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Proof of a McKean conjecture on the rate of convergence of Boltzmann-equation solutions

The present work provides a definitive answer to the problem of quantifying relaxation to equilibrium of the solution to the spatially homogeneous Boltzmann equation for Maxwellian molecules. The beginning of the story dates back to a pioneering work by Hilbert, who first formalized the concept of linearization of the collision operator and pointed out the importance of its eigenvalues with respect to a certain asymptotic behavior of the Boltzmann equation. Under really mild conditions on initial data - close to being necessary - and a weak, physically consistent, angular cutoff hypothesis, our main result (Theorem 1.1) contains the first precise statement that the total variation distance between the solution and the limiting Maxwellian distribution admits an upper bound of the form C e^{Lambda_b t}, Lambda_b being the least negative of the aforesaid eigenvalues and C a constant which depends only on a few simple numerical characteristics (e.g. moments) of the initial datum. The validity of this quantification was conjectured, about fifty years ago, in a paper by Henry P. McKean but, in spite of several attempts, the best answer known up to now consists in a bound with a rate which can be made arbitrarily close to Lambda_b, to the cost of the"explosion"of the constant C. Moreover, its deduction is subject to restrictive hypotheses on the initial datum, besides the Grad angular cutoff condition. As to the proof of our results, we have taken as point of reference an analogy between the problem of convergence to equilibrium and the central limit theorem of probability theory, highlighted by McKean. Our work represents in fact a confirmation of this analogy, since the techniques we develop here crucially rely on certain formulations of the central limit theorem.


Introduction and new results
On the basis of an analogy pointed out by McKean in [50,51], a few years ago we started a program which aims at studying the long-time behavior of solutions of some kinetic equations, by means of representations which connect these solutions to probability laws of certain weighted sums of independent and identically distributed (i.i.d.) random variables. The discovery of the right representation is comparatively simple for the solution of the spatially homogeneous one-dimensional Kac equation. This fact has produced both new results and improvements on the existing ones concerning the Kac equation. See [3,4,5,17,32,33,38,39].
Our goal in the present paper is to go back to the original kinetic model, the spatially homogeneous Boltzmann equation for Maxwellian molecules (SHBEMM), which had inspired the aforesaid one-dimensional model. The reason for having deferred its treatment is connected, on the one hand, with the mathematical complexity of the subject and, on the other hand, with the hope that useful insights could be derived from the study of simpler allied cases. More specifically, we discuss here the problem of quantifying the "best" rate of relaxation to equilibrium. The starting point of the argument is the new probabilistic representation exhibited in Subsection 1.5 of the present paper.
The last part of the program, to be developed in forthcoming papers, is concerned with the inhomogeneous Boltzmann equation for Maxwellian molecules. Although the assumption of spatial homogeneity adopted here may seem a strong restriction, it is nonetheless proving an interesting and inspiring basis for studying qualitative properties of the complete model.

The equation
In classical kinetic theory, a gas is thought of as a system of a very large number N of like particles, described by means of a time-dependent sta-tistical distribution µ(·, t) on the phase space X × R 3 , where X stands for the spatial domain. Then, for any subset A of X × R 3 , µ(A, t) provides an approximation, independent of N , of the statistical frequency of particles in A, at time t. It is worth noting that µ(·, t) can be also interpreted, consistently with its statistical meaning, as a genuine probability distribution (p.d.) by arguing about µ(A, t) as probability that the position-velocity of a randomly selected particle, at time t, belongs to A. See the discussion in Subsection 2.1 in Chapter 2A of [65]. The basic assumptions for the derivation of the classical equation which governs the evolution of µ(·, t) are that the gas is dilute (Boltzmann-Grad limit) and that the particles interact via binary, elastic and microscopically reversible collisions. Particles which are just about to collide are viewed as stochastically independent (Boltzmann's Stosszahlansatz ). See [22,23,63] for a comprehensive treatment. In this work, we also assume spatial homogeneity, so that the phase space reduces to R 3 and the SHBEMM can be written as where (v, t) varies in R 3 × (0, +∞), f (·, t) stands for a density function of µ(·, t) and u S 2 for the uniform p.d. (normalized Riemannian measure) on the unit sphere S 2 , embedded in R 3 . The symbols v * and w * denote postcollisional velocities which, according to the conservation laws of momentum and kinetic energy, must satisfy v + w = v * + w * and |v| 2 + |w| 2 = |v * | 2 + |w * | 2 .
Throughout the paper, v * and w * are parametrized according to the ω- where · denotes the standard scalar product. The angular collision kernel b is a non-negative measurable function on [−1, 1]. Henceforth, for the sake of mathematical convenience, it will be assumed that b meets the symmetry for all x in (−1, 1), an assumption which does not reduce the generality of (1), as explained in Subsection 4.1 in Chapter 2A of [65]. 1 In presence of a general interaction potential governing the mechanism of binary collisions, b is replaced by a more complex function called collision kernel. See Section 3 in Chapter 2A of [65]. Maxwell [49] was the first to study particles which repel each other with a force inversely proportional to the fifth power of their distance, named Maxwellian molecules after him. In this particular circumstance, the resulting collision kernel turns out to be a specific function only of w−v |w−v| · ω, as in (1), with a non-summable singularity near 0. It is customary, as we do here, to call Maxwellian any collision kernel which is a function only of w−v |w−v| · ω, and to distinguish Maxwellian kernels depending on whether they are summable or not. The former case corresponds to a SHBEMM with Grad (angular) cutoff. Without any loss of generality, this condition can be formalized assuming that since any SHBEMM with cutoff can be reduced, via a time-scaling, to a SHBEMM with a kernel satisfying (3). The case when b is not summable corresponds to a SHBEMM of the non-cutoff type. We shall confine ourselves to considering the weak (angular) cutoff, i.e.
This condition is actually satisfied by the explicit form of b given by Maxwell, namely the only form of b that has been justified from a physical standpoint.
The first rigorous results on existence and uniqueness, given a probability density function f 0 on R 3 as initial datum, were obtained in [53,67] under the validity of (3). To discuss this question about the SHBEMM with or without cutoff within a unitary framework, one needs a reformulation of the problem. Accordingly, the weak version of (1) used throughout this paper reads d dt where ψ varies in BL(R 3 ), the space of all bounded and Lipschitz-continuous functions defined on R 3 . This formulation enables us to consider any p.d.  ii) t → R 3 ψ(v)µ(v, t) belongs to C([0, +∞)) ∩ C 1 ((0, +∞)) for all ψ in BL(R 3 ); iii) R 3 |v|µ(v, t) < +∞ for all t ≥ 0, if b is not summable but obeys (4); iv) µ(·, t) satisfies (5) for all t > 0 and for all ψ in BL(R 3 ).
From now on, the term solution of (1) has to be meant as weak solution, according to the above definition, of the Cauchy problem with initial datum µ 0 . Tanaka [61] gave a rigorous result of existence and uniqueness for weak solutions by probabilistic arguments. The sole assumption required on µ 0 , only if b is not summable but obeys (4), is the finiteness of the absolute first moment. Apropos of the uniqueness, see also [62].
It should be recalled that, in the non-cutoff case, existence can be recovered from the existence of the solution to the SHBEMM with cutoff, via a truncation procedure originally introduced in [1]. More precisely, given a non-summable b satisfying (4) (3), one can find the solution µ n (·, t) to (1), with b replaced by [b(x) ∧ n]/B n and initial datum µ 0 . Following [1,31], it can be shown that µ n (·, B n t) converges weakly as n goes to infinity to some limit µ(·, t), for every t ≥ 0, and that µ(·, t) turns out to be the solution to the original Cauchy problem. Recall that a sequence {P n } n≥1 of p.d.'s on some topological space T, endowed with its Borel σ-algebra, converges weakly to a p.d. P on the same space if and only if lim n→∞ T hdP n = T hdP , for every bounded and continuous function h on T. Henceforth, this kind of convergence will be denoted with P n ⇒ P .

