Wishart processes : mean-field limit, long time behavior, and free probability
aa r X i v : . [ m a t h . P R ] S e p About the eigenvalues of Wishart processes
Ezéchiel Kahn ∗ September 22, 2020
Abstract
This paper is devoted to the study of the eigenvalues of the Wishart process which are the analogof the Dyson Brownian Motion for covariance matrices. Such processes were in particular studied byBru. The mean field convergence of the empirical measure of these eigenvalues was proved Malecki andPerez. In this paper, we provide a new approach to the mean field convergence problem using toolsfrom the free rectangular convolution theory developed by Benaych-Georges, which in particular allowsto compute explicitly the limit measure valued flow. We highlight the link with the integro-differentialequation related to the mean field limit and its translation into a complex Burgers partial differentialequation.
Contents
We will denote by N = { , , , . . . } the set of non-negative integers, and by N ∗ = { , , , . . . } the set ofpositive integers. Let n, m ∈ N ∗ such that n ≤ m and let ( M t ) t be a stochastic process taking its values inthe space of n × m matrices with real entries verifying the following stochastic differential equation (SDE) dM t = κdW t − γM t dt (1)where W is a n × m matrix filled with independent Brownian motions, γ ≥ κ ≥ M is thus amatrix whose entries are independent Ornstein-Uhlenbeck processes just as the one considered in [Bru91].The reader will find in [KO01] an analysis of the complex analog of Bru’s model. ∗ CERMICS, Ecole des Ponts, INRIA, Marne-la-Vallée, France. Email: [email protected] t ≥ x ,nt , . . . , x n,nt ) and ( λ ,nt , . . . , λ n,nt ) be respectively the eigenvalues of the n × n squarematrices q m M t M ∗ t and m M t M ∗ t where ∗ is the transposition operator.It is proved in [Bru89] and [Bru91] that the eigenvalues ( λ ,nt , . . . , λ n,nt ) satisfy the system of SDEs dλ i,nt = 2 κ √ m q λ i,nt dB it + κ dt − γλ i,nt dt + κ m X j = i λ i,nt + λ j,nt λ i,nt − λ j,nt dt for all i ∈ { , . . . n } (2)0 ≤ λ ,nt < · · · < λ n,nt a.s. dt-a.e.where B , . . . , B n are independent Brownian motions. It is proved for instance in [GM14] that the SDE (2)admits a strong pathwise unique solution.If ( λ ,nt , . . . , λ n,nt ) t is solution to the SDE (2), then a formal calculus gives for x i,n = √ λ i,n : dx i,nt = κ √ m dB it + (cid:18) − m (cid:19) κ x i,nt − γx i,nt + κ mx i,nt X j = i ( x i,nt ) + ( x j,nt ) ( x i,nt ) − ( x j,nt ) dt ≤ x ,nt < · · · < x n,nt a.s. dt-a.e. . We respectively define the empirical measure of the eigenvalues of m M t M ∗ t and the symmetrized empiricalmeasure of the eigenvalues of q m M t M ∗ t by ν nt = 1 n n X i =1 δ λ i,nt and µ nt = p ν nt = 12 n n X i =1 (cid:16) δ x i,nt + δ − x i,nt (cid:17) where here and in the rest of the paper, for any probability measure µ on [0 , + ∞ ), √ µ = sym( √ .♯µ ) denotesthe symmetrization of the push-forward of µ by the map v
7→ √ v . The application sym is fully defined inSection 4. These measure sequences are random variables sequences in the space of probability measures onrespectively R + and R . We will thus speak about weak convergence in probability or almost surely throughoutthe paper in the sense that the sequence converges in probability or almost surely in the space of probabilitymeasures on R + (or R ) with the distance of Prokhorov (see for instance [Bil13]). Let us note that by nonnegativeness of the ( λ ,nt , . . . , λ n,nt ) t ≥ , symmetry of µ nt and continuity of the functions v v and v
