Infinite energy solutions to inelastic homogeneous Boltzmann equation

This paper is concerned with the existence, shape and dynamical stability of infinite-energy equilibria for a general class of spatially homogeneous kinetic equations in space dimensions $d \geq 3$. Our results cover in particular Bobyl\"ev's model for inelastic Maxwell molecules. First, we show under certain conditions on the collision kernel, that there exists an index $\alpha\in(0,2)$ such that the equation possesses a nontrivial stationary solution, which is a scale mixture of radially symmetric $\alpha$-stable laws. We also characterize the mixing distribution as the fixed point of a smoothing transformation. Second, we prove that any transient solution that emerges from the NDA of some (not necessarily radial symmetric) $\alpha$-stable distribution converges to an equilibrium. The key element of the convergence proof is an application of the central limit theorem to a representation of the transient solution as a weighted sum of i.i.d. random vectors.

1. Introduction 1.1. The equation. In this paper, we analyzed the long-time asymptotic of the velocity distribution in kinetic models for spatially homogeneous inelastic Maxwellian molecules [14], and certain generalizations. We assume that the space dimension d is at least two, with the physical situation d = 3 being the most interesting choice. Under the cut off assumption and after proper normalization of the collision frequency, the evolution equation for the time-dependent velocity distribution µ : R + → P(R d ) is given by ∂ t µ(t) + µ(t) = Q + (µ(t), µ(t)) (t > 0) µ(0) = µ 0 (1) where the collisional gain operator Q + has the weak formulation Above, the expectation E is taken for the post-collisional velocities v ′ , v ′ * , which are random vectors whose distribution is determined from the pre-collisional velocities v, v * by means of collision rules. Later, we formulate our hypotheses on the collision rules under which we are able to prove existence and stability of stationary solutions. For the original inelastic Maxwell molecules from [14], these rules read as where δ ∈ (0, 1/2) is the modulus of inelasticity, and n is a random unit vector of prescribed distribution on the unit sphere S d−1 : there is a properly normalized (see (7) below) density function (cross section) b ∈ L 1 (−1, 1), such that n has law where u S is the normalized surface measure (uniform probability) on S d−1 . The characteristic property of Maxwellian molecules -in contrast to more general ideal gases -is that this density does not explicitly depend on the norm |v − v * | of the relative velocity. This property allows to restate (1) as an evolution equation for the characteristic functionμ(t; ξ) = exp(iξ · v)µ(t; dv) of µ(t), with an explicit form of the Fourier transformed collision operator: there are non-negative random variables r ± and random rotations R ± in SO(d), such that holds for all ρ ∈ R + and all O ∈ SO(d). This special form of Q + is of crucial importance for our analysis of (1) by probabilistic tools. The existence of such a representation (5) is by no means obvious. A similar expression has been given for the collision operator modelling fully elastic Maxwell molecules recently [27]. For inelastic molecules, it is proven in Proposition 7 below.
Notice: In the following, we assume that the reader is familar with basic notions of the central limit theorem, in particular with the Lévy representation of multi-dimensional α-stable distributions and their normal domain of attraction (NDA). A brief introduction to this topic is included in Appendix A.
1.2. Related results. In the rich literature on long-time asymptotics for (1), both solutions with finite (kinetic) energy, that is, and with infinite energy have been studied. In order to relate our own results to the existing literature, we briefly recall a small selection of results on convergence to equilibrium for elastic and inelastic Maxwell molecules; the following summary is focussed on weak convergence results under minimal hypotheses on the initial conditions. • Finite energy solutions for fully elastic collisions. The only stationary solutions of finite energy to the fully elastic Maxwell model [11] are Gaussians, and these attract all solutions of finite energy. This is known as Tanaka's theorem [44]. Various simple proofs are available, see e.g. [45]. • Infinite energy solutions for fully elastic collisions. The elastic Maxwell model does not admit stationary solutions of infinite energy [22]. However, Bobylëv and Cercignani [12] have identified for every α ∈ (0, 2) a family of self-similar solutions for which the αth moment is marginally divergent. These self-similar solutions converge vaguely to zero as time goes to infinity, i.e., the velocities concentrate at infinity. It has been shown recently [20] that the self-similar solutions for a given α attract all transient solutions (of infinite energy) whose initial condition's characteristic functionμ 0 satisfies lim |ξ|→0μ 0 (ξ) − 1 |ξ| α = K for some K < 0.