The conjecture and its motivations
Relaxation to equilibrium of solutions to the Boltzmann equation is at the core of kinetic theory ever since the works of Boltzmann himself. The importance of accurate estimates of the rate of convergence is tightly connected with the issue on the physical value of any convergence statement of Boltzmann-equation solutions w.r.t. the time scale on which the Boltzmann description may be relevant. See, for example, Section 2 in Chapter 2C of [65]. Within this framework, a first preliminary question arises apropos of the choice of the topology in which this convergence ought to take place, keeping in mind that one is dealing with convergence of probability measures (p.m.'s). In fact, the literature has dealt with a variety of probability metrics, but no doubt the total variation distance (t.v.d.) continues to be a formidable reference for the study of relaxation to equilibrium in kinetic models. Recall that, for any pair (α, β) of p.d.'s on some measurable space (S, S ), such a distance is defined by and that it can be written as when λ is any σ-finite measure dominating both α and β, and p, q are probability density functions w.r.t. λ of α and β, respectively. See Chapter III of [60] for more information. Once the right metric has been singled out, the problem of convergence to equilibrium is greatly enhanced by the knowledge of the rate of approach to the limiting distribution and even more so by a precise bound on the error in approximating the limit for each fixed instant. To introduce the reader to the essential part of the problem, we recall that, for an entire class I of initial data µ 0 , one can prove that is met with a suitable constant C * and This result can be reached from a well-known statement by Ikenberry and Truesdell [46], according to which holds true with suitable constants C α , for any multi-index α such that Recently, it has been proved that I contains all the p.d.'s µ 0 satisfying R 3 e iξ·v µ 0 (dv) = R e i|ξ|x ζ 0 (dx) for every ξ in R 3 , where ζ 0 is a symmetric p.d. on (R, B(R)) with non-zero kurtosis coefficient. See [33]. Such being the case, inequality (9) is conducive to checking whether it is possible to establish also the reverse relation Actually, when (9) and (12) are in force simultaneously, Λ b can be viewed as the best rate of exponential convergence of µ(·, t) to equilibrium. The characterization of the largest class of initial data for which (12) is valid is commonly referred to as McKean's conjecture. The reference to McKean is due to the fact that, relative to the solution µ(·, t) of the well-known Kac's simplification of the SHBEMM, he was the first to prove rigorously, in [50], that d TV (µ(·, t), γ v 0 ,σ 2 ) ≤ C ′ e λt holds true with λ ≈ −0.016 and for a suitable constant C ′ . However, this value of λ is strictly greater than Λ b , equal to −1/4 in the case of Kac's equation. See [32].
As a completion of the argument, it is interesting to point out the meaning of Λ b w.r.t. the asymptotic behavior of µ(·, t). Besides the important role played in (11), Λ b represents also the least negative eigenvalue of the linearized collision operator defined on H := L 2 (R 3 , γ v 0 ,σ 2 (dx)). Hilbert [44] was the first to derive this operator from a linearization of (1) and to highlight the opportunity of choosing the domain H with a view to carrying out the spectral analysis. In the Hilbert setting, L b turns out to be self-adjoint and negative with discrete spectrum and |Λ b | represents the spectral gap. See [28]. Finally, it is worth recalling that Λ b arises also in Kac-like derivations of the SHBEMM [47], based on a stochastic evolution of an N -particle system. See [15,19].

A glance at the literature on McKean's conjecture
The formulation of the Boltzmann H-theorem originated a significant mathematical research, aimed at studying the convergence to equilibrium in total variation, whose first rigorous outcomes are in [10,54]. In any case, in spite of the huge literature on this subject, the number of works which expressly pursued the validation of the conjecture is small. Essentially, four lines of research have been followed to achieve the goal, based on: 1) use of contractive functionals or probability metrics; 2) entropy methods; 3) linearization; 4) central limit theorem. 1) As for the first line of research, the papers [18,40,56,61,64] are worth mentioning. In particular, Theorem 1.1 in [18] constitutes the closest result to the McKean conjecture obtained so far. It is valid only under (3) and states that, for every ε > 0, there is C ε (µ 0 , b) such that holds for every t ≥ 0, but this C ε goes to infinity as ε goes to zero. Therefore, the presence of ε, together with such a behavior of C ε (µ 0 , b), defeats the hope of extending (13) to the solution of the SHBEMM of non-cutoff type through the truncation argument explained in Subsection 1.1: A strong motivation for the pursuit of a bound with ε = 0 and of a constant C(µ 0 ) depending only on µ 0 , in the place of C ε (µ 0 , b). Moreover, (13) has been deduced thanks to rather strong conditions on µ 0 (dx) = f 0 (x)dx, such as finiteness of all absolute moments, Sobolev regularity and finiteness of the Linnik functional. 2) Entropy methods aim at proving quantitative H-theorems, on the basis of the seminal ideas introduced in [11,12]. An attempt to improve this strategy, towards the achievement of the McKean conjecture, was represented by the Cercignani conjecture which, however, proved to be false in the case of Maxwellian molecules. See [8,66]. Anyway, quantitative Htheorems are still considered as conducive to the most powerful strategy to study relaxation to equilibrium in non-homogeneous frameworks. See [26].
3) The strategy of the linearization is outlined in [29,42]. It gives general positive answers to the problem of quantifying the relaxation to equilibrium only when the solution enters a small neighborhood of the equilibrium itself, so that the spectral analysis of L b , as an operator on H, becomes relevant to the study of the nonlinear problem. It is only recently that, in the case of the homogeneous Boltzmann equation with hard potentials, the linearization has been used successfully to prove the conjecture. See [55]. However, the radical difference between the situation of hard potentials and that of Maxwellian molecules hampers a direct extension of the positive conclusion from the former to the latter. 4) Finally, the link with the central limit theorem discovered by McKean in [50,51] has been taken into serious consideration only recently in [13,14], two works which have strongly inspired and motivated our program.