7→ √ v ,there is equivalence between the convergence of ( ν n ) n and the convergence of ( µ n ) n .In this note, we are interested in the convergence and limit of these measure processes when n and m goto infinity at a rate n/m → α ∈ (0 , µ n ) n problem, using free probability tools (inparticular the rectangular free convolution of index α denoted by ⊞ α defined in Section 4). Let us first denoteby µ MP ρ,σ the Marcenko-Pastur distribution with shape parameter ρ and scale parameter σ which admitsthe density with respect to the Lebesgue measure : x p ( a + − x )( x − a − )2 πρxσ [ a − ,a + ] where a ± = σ (1 ± √ ρ ) . Theorem 1.1 (Mean-field limit) . Let us assume n ≤ m , and that when n grows to infinity, m grows toinfinity too, with nm → α ∈ (0 , . Let us also assume that ( µ n ) n converges weakly in probability towards anon random probability measure µ . Then for all t ≥ , ( µ nt ) n ≥ converges weakly in probability and µ t := lim n →∞ µ nt = ( e − γt µ ) ⊞ α q µ MP α,σt here ⊞ α is the rectangular free convolution of parameter α and e − γt µ denotes the push-forward of µ bythe map v e − γt v , and where for all t ≥ : σ t = κ γ (1 − e − γt ) if γ = 0 κ t if γ = 0 . Moreover, if γ = 0 and with σ ∞ = q κ γ , lim t → + ∞ µ t = p µ MP α,σ ∞ . This last result is in fact true in a complex framework (with complex Brownian motions of the form( B t + i B t ) t ≥ where ( B t ) t ≥ and ( B t ) t ≥ are independent real Brownian motions), as the reader will seein Section 2. This result might also be extended to a quaternionic framework but the rectangular freeconvolution theory is not developed yet in this case. We moreover proved the following result. Theorem 1.2 (Commutativity of the limits) . Under the assumption of Theorem 1.1, let us moreover suppose γ = 0 and that for all n ∈ N ∗ : P ni =1 E [ λ i,n ] < + ∞ . Let us note σ ∞ = q κ γ . Then we have in the sense ofweak convergence lim n →∞ n/m → α lim t →∞ ν nt = lim t →∞ lim n →∞ n/m → α ν nt = µ MP α,σ ∞ . The SDE (2) is related to the β - Wishart (or β - Laguerre ) process, the system of particles defined by theSDE dλ i,nt = 2 q λ i,nt dB it − ηλ i,nt dt + δdt + β X j = i λ i,nt + λ j,nt λ i,nt − λ j,nt dt for all i ∈ { , . . . n } ≤ λ ,nt < · · · < λ n,nt a.s. dt-a.e.with β, δ, η > β = 1, η = mγ/κ , δ = m , and after the change of time t → κ m t , correspondsto the dynamics of the process followed by the eigenvalues of m M t M ∗ t . Sometimes in the literature, the β - Wishart process is considered with η = 0 (see for instance [Dem09]) or with η = 1 / β -Wishart process is also found in the literature as the system of particles givenby the SDE dλ i,nt = 2 q λ i,nt dB it − ηλ i,nt dt + β δ + X j = i λ i,nt + λ j,nt λ i,nt − λ j,nt dt for all i ∈ { , . . . n } ≤ λ ,nt < · · · < λ n,nt a.s. dt-a.e.,as for instance in [ABMV13].The reader will find in Section 3 an extension of Theorems 1.1 and 1.2 to the β -Wishart case.3imilar problems where tackled before, often by the direct study of the limit of empirical measure processesof a system of interacting diffusive particles. In [RS93], the authors studied the empirical measure of diffusingparticles with electrostatic repulsion and a negative linear elastic force. They proved that the accumulationpoints of the sequence of empirical measures of the particles sequence (the sequence being indexed by thenumber of particles) satisfy an integro-differential equation. The method of their proof was used in otherpapers and is reproduced here in our context. They proved the uniqueness of the limit by translating theintegro-differential equation problem into a complex partial differential equation (PDE) problem, a Burgersequation, which they could solve using the method of characteristics.In [CL97] was studied a more general system of diffusing particles with electrostatic repulsion, a linearelastic force, and an additional constant drift. In this case, an integro-differential equation for the limitof the empirical measure of the particles was found and translated into a Burgers-like complex PDE usingthe methods developed in [RS93]. However, the existence and uniqueness of the solution to the PDE wereproved using Fourier transforms methods. A second article from the same authors [CL01] followed, studyingBrownian particles with electrostatic repulsion on the circle. There, the same procedure allowed the authorsto get another Burgers-like PDE of order two, the existence and uniqueness of a solution to which were thistime proved using the Hopf-Cole transformation. This allowed the authors to find an explicit expression ofthe limit measure flow when each particle starts from the position 