• Finite energy solutions for inelastic collisions. Inelastic Maxwellian molecules lose kinetic energy in every collision. If the energy is finite initially, then it converges to zero exponentially fast in time [14]. As was conjectured by Ernst and Brito [29], this collapse happens in a self-similar way. More precisely, there is a time-dependent rescaling of the velocity variable such that the rescaled Boltzmann equation possesses a family of nontrivial stationary solutions, the so-called homogeneous cooling states. It has further been proven [10,13,17,18] that any solution of finite energy to the rescaled equation eventually converges towards one of these cooling states. • Infinite energy solutions for inelastic collisions. This case has received less attention than the aforementioned situations. Some results are available for the inelastic Kac model [43], which is a one-dimensional caricature of inelastic Maxwell molecules: for each inelastic Kac model, there is precisely one α ∈ (0, 2), such that the symmetric α-stable laws are stationary solutions and attract all transient solutions that start in their respective NDA [6]. A generalization of this result has been obtained by the authors [5] for Kac-type models with more complicated collisions and a richer class of stationary states. A related generalization [15,16] also covers the case of radially symmetric solutions to the inelastic Kac model in multiple space dimensions. The existence of a family of stationary solutions is proven, and the αth moment of that solutions is marginally divergent, where α ∈ (0, 2) is specific for the considered model. It is shown that the self-similar solutions attracts all radially symmetric solutions whose initial condition satisfies a condition that is slightly more restrictive than (6) as above. Using the results contained in [4], it can be proved that the same conclusions hold under condition (6). Various of these fundamental weak convergence results have been made quantitative (e.g. in terms of estimates on convergence rates) and improved qualitatively (by proving e.g. convergence in strong topologies). Naturally, such improvements require additional hypotheses on the initial data (like higher moments or finite entropy) and are not of interest here. We refer the reader to the reviews [23,46], and to the more recent results on self-similar asymptotics for inelastic Maxwell molecules [21] and for inelastic hard spheres [41].
1.3. Results and Method. In the present paper, we give a refined analysis of infinite energy solutions for kinetic equations with collision kernel of the form (5) in general, and for inelastic Maxwell molecules in particular. We show the existence of a family of stationary solutions and we give a representation for them as scale mixtures of radially symmetric α-stable laws. Our main result is the dynamic stability of stationary solutions under assumptions on the initial conditions that we expect to be minimal. The full statement is given in Theorem 3. In the special case of inelastic Maxwellian molecules, it reduces to the following. (2), where δ ∈ (0, 1/2), and the unit vector n has law (3) with cross section b, which is such that

Theorem 1. Consider equation (1) with collision rules
Then there is a unique exponent α ∈ (0, 2) and a probability measure m on R + -both computable from δ and b in principle -such that the following is true.
A one-parameter family (µ c ∞ ) c>0 of stationary solutions to (1) is given in terms of their characteristic functionsμ c ∞ bŷ If µ 0 belongs to the NDA of a full α-stable distribution (centered, if α > 1, and an additional condition is needed if α = 1 -see (13) in Section 2.2 ), then the corresponding solution µ to (1) converges weakly to a stationary solution µ c ∞ , where c ∈ R + is computable in terms of µ 0 . In particular, the µ c ∞ are the only stationary solutions that belong to the NDA of some α-stable distribution on R d .
Apparently, these are the first results on the stability of stationary solutions in the inelastic Maxwell model without the assumption of radial symmetry. Indeed, it seems that the approach to derive long-time asymptotics directly from contraction estimates on the Fourier transform of the transient solutions like in [16] or [20], needs an hypothesis on the initial datum of the form (6). This hypothesis is significantly stronger than ours, as can be seen from the characterization of NDAs by means of characteristic functions, see e.g. [2]. For instances, (6) implies that µ 0 belongs to the NDA of a radially symmetric α-stable law, which further implies that µ 0 is radially symmetric "asymptotically" on the complement of large balls. Hence, our condition that µ 0 belongs to the NDA of some full α-stable law is much weaker. In fact, we expect that the NDA is a sharp characterization of the basin of attraction for the kinetic equation in the sense that all other transient solutions either concentrate at the origin or vaguely converge to zero as time tends to infinity.
The key element in our proof is a probabilistic representation of the solution µ to (1). First, µ can be written as a Wild sum, Now each of the µ n ∈ P(R d ) is the law of a random vector V n in R d , and the V n are characterized as follows. There is an array (β k,n , O k,n ) 1≤k≤n+1 of non-negative random numbers β k,n and random rotations O k,n in SO(d), such that for every O in SO(d) where the X k are i.i.d. (independent and identically distributed) random variables, independent of β k,n and O k,n , with distribution µ 0 . Now, the techniques pertaining to the central limit theorem are adapted to conclude weak convergence of the µ n to some µ c ∞ , and this implies via (8) weak convergence of µ(t) to the same limit as t → ∞.