The main result
A precise and complete formulation is encapsulated in the following theorem whereμ stands for the Fourier transform of the p.d. µ on (R 3 , B(R 3 )), Theorem 1.1. Assume that (2) and (4) are in force and that the initial datum µ 0 satisfies and for some strictly positive p. Then, the solution µ(·, t) meets for every t ≥ 0, where Λ b is given by (10) Now, the combination of this inequality with leads to the desired conclusion.
2. Let us now discuss assumption (14). It is interesting to recall that, under the cutoff condition, convergence in the total variation metric to the Maxwellian holds under (7). See [20]. The necessity of this condition, in a cutoff setting, is stated both in [16] and in Theorem 1.3 of the present paper. In [20] it is also shown that convergence to equilibrium, under the sole assumption of finiteness of the second moment of µ 0 , could be arbitrarily slow, whereas the finiteness of the (2 + δ)-th absolute moment, for some δ > 0, is enough to get exponentially decreasing bounds. Nevertheless, if δ < 2, these bounds can be worse than that conjectured by McKean.
Here is an example which shows that, even if the tail condition (15) is fulfilled, the desired bound is not achieved because of "infinitesimal" deviations from hypothesis (14). Consider the class of initial data µ (q) for q in (3,4). The Fourier transform of this density at ξ is which meetsμ (q) 0 (ξ) = O(|ξ| −1 ) when |ξ| goes to infinity. Then, µ (q) 0 satisfies (15) and has finite absolute moment of order (3 + δ) for every δ in (0, q − 3), but has infinite absolute fourth moment. Denoting by µ (q) (·, t) the solution of (1) relative to µ (q) 0 , one can mimic the argument explained in [33] to prove that 3. As far as the tail assumption (15) is concerned, it is worth noting that it is implied by the finiteness of the Linnik functional, according to Lemma 2.3 in [18]. Also the relationship between (15) and certain regularity conditions adopted to guarantee the validity of classical local limit theorems of probability theory are worth noting. See, for example, Theorem 19.1 in [7].

A probabilistic representation of the solution
The proof of Theorem 1.1 relies on a representation of the solution µ(·, t) -already proposed and studied in [27] -which is valid under the cutoff condition (3). The motivation for this representation is twofold. On the one hand, it leads us to study the problem of convergence to equilibrium from the standpoint of the central limit problem of probability theory. On the other hand, it lends itself to computability of certain derivatives of the Fourier transform of µ(·, t) involved in the first steps of the proof of Theorem 1.1. See, for example, (58) below. It should be mentioned that the existing representations, essentially based on the Bobylev identity (see Section 3 of [9]), turn out to be unfit for the aforesaid computations.
In a nutshell, the probabilistic representation at issue states that for every t ≥ 0 and every B in B(R 3 ), where E t is an expectation and M is a random p.m. connected with a distinguished weighted sum of random vectors, to be defined below. Here and in the rest of the paper we use the term random p.m. to designate any measurable function from some measurable space into the space P(R 3 ) of all p.m.'s on (R 3 , B(R 3 )), endowed with the Borel σ-algebra of weak convergence of p.m.'s. See, e.g., Chapters 11-12 of [48] for further details. Then, to carry out our programme, it remains to provide the reader with those definitions and preliminary results which are necessary to understand (17). In this way, we shall present also the core of the notation used in the rest of the paper.
The starting point is the introduction of the sample space where: For any nonempty set X, X ∞ stands for the set of all sequences (x 1 , x 2 , . . . ) whose elements belong to X; T := X n≥1 T(n) and T(n) is the (finite) set of all McKean binary trees with n leaves. See, e.g., [37] for details.
On writing t n to denote an element of T(n), t n,k indicates the germination of t n at its k-th leaf (cf. Fig. 1), while t l n and t r n symbolize the two trees, of n l and n r leaves respectively, obtained by a split-up of t n (cf. Fig. 2). A recent and comprehensive treatment of random trees can be found in [34].
Then, associate with Ω the σ-algebra where 2 X stands for the power set of X and B(X) for the Borel class on X. Define to be the coordinate random variables of Ω and, by them, generate the σ-algebras Now, for every t ≥ 0, consider the unique p.d. P t on (Ω, F ) which makes the random coordinates stochastically independent, consistently with the following marginal p.d.'s: with the proviso that 0 0 := 1.
b) {τ n } n≥1 is a Markov sequence driven by for every n in N and t n in T(n), where, for a given t n , G(t n ) is the subset of T(n + 1) containing all the germinations of t n .
c) The elements of {φ n } n≥1 are i.i.d. random numbers with p.d.
e) The elements of {V n } n≥1 are i.i.d. with p.d. µ 0 , the initial datum of the Cauchy problem relative to (1). According to the above notation, E t denotes expectation w.r.t. P t .
An induction argument shows that n j=1 π 2 j,n = 1 (23) for every n in N. It is also worth recalling the identity valid for every t, s > 0, with l s (b) : The original derivation is in [37] but, for the sake of completeness, we have included its proof in A.1. Throughout the paper, A.n designates the n-th subsection of the Appendix. With a view to the proof of Theorem 1.1, it is interesting to point out that M l (ϕ n−1 , θ n−1 )O * j,n l (t l n , ϕ l , θ l ) for j = 1, . . . , n l M r (ϕ n−1 , θ n−1 )O * j−n l ,nr (t r n , ϕ r , θ r ) for j = n l + 1, . . . , n for every t n in T(n), ϕ in [0, π] n−1 and θ in (0, 2π) n−1 . Here θ l := (θ 1 , . . . , θ n l −1 ) and θ r := (θ n l , . . . , θ n−2 ) and, finally, Working out the recursion formula (26) gives where n h=1 A h := A 1 × · · · × A n and δ j (t n ) indicates the depth of the j-th leaf of t n , that is the number of generations separating this leaf from the root (the top node of the tree). The ǫ h (t n , j)'s take values in {l, r} and, in particular, ǫ 1 (t n , j) equals l (r, respectively) if j ≤ n l (j > n l , respectively).
for j = 1, . . . , n and n in N, with e 3 := t (0, 0, 1). It should be noticed that such a B actually exists and that it cannot be continuous. See, e.g., Chapter 5 of [45].
The central object of our construction is the random sum which, as a function of ξ, is not c.f. and depends on the choice of B.
One of the merits of representation (17) is that it allows the formulation of a central limit-like theorem for the asymptotic behavior of the solution of the SHBEMM with cutoff, condensed in the following to infinity if and only if (7) holds true. Moreover, in case this condition is satisfied, the limiting distribution is given by (8).
As already mentioned at the beginning of Remark 2, this theorem is well-known. In fact, the "if part" was proved in [20,61], while the "only if" part was proved, in a quite different way, in [16]. What is new is the proof we develop in Subsection 2.3 on the basis of (17).

