Generalised stable Fleming-Viot processes as ﬂickering random measures

We study some remarkable path-properties of generalised stable Fleming-Viot processes (includ-ing the so-called spatial Neveu superprocess), inspired by the notion of a “wandering random measure” due to Dawson and Hochberg (1982). In particular, we make use of Donnelly and Kurtz’ (1999) modiﬁed lookdown construction to analyse their longterm scaling properties, ex-hibiting a rare natural example of a scaling family of probability laws converging in f.d.d. sense, but not weakly w.r.t. any of Skorohod’s topologies on path space. This phenomenon can be explicitly described and intuitively understood in terms of “sparks”, leading to the concept of a “ﬂickering random measure”. In particular, this completes results of Fleischmann and Wachtel (2006) about the spatial Neveu process and complements results of Dawson and Hochberg (1982) about the classical Fleming Viot process.


Classical and generalised Fleming-Viot processes
In 1979, Fleming and Viot introduced their now well-known probability-measure-valued stochastic process as a model for the distribution of allelic frequencies in a selectively neutral genetic population with mutation (cf. [FV79]). More formally, they introduced a Markov process {Y δ 0 ,∆ t , t ≥ 0}, with values in 1 ( d ) (denoting the probability measures on d ), such that for functions F of the form (1.1) where n ∈ and φ i ∈ C 2 c ( d ), the generator of {Y δ 0 ,∆ t , t ≥ 0} can be written as with ∆ the Laplace operator. The meaning of the superscripts in {Y δ 0 ,∆ t , t ≥ 0} will become clear once we identify this process as a special case of a much larger class of processes.
It is well known (cf. [DH82]) that the classical Fleming-Viot process is dual to Kingman's coalescent, introduced in [K82], in the following sense (our description being rather informal). For t ≥ 0, if one takes a uniform sample of n individuals from Y δ 0 ,∆ t and forgets about the respective spatial positions of the n particles, then their genealogical tree backwards in time can be viewed as a realisation of Kingman's n-coalescent. That means, at each time t − s, where s ∈ [0, t] (hence backwards in time), the ancestral lineages of each particle merge at infinitesimal rate k 2 , where k ∈ {2, . . . , n} denotes the number of distinct lineages present at time t − s(−). This can be made rigorous, for example, using Donnelly and Kurtz' lookdown construction ( [DK96]), and spatial information may also be incorporated, see e.g. [Eth00], Section 1.12.
Since its introduction, the Fleming-Viot process received a great deal of attention from both geneticists and probabilists. One reason is that it is the natural limit of a large class of exchangeable population models with constant size and finite-variance reproduction mechanism, in particular the so-called Moran-model, and can be viewed as the infinite-dimensional analogue of the Wright-Fisher diffusion. See [Eth00] for a good overview.
More general limit population processes describing situations where, from time to time, a single individual produces a non-negligible fraction of the total population have been introduced in [DK99] (see also [BLG03] for a different approach). We follow [BLG03] in calling such processes generalised Fleming-Viot processes. The limits of their dual genealogical processes have been classified in [Sa99], [MS01]. See [BB09] for an overview. Generalised Fleming-Viot processes are probability measure valued Markov processes Y Λ,∆ α whose generator acts on functions F of the form (1.1) with φ i in the domain of ∆ α as with Λ a finite measure on [0, 1], and ∆ α = −(−∆) α/2 is the fractional Laplacian of index α ∈ (0, 2], see e.g. [Y65], Chapter IX.11, or [Fe66], Chapter IX.6, i.e., ∆ α is the generator of the semigroup (P Note that for notational convenience, we denote by (P (2) t ) t≥0 the semigroup of d-dimensional Brownian motion with covariance matrix 2Id at time 1.
Remark 1.1. The special form (1.2) which the generalized Fleming-Viot generator takes when acting on functions of type (1.1) highlights its connection to the corresponding dual coalescent processes. Note that (1.2) has been derived e.g. in the proof of Theorem 3 in [BLG03], characterizing the Λ-Fleming-Viot process as solution to a well-posed martingale problem (which implies the strong Markov property); see also [DK99,Thm. 4.3]. For the 'general form' of the generator of the Λ-Fleming-Viot process and its construction as flow of bridges see Section 5.1 and (16) in [BLG03], or, alternatively, the explicit construction via particle systems in [DK99] or [BBM + 09], where the latter reference also provides a classical construction via the Hille-Yosida Theorem.
We endow 1 ( d ) with the topology of weak convergence, which we think of being induced by the Prohorov metric d , the space of càdlàg paths, endowed with the usual Skorohod (J 1 -)topology (cf. [S56], or [Bi68], Chapter 3).
In [BBC + 05], the relation between stable continuous-mass branching processes {Z t , t ≥ 0} and Beta(2 − β, β)-Fleming Viot processes, for β ∈ (0, 2], (with a "trivial" spatial motion) has been explored. Informally, Z t /〈1, Z t 〉, time-changed with the inverse of (1.5) is a Beta(2 − β, β)-Fleming Viot process. This can be viewed as an extension of Perkins' classical disintegration theorem ( [EM91], [Pe91]) to the stable case. It is in principle easy to include a spatial motion component, but note that then the corresponding Fleming-Viot process uses a timeinhomogeneous motion, namely an α-stable process time-changed by the inverse of (1.5). However, Neveu's branching mechanism is stable of index β = 1, so that the time change induced by (1.5) becomes trivial. Thus we obtain Proposition 1.2 (Normalised spatial Neveu branching process as generalised Fleming-Viot process).
Under the above conditions, we have Note that in particular in this situation, the (randomly) renormalised process {X t /〈1, X t 〉, t ≥ 0} is itself a Markov process. In fact, as observed in [BBC + 05], it is the only "superprocess" with this property. This observation was the starting point of our investigation.
By considering F as in (1.1) with n = 1 resp. n = 2, it follows from the martingale problem for (1.2) that the first two moments of a generalised Λ-Fleming-Viot process only depend on the underlying motion mechanism and the total mass Λ([0, 1]).