The general idea of a probabilistic representation of Boltzmann like equations goes back essentially to McKean [38,39], who applied it to the Kac equation, a caricature of the homogeneous Boltzmann equation in dimension d = 1. The idea has since then been extended and refined, for instance in [25,26,32,33] (for the Kac equation) and [4,5,6] (for various one-dimensional Kac-type kinetic equations).
The extension to dimension d > 1 is by no means straightforward. Only in the very recent paper [27], Dolera and Regazzini derived a suitable probabilistic representation of the solution of the homogeneous Boltzmann equation in dimension d = 3, using particular coordinates on R 3 and its rotation group. Here, we extend the Dolera-Regazzini probabilistic representation to equation (4) with kernels of the form (5), in arbitrary dimensions d ≥ 3. Our probabilistic representation is summarized in Proposition 5, which should be an interesting result in itself.
1.4. Plan of the paper. In Section 2 below we state our hypotheses and formulate the main result about the general kinetic model with collision kernel of type (5). We also introduce the main tool for the proof: the probabilistic representation of transient solutions. In Section 3, we prove that inelastic Maxwell molecules, that is (1) with collision rules (2) and an arbitrary choice for the density b of the cross section (3), fit into the general framework provided in the previous section. Sections 4 and 5 contain the proof of the main result, which is naturally divided into two parts: Section 4 is concerned with contraction estimates on a random walk in the rotation group, which is induced by our probabilistic representation. In Section 5, we apply the machinery of the central limit theorem to the representation (9) to obtain the long-time asymptotics of transient solutions to (1). The Appendix contains a summary of various results on α-stable distributions that are relevant to our proofs.

An abstract Boltzmann-like equation
In this section, we formulate our hypotheses and state our results for the general kinetic equation (4) with collision kernel (5). We will see in Section 3 that inelastic Maxwell molecules fall into this model class, so Theorem 1 from the introduction follows as a corollary from the general Theorem 3 below.
2.1. Notations. Denote by SO(d) the usual orientation-preserving rotation group in R d , andby abuse of notation -by SO(d − 1) its subgroup that acts on R d−1 ⊂ R d only, i.e., that leaves the "last" unit vector e d := (0, . . . , 0, 1) ∈ R d invariant. As usual, Accordingly, we define powers B ⋆2 = B ⋆ B etc. Finally, let H be the Haar measure on SO(d).
Under hypothesis (H2), the following defines probability measures B + and B − on SO(d): In addition to (H1)-(H2), we shall assume further: (H3) The probability measures B ± are non-singular with respect to the Haar measure, i.e. they have a non-trivial absolutely continuous component with respect to H. Before stating the general form of our main result, we briefly comment on the role of assumptions (H2) and (H3). Assumption (H2) is a classical hypothesis which guarantees the existence of a (unique up to scaling) fixed point of the smoothing transformation associated with (r − , r + ). The respective result is the following.
Assumption (H3) entails the convergence of the n-fold convolution (B ± ) ⋆n to the Haar measure H. See e.g. [8] for a proof of exponentially fast convergence in total variation. We only need a corollary of that result, which is formulated in Proposition 18.
With the notations and preliminary results at hand, we can formulate our main theorem.
Theorem 3. For a given random element (r − , r + , R − , R + ), define a collision operator Q + by means of (5). Assume that there is an α ∈ (0, 2) such that hypotheses (H1)-(H3) hold, and consider the initial value problem (1) with an initial condition µ 0 that belongs to the NDA of a full α-stable distribution with Lévy measure φ. If α > 1, assume also that µ 0 is centered, while if α = 1, assume that there is some Then the unique solution µ(t) to (1) converges weakly to the probability distribution µ c ∞ that has the characteristic function where the probability measure m is defined in Proposition 2, and In particular, the µ c ∞ are the only stationary solutions of (1) that belong to the NDA of some α-stable distribution on R d .
The proof of Theorem 3 is given in Section 5.
2.3. A probabilistic representation. As already mentioned in the introduction, the key element in our proof of Theorem 3 is a suitable stochastic representation of µ(t) connected to a randomly weighted sum of i.i.d. random vectors. This probabilistic representation enables us to study the long-time asymptotics of µ(t) by methods related to the central limit theorem.