0, i.e. when the initial condition of theempirical measure is δ .More recently, these questions regained in interest and two articles, [SYY19] and [MP19] tackled moregeneral problems. They got interested into the empirical (spectral) measure of the first introduced in [GM11]general matrix valued stochastic differential equations on H n , the space of Hermitian n × n matrices, of theform dX t = g ( X t ) dW t h ( X t ) + h ( X t ) d ( W t ) † g ( X t ) + 1 n b n ( X t ) dt, X ∈ H n , where † is the conjugate transpose operator, and where the continuous functions g, h, b n : R → R act spectrallyon X t , i.e. if X ∈ H n , U unitary and D diagonal are such that X = U † DU , then g is identified with themap H n ∋ X U † Diag( g ( D , ) , · · · , g ( D n,n )) U . Here W = ( W t ) t stands for a n × n complex-valuedBrownian motion, i.e. the matrix valued process with entries being independent one-dimensional complex-valued Brownian motions. This general formulation encompasses the β -Wishart (or β -Laguerre) model andthe β -Jacobi model, as it is stated in [GM11]. After proving the tightness of the family of measure valuedprocesses, these two papers use a [RS93]-like method to get the integro-differential equation that is satisfiedby the accumulation points of the measure valued processes family. In [MP19], the more general of the twoarticles, the authors derive existence and uniqueness of the solution to the integro-differential equation, andthus the convergence of the empirical measure valued process when n → ∞ using the fact that the momentsof the limit measure valued flow are uniquely determined by the integro-differential equation. Moreover, insome peculiar cases, they give an explicit expression of the limit, but only for the initial condition X = 0. In[SYY19], rather than using a moment approach, they translate as in [RS93] the integro-differential equationproblem into a Burgers equation, which they do not solve in general, but only for the initial condition X = 0, using scaling properties of the initial SDE problem. For this initial condition, the Marcenko-Pasturdistribution appears naturally in the expression of the measure valued flow limit.The β -Wishart (or β -Laguerre) problem was also tackled in [ABMV13]. They compute there in the hightemperature regime (when βn −→ n →∞ c ∈ (0 , ∞ )) the limit integro-differential equation, parametrized by c ,the stationary probability measure of this equation, and remark that this limit distribution interpolates theMarcenko-Pastur distribution on the c → + ∞ limit and the Gamma distribution on the c → β -Wishart(or β -Laguerre) and β -Jacobi processes using a moment based method as in [MP19] to prove the uniqueness4he limit measure process. The authors moreover compute, in the high temperature regime (when βn −→ n →∞ c ∈ (0 , ∞ )), the long-time behaviour of the limit measure flow for both process families.The interest of this note is to understand the limit behaviour of the empirical spectral measure of m M t M ∗ t and q m M t M ∗ t both in a real and a complex framework, using rectangular free probability tools. This pointof view, more random matrix theory and free probability oriented, sheds a new light on the problem of thelimit behaviour of the empirical measure process of β -Wishart particles for all β >
0, and allows to computethe limit measure valued flow for any initial condition, and to recover the long-time behaviour of the limitmeasure valued flow. For the sake of completeness, we also derive the integro-differential equation and thecomplex PDE approach of the problem.The note is organized the following way. After the Introduction where the main results are given, thecomplex version of these results are given in Section 2. The reader will find in Section 3 an application ofTheorem 1.1 to the general β -Wishart processes. Preliminary results about the Cauchy-Stieltjes transformand about the free and the rectangular free convolution are given in Section 4, and the results are proven inSection 5. Acknowledgement : I thank Djalil Chafai and Benjamin Jourdain for numerous fruitful discussions.