Proposition 1.3 (First and second moment measure).
and for t 1 ≤ t 2 , writing ρ := Λ([0, 1]), and ϕ, ϕ 1 , ϕ 2 ∈ C 2 b , (1.7) In particular, the first two moment measures agree with those of the classical Fleming-Viot process, which explains Proposition 3 in [FW06]. Note that in order to establish (1.6) and (1.7), one cannot apply the Laplace-transform method as in [FW06] since the branching property does not hold in general. A proof can be found in Section 3.
Note that a simple explanation can be given in terms of the dual coalescent process mentioned before. Indeed, in [BLG03] it is shown that generalised Fleming-Viot processes are moment dual to Λcoalescents (see also [BB09]). Since the first two moments do not involve multiple coalescent events, they cannot "feel" the finer properties of the measure Λ. Of course, for moments greater than two, the moment formulae cannot be expected to agree.

Coherent wandering random measures
In the terminology of [DH82], the classical Fleming Viot process Y δ 0 ,∆ is a (compactly) coherent wandering random measure, which means that there is a "centring process" {x(t), t ≥ 0} with values in d and for each ǫ > 0 a real-valued stationary "radius process" {R ǫ (t), t ≥ 0} and an a.s. finite where B x (r) is the closed ball of radius r around x ∈ d . One natural choice for {x(t), t ≥ 0} is the "centre of mass process" Equation 3.10. However, in the context of the lookdown construction, a more convenient choice is x(t) = ξ 1 t , the location of the so-called "level-1 particle" (see Section 2). With this choice, an obvious extension of [DK96], Theorem 2.9, shows that any Y Λ,∆ α is a coherent wandering random measure. If the process Y Λ,∆ α has the compact support property, i.e., almost surely, this will also yield compact coherence, i.e. one can choose ǫ = 0 in (1.8), see [DH82], Theorem 7.2.
It is interesting to see that generalised Fleming-Viot processes need not have the compact support property, even if the underlying motion is Brownian and the initial state has compact support.
Recall that if the standard Λ-coalescent does not come down from infinity (a necessary and sufficient condition for this can be found in [Sc00]), it either has a positive fraction of singleton classes (socalled "dust"), or countably many families with strictly positive asymptotic mass adding up to one (so called "proper frequencies"), cf. [Pi99], Lemma 25. Using the pathwise embedding of the standard Λ-coalescent in the Fleming-Viot process provided by the modified lookdown construction (see (2.7) below) we see that in the first case, the positive fraction of singletons contributes an α-heat flow component to Y Λ,∆ α t , whereas in the latter case there are infinitely many independent families of strictly positive mass, so that by the Borel-Cantelli Lemma any given open ball in d will be charged almost surely.
Combining this with Proposition 1.2, we recover Proposition 14 of [FS04] on the instant propagation of the spatial Neveu branching process.