Proofs
In this section, we present the skeleton of the proofs of Theorems 1.1, 1.2 and 1.3. Some technical issues are deferred to the Appendix and to [30,31].
We start from the basic representation formulated in Theorem 1.2.

Proof of Theorem 1.2
When (2)-(3) are in force, recall that µ(·, t) can be expressed by means of the so-called Wild-McKean sum [51,67], namely for every t ≥ 0 and B in B(R 3 ). According to McKean, the weights p n (t n ) are defined inductively starting from p 1 (t 1 ) := 1 and then putting for every n ≥ 2 and t n in T(n). These p n 's are connected with the p.d. of valid for every n in N and t n in T(n). See A.2 for the proof. As far as the Q tn 's are concerned, where Q is an operator which sends a pair (ζ, η) belonging to P( into a new element Q[ζ, η] of P(R 3 ) according to the following rule. First, take two sequences {ζ n } n≥1 and {η n } n≥1 of absolutely continuous p.m.'s such that ζ n ⇒ ζ and η n ⇒ η, and denote with p n (q n , respectively) the density of ζ n (η n , respectively). Then, denoting limit w.r.t. weak convergence by where , as a consequence of (2). As shown in [31], the limit in (35) exists and is independent of the choice of the approximating sequences {ζ n } n≥1 and {η n } n≥1 .
To carry on with the proof, consider the Fourier transform and apply the well-known Bobylev formula, as in [31], to get for every ξ in R 3 \ {0}. This, by the change of variable ω = ω(ϕ, θ, ξ) = sin ϕ cos θa(u) + sin ϕ sin θb(u) + cos ϕu, becomeŝ where ρ = |ξ|, u = ξ/|ξ| and ψ l , ψ r are abbreviations for the quantities which depend on the choice of the orthonormal basis The components of this basis are exactly the columns of the matrix B introduced in (29). The inner integral in (36), that is has interesting properties, which are at the basis of the new representation (17). In particular, I is a measurable function of (ξ, ϕ) independent of the choice of {a(u), b(u), u}. Moreover, for every fixed ϕ in [0, π], I(·, ϕ) is the Fourier transform of a p.m. on (R 3 , B(R 3 )), say C[ζ, η; ϕ], that is The link with Q is given by for every B in B(R 3 ). The proof of these facts is contained in A.3. At this stage, mimicking the iteration procedure developed for Q leads to the following definition for every t n in T(n) and ϕ in [0, π] n−1 , with the proviso that ϕ l (ϕ r , respectively) is void when n l (n − n l , respectively) is equal to one. For every n ≥ 2 and t n in T(n), the mapping ϕ → C tn [µ 0 ; ϕ] is a random p.m. and holds true for every B in B(R 3 ), as proved in A.4. In view of this link, the Wild-McKean can be re-written as . Therefore, to show the validity of (17), it is enough to verify that hold true for every n ≥ 2, t n in T(n), ϕ in [0, π] n−1 and ξ = 0. The q j,n 's are defined inductively starting from q 1,1 (t 1 , ∅, ∅, u) := u and then putting for every n ≥ 2, t n in T(n), ϕ in [0, π] n−1 and θ in (0, 2π) n−1 .
As far as the proof of (42) is concerned, start by noting that q j,2 (t 2 , ϕ, θ, u) equals B(u)O * j,2 (t 2 , ϕ, θ)e 3 for j = 1, 2, for every ϕ in [0, π], θ in (0, 2π) and u in S 2 , provided that the basis {a(u), b(u), u} in (37) is formed by the three columns of B(u). Then, assume n ≥ 3 and argue by induction starting from (41), definitions (26) and (43). Whence, and P r j,n := n j=n l +1μ For the sake of clarity, the integral (0,2π) n l −1 ( (0,2π) nr −1 , respectively) in (45) should not be written if n l = 1 (n r = 1, respectively) since θ l (θ r , respectively) corresponds to the empty set. At this stage, it will be proved holds for every ρ in R, u in S 2 , ϕ in [0, π] n−1 and θ n−1 in (0, 2π). If n l = 1, the proof of (46) reduces to verify that To proceed, since the third column of B(ψ l (ϕ n−1 , θ n−1 , u)) is the same as that of B(u)M l (ϕ n−1 , θ n−1 ), then there exists an orthogonal matrix is valid for s = l, r and for every ϕ and θ. Then, when n l ≥ 2, consider the definition of P l j,n , recall (28) and take account that the product which, in view of (26), turns out to be the same as the RHS of (46). The proof of (42) is completed using (45), after noting that an equality similar to (46) can be stated by changing subscripts and superscripts from l to r, and replacing O * j,n with O * j+n l ,n . Finally, the invariance of M w.r.t. B is equivalent to the invariance of representation (42) when B(u) is replaced by any matrix B ′ (u) having the same characteristics as B(u). Anyway, such an equivalence follows from the above reasoning.

Proof of Theorem 1.1
In the first place, we recall that the entire proof will be developed under hypotheses (2)-(3) on b, in view of Remark 1 in Subsection 1.4. Then, we set a few conditions on µ 0 to simplify a number of arguments without loss of generality. In this sense, we make use of (6) to assume, from now on, implying that the limiting Maxwellian is γ := γ 0,1 . We also assume that [31] for a more detailed explanation. Hence, we suppose that is valid for every ξ in R 3 , with q = 1/(2⌈2/p⌉).
Here, ⌈x⌉ indicates the least integer not less than x, while p is the same as in (15). As to the numerical evaluation of λ, the reader is referred to the proof of the proposition in A.5.
As first step of the real proof, an application of (17) yields After introducing the random number we put r := 11⌈2/p⌉ and a * := (2 r r!) −1 (51) to define the partition {U, U c } of Ω by This can be used to write where E t [X; S] denotes S XdP t . The former summand on the right of (52) will be bounded by utilizing the fact that U has "asymptotically small" probability. As to the latter, it will be shown that M(·; ω) has nice analytical properties for each ω in U c , so that a proper bound will be derived from these very same properties. In fact, as d and the RHS is equal to zero since each φ j has an absolutely continuous law.
To complete the evaluation of P t (U ), it is enough to combine the Markov Whence, The argument to deduce a bound for the expectation over U c occupies the rest of this subsection. It is based on the following multidimensional extension of a result by Beurling [6].
The proof is deferred to A.6. The applicability of this proposition to for h = 1, . . . , 2k and some integer k ≥ 2, then there are positive constants for h = 1, . . . , 2k and any choice of B, P t -almost surely. Moreover, ρ → ∂ h ∂ρ hM (ρu) exists for every u in S 2 and P t -almost surely with h = 1, . . . , 2k, which entails See A.7 for the proof and the numerical evaluation of the constants g h .
At this stage, Proposition 2.2 yields To evaluate the integrals on the RHS, we change the variables according to become integrals with respect to 4πρ 2 L 1 ⊗ u S 2 (dρdu) and the standard and then we define the random functions Hence, for the sum of the two integrals on the RHS of (58) we obtain where R := 1 2 In the following sub-subsections we analyze the integrals appearing in (59), calling inner (outer, respectively) any integral on [0, R] × S 2 ((R, +∞) × S 2 , respectively).

