1.7)], the latter is where φ is the unique positive, non-increasing solution to the Sturm-Liouville equation This solution is convex and converges at ∞ to some φ(∞) ∈ (0, 1]. We compute the Laplace functional in the setting of Proposition 1 by conditioning first on ζ (v) = x and then on Y (x) = y. We use notation P (γ) y := BESQ y (γ) and also P (γ),x a,b for the distribution of a BESQ(γ) bridge of length x from a ≥ 0 to b ≥ 0. Specifically, for δ > 0, for each a > 0 we define P (−δ ),x a,0 for x ≥ 0 to be the first passage bridge obtained as the weakly continuous conditional distribution of P (−δ ) a ( · | T 0 = x). By duality, this equals P (4+δ ),x a,0 , which is the time reversal of P (4+δ ),x 0,a . See [23]. We need several expectations of quantities of the form L(f, w) := exp(− w 0 Y (u)f (u)du) and also use notation θ x (f ) = f (x + ·). In this notation, we want to compute The key technical formula is a generalisation of [24, Theorem XI.(3.2)] from unit-length bridges to bridges of length x, which we express in terms of the solution of φ = 2f φ without restricting f to support in [0, x]. We obtain for all γ > 0, where for the first, this yields the two bridge functionals of (17) while the remaining functional can be obtained from [24, Theorem XI.(1.7)]. We leave the remaining details to the reader.

Related developments in the literature
We discuss three related developments in the literature. These are interval partition diffusions, an instance of a weaker form of additivity related to sticky Brownian motion, and local time flows generated by skew Brownian motion.
Motivated by Aldous's conjectured diffusion on a space of continuum trees, the authors of [8] study interval partition diffusions in which interval lengths evolve as independent BESQ(−1) processes until absorption at 0, while new intervals are created according to a Poisson point process of BESQ(−1) excursions. See also [9] and further references there. Specifically, the construction for one initial interval of length v is illustrated in Figure  2 (left). Next to a BESQ v (−1) process Y with absorption time ζ (v), the total sums of all other interval lengths form a process Y that is shown to be BESQ 0 (1) up to ζ (v) continuing as BESQ(0), as in Corollary 2. See [8, Theorem 1.5 and Corollary 5.19]. A generalisation to BESQ(−δ) for δ ∈ (0, 2) is indicated in [10, Section 6.4], to be taken up elsewhere.
Shiga and Watanabe [26] showed that families of one-dimensional diffusions with the additivity property can be parameterised by three real parameters, one of which corresponds to a linear time-change parameter affecting the diffusion coefficient, which we fix here without loss of generality. The family formed by the other two parameters, δ and µ, are the strong solutions to the stochastic differential equation We observe that these families may be extended to δ < 0 with absorption at 0 much as in (1), and that the statement and proof of Proposition 1 generalizes straightforwardly  Figure 2. Left: The left-most shaded area is Y ∼ BESQ v (−δ) for some δ ∈ (0, 2), represented in the widths of a symmetric "spindle" shape.
Other shaded areas form a Poisson point process of BESQ(−δ) excursions placed on a "scaffolding", an induced Stable(1+δ/2) process whose jump heights are the excursion lifetimes. Simulation, as in [9], due to [7]. Right: Simulation of Brownian motion split along two increasing paths.
to this case. Warren [29,Proposition 3] establishes an additivity in the case δ = 0 that involves the parameter µ of (19), where the second process Y = (Y (t), t ≥ 0) is driven by a Brownian motion (B (t), t ≥ 0) independent of the first process Y , but the first process (Y (t), t ≥ 0) appears in the dt-part of its stochastic differential equation: Specifically, Y + Y d = BESQ y (0). Furthermore, [29, Theorems 10-11 and Proposition 12] demonstrate how to find this decomposition embedded in the local times of a given Brownian motion, using the Brownian motion to drive a stochastic differential equation whose strong solution is sticky Brownian motion of parameter µ ≥ 0.
Burdzy et al. [3,4] treat other aspects of what they call the local time flow generated by skew Brownian motion. They study solutions to uncountably many coupled variants of (5) jointly. Specifically, [4] focusses on (X s,x γ (t), L s,x γ (t)), t ≥ s, x ∈ R, for B(t) in (5) replaced by x + B(t) − B(s), while [3] exhibits various one-dimensional families indexed by x or by γ that form Markov processes in a way reminiscent of Ray-Knight theorems.