Remark 1.4.
Observe that for continuous test functions ϕ with compact support, . This is essentially Corollary 6 of [FW06], which was formulated for Λ = U only. In the subsequent Remark 7, Fleischmann and Wachtel ask about convergence of t d/α 〈ϕ, Y U,∆ α 〉. With the lookdown construction in mind, (1.9) can be understood as follows: without loss of generality assume that ϕ has support in the unit ball, put t , t ≥ 0}, the position of the level-1 particle. Then Y Λ,∆ α t (· − ξ 1 t ) converges to some stationary distribution (again, as in [DK96], Theorem 2.9). Thus if ξ 1 t is "close" to the origin, an event of probability ≈ t −d/α , C t is substantial, whereas otherwise it is essentially zero. The terms balance exactly, so that the lefthand side of (1.9) converges, but in fact as {B (α) t , t ≥ 0} is not positive recurrent, C t converges to zero in distribution (and even a.s. if α < d, i.e. if ξ 1 t is transient).

Main results: Longterm-scaling, existence of sparks, and flickering random measures
The long-time behaviour of a generalised Fleming-Viot process reflects the interplay between motion and resampling mechanism. If one attempts to capture this via a space-time rescaling, the scaling will be dictated by the underlying (stable) motion process: With these definitions, one readily expects the following convergence of finite-dimensional distributions (f.d.d.) to hold: , t ≥ 0}, this is essentially Theorem 8.1 in [DH82]. Combining Proposition 1.2 and Proposition 1.5, we recover Part (a) of Theorem 1 in [FW06].
In addition to f.d.d.-convergence, Part (b) of Theorem 1 in [FW06] provides weak convergence on D [0,∞) ( 1 ( d )) if the underlying motion of the spatial Neveu process (i.e. Λ = U) is Brownian. However, the question whether this holds in general seems to be inaccessible to the Laplacetransform and moment-based methods of [FW06], and therefore had been left open.
Rather surprisingly, it turns out that pathwise convergence does not hold if α < 2, and that, with the help of Donnelly and Kurtz' modified lookdown construction ([DK99]), it is possible to understand explicitly "what goes wrong". To this end, we introduce the concept of "sparks" and of a family of "flickering random measures". The space-time scaling family of many generalised Fleming-Viot processes satisfies this definition: Lemma 1.8 (Generalised Fleming-Viot processes as flickering random measures). If α < 2 and Λ((0, 1)) > 0, there exists ǫ > 0 such that We will see below that the behaviour of {Y Λ,∆ α [k], leading to an ǫ-1/k-spark described by condition (1.12) typically arises as follows: At times t 1 and t 3 , Y Λ,∆ α [k] is (almost) concentrated in a small ball with (random) centre x, say. At time t 2 , suddenly a fraction ǫ of the total mass appears in a remote ball with centre y, where |x − y| ≥ 1, and vanishes almost instantaneously, i.e., by time t 3 .
Technically, we see that Lemma 1.8 shows that the modulus of continuity w ′ (·, δ, T ) of the processes Y Λ,∆ α [k], see (3.5) below, does not become small as δ → 0, contradicting relative compactness of distributions on D [0,∞) ( 1 ( d )). Intuitively, at each infinitesimal "spark", a limiting process is neither left-nor right-continuous. We will see below how this intuition can be made precise in the framework of the (modified) lookdown construction.
The situation is different if Λ = cδ 0 for some c > 0 and α < 2. Here, each Y cδ 0 ,∆ α [k] a.s. has continuous paths, so that any limit in Skorohod's J 1 -topology would necessarily have continuous paths. However, the f.d.d. limit {δ B (α) t , t ≥ 0} has no continuous modification. Intuitively, there is no "flickering", but an "afterglow" effect: >From time to time, a very fertile "infinitesimal" particle jumps some distance, and then founds an extremely large family, so that the population quickly becomes essentially a Dirac measure at this point, while at the same time the rest of the population (continuously) "fades away".
To complete the picture, we are finally able to provide the full classification of the scaling behaviour of generalised α-stable Fleming-Viot processes. Remark 1.10 (Other Skorohod topologies). Note that the above-mentioned "afterglow"phenomenon in the case Λ = cδ 0 and α < 2 fits well to Skorohod's M 1 -topology (see [S56], Definition 2.2.5), which is tailor-made to establish convergence in situations in which a discontinuous process is approximated by a family of continuous processes. However, in the situation of Lemma 1.8, Condition (1.12) implies that the distributions of the processes Y Λ,∆ α [k] cannot converge with respect to any of the topologies considered in [S56]. We are not aware of a reasonable topology on Remark 1.11 (The case of star-shaped genealogies). Note that the case Λ = cδ 1 , c > 0, has been excluded from the setup of this subsection. Here, occasionally (i.e. with rate c), the whole population always jumps to the position of the level-1-particle, producing a star-shaped genealogy. Inbetween, the α-heat flow acts so that the process is continuous between consecutive jumps. Hence, it is clear that Y cδ 1 ,∆ α [k] converges weakly to the stable unit motion δ B α , approximated by the rescaled path of the level-1-particle.