The starting point is the Wild sum representation (8) of solutions to (1). Equivalently, the time-dependent characteristic functionμ satisfying (4) can be written aŝ where the charcteristic functionsμ n of the probability measures µ n are defined inductively from the initial conditionμ 0 as follows: The probabilistic representation we introduce now gives a meaning to the measures µ n -or rather, to their characteristic functionsμ n -in terms of randomly weighted sums of i.i.d. random vectors.
Proof. For n = 0 there is nothing to prove. For n = 1 the statement reduces to the definition of Q + in (5). We proceed by induction on n. Fix n ≥ 1 and assume that (16) is true for all k = 0, . . . , n − 1 in place of n. By construction, with a random index J ∈ {1, . . . , n} depending on ℓ 1 to ℓ n . The factorization (17) corresponds to splitting the nth binary tree at the root into a left tree (with J leaves) and a right tree (with n + 1 − J leaves). It is easy to see that J is uniformly distributed on {1, . . . , n}, see e.g. [5]. It is further easy to see that, given (J, r − 1 , r + 1 , R − 1 , R + 1 ), the random elements (β ′ j,n , O ′ j,n ) j=1,...,J and (β ′′ j,n , O ′′ j,n ) j=J+1,...,n+1 are conditionally independent. Their conditional distribution, given the event {J = k}, satisfies Thus, if (r − , r + , R − , R + ) is defined as above and it is assumed independent of all the rest, using the induction hypothesis, one can write which, by (5) and (15), equals toμ n .
We can now formulate the above mentioned probabilistic representation. The first representation of this type has been derived in [27] for the fully elastic Boltzmann equation in R 3 .
Proof. We calculate the characteristic function of the sum given in (18) at ρ ∈ R + : where we have used (16). Since two characteristic functions that coincide on the positive real axis are equal, the first claim follows.
is a random process with (marginal) distribution µ(t) for every t > 0 and (N t ) t≥0 is a random process with values in {0, 1, . . . , } and independent of (β k,n , O k,n ) k,n and (X k ) k≥1 , such that for every O ∈ SO(d) and all t ≥ 0. Indeed, it suffices to observe that by (18) above and the Wild representation (14).

The inelastic Maxwell models as a special case
The aim of this section is to show that the homogeneous Boltzmann equation with collision rules (2) is indeed a special case of the more general equation considered here. Theorem 1 then follows as a corollary of Theorem 3.
Our starting point is the equation in its Fourier representation (4), which has been derived in [14], with the collision kernel where, for any ξ ∈ R d , the two random vectors Y − ξ and Y + ξ in R d are given by with a random unit vector n which has law b σ · ξ/|ξ| du S (σ).
3.1. Preliminaries on rotation groups. We start by recalling some well-known facts about the Haar distribution. If the random matrix O has Haar distribution on SO(k), then for every orthogonal matrix G ∈ SO(k); see, e.g., Theorem 5.14 in [42]. Moreover, for any e ∈ S k−1 , A random matrix U with values in SO(d) will be called uniformly distributed on which is easily verified by the change of variables formula. Finally, we denote by Z ψ ∈ SO(d) the (positive) rotation in the e 1 − e d -plane about the angle ψ ∈ [0, π], that is and all other entries of Z ψ are zero. The following probabilistic interpretation of Hurwitz's [35] representation of the Haar measure will be of importance.
Theorem 6. There are random rotations U 1 , U 2 in SO(d) and a random angle ψ * in [0, π] such that • ψ * has a continuous probability density function that is positive on (0, π), • the law of U 1 Z ψ * U 2 is the Haar measure on SO(d).
Sketch of the proof. In [35] it is shown that an arbitrary d-dimensional rotation matrix may be written as a product of d(d − 1)/2 elementary rotations in two-dimensional subspaces. Denote by Z i,j (ψ) the matrix of an elementary rotation in the plane e i − e j of an angle ψ, i.e. the only nonzero elements of Z i,j are . Then, any rotation matrix O can be represented as The Haar distribution on SO(d) is obtained if the generalized Euler angles ψ j,i are independent, ψ 0,i are uniformly distributed on [0, 2π) for i = 1, . . . , d − 1 and ψ r,s for r = 1, . . . , s − 1 are absolutely continuous with density sin(ψ) r I [0,π) (ψ). It is then easy to see that, as a consequence of the above representation, one obtains the result.

3.2.
