Chaoticity of the stationary distribution of rank-based interacting diffusions

The mean-field limit of systems of rank-based interacting diffusions is known to be described by a nonlinear diffusion process. We obtain a similar description at the level of stationary distributions. Our proof is based on explicit expressions for the Laplace transforms of these stationary distributions and yields convergence of the marginal distributions in Wasserstein distances of all orders. We highlight the consequences of this result on the study of rank-based models of equity markets, such as the Atlas model.

For all n ≥ 1, consider the system of rank-based interacting diffusions, or particles, (1) ∀i ∈ {1, . . ., n}, By the Girsanov theorem, the stochastic differential equation (1) possesses a unique weak solution, and actually a unique strong solution, see [13] for instance.When the number n of particles grows to infinity, propagation of chaos results toward the unique weak solution to the nonlinear (in McKean's sense) stochastic differential equation ( 2) dX(t) = b(F t (X(t)))dt + σdW (t), where obtained in [6,7]; we also refer to [9,3] for nonconstant diffusion coefficients.These propagation of chaos results are not uniform in time, and therefore do not provide any indication on the link between the long time behaviour of the particle system, which was studied in [10,7], and the long time behaviour of the nonlinear diffusion process, which was described in [7,9].The purpose of this note is to clarify this link by showing that the stationary distribution of a suitably modified version of the particle system is chaotic with respect to the stationary distribution of the nonlinear process.1.2.Projected particle system and nonlinear diffusion process.As was remarked in [7], the solution to (1) cannot converge to an equilibrium, since its projection along the direction (1, . . ., 1) is a drifted Brownian motion.One can however address the long time behaviour of the projection onto the hyperplane which is orthogonal to the singular direction (1, . . ., 1).The resulting process is called the projected particle system, it is the M n -valued diffusion process solving where b := 1 n n k=1 b n (k) = B (1).Propagation of chaos for the projected particle system toward the nonlinear diffusion process (2) was established in [7].The long time behaviour of the projected particle system was addressed by [10,7], under the following natural assumption: (E) b = B(1) = 0, and the function b is decreasing on [0, 1].The following proposition is due to Pal and Pitman [10, Theorem 8]; see also Jourdain and Malrieu [7,Theorem 2.12].We use the notation z (1) ≤ • • • ≤ z (n) to refer to the order statistics of a vector (z 1 , . . ., z n ) ∈ R n .Proposition 1.Under Assumption (E), for all n ≥ 1, and the probability distribution with density with respect to the surface measure dz on M n is the unique stationary distribution of the process (Z n 1 (t), . . ., Z n n (t)) t≥0 .Let us remark that the density p n ∞ (z) only depends on the order statistics of z, and therefore is invariant under the permutations of the coordinates of z.As a consequence, the probability distribution P n ∞ := p n ∞ (z)dz is a symmetric probability distribution on R n , which gives full measure to M n .
On the other hand, the stationary distributions of the nonlinear diffusion process were described in [9].This description relies on the function Φ introduced in Lemma 2 below.
Assumption (E) combined with the continuity of b also implies that therefore the integrals in the right-hand side of (3) are finite, and the function Φ is C 2 and increasing on (0, 1), and satisfies (4).The integrability of Φ on [0, 1] follows from (4) and the continuity of Φ on (0, 1), and by the Fubini-Tonelli theorem, whence (5).
Note that the inverse function Φ −1 of the function Φ defined in Lemma 2 is the cumulative distribution function F ∞ of a probability distribution P ∞ on R, which is such that Besides, since Φ is C 2 and Φ ′ (u) > 0 for all u ∈ (0, 1), we deduce that F ∞ possesses a density p ∞ with respect to the Lebesgue measure on R, which writes p ∞ (x) = 2 σ 2 B(F ∞ (x)).We can now recall the description of the set of stationary distributions of the nonlinear process, which follows from [9, Proposition 4.1].Proposition 3.Under Assumption (E), the stationary probability distributions for the nonlinear process (X t ) t≥0 are the translations of the probability distribution P ∞ ; that is to say, the probability distributions with cumulative distribution function x → F ∞ (x + x) for some x ∈ R.
Ergodicity results for the nonlinear diffusion process were obtained in [7,9].Let us precise that in [9], the stationary distributions are proven to be the translations of the function Ψ defined on (0, 1) by dv.
Since Φ and Ψ have the same derivative, it is clear that the set of translations of Φ −1 coincides with the set of translations of Ψ −1 .
As a consequence of Proposition 3, a stationary distribution for the nonlinear process is characterised by its expectation.In particular, P ∞ is the unique centered stationary distribution of the nonlinear process.
1.3.Main results and outline.We are now ready to state our main results and detail the outline of the paper.
Let us first recall the definition of the notion of chaoticity [12, Definition 2.1, p. 177].If P n is a probability distribution on R n and k ∈ {1, . . ., n}, we denote by P k,n the marginal distribution of the k first coordinates under P n .Definition 4. For all n ≥ 1, let P n be a symmetric probability distribution on R n , and let P be a probability distribution on R. The sequence (P n ) n≥1 is said to be P -chaotic if, for all k ≥ 1, P k,n converges weakly to the product measure P ⊗k .