Donnelly and Kurtz' lookdown construction 2.1 A countable representation for generalised Fleming-Viot processes
We consider a countably infinite system of individuals, each particle being identified by a level j ∈ . We equip the levels with types ξ j t in d , j ∈ . Initially, we require the types ξ 0 = (ξ j 0 ) j∈ to be an i.i.d. vector (in particular exchangeable), so that for some probability measure µ ∈ 1 ( d ), which will be the initial condition of the generalised Fleming-Viot process constructed below via (2.6). The point is that the construction will preserve exchangeability.
There are two "sets of ingredients" for the reproduction mechanism of these particles, one corresponding to the "finite variance" part Λ({0}), and the other to the "extreme reproductive events" described by Λ 0 = Λ − Λ({0})δ 0 . Restricted to the first N levels, the dynamics is that of a very particular permutation of a generalised Moran model with the property that always the particle with the highest level is the next to die.
For the first part, let {L i j (t), t ≥ 0}, 1 ≤ i < j < ∞, be independent Poisson processes with rate Λ({0}). Intuitively, at jump times of L i j , the particle at level j "looks down" to level i and copies the type from there, corresponding to a single birth event in a(n approximating) Moran model. Let ∆L i j (t) = L i j (t) − L i j (t−). At jump times, types on levels above j are shifted accordingly, in formulas if ∆L i j (t) = 1. This mechanism is well defined because for each k, there are only finitely many processes L i j , i < j ≤ k at whose jump times ξ k has to be modified.
For the second part, which corresponds to multiple birth events, let n be a Poisson point process on + × (0, 1] × [0, 1] with intensity measure d t ⊗ r −2 Λ 0 (d r) ⊗ (du) ⊗ . Note that for almost all realisations {(t i , y i , (u i j ))} of n, we have The jump times t i in our point configuration n correspond to reproduction events. Define for l ∈ , l ≥ 2 and J ⊂ {1, . . . , l} with |J| ≥ 2, so that L l J (t) is a.s. finite. Intuitively, at a jump t i , each level performs a "uniform coin toss", and all the levels j with u i j ≤ y i participate in this birth event. Each participating level adopts the type of the smallest level involved. All the other individuals are shifted upwards accordingly, keeping their original order with respect to their levels (see Figure 1). More formally, if t = t i is a jump time and j is the smallest level involved, i.e. u i j ≤ y i and u ik > y i for k < j, we put is the σ-algebra describing all "genealogical events" between times u and v.
So far, we have only treated the reproductive mechanism of the particle system. In-between reproduction events, all the levels follow independent α-stable motions. For a rigorous formulation, all three mechanisms together can be cast into a suitable countable system of stochastic differential equations driven by Poisson processes and α-stable processes, see [DK99], Section 6.

Pathwise embedding of Λ-coalescents in Λ-Fleming-Viot processes
Note that for each t > 0 and s ≤ t, the modified lookdown construction encodes the ancestral partition of the levels at time t with respect to the ancestors at time s before t via N t i (s) = level of level i's ancestor at time t − s.
For fixed t, the vector-valued process {N t i (s) : i ∈ } 0≤s≤t satisfies an "obvious" system of Poissonprocess driven stochastic differential equations, see [DK99], p. 195, (note that we have indulged in a time re-parametrisation), and the partition-valued process defined by is a standard Λ-coalescent with time interval [0, t]. This implies in particular by Kingman's theory of exchangeable partitions (see [K82], or, e.g., [Pi06] for an introduction), that the empirical family sizes exist a.s. in [0, 1] for each j and s ≤ t, describing the relative frequency at time t of descendants of the particle at level j at time t − s.