Definition of the probabilistic representation. Given the cross section b on (−1, 1), define the projected density Πb according to (23). Since b is normalized as stated in (7), Πb is a probability density. Let ψ be a random angle in (0, π) such that cos ψ has Πb as density, which is equivalent to saying that ψ itself is distributed with law Further, let U 1 , U 2 be random rotations in SO(d − 1) -independent of each other and independent of ψ -with the properties from Theorem 6. In particular, U 1 is uniformly distributed on SO(d − 1). From that, define two further random angles in ψ ± in (0, π) implicitly by Now set Proposition 7. For every vector ξ and every O ∈ SO(d) such that ξ = |ξ|Oe d one has The essential ingredient of the proof is the following.
Proof. We need to show that the law λ of U 1 Z ψ U 2 e d is the same as the law λ ′ of n. Both λ and λ ′ are invariant under SO(d − 1): for λ ′ , this is clear by definition in (21). For λ, this follows since U 1 , ψ and U 2 are independent, and GU 1 L = U 1 for every G ∈ SO(d − 1). By our considerations on SO(d − 1)-invariant measures above, it therefore suffices to show that the projected measures are equal, Πλ = Πλ ′ .
For λ ′ , we obtain from the definition of n and formula (23) that n · e d has law Πb. Concerning λ, recall that U 1 and U 2 take values in SO(d − 1) a.s., which implies that Z ψ e d ) = cos ψ, using the definition of Z ψ . The claim now follows since cos ψ has law Πb by definition.
Proof of Proposition 7. Let ξ = |ξ|Oe d be given. For any bounded continuos function f where we have used (20)-(21) and a change of variables in the integral. Hence . It is thus sufficient to prove the claim for ξ = e d and O = 1 d . By Lemma 8, we have To finish the proof, observe that we have which easily follows from our definitions of r ± and R ± by elementary geometric considerations.
The validity of (H2) is a consequence of the following.
Proof. Recall the convex function S defined in (10). We have On one hand, S(0) = 1, because ψ is an absolutely continuous random variable. On the other hand, since 0 < r ± < 1 almost surely, it follows that lim s→+∞ S(s) = −1. Finally, at s = 2, we have  Proof. Recall Theorem 6, and let U 1 , U 2 and ψ * , ψ be chosen as indicated above. Further, observe that, since U 1 , U 2 , ψ are independent, and since the law of ψ is given in (24), one can write, for every f ∈ C b 0 (SO(d)), where ψ ± (η) and r ± (η) are defined as functions of η via (25)- (27) using η in place ψ. Hence whereψ ± are defined via (25) from a random angleψ -being independent of U 1 and U 2 -in (0, π) with law It thus suffices to show that the laws of the random rotations U 1 Zψ ± U 2 are absolutely continuous with respect to the law of U 1 Z ψ * U 2 . Sinceψ has a density on (0, π), also cosψ ± given via (25) have densities on (−1, 1), and thusψ ± themselves have densities on (0, π), all with respect to the Lebesgue measure on the respective intervals. Since further the density of ψ * is positive on (0, π), it follows that the laws ofψ ± are absolutely continuous with respect to that of ψ * . Then also the law of the triple (U 1 ,ψ ± , U 2 ) is absolutely continuous with respect to the law of (U 1 , ψ * , U 2 ) on . And the respective images in SO(d) under the continuous map (G 1 , θ, G 2 ) → G 1 Z θ G 2 inherit the absolute continuity.

Study of an instrumental process on C 0 (SO(d))
This section is devoted to the proof of convergence of the following auxiliary random processes (Ψ n ) n≥0 taking values in C 0 (SO(d)). Given a continuous function Ψ 0 ∈ C 0 (SO(d)), define for all n ≥ 1: Throughout this section, we continue to assume hypothesis (H1)-(H3). The ultimate goal is to show convergence of Ψ n to a (random) constant function in the sense made precise in Proposition 12 below. In order to characterize the limit, we start with an auxiliary result.
have the following properties: n ] = 1 for every n. For the sake of simplicity the proof of Proposition 12 is split into several steps. Some of them use techniques developed in [7]. 4.1. Basic properties of Ψ n . Introduce the L p -norms with respect to the Haar measure H on measurable functions f : SO(d) → R as usual: Lemma 13. For every n ≥ 0, where m 0 is given in (31), and Proof. Since H is right invariant, Now (32) follows by means of (i) in Lemma 11. Another application of that property yields (33): Lemma 14. The laws of Ψ n form a tight sequence of probability measures on C 0 (SO(d)) and hence they are relatively sequentially compact.