Outer integral of I 1 (ρ, u)
An application of the inequality and a first proposition is given to analyze those summands which contain the Gaussian c.f..
Then we study those terms on the RHS of (61) which depend on M making use of the next proposition, whose statement involves the random with λ and q as in Proposition 2.1. and hold for every ρ in [0, R], with the exception of a set of P t -probability zero.
Moreover, there are two non-random polynomials ℘ 1 and ℘ 2 of degree 2 and 4 respectively, with positive coefficients depending only on µ 0 , such that and hold for every ρ in [0, R], with the exception of a set of P t -probability zero.
A complete characterization of ℘ 1 and ℘ 2 is given in the course of the proof of this proposition, in A.9. For the sake of completeness, we observe that (64) and (66) hold true for any choice of B in (29).
One of the advantages of the splitting (52) consists in the fact that all the realizations of Ψ on U c share a property of uniform integrability, as shown in the following Proposition 2.6. Over U c , the inequality is valid for every x > 0, with ǫ := (2r!) −1 and r given by (51). Therefore, holds true for every x > 0, s > 0 and m < (2rqs − 1).
See A.10 for the proof. We are now in a position to complete the study of the outer integral of I 1 (ρ, u). First, taking into account that 2rq = 11, combination of (65) with (69) yields The applicability of (69) is guaranteed by the fact that, when s = 2 and m = 2, one has m < 4rq − 1 = 21. Second, since (56), (65) and (68) entail on U c , after integrating by parts we get Thus, (68)-(69) with s = 1 and m = 0 lead to To study the last integral, we recall that ν j=1 π j,ν = 0 on U c and then combine (65) with (67) At this stage, after two integrations by parts, we have and, in view of Proposition 2.6, the above RHS is bounded by The final bound can be obtained, via the Tonelli theorem, by starting from (61) and collecting the upper bounds in (62) and (70)-(72). Indeed, these last upper bounds are independent of u and are expressed as sums of powers of R, of order less than or equal to −8. Therefore, recalling (60) and the inequality W ≤ 1, we obtain 4π (R,+∞)×S 2 As first step, we use the Tonelli theorem to write the outer integral of I 2 (ρ, u) Then, we apply Theorem 3.16 in [41] to obtain which, by virtue of (65), yields At this stage, to handle the computations involving the Laplace-Beltrami operator, we define the following plane domains to form the atlas A on S 2 composed by the charts Ω k := h k (D k ) ⊂ S 2 for k = 1, . . . , 4. Then, ∆ 2 S 2 can be expressed in local coordinates as ∆ 2 (u,v) = ∂ uuuu + 2 cot u∂ uuu − sin −2 u∂ uu + sin −2 u cot u∂ u + sin −4 u∂ vvvv − 2 sin −4 u(2 − sin 2 u)∂ vv + 6 sin −2 u cot u∂ uvv + 2 sin −2 u∂ uuvv by virtue of (3.84) in [41], and hence sup u∈S 2 where α indicates the multi-index (α 1 , α 2 ), ∂ α stands for the partial derivative ∂ α 1 +α 2 ∂u α 1 ∂v α 2 , and ∆ = 4(2 + √ 3) 2 (6 + √ 3) is the maximum absolute value of the coefficients of ∆ 2 (u,v) . To study ∂ αM (ρh k (u, v)) we resort to the multi-dimensional Faà di Bruno formula stated and proved in [25]. Therefore, taking into account that ∂ α h k (u, v) ≤ 1 for every multi-index α, we where the a h,l 's are constants specified in [25], and M l := R 3 |v| l M(dv).
The analysis is essentially based on certain new Berry-Esseen-type inequalities presented in [30], after observing the analogy between ρ →M(ρu) and the c.f. ϕ n (t) therein. Indeed, for any u in S 2 and for every choice of B in (29), each realization ofN (ρ; u), as a function of ρ, coincides with the c.f. of a weighted sum of independent random numbers, according to (32). Moreover, the definition of R in (60) corresponds to the upper bound τ appearing in the Berry-Esseen-type inequalities proved in [30]. To implement the aforesaid inequalities within the present framework, it is worth introducing the following entities where T (u) belongs to H . With this new notation at hand, the Berry-Esseen-type inequality can be re-written as for l = 0, 1, 2, ρ in [0, R] and u in S 2 , u 2,l , u 3,l , u 4,l , v l being non-random rapidly decreasing continuous functions depending only on µ 0 . See [30] for their definition. The above inequality yields the latter inequality following from the conditional Markov inequality. Now, we apply (64) and (66) and, after observing that the upper bounds provided therein are G -measurable, we obtain Since the inner integral on the RHS has already been studied, it remains to explain how it depends on u. For this, a fundamental role is played by B, which appears in the RHS of (85) through the random variables X, Y and Z.
Apropos of this, it should be recalled that the so-called hairy ball theorem - where X k , Y k , Z k are the same as in (82) for k = 1, 2 and for every u in Ω k , where N k and T k (u) are the same as N and T (u), respectively, with B = B k . To bound the former summand we make use of the following Proposition 2.7. Assume that the tail condition (15) is in force together with the moment assumptions (14) and (47) holds for every ρ in [0, R], with the exception of a set of P t -probability zero.
The proof is deferred to A.9, where ℘ L is given explicitly. At this stage, we note that the upper bound in (93) is G -measurable and, afterwards, we apply (86) to obtain If we consider the random variable +∞ 0 ρ 2 ℘ 2 L (ρ)Ψ 2 (ρ)dρ on U c , then Proposition 2.6 can be used to conclude that this random variable is bounded by In the final part of this sub-subsection we provide an upper bound for the latter summand in the RHS of (92), by means of the following statement which involves new random quantities such as where the M (m) j,n (u)'s are the same as in (81), ∇ S 2 is the Riemannian gradient on S 2 and || · || S 2 the Riemannian length.
holds for every u in Ω k and ρ in [0, R], with the exception of a set of P tprobability zero. X L,k , Y L,k , Z k , Z G,k and Z L,k are defined as in (95)-(98) and (84) with B k in place of B.
For the proof and the definition of the z i 's see A.11. Now, a straightforward application of the above proposition yields where B i,L := 6 +∞ 0 z 2 i (ρ)ρ 2 dρ for i = 1, . . . , 6. The final bound is achieved by collecting inequalities (92), (94) and (100)