Theorem 1.1 in fact true in a complex framework, i.e. for ( M t ) t ≥ being a complex valued matrix process,following the SDE dM t = κdW t − γM t dt with W a n × m matrix filled with independent complex Brownian motions (of the form ( B t + i B t ) t ≥ where( B t ) t ≥ and ( B t ) t ≥ are independent real Brownian motions), M a complex valued random matrix. In thisSection, we keep the definitions of the ( λ ,nt , . . . , λ n,nt ) t and ( x ,nt , . . . , x n,nt ) t as the eigenvalues of respectively m M t M ∗ t and q m M t M ∗ t but with the symbol ∗ denoting the conjugate transpose rather than the transposeoperator. Theorem 2.1 (Complex case) . Theorem 1.1 applies in this framework with for all t ≥ : σ t = κ γ (1 − e − γt ) if γ = 02 κ t if γ = 0 . and with σ ∞ = q κ γ . Let us note that in this context, the eigenvalues ( λ ,nt , . . . , λ n,nt ) t follow the SDE (see [KO01]) : dλ i,nt = 2 κ √ m q λ i,nt dB it + 2 κ dt − γλ i,nt dt + 2 κ m X j = i λ i,nt + λ j,nt λ i,nt − λ j,nt dt for all i ∈ { , . . . n } ≤ λ ,nt < · · · < λ n,nt a.s. dt-a.e.where B , . . . , B n are independent Brownian motions. It corresponds to a β -Wishart process with β = 2, orto a W (2 ,
1) process, which is defined in the next Section.5
Application to more general Wishart processes
For β , β >
0, let us consider the changed of time general Wishart process defined by the SDE : dλ i,n,Wt = 2 κ √ m q λ i,n,Wt dB it − γλ i,n,Wt dt + β κ β m X j = i λ i,n,Wt + λ j,n,Wt λ i,n,Wt − λ j,n,Wt dt for all i ∈ { , . . . n } (3)0 ≤ λ ,n,Wt < · · · < λ n,n,Wt a.s. dt-a.e.where B , . . . , B n are independent real Brownian motions. We will refer to this SDE by W ( β , β ). Asstated in the introduction, the classical β -Wishart processes found in the literature are of the form W (1 , β )or W ( β,
1) (after the change of variables t mκ t ). We also remark that the SDE (2) corresponds to W (1 , t mκ t to the SDE(3)) that this SDE admits a strong pathwise unique solution defined on R + as soon as m − ( n − β > • β β ≥ • or if 0 < β β < mβ + (2 − n ) β β ≥ t mκ t in the SDE (3) coupled with the results [JK20, Lemma 3.1 and Proposition 2.8] prove that, givenintegrable initial conditions, i.e. P ni =1 E [ λ i,n,W ] < + ∞ , the distribution of ( λ ,n,Wt , . . . , λ n,n,Wt ) convergesweakly when t → + ∞ to a unique stationary probability measure with density with respect to the Lebesguemeasure ( λ , . . . , λ n ) Z n Y i =1 ( λ i ) β m − ( n − β β − e − mγκ λ i Y j = i | λ j − λ i | β β / ≤ λ ≤···≤ λ n (4)where Z is a normalizing constant. For β = 1, this Gibbs measure is the β -Laguerre ensemble, on realsymmetric matrices for β = 1, on complex hermitian matrices for β = 2 and on quaternion self-dualmatrices for β = 4, see for instance [For10]. This distribution is moreover related to the distribution of thesingular values of n × m random matrix with independent identically distributed real (for β = 1) or complex(for β = 2) Gaussian entries.It is a well known fact that, for β = 1, if ( λ , . . . , λ n ) is a random vector distributed according to (4),then ν n,W = 1 n n