Recall that we denote by P n
∞ the unique stationary distribution of the projected particle system; it is the probability distribution on R n with density p n ∞ (z) with respect to the surface measure dz on M n .On the other hand, P ∞ refers to the unique centered stationary distribution of the nonlinear diffusion process; it is the probability distribution with density p ∞ with respect to the Lebesgue measure on R. Of course, our purpose is to establish the P ∞ -chaoticity of the sequence (P n ∞ ) n≥1 .Our proof is based on the study of the Laplace transform L exp(rΦ(u))du.
Besides, under Assumption (E), the point (4) in Lemma 2 ensures that as soon as r is taken in the set then L ∞ (r) < +∞.This is the first part of Theorem 5.
Theorem 5.Under Assumption (E), (i) for all r ∈ V, L ∞ (r) is finite and writes exp(rΦ(u))du; (ii) for all (s, t) taken in the set where, for all 1 ≤ i < j ≤ n, The point (ii) is proved in Section 2, while the point (iii) is detailed in Section 3.
As is stated in Corollary 8 below, Theorem 5 implies the P ∞ -chaoticity of P n ∞ , and actually yields the convergence of P k,n ∞ in a stronger sense than in Definition 4; namely, in Wasserstein distance [14].Definition 6.Let k ≥ 1 and q ∈ [1, +∞).The Wasserstein distance of order q between two probability distributions µ and ν on R k is defined by where Π(µ, ν) refers to the set of pairs of random variables with marginal distributions µ and ν.
Remark 7. The definition of W q depends on the choice of the norm | • | on R k .But since all norms are equivalent on R k , all the associated distances W q are also equivalent.Therefore, convergence results for the W q topology do not depend on the choice of the underlying norm.In this paper, we take the convention that the Wasserstein distance of order q is defined with respect to the ℓ q norm |x| := (|x We derive chaoticity and convergence in Wasserstein distance as a corollary of Theorem 5. Corollary 8.Under Assumption (E), (i) the sequence of stationary distributions P n ∞ of the projected particle system is P ∞ -chaotic, (ii) for all k ≥ 1, for all q ∈ [1, +∞), The proof of Corollary 8 is postponed to Appendix A. A summary of the long time and large scale behaviour of the projected particle system is detailed on Figure 1.

Projected particle system
Stationary distribution P n ∞ Nonlinear diffusion process Stationary distribution P ∞ n → +∞: propagation of chaos [7] n → +∞: chaoticity (Corollary 8) t → +∞: ergodicity [10] t → +∞: ergodicity [7] Figure 1.A summary of convergence results, in long time as well as for a large number of particles, for the projected particle system.
Figure 1 illustrates the fact that, when it makes sense, the interversion of the limits 'n → +∞' and 't → +∞' is generically correct for functionals of systems of rank-based interacting particles.This remark is of interest in the study of rank-based models of equity markets, such as the Atlas model introduced by Fernholz in the framework of Stochastic Portfolio Theory [4,1,5].Indeed, in this context, relevant quantities such as capital distribution curves or growth rates of portfolios are expressed in terms of the stationary distribution P n ∞ described in Proposition 1.The asymptotic behaviour of these quantities when the size of the market grows to infinity where investigated in [1,2,11].On the other hand, it was suggested in [8] to use the propagation of chaos results of [9] to obtain a functional description of an infinite market first, and then apply the available ergodicity results on the nonlinear diffusion process to derive closed formulas for these relevant quantities.Theorem 5 is a first step toward the validation of the equivalence of both approaches, and we refer to [8] for a detailed account.