Proofs
Fix µ ∈ 1 ( d ) as the initial condition of the un-scaled process Y Λ,∆ α . We begin with the useful observation that, due to the scaling properties of the underlying motion process, for each k, the process {Y starting from the image measure of µ under x → x/k 1/α , has the same distribution as {Y Λ,∆ α t [k], t ≥ 0} defined in (1.10). It will be convenient to work in the following with a version of Y (k) which is obtained from a lookdown construction with "parameter" kΛ, in particular, we have Note that the family ξ i , i ∈ , used to construct Y (k) depends (implicitly) on k, but for the sake of readability, we suppress this in our notation.
Instead of following the arguments of [FW06,Lemma 19] (which only make use of the formulas for the first two moments and thus in view of Prop. 1.3 could be adapted as well), we give a proof which is directly based on the lookdown-construction. Indeed, we show that the path of the unit mass {δ B (α) t , t ≥ 0} can be viewed as the trail of the level-1-particle. To this end, we first show convergence in law of the one-dimensional distributions, i.e.
Since the motion of the level-1-particle {ξ 1 t , t ≥ 0} is a symmetric α-stable process, i.e. B (α) t = d ξ 1 t , it suffices by the special form of the limit variable to check that for each t and ǫ, where B ξ 1 t (ǫ) denotes the ball centred in ξ 1 t and radius ǫ. The latter will be implied by lim In order to check this, let Φ ǫ be a "mollified" (continuous) indicator of B ξ 1 t (ǫ) c , and note, by dominated convergence, that for any δ > 0 The second term in the last line, for each δ > 0, converges to 0 as k → ∞, cf. [Pi99], Prop. 30. Conditioning on t−δ,t , the genealogical information as defined in (2.5), we estimate the first term as follows: which for fixed ǫ tends to 0 as δ → 0.
For n time points t 1 < t 2 < · · · < t n observe that which converges to 0 by (3.2) and the Markov inequality.

Proof of Theorem 1.9
In the case α = 2, using Proposition 1.3, tightness on the space D [0,∞) ( 1 ( d )) can be proved by inspection, literally tracing through the corresponding arguments of [FW06], Lemma 20 and 21 (note that even though Equations (133)-(137) in [FW06] estimate a fourth moment, this refers only to an increment of a d-dimensional Brownian motion).

d( y(s), y(t)), (3.5)
and {t i } ranges over all finite partitions of [0, T ] such that t i − t i−1 > δ for all i.
By Lemma 1.8, there is an ǫ > 0 such that for k 0 ∈ and δ > 1/k 0 is bounded away from 0 uniformly in k ≥ k 0 .
Finally, in the case α < 2 and Λ = cδ 0 for some c > 0, note that, due to the absence of macroscopic birth events, each Y cδ 0 ,∆ α [k] a.s. has continuous paths (formally, this follows e.g. from Theorem 4.7.2 in [Da93] and the standard disintegration result, see [EM91], [Pe91]). Let µ k denote the distribution of Y cδ 0 ,∆ α [k]. Since the set of continuous paths by the Portmanteau Theorem. However, the f.d.d. limit {δ B (α) t , t ≥ 0} has no continuous modification for α < 2, and we arrive at a contradiction.