X i =1 δ λ i −→ n →∞ n/m → α µ MP α,σ ∞ (5)with σ ∞ = q β κ γ , see for instance [HP06, Theorem 5.5.7].Let us define for all t ≥ ν n,Wt = 1 n n X i =1 δ λ i,n,Wt , µ n,Wt = q ν n,Wt = 12 n n X i =1 (cid:16) δ √ λ i,n,Wt + δ − √ λ i,n,Wt (cid:17) . The next result tackles the limit of the sequence ( ν n,Wt ) n . As written in the introduction, it is a peculiarcase of the results of [MP19] reproduced here partly for the sake of completeness. The limit of the empiricalmeasure problem is transformed into a complex PDE problem, a complex Burgers equation, to which theCauchy-Stieltjes transform of the limit measure process, if it exists, must be solution.Definitions and properties about the Cauchy-Stieltjes transform are given in Section 3.6 heorem 3.1 (Complex Burgers and mean field limit : weak formulation) . Let us assume that the measurevalued sequence ( ν n,W ) n converges weakly in probability when n → + ∞ with nm → α ∈ (0 , to a limitprobability measure denoted by ν W and that sup n Z x ν n,W ( dx ) < ∞ . Then the family { ( ν n,Wt ) t ≥ , n ∈ N ∗ } is tight and any limiting measure valued flow when n goes to infinitywith nm → α ∈ (0 , satisfies for all twice continuously differentiable real test function f the equation : h ν t , f i = h ν , f i + Z t h ν s , ( β κ − γ Φ) f ′ i ds + αβ β κ Z t (cid:18)Z Z ( x + y ) f ′ ( x ) − f ′ ( y ) x − y ν s ( dx ) ν s ( dy ) (cid:19) ds (6) where Φ : x x and with the convention f ′ ( x ) − f ′ ( y ) x − y = f ′′ ( x ) when x = y . If ν admits a characteristicfunction, and if this function is analytic on a neighborhood of the origin, then this equation admits a uniquesolution.Moreover, if ( ν t ) t ≥ is a solution to equation (6) and if for all t ≥ , G t is the Cauchy-Stieltjes transformof ν t , then G satisfies the complex Burgers PDE ∂∂t G t ( z ) = ( αβ β κ − β κ + 2 γz ) ∂∂z G t ( z ) − αβ β κ zG t ( z ) ∂∂z G t ( z ) − αβ β κ G t ( z ) + 2 γG t ( z ) ,G ( z ) = Z ν W ( dv ) z − v = φ ( z ) . (7) Remark 3.2 (Moments dynamics) . Let ν = ( ν t ) t be a real measure valued flow verifying equation (6), andlet us denote for all k ∈ N ∗ , t ≥ : m kt = E [ ν kt ] . Then for all t ≥ :(i) m t = β κ γ + (cid:16) m − β κ γ (cid:17) e − γt ;(ii) m t = γ h ( β κ + ( αβ β κ − γm ) β κ + 4 γ m − αβ β κ γm )e − γt +4 β κ ( β α + 1) (cid:16)(cid:16) γm − β κ (cid:17) e − γt + β κ (cid:17) i ;(iii) ∀ k ≥ , ddt m kt = kβ κ m k − t − kγm kt + kαβ β κ P k − j =0 (cid:16) m j +1 t m k − − jt + m jt m k − j − t (cid:17) . These equations are used in [MP19, TT20b] to prove existence and uniqueness of the solution to the integro-differential equation (6).
Remark 3.3.
The PDE (7) satisfies the assumptions of the Cauchy-Kowalevski Theorem, see for instance[Fol95, Theorem 1.25]. However, this theorem only gives local existence and uniqueness of an analytic solution(both in space and time) to the PDE. We did not manage to expand this approach to conclude to global existenceand uniqueness of a solution.
The next Proposition gives results on the PDE (7) using a different method than the ones used in [MP19].