Proof. By the classical tightness criterion for sequences of random continuous functions, see e.g. Theorem 16.5 in [37], it suffices to show that where · * is the matrix (operator) norm induced by the euclidean norm on R d , satisfies Observe that for arbitrary O 1 , O 2 ∈ SO(d), The expectation value on the right-hand side equals to one, independently of n, by Lemma 11 (i). The supremum, which is also independent of n, tends to zero for δ ↓ 0, since the continuous function Ψ 0 on the compact manifold SO(d) is automatically uniformly continuous. Since C 0 (SO(d)) is a Polish space, the last part of the statement follows from Prohorov's Theorem, see e.g. Thm. 17, Chapter 18 in [31].

Definition of the recursion operator T . Given A ∈ SO(d) and a function f on SO(d), we denote by A # f and A # f the functions given by
With these notations, Introduce a sequence (ν n ) of probability measures on C 0 (SO(d)) by ν 0 := δ Ψ0 , and for every n ≥ 1, ν n := Law Next, define a recursion operator T on the set P(C 0 (SO(d))) of all probability measures on C 0 (SO(d)) as follows. Given ν ′ , ν ′′ ∈ P(C 0 (SO(d))), let Ψ ′ and Ψ ′′ be two independent random functions with distributions ν ′ and ν ′′ , respectively, which are also independent of (r − , r + , R − , R + ). Then define T has a fixed point: set Ψ ∞ := m 0 M (α) ∞ and ν ∞ := Law(Ψ ∞ ). Using Lemma 11, it is easy to see that In the following, we shall show that this fixed point is attractive in a suitable metric.
Lemma 15. For each n ≥ 1, the following recursion relation holds: Proof. The proof is similar to the one of Proposition 4. With the notations (17), we can write The goal for the rest of this section is to show that the map T is a contractive in an appropriate metric. Once this is shown, the proof of Propostion 12 follows easily.

Contraction in Fourier distance. Recall that L 2 (SO(d), H) is a real Hilbert space with respect to the scalar product
For a probability measure ν on C 0 (SO(d)), define its L 2 -characteristic functional (or Fourier transform)ν : where Ψ is a random function with law ν. Now, given γ > 1, introduce the Fourier distance between any ν ′ , ν ′′ ∈ P(C 0 (SO(d))) by where Ψ ′ and Ψ ′′ are two random functions distributed according to ν ′ and ν ′′ . This is a variant of the original Fourier metric, which was first introduced in the context of kinetic equations in [34], and has since then been generalized in manifold ways, see e.g. [7] for another application to measures on matrices. Strictly speaking, this distance is not a metric since it might attain the value infinity. Notice further that ν ′ and ν ′′ might be "close" with respect to d γ even if their expectation values differ significantly.
Lemma 16. Let ν ′ 1 , ν ′ 2 and ν ′′ 1 , ν ′′ 2 probability measures on C 0 (SO(d)) and Ψ ′ 1 , Ψ ′′ 1 , Ψ ′ 2 , Ψ ′′ 2 be independent random functions with laws ν ′ 1 , ν ′ 2 and ν ′′ 1 , ν ′′ 2 , respectively. Then, for a given γ ∈ (1, 2), and (r − , r + , R − , R + ) be independent. We proceed in full analogy to the proof of Lemma 6 in [7]. Set ν j := T [ν ′ j , ν ′′ j ] and let Ψ j be distributed with laws ν j , respectively. Recalling the definition of R # and R # from (35), we obtain , and observe that Since R − and R + are orthogonal matrices, we have and so By definition of the Fourier transform, and since |1 − e ix | ≤ |x|, it follows that and similarly for the other supremum. To finish the proof, observe that by Young's inequality To prove Proposition 17 we need some preliminary results. Recalling the definition of B ± from (11), introduce continuous linear operators L ± on L 2 (SO(d), H) by Since for every f, g ∈ L 2 (SO(d), H), we have that it follows that the adjoint operator (L ± ) * of L ± is given by Consider the symmetric operator (L ± ) * L ± on L 2 (SO(d), H), which can be written as where we defineB ± as the law of the random rotation R T 2 R 1 for independent R 1 , R 2 with distribution B ± each. It is easy to see that the powers of (L ± ) * L ± admit the representations where ⋆n denotes the n-fold convolution of a measure. The following result is essential for the proof of Proposition 17.
Proposition 18 (Bhattacharya). Let G be a compact, connected, Hausdorff group and let β be a probability measure on G such that β has a nonzero absolutely continuous component with respect to the normalized Haar measure H on G. Then there is n ≥ 1 and 0 < c ≤ 1 such that for every measurable B ⊂ G.