The final step
With a view to bounding the RHS of (58) (101) respectively, to write Then, we proceed by taking expectation of both sides of (102). Apropos of this computation it is worth noting that, if µ 0 meets the additional condi- As for the random variables X k and X L,k , a key role is played by the valid for j = 1, . . . , n, n in N and u in S 2 , independently of the choice of B in (29). The ζ j,n 's are given by ζ j,n := ζ * j,n (τ n , (φ 1 , . . . , φ n−1 )) (104) and the ζ * j,n 's are defined on T(n) × [0, π] n−1 as follows. Put ζ * 1,1 ≡ 1 and, for n ≥ 2, for every ϕ in [0, π] n−1 . The reader is referred to A.12 for the proof of (103).
Unfortunately, unlike the X's and the Y's, it is not possible to write the random variables Z, Z G and Z L as product of a given function of u by some other random variable independent of u and "contracting" in some sense.
Nevertheless, such a contraction property can be found on the integrals of the Z's over Ω k . Accordingly, we show that the expectations of the last three random variables in (102) admit bounds like Ce Λ b t with C depending only on µ 0 . To prove this, we apply the Jensen inequality and exploit (48) to get where ψ j,n;k is the analog of (29) when B is replaced by B k , ψ j,n;k,s denotes its s-th component and S k,s := ν j=1 π 2 j,ν 3ψ 2 j,ν;k,s −1 . Whence, by a further application of Jensen's inequality and of an obvious inequality concerning the square root of a sum, Both the square roots and the squares after the brackets constitute an obstacle for the interchange of the integral with the expectation E t and for the consequent application of useful properties of conditional expectation.
To overcome this difficulty, we resort to the imbedding of the Sobolev space W 1,1 (Ω k ) into L 2 (Ω k ). See, e.g., Chapter 2 of [2]. Taking the same constants A 1 (0) and K(2, 1) as in Theorem 2.28 therein, we write where D can be Id, ∇ S 2 , ∆ S 2 , and DS k,s is to be interpreted in accordance with the meaning of D. To work out the last term in the above inequality, we use (5.1.25) in [60] to say that holds true P t -almost surely. Moreover, when D is Id or ∆ S 2 , the Leibnitz rule for the gradient entails When D is ∇ S 2 , the definition of the Hessian as symmetric bilinear form leads to for every vector field V , D standing for the Levi-Civita connection. See Exercise 11 in Chapter 6 of [21]. Whence, where || · || * denotes the L 2 -norm of the Hessian given by || Hess S 2 [S k,s ] || 2 * := ij [Hess S 2 [S k,s ](V i , V j )] 2 for some orthonormal basis {V 1 , V 2 } of vector fields. At this stage, it comes in useful to emphasize the fact that, in view of (121)-(123), the latter summand in (120) can be bounded by a sum of terms sharing the same structure of the former summand. Then, to provide an effective bound for the RHS of (120) it is enough to prove that holds for some suitable constant c(D ′ ), D ′ being one of the following op- For the proof of (124), cf. A.13. Now, we are in a position to write explicit bounds for the last three terms in (102), which read To conclude, we gather (24) and we resort to (58) and (102) to obtain with C(U c ) := (C * ,r + C * ,s + B 4 m 2 4 + 96πB 1,L ) 1/2 Finally, we recall (52) and combine the last inequality with (53).
Then, one has σ 2 * := min{σ 2 1 , σ 2 2 , σ 2 3 } ≤ T 2 ≤ 3 and 1 8 holds true for every strictly positive ε. At this stage, take ε = √ π o and combine (130)-(132) to get To complete the analysis of the first summand in the RHS of (129), one shows that the expectation of the RHS of (133) approaches zero as t goes to infinity, for every ρ in [0, +∞). Indeed, for any monotonically increasing bounded function g : (0, ∞) → (0, ∞) satisfying lim x↓0 g(x) = 0, one has for every z in (0, ∞). By virtue of (24)- (25), E t [π 4 o ] ≤ e Λ b t and, after choosing z = −Λ b /8, one obtains lim t→+∞ E t [g(π o )] = 0. This argument, applied with x 2 ρ 4 , leads to the desired result. As far as the latter summand in (129) is concerned, a plain application of (17) implies that E t [e −T 2 ρ 2 /2 ] can be thought of as the Fourier transform of the solution of (1) when the initial datum coincides with 3 the σ i 's have been fixed initially. Now, in view of (128), this initial datum belongs to a convenient neighborhood of the equilibrium γ -according to holds true for every t ≥ 0 with the same C * as in the above-quoted theorem.
As to the necessity of (7), suppose that µ(·, t) converges weakly to some limit as t goes to infinity. Following a technique developed in [35], the argument starts with the introduction of the random vector ii) Λ to be the random p.d. on (R, B(R)) obtained as convolution of all elements of λ, i.e. Λ = λ 1 * · · · * λ ν .
iii) U := {U 1 , U 2 , . . . } to be the sequence of random numbers defined by To grasp the usefulness of W , one can note that its components are the essential ingredients of the central limit problem for independent uniformly asymptotically negligible summands. See Sections 16.6-9 of [36]. Apropos of the negligibility condition, it is easy to prove that lim t→+∞ P t [U k > α] = 0 holds for every k in N and for every α in (0, +∞). In fact, the inclusion To conclude, apply the argument used to prove Lemma 2 in [38]. Here, P(R) is metrized, consistently with the topology of weak convergence, in such a way that it turns out to be a separable, compact and complete metric space. Cf. Section 6.II of [57]. Then, S is a separable, compact and complete metric space w.r.t. the product topology and so the family of probability distributions {P t • W −1 } t≥0 is tight. This implies that any sequence {P tm • W −1 } m≥1 , when t m strictly increases to infinity, contains a subsequence {Q l } l≥1 , with Q l := P tm l • W −1 , which converges weakly to a p.d. Q. It is worth noting that, thanks to the weak convergence of µ(·, t), Q is supported by This claim can be verified by recalling Lemma 3 in [38]. Since S is Polish, one can now invoke the Skorokhod representation theorem (see Theorem 4.30 in [48]). Therefore, there are a probability space (Ω,F ,P) and S-valued ran- with the exception of a set of pointsω ofP-probability 0. The final argument is split into the following steps. First, where theπ's andψ's denote the counterparts, in the Skorokhod representation, of the π's and ψ(u 0 )'s,π l,o := max 1≤j≤ν l |π j,ν l | and the inequality is a consequence of the inclusion {v ∈ R 3 |π j,ν lψj,νl · v| ≤ ε} ⊃ {v ∈ R 3 π l,o |v| ≤ ε}. Second, define d = d(ω; j, l) to be an element of {1, 2, 3} for whichψ 2 j,ν l ;d = max 1≤i≤3ψ 2 j,ν l ;i . Note thatψ 2 j,ν l ;d must be greater than 1/3 sinceψ j,ν l belongs to S 2 , for everyω inΩ, l in N and j = 1, . . . ,ν l . Then, wheres h,l denotes the sum of thoseπ 2 j,ν l for which d(ω; j, l) = h. At this stage, observe thatπ l,o goes to zero with probability one as l goes to infinity, in view of Lemma 1 in [38]. Since 3 h=1s h,l = 1 with probability one, there are someω and h, sayω * and h * , such thatπ l,o (ω * ) → 0 and lim lsh * ,l (ω * ) is strictly positive. Then, which shows that the h * -th marginal of µ (s) 0 -and hence also the h * -th marginal of µ 0 -has finite second moment. To complete the proof, observe that h * can be determined independently of µ 0 and that weak convergence of µ(·, t) entails weak convergence of µ(·, t) • f −1 Q , f Q being the map v → Qv and Q an orthogonal matrix. Hence, since µ(·, t) • f −1 Q turns out to be the solution of (1) with initial datum µ 0 • f −1 Q (cf. [31]), the above argument can be used to prove that