Proposition 3.4 (Marcenko Pastur stability along the Burgers PDE and stationarity) . i) Let ρ , σ ∈ R + . Let us assume κ = 0 and that ν W = lim n →∞ ν n,W follows a Marcenko-Pastur distributionof shape parameter ρ and of scale parameter σ .Then, the PDE (7) admits a solution which is the Cauchy-Stieltjes transform of a Marcenko-Pasturdistribution for all t > if and only if ρ = β α . In this case, this solution can be written as theCauchy-Stieltjes transform of the Marcenko-Pastur distribution µ MP β α,σ ( t ) with σ : t ∈ R + s(cid:18) σ − β κ γ (cid:19) e − γt + β κ γ if γ = 0 q σ + β κ t if γ = 0 (ii) Let us assume γ = 0 and let σ = √ β κ √ γ . Then G µ MPβ α,σ is the unique stationary solution to theequation (7) corresponding to the Cauchy-Stieltjes transform of a probability measure on R . Moreover,under the assumptions of Theorem 3.1 and if ν = µ MP β α,σ , then for all t ≥ , ν Wt = lim n →∞ ν n,Wt = µ MP β α,σ . To our knowledge, the assertion ( i ) was not remarked yet in the past literature, and ( ii ) can be seenas in the continuity of the results of [ABMV13]. Indeed, their computations are in the high temperatureregime (when βn −→ n →∞ c ∈ (0 , ∞ )) so that the limit of their integro-differential equation and its stationaryprobability measure are parametrized by c , and they remark that this limit stationary probability measureconverges to the Marcenko-Pastur distribution in the limit c → + ∞ .The next result gives a complete answer to the limit of ( µ n,W ) n problem, using free probability tools(in particular the rectangular free convolution defined in Section 3). It is the analog of Theorem 1.1 in thiscontext. Theorem 3.5 (Mean-field limit for the general Wishart process) . Under the assumptions of Theorem 3.1,let us moreover assume β α ≤ , that ( λ ,n,Wt , . . . , λ n,n,Wt ) t is defined on R + and that µ W = p ν W . Let usmoreover assume that ν W has a characteristic function which is analytic on a neighborhood of the origin.Then for all t ≥ , ( µ n,Wt ) n ≥ converges weakly in probability and µ Wt := lim n →∞ µ n,Wt = ( e − γt µ W ) ⊞ β α q µ MP β α,σt where ⊞ β α is the rectangular free convolution of parameter β α , where e − γt µ W denotes the push-forward of µ W by the map v e − γt v , and where for all t ≥ : σ t = β κ γ (1 − e − γt ) if γ = 0 β κ t if γ = 0 . Moreover, if γ = 0 and with σ ∞ = q β κ γ , lim t → + ∞ µ Wt = q µ MP β α,σ ∞ . Theorem 3.5 coupled with the limit (5) allows to show the following result.8 heorem 3.6 (Commutativity of the limits) . Under the assumptions of Theorem 3.5, let us moreover suppose γ = 0 , β = 1 and that for all n ∈ N ∗ : P ni =1 E [ λ i,n,W ] < + ∞ . Let us note σ ∞ = q β κ γ . Then we have inthe sense of weak convergence lim n →∞ n/m → α lim t →∞ ν n,Wt = lim t →∞ lim n →∞ n/m → α ν n,Wt = µ MP α,σ ∞ . Let us denote C ± = { z ∈ C , ±ℑ ( z ) > } . Let µ be a probability measure on R . Its Cauchy-Stieltjes transform is defined by z ∈ C + G µ ( z ) = Z R µ ( dv ) z − v ∈ C − . The next result shows that this transformation characterizes the probability measure on R . Theorem 4.1 ([SM17]) . (1)Let µ be a probability measure on R . Then,(i) G µ is analytic on C + ,(ii) we have lim y →∞ i yG µ (i y ) = 1 . (2) Any probability measure on R can be recovered from its Cauchy-Stieltjes transform G µ via the Stieltjesinversion formula : for all a, b ∈ R with a