Actually, in the proof of Theorem 3 in [8] it is shown that there are a set A ⊆ G of positive Haar measure, a positive numberc > 0 and an index N 0 ∈ N such that, for every g in G, where h denotes the density of the absolutely continuous component of β, and ⋆ is the convolution of functions. Here clearly N 0 can be replaced by any power of two that is larger or equal, at the possible expense of diminishingc to another (still positive) constant c. This obviously implies our assertion (40).

Lemma 19.
There areκ ± < 1 and n ≥ 1 such that Proof. We follow the lines of the proof of Theorem 2 in [8]. Assumption (H3) implies that the probability measuresB ± s have nonzero absolutely continuous component with respect to the Haar measure. Hence we can apply Lemma 18. If (B ± ) ⋆2 n = H then [(L ± ) * L ± )] 2 n f L 2 = 0, and there is nothing to be proved. If instead (B ± ) ⋆2 n = H, then c < 1 in (40), and hence one can write Since f is such that f (O)H(dO) = 0, then, using also Jensen inequality, This shows the desired inequality, withκ ± = (1 − c) < 1.
Proof of Propostion 17. Observe that and iteration of these estimates leads to for arbitrary n ≥ 0. We combine this estimate with Thus, by Lemma 19, we arrive at Taking the 2 n+1 th root, the hypothesis follows with κ ± := (κ ± ) 1/2 n+1 < 1.
Remark 2. Note that in order to prove Proposition 17 one only need the assumption (H3') The probability measuresB ± are non-singular with respect to the Haar measure, i.e. they have a non-trivial absolutely continuous component with respect to H.

4.5.
Convergence of Fourier transforms. Out of the Fourier distance d γ , we define yet another distance on P(C 0 [SO(d)]) by where Ψ ′ , Ψ ′′ have law ν ′ , ν ′′ , respectively. Here a is a positive constant to be determined later. Clearly, this distance satisfies the convexity inequality Proposition 20. D γ,a (ν n , ν ∞ ) → 0 as n → ∞ for an appropriate choice of γ > 1, a > 0.
Using the definitions of T and of D γ,a , the terms on the right-hand side can be estimated as follows: thanks to (32). Hence Proposition 17 is applicable to estimate the last term on the right-hand side in (44). In combination with an estimate of the first term by means of Lemma 16 -which applies because of (33) -we arrive at Further, recalling that κ − +κ + < E[(r − ) α ]+E[(r + ) α ] = 1, we can choose a > 0 such that a(κ − +κ + )+2C ′ < a. Thus we have shown (42), with λ := max{λ γ , κ − + κ + + 2C ′ /a} < 1.
4.6. Proof of Proposition 12. By Proposition 20 one getŝ for every g in L 2 (SO(d); H). According to Lemma 14, (Ψ n ) n is a tight sequence in C 0 (SO(d)). Assume that a subsequence Ψ n ′ converges weakly in C 0 (SO(d)) to a limit Y . Since f → exp{i g, f L 2 } is a continuous function on C 0 (SO(d)) for any g in L 2 , one gets thatν n ′ (g) → E[e i g,Y L 2 ], and hence E[e i g,Ψ∞ L 2 ] = E[e i g,Y L 2 ] for every g in L 2 . Using the previous identity it is easy to see that the finite dimensional law of Y and Ψ ∞ are the same and hence they have the same distribution as processes (see, e.g., Proposition 3.2 [37]). The last part of the proof follows by the continuous mapping theorem, since point evaluation is a continuous functional on C 0 [SO(d)].

5.
Proof of the main theorem 5.1. Preliminary weak convergence results. Recall that we deal with initial conditions µ 0 belonging to the NDA of a (full) α-stable law with Lévy measure φ. Let X 0 be a random variable with probability distribution µ 0 . Moreover, for every u) and Let B n denote the σ−field generate by the β j,n 's and O j,n , i.e. β α j,n φ(B −̟j,n )), the random function y → (Q 1,n (y), Q 2,n (y)) is a càdlàg (i.e. right continuos with left-hand limits) function from [0, +∞) to R 2 . Since, clearly, all the finite dimensional components are measurable, (Q 1,n , Q 2,n ) can be seen as process taking values in the space D(R + , R 2 ) of càdlàg functions with the Skorohod topology (see, e.g., [36] and Thm. 4.5 in [9]). Furthermore, given any γ 0 ∈ R d and O ∈ SO(d), define and observe that, for 0 < y ≤ δ, one has Since Oe d is uniformly distributed on S d−1 whenever O has Haar distribution on SO(d) (see (22)), then This yields that the vector n , Q 1,n (y), Q 2,n (y), Q 3,n (O e )) = Z n + (0, R 1,n (y), R 2,n (y), 0) using (47) and Lemma 3.31 Chapter VI of [36] one obtains the thesis.