Proof of Lemma 1.8
The intuitive mechanism behind a "spark" obtained from the lookdown construction is as follows: Typically when k is large, most of the total mass of Y (k) as defined in (3.1) will be in the immediate vicinity of the location of the level-1 particle. A "spark" arises if the level-2 particle jumps to a remote position and shortly afterwards participates in an extreme reproduction event involving a positive fraction of the current population, but not the level-1 particle. In this situation, a new atom appears in the support of Y (k) , which is then removed quickly, since mass is attracted rapidly towards the position of the level-1 particle. Note that corresponding phenomena will occur on any level j ≥ 2.
First, we collect some useful notation. Without loss of generality assume T = 1. The following choices for the constants δ 2 and ǫ are justified by Lemma 4.1 and Lemma 4.2 from the Appendix. Indeed, choose δ 1 ∈ (0, 1) with Λ((δ 1 , 1)) > 0, and then ǫ = ǫ(δ 1 ) > 0 such that for any y ∈ d and any pair µ, µ ′ ∈ 1 ( d ), and choose δ 2 = δ 2 (ǫ, δ 1 ) ∈ (0, δ 1 ] such that for any x, For k ∈ , we split the time interval In order to cook up a "spark" within (a i , a i+1 ], we collect the following "ingredients": • Within the time-interval (a i , b i ], consider the event (k) i that at time b i most of the population (including the level-2 particle) is sufficiently closely related to the level-1 particle and has not moved too far away in space, more precisely, recalling (2.8), requires that the level-2 particle jumps to a sufficiently remote position and there is no subsequent lookdown-event involving level-1 and level-2, more precisely, Hence, an application of (3.7) using the observation that |ξ 1 [k]) ≤ ǫ. that the level-2-particle is sufficiently far away from the location of the level-1-particle in (b i , c i ], stays away more than 2 units during (c i , d i ] and at some time τ ∈ (c i , d i ] it accumulates a proportion of mass at least δ 1 . Hence, and we know from above that under the assumptions as before (again, given 2 by our choice of δ 1 ≥ δ 2 . Using (3.6) with y = ξ 1 τ we arrive at guarantees that most of the mass returns to the close vicinity of the level-1-particle at time a i+1 , which is, due to i , more than 2 units away from the position of the level-2-particle at time τ, but still close to its own position at time τ. Hence, as before, [k](B ξ 1 τ (1)) ≥ 1 − δ 1 /2. Using (3.6) again, we finally arrive at [k]) > 2ǫ. (3.13) Recalling Definition 1.6, we see that (3.11), (3.12) and (3.13) guarantee the existence of an ǫ- It remains to show that inf which yields the claim. In order to verify (3.14), note that by (3.9), for all i < k, and that for fixed k, the family of events is independent (by the usual independence properties of the driving Poisson processes and the increments of the motion processes for each level). Moreover, for some C = C(α, Λ, δ, (δ m )) > 0, which together with the independence properties implies (3.14). To see (3.15), note that the 'crucial' component in i , which satisfies by the scaling properties and tail behaviour of symmetric α-stable processes, see e.g. [Be96, Chapter VIII], The probability of the genealogical components of (k) i does not converge to zero, since smaller time intervals (of length 1/k) are compensated by higher resampling intensities due to the time rescaling (proportional to k) in each step. Finally, the probability of the remaining requirements (upper bounds on the distance that the level-1-particle or the level-2-particle are allowed to travel) do not converge to zero either by (3.9) or the fact that these upper bounds are fixed values, independent of k.

Proof of Proposition 1.3
The proof is a straightforward modification of arguments in [Eth00], Prop. 2.27. Indeed, it is easy to read off the function-valued dual process from (1.2), which is an obvious extension of [Da93], Section 5.6 (see [BBM + 09, Section 5.2] for an explicit description in the context of Ξ-Fleming-Viot processes, which contains Λ-Fleming-Viot processes as a special case).
For the first moment, the function valued dual consists of one component only, to which the α-heat flow is applied. This gives, for t > 0, (3.16) For the second moment, the function valued dual consists of two components, and the formula follows from considering the exponential jump time at which both components merge, yielding, for t > 0, Note that this can easily be extended to distinct time points t 1 , t 2 using the tower property and the strong Markov property. Indeed, w.l.g. assume t 1 < t 2 and write t := t 1 , t + s = t 2 for s := t 2 − t 1 . Then, using (3.16), is an independent copy of Y Λ,∆ α s . Denotingφ 2 := P (α) s ϕ(x) and observing that ϕ 2 ∈ C 2 b impliesφ 2 ∈ C 2 b , we obtain from (3.17) as required.

Appendix
Consider the metric space of probability measures ( 1 ( d ), d 1 ) on d and recall that, for µ, ν ∈ 1 ( d ), d 1 (µ, ν) := inf ǫ > 0 : µ(B) ≤ ν(B ǫ ) + ǫ for all closed sets B ⊂ d , (4.1) where B ǫ is the usual ǫ-enlargement of the set B ⊂ d . Let B y (r) be the d-dimensional closed ball centered in y with radius r. We collect two technical lemmas.