The R-tranform linearizes the free convolution : for µ and ν probability measures on the realline, and for z in a neighbourhood of zero, R µ ⊞ ν ( z ) = R µ ( z ) + R ν ( z ) and µ ⊞ ν is the unique probability measure verifying this relation. Let us denote by B ( R ) the Borel sets of R . Let us define the application from M ( R + ), the set ofprobability measures on R + , to M S ( R ) = { µ ∈ M ( R ) , ∀ A ∈ B ( R ) , µ ( A ) = µ ( − A ) } , the set of symmetricprobability measures on R :sym : µ ∈ M ( R + ) (cid:18) sym( µ ) : A ∈ B ( R ) µ ( A ∩ R + ) + µ ( − A ∩ R + )2 (cid:19) ∈ M S ( R ) . This application is bijective from M ( R + ) to M S ( R ) and admits the inverse:sym − ( ν ) : A ∈ B ( R + ) A (0) ν ( { } ) + 2 ν ( A \{ } ) . (8)For any α ∈ [0 , ⊞ α can be defined the following way. Theorem 4.4 (Additive free rectangular convolution of ratio α for random matrices [BG09]) . For all n, m ∈ N ∗ let us define M n,m and N n,m two independent n × m random matrices, such that the distribution of M n,m is invariant under the action of the unitary group by conjugation on any side, • the empirical measures sequences ( µ √ M n,m M ∗ n,m ) n,m and ( µ √ N n,m N ∗ n,m ) n,m respectively converge inprobability, when n and m go to infinity with n/m tending to α ∈ (0 , , to non-random probabilitymeasures µ M ∞ and µ N ∞ .Then in the sense of weak convergence in probability, sym (cid:16) µ √ ( M n,m + N n,m )( M n,m + N n,m ) ∗ (cid:17) −→ n,m →∞ n/m → α sym( µ M ∞ ) ⊞ α sym( µ N ∞ ) . This operation can also be equivalently defined in reference to free elements of a rectangular non commu-tative probability space.Let µ be a symmetric probability measure on R . Its rectangular Cauchy transform with ratio α is definedby H µ : z ∈ C \ [0 , + ∞ ) z ( αM µ ( z ) + 1)( M µ ( z ) + 1)where M µ : z ∈ C \ [0 , + ∞ ) Z R zv − zv dµ ( v ) = 1 √ z G µ (cid:18) √ z (cid:19) − . This equality can be derived the following way in a neighbourhood of zero : G µ (cid:18) √ z (cid:19) = z Z R − z v dµ ( v )= z X k ≥ Z R z k v k dµ ( v )= z zv X k ≥ Z R z k v k dµ ( v ) where we use in the last equality the fact that µ is symmetric.The rectangular R-transform with ratio α of µ is defined on a neighbourhood of zero by C µ ( z ) = U (cid:18) zH − µ ( z ) − (cid:19) where on a neighbourhood of zero U ( z ) = − α − α + 1) + 4 αz ] / α if α = 0 z if α = 0 . Theorem 4.5 ([BG09]) . The rectangular R-transform with ratio α linearizes the rectangular free convolutionwith ratio α : for µ and ν symmetric probability measures on the real line, and for z in a neighbourhood ofzero, C µ ⊞ α ν ( z ) = C µ ( z ) + C ν ( z ) and µ ⊞ α ν is the unique symmetric probability measure verifying this relation. heorem 4.6 (Injectivity of the rectangular R-transform, [BG09]) . If the rectangular R-transforms with ratio α of two symmetric probability measures coincide on a neighbourhood of in ( −∞ , , then the measures areequal. The convergence results with free convolution and rectangular free convolution overlap in the case of M n , N n n × n square symmetric semi-definite positive codiagonalizable matrices. Then we havesym (cid:16) µ √ ( M n + N n )( M n + N n ) ∗ (cid:17) = sym( µ M n + N n ) y n → ∞ y n → ∞ sym( µ M ∞ ) ⊞ sym( µ N ∞ ) = sym( µ M ∞ ⊞ µ N ∞ ) . Proof of Theorem 1.1.
The entries of the matrix M are independent