5.2.
Proof of Theorem 3. The proof is split into three steps. In the first step we introduce a Skorohod-type representation which is inspired to the one used in [30] as an essential ingredient to prove central limit theorem for array of partially exchangeable random variables. This technique has been already employed in a fruitful way in the context of the asymptotic study of kinetic equations, see e.g. [6,19,27,32]. In the second step we prove that the classical conditions for the convergence to a (one-dimensional) stable law hold almost surely in the Skorohod representation.
In the third step we conclude the proof.
Step 1: Skorohod representation. For every n ≥ 1 and for j > n + 1, let us define β j,n = 0 and ̟ j,n = e d , while for j ≤ n + 1 they are defined as in the previous sections.
Step 2: sufficient conditions for the convergence to a stable law. The next step is to prove that the following conditions holdP−a.s.: x > 0, with c as in (46); In view of the well-known criteria for the convergence to a (one-dimensional) stable law -see, e.g., Theorem 30 in Section 16.9 and in Proposition 11 in Section 17.2 of [31] -the previous conditions yield thatP−a.s.
and this will lead easily to the conclusion.
Let us first prove i). Recall that from Lemma 24 we know that and hence, in particular, Since for every u ∈ S d−1 , one has φ(B u ) ≤ φ{y : |y| ≥ 1} < +∞, then (51) yields In view of (48) we havê ǫ α and the last term converges to zero for n ′ → +∞.
As for ii), if x > 0 x .
Since β (n ′ ) → 0 andM n ′ →M it follows from assumption (13) that lim n ′ →+∞ E n ′ = 0 in the case α = 1 too. At this stage the proof of iv) is completed.
Step 3: conclusion of the proof. By (49) and dominated convergence theorem one has whereÊ denotes the expectation with respect toP and the last equality is due to the fact that we proved that M (α) ∞ andM have the same probability distribution. In particular we have stated that the limit does not depend on the subsequence (n ′ ) and hence the convergence is true for the entire sequence (n). Hence, using also Proposition 5, one has that for every e ∈ S d−1 and any ρ > 0 lim n→∞μ n (ρe) = E[e −cM (α) ∞ |ρ| α ].
At this stage, the convergence of µ(t) to µ c ∞ follows from (14). In order to prove the last part of the theorem it is enough to check that since µ c ∞ is a scale mixture of a spherically symmetric stable law, it belongs to NDA of the same stable law.
A random vector Z has a centered α-stable spherically symmetric distribution if for some c > 0. Clearly, in this case, Λ(A) ∝ |A|.
As in the one-dimensional case, one says that: A random vector X 0 (or equivalently its law µ 0 ) belongs to the normal domain of attraction (N DA, for short) of an α-stable law if for any sequence (X i ) i≥1 of i.i.d. random vectors with the same law of X 0 , there is a sequence of vectors (b n ) n≥1 such that n −1/α n i=1 X i − b n converges in law to an α-stable random vector.
A stable law is said to be full if it is not supported on any d − 1 dimensional subspace of R d . In this case, it is possible to characterize the N DA in terms of the tails of µ 0 in the following way: X 0 belongs to the N DA of a stable law with Lévy measure φ = φ Λ if and only if for every r > 0 and every Borel set B ⊂ S d−1 such that Λ(∂B) = 0 with k α = 2Γ(α) sin(απ/2) π .
See Theorems 6.20 and 7.11 in [3]. We collect some results on the NDA of an α-stable law, which are used in Section 5.

Lemma 22.
If a stable law is full, then the corresponding Lévy measure φ is full, that is φ is not supported on any d − 1 dimensional subspace of R d .
Proof. The thesis can be deduced combining Proposition 3.1.20 and Theorem 7.3.3 in [40].
Recall that, for every x ∈ R d , Lemma 23. Let φ be a full Lévy measure, then is a continuous function on R d \ {0}.
Proof. The proof is essentially the same as the proof of Lemma 6.1.25 in [40] and it is left to the reader. Proof. The proof of this result can be obtained with minor modifications from the proof of a similar result contained in Lemma 6.1.26 of [40]. The details are left to the reader.