A Appendix
Gathered here are the proofs of unproved propositions and formulas scattered throughout Sections 1 and 2.
A.1 Proof of (24), (106) and (114) Fix s > 0 and define These functions satisfy the relations for every n in N, t n in T(n) and for some suitable constant α. This claim is checked for each of them, following a common scheme of reasoning. First, (1, t 1 ) = 1 holds by definition. Then, to obtain the latter identity in (137) as regards A (s) 1 , utilize (22) in the equality , where the validity of the exchange of cos with sin is a consequence of (2). As to A 2 , use (22) and (105) in to show that A 2 satisfies the latter identity in (137) with α = f (b). Passing to A 3 , consider (22) and (113) in conjunction with A 3 (n, t n ) = [0,π] n−1 n j=1 |π * j,n (t n , ϕ)| 3 |η * j,n (t n , ϕ)|β ⊗ n−1 (dϕ) to verify that A 3 meets the latter identity in (137) with α = g(b).

A.2 Probability law of {τ n } n≥1
The aim is to show that the coefficient p n (t n ) in the Wild-McKean sum is equal to P t [τ n = t n ] for every n. Proceeding by mathematical induction, observe that the assertion is trivially true for n = 1, 2. To treat the case n ≥ 3, introduce the symbol P(t n ) to denote the subset of T(n − 1) of the trees which are able to produce t n by germination. Whence, p nr−1 (s nr−1 ) = (n r − 1) and that the same procedure yields s n−1 ∈P(tn) s r n−1 =t r n p n l −1 (s l n−1 ) = (n l −1)p n l (t l n ).
To complete the proof it is enough to combine the previous equations and to recall that n = n l + n r .

A.3 A few interesting characteristics of C[ζ, η; ϕ]
The first point concerns the invariance of (38)  After substituting these expressions in (38), the desired conclusion follows from an obvious change of variable.
To prove the measurability of (ξ, ϕ) → I(ξ, ϕ), resort to Proposition 9 in Section 9.3 of [36], so that it is enough to verify the continuity of ϕ → I(ξ, ϕ) for each fixed ξ and the measurability of ξ → I(ξ, ϕ) for each fixed ϕ. The former claim follows from the form of the dependence on ϕ in (37)- (38). To verify the latter, one can show that also ξ → I(ξ, ϕ) is continuous for each fixed ϕ. Continuity at ξ = 0 can be derived from the relation |ψ l | = |ψ r | = 1 and an ensuing application of the dominated convergence theorem. To check continuity at ξ * = 0, take a sequence {ξ n } n≥1 converging to ξ * and observe that |ξ n | → |ξ * | and u n := ξ n /|ξ n | → u * := ξ * /|ξ * |. Fix a small open neighborhood Ω(u * ) ⊂ S 2 of u * in such a way that S 2 \ Ω(u * ) contains at least two antipodal points. In view of the first part of this appendix, choose a distinguished basis in such a way that the restrictions of u → a(u) and u → b(u) to Ω(u * ) vary with continuity. As a consequence, ψ l (ϕ, θ, u n ) converges to ψ l (ϕ, θ, u * ) and ψ r (ϕ, θ, u n ) converges to ψ r (ϕ, θ, u * ) for every ϕ in [0, π] and θ in (0, 2π), and the convergence of I(ξ n , ϕ) to I(ξ * , ϕ) follows again from the dominated convergence theorem.
To show that ξ → I(ξ, ϕ) is a c.f. for every ϕ in [0, π], resort to the multivariate version of the Bochner characterization. See Exercise 3.1.9 in [60].
The only point that requires some care is positivity. If this property were not in force, one could find a positive integer N , two N -vectors (ω 1 , . . . , ω N ) and (ξ 1 , . . . , ξ N ) in C N and (R 3 ) N respectively, and some ϕ * in [0, π] in such a way that N j=1 N k=1 ω j ω k I(ξ j − ξ k , ϕ * ) < 0. Note that the LHS of this inequality is a real number since I(−ξ, ϕ) = I(ξ, ϕ) for any ξ and ϕ. Hence, by continuity of ϕ → I(ξ, ϕ), there exists an open interval J in [0, π] containing ϕ * such that ϕ → N j=1 N k=1 ω j ω k I(ξ j −ξ k , ϕ) is strictly negative on J. Now, choose a specific b * for which the resulting p.m. in (20), say β * , is supported by J. By construction, L := π 0 N j=1 N k=1 ω j ω k I(ξ j − ξ k , ϕ)β * (dϕ) is a strictly negative number, a fact which immediately leads to a contradiction. Indeed, denote by Q * [ζ, η] the RHS of (35) when b * replaces b in the definition of Q[p, q]. Observe that Q * [ζ, η] is in any case a p.m. even if b * does not meet (2). Now, L must be equal to N j=1 N k=1 ω j ω kQ * [ζ, η](ξ j − ξ k ), thanks to (36), and this quantity must be non-negative, from the Bochner criterion again.
To prove (39), start by verifying that ϕ → C[ζ, η; ϕ] is measurable, which is tantamount to checking that ϕ → C[ζ, η; ϕ](K) is measurable for every This paves the way for writing the integral in (39), and the equality therein follows from (36) and (38) in view of the injectivity of the Fourier-transform operator.

A.6 Proof of Proposition 2.2
Initially, suppose that χ(dx) = f (x)dx for some f in L 1 (R 3 ). Therefore, ∆χ(ξ) = R 3 |x| 2 f (x)e ix·ξ dx and then, by the Plancherel identity, Now, note that |χ|(R 3 ) = R 3 |f (x)|dx and apply the Cauchy-Schwartz inequality to get For a general χ, consider the convolution χ ǫ of χ with the Gaussian distribution of zero mean and covariance matrix ǫ 2 I.