Expression of the Laplace transforms
This section is dedicated to the proof the point (ii) of Theorem 5. We first collect preliminary estimates in Subsection 2.1, and then compute the Laplace transform 2.1.Preliminary estimates.Under Assumption (E), for all r ∈ V, for all n ≥ 1, for all k ∈ {1, . . ., n − 1}, let us define so that the quantities J n i,j (s, t) introduced in Theorem 5 rewrite In this subsection, we exhibit upper bounds on the quantities f + k,n (r) and f − k,n (r), for r ∈ {s, t, s + t}, at least for n large enough, which ensure that the quantities J n i,j (s, t) are well defined.We roughly proceed as follows: when k/n is far from 1, then k/n B(k/n) remains bounded by above, so that f + k,n (r) is arbitrarily small for n large enough.On the contrary, when k/n is close to and the fact that r ∈ V provides natural upper bounds on the right-hand side.The same ideas allow to obtain similar bounds on f − k,n (r).We now give a rigorous formalisation of these arguments.Under Assumption (E), for all ǫ > 0 such that ǫ < b(0) ∧ (−b(1)), we introduce Let us now fix (s, t) ∈ V ǫ 2 .Then, for δ ∈ (0, 1/2) small enough, Besides, by Assumption (E), we have The heuristic arguments detailed at the beginning of the subsection translate into the following precise estimates: for all r ∈ {s, t, s + t}, for all n ≥ 1, for all k ∈ {1, . . ., n − 1}, Similarly, In particular, if n is chosen so that then we deduce from (6) that, for all k ∈ {1, . . ., n − 1}, Similarly, if n is chosen so that then we deduce from ( 6) and ( 7) that, for all k ∈ {1, . . ., n − 1}, These results are gathered together in the following lemma.Lemma 9. Let (s, t) ∈ V 2 .Under Assumption (E), there exists ǫ > 0 such that (s, t) ∈ V ǫ 2 .Let δ ∈ (0, 1/2) satisfying the conditions above.Let us define ᾱ ∈ (0, 1) by Then, there exists n 0 ≥ 2 such that, for all n ≥ n 0 , for all k ∈ {1, . . ., n − 1}, the quantities ∞ (s, t).Let us first note that, since p n ∞ (z)dz is a symmetric probability distribution on R n , then for all symmetric and nonnegative function f : R n → R, where, for all z = (z 1 , . . ., Using the symmetry of p n ∞ (z)dz again, we deduce that, for all (s, t Let us now fix i ∈ {1, . . ., n} and j = i, and define Note that, at this stage, nothing prevents J n i,j (s, t) from being infinite.Using the parametrisation of M n by the n − 1 coordinates x 1 = z 1 , . . ., x i−1 = z i−1 , x i+1 = z i+1 , . . ., x n = z n , so that the surface measure dz on M n rewrites We denote and let y k := x k + S, for all k = i, in the right-hand side above.Then we have Let n 0 ≥ 2 and ᾱ ∈ (0, 1) be given by Lemma 9. We deduce from the definition of γ k,n i,j (s, t) that, if n ≥ n 0 , then, for all i, j ∈ {1, . . ., n} such that i = j, for all k ∈ {1, . . ., i − 1}, and similarly, for all k ∈ {i + 1, . . ., n}, which ensures that J −,n i,j (s, t) and J +,n i,j (s, t) are finite and write thanks to successive integrations.This finally gives To complete the proof, we remark that , which allows us to get rid of the constant term n! √ n nZn and to obtain the expected expression of J n i,j (s, t) in the point (ii) of Theorem 5, for (s, t) ∈ V 2 .

Convergence of the Laplace transforms
The proof of the point (iii) of Theorem 5 works in two steps: first, we prove that, for all t ∈ V, the Laplace transform . Second, we check that, for (s, t) ∈ V 2 , the difference between L 2,n ∞ (s, t) and the product L 1,n ∞ (s)L 1,n ∞ (t) vanishes.These two steps are addressed in the respective Subsections 3.2 and 3.3.The preliminary Subsection 3.1 gathers useful elementary results.
In particular, we deduce from (TL1) that, for all C ∈ [0, +∞), By the results of Section 2, there exists n 0 ≥ 2 such that, for all n ≥ n 0 , the Laplace transform Then, we have Let ǫ > 0 and δ ∈ (0, 1/2) be given by Subsection 2.1 for the pair (t, 0) ∈ V 2 .We split the sum appearing in the right-hand side of (9) into boundary terms, corresponding to i ≤ nδ and i ≥ n(1 − δ), and a central term, corresponding to nδ < i < n(1 − δ).These terms are addressed separately, in the respective §3.2.1 and §3.2.2.

Boundary terms. For all
It is an easy consequence of the point (i) in Theorem 5 that the integral in the right-hand side above vanishes with δ.The purpose of this paragraph is to show that ( 10) Let us first assume that t ≥ 0.Then, for all i ≤ nδ, We now use the fact that, if i ≤ nδ, then for all k ∈ {1, . . ., i − 1}, as soon as n is large enough to ensure that tσ 2 /(2n(b(0) − ǫ) < 1.Using (8), we deduce that and (10) easily follows.

Convergence of
Let n 0 ≥ 2 be given by Lemma 9. Then for all n ≥ n 0 , By the results of Subsection 3.2, the last term in the right-hand side above vanishes when n grows to infinity.The diagonal term To this aim, we write, for all i ∈ {1, . . ., n}, , and note that if st ≥ 0, then I n i (s)I n i (t) ≤ I n i (s + t), so that (12) follows from the results of Subsection 3.2.On the other hand, if st < 0, say s < 0 < t, then Let us fix ǫ > 0 and δ ∈ (0, 1/2) as in Subsection 2.1.Arguing as in §3.2.1, we obtain and the same arguments apply to the sum for i ≥ n(1 − δ).On the other hand, combining the estimates ( 6), ( 7) with (8) yields .
We deduce that We deduce from the estimates ( 6) and ( 7) that For n large enough, the right-hand side above is lower that 3C, so that (TL2) yields As a consequence, there exist a nonnegative and finite constant M ′ (δ), that depends on δ, and a nonnegative and finite constant C ′ , that does not depend on δ, such that, for n large enough, for all i = j in {1, . . ., n}, |J n i,j (s, t) − I n i (s)I n j (t)| ≤ I n i (s)I n j (t) To complete the proof of (13), we now check that