A.7 Proof of Proposition 2.3
Taking account of (47), note that with m 2 = 3. The equality emanates by virtue of the stochastic independence of the V j 's while the inequality follows from the combination of the Cauchy-Schwartz inequality with (23) and the identity |ψ j,n (u)| = 1. Thus, (55) holds true for h = 2 with g 2 = m 2 . The case h = 1 can be derived from the case h = 2 thanks to the conditional Lyapunov inequality after putting When h ≥ 3, an inequality due to Rosenthal (see Section 2.3 in [58]) yields

A.8 Proof of Proposition 2.4
The definition of the k-th Hermite polynomial shows that d k dρ k e −ρ 2 /2 = 2 −k/2 H k ρ √ 2 e −ρ 2 /2 for every k in N 0 and ρ in R. See, for example, (1) in Section 2.IV of [59]. Moreover, according to (9 2 ) therein, where [n] stands for the integral part of n, and hence Now, take account of the following elementary inequalities: x e −x 2 /2 for x > 0, and ρ t e −ρ 2 /2 ≤ (t/e) t/2 for ρ ≥ 0 and t ≥ 0, with the proviso that 0 0 := 1 when t = 0. Whence, A.9 Proof of Propositions 2.5 and 2.7 The main task is to prove (64), (66) and (93). The remaining inequalities (65) and (67) can be derived by interchanging derivative with expectation in the equalityM(ρu) = E t N (ρ; u) | G , since Ψ(ρ) is a G -measurable random variable for every fixed ρ. To start, (64) follows from the combination of (32), (49) and (63), upon recalling that |ψ j,ν | = 1. With a view to proving (66) and (93), it is worth noting that 0 ≤ ρ ≤ R entails sup u∈S 2 μ 0 (ρπ j,ν ψ j,ν (u))− 1 ≤ 19/128 for j = 1, . . . , ν and for every choice of B in (29), as shown in [30]. This paves the way for considering the principal value of the logarithm and then for writinĝ The next step concerns the computation of certain derivatives ofN (ρ; u) by means of the above identity. To this aim, the system of coordinates introduced in Sub-subsection 2.2.2 comes now in useful. Then, for k = 1, . . . , 4, where x can be ρ, u or v. To bound each of these products, use (64) as far asN (ρ; h k ) is concerned, and proceed with the detailed computation of bounds for the derivatives of the logarithms. As a starting point for all these calculations, consider the following equalities from [30]: and Log[μ 0 (ρπ j,ν ψ j,ν (u))] = − 1 2 Here, w j (ρ, u) :=μ 0 (ρπ j,ν ψ j,ν (u)) − 1, and the remainder R j (ρ, u) can assume one of the following forms: The aim is now to show that ν j=1 ∂ l ∂x l Log[μ 0 (ρπ j,ν ψ j,ν )] admits, for l = 1, 2, an upper bound presentable as a non-random polynomial in ρ, independent of u.
As far as the derivatives w.r.t. ρ are concerned, for the first two terms on the RHS of (145) one gets for l = 0, 1, 2, thanks to the fact that |ψ j,ν | = 1. Moreover, recall that m 2 = 3 in view of (47). Standard manipulations of the above expressions of for l = 0, 1, 2, with c 0 (R) = 1/24, c 1 (R) = 1/6 and c 2 (R) = 1/3. See [30] for the details. After recalling (60), this last inequality plainly entails As to the last term in (145), one has and This completes the proof of (66), showing also that the bound therein is independent of the choice of B in (29).
As to the derivatives w.r.t. x, one evaluates the expression of ∂ l ∂x l [ψ j,ν;k · v] m for l = 1, 2, and applies the Cauchy-Schwartz inequality. Whence, after recalling (29) and introducing the L 2 norm || · || * of matrices, one gets for l = 1, 2. Then, one proceeds with the study of the derivatives of the third term in the RHS of (145). As far as the first order derivative is concerned, one resorts to the second of the expressions of R j , given in the first part of this appendix, to write By virtue of (154), one gets which, recalling (60), entails To compute the second order derivatives of R j one employs the third of its expressions to write From (154) and the inequality |e ix − N −1 r=1 (ix) r /r!| ≤ |x| N /N ! one obtains the bound which, taking account of (60), becomes Finally, as to the remaining term in the RHS of (145), one utilizes (149)-(150) with x = u, v. Then, combining (144) with (155) and (156) gives Fix the sample point ω in U c and denote by n the value of ν at ω. Then, designate the values of π 2 1,ν , . . . , π 2 ν,ν at ω by a 1 , . . . , a n respectively, so that each a j belongs to [0, 1] and n j=1 a j = 1 in view of (23). The argument continues by resorting to the following combinatorial tools: i) The k-th elementary symmetric function S k (a 1 , . . . , a n ) defined by S k (a 1 , . . . , a n ) := ii) The k-th Newton symmetric function given by N k (a 1 , . . . , a n ) := n j=1 a k j .
Finally, recall the relation between r and a * given by (51) and apply (169) to obtain To prove (69), an obvious change of variable entails A.11 Proof of Proposition 2.8 The analysis to be developed is concerned with each of the charts Ω 1 , . . . , Ω 4 , but it is of the same kind for all of them. Therefore, even if the notation agrees with that introduced in A.9, the subscript k referring to the k-th chart will be dropped. The computation of the Laplacian in (141) yields for every fixed ρ. It is worth mentioning that here any differential operator D, applied to the complex-valued function h = f + ig, must be intended as Dh := Df + iDg and that the scalar product U 1 + iU 2 , V 1 + iV 2 is defined, by linearity, as U 1 , V 1 − U 2 , V 2 + i U 1 , V 2 + i U 2 , V 1 for the vector fields )|k 0 (w)}. As to the term containing the gradient, an application of the triangular inequality in (145) shows that for (ρ, u) in [0, R] × Ω, where E 1 and E 2 are conditional expectations, to be specified below, involving the remainders R j and Φ(w j )w 2 j respectively. As to the second summand, the convexity of the square of the Riemannian length entails To evaluate the last integral, recall that ψ j,ν is given by (29) with the proper choice of B as in (90)-(91), which makes ψ j,ν : Ω → S 2 smooth. Writing the gradient in coordinates yields The terms E 1 and E 2 in (172) can be derived as uniform bounds w.r.t. u by writing ∇ S 2 LogN (ρ; u) 2 S 2 in coordinates, according to As for E 1 , (156) and (157) give for every ρ in [0, R], the RHS being a G -measurable function. Apropos of E 2 , start from (149) and notice that, in view of (153) and the complete monotonicity of Φ, respectively. These last inequalities, in combination with (153) and (162), for every ρ in [0, R] with This concludes the analysis of the first summand in the RHS of (170), after noting that the upper bound provided by (177) is G -measurable. To proceed, for notational simplicity put so that (141) can be re-written asN = exp{−ρ 2 /2 + Aρ 2 + iBρ 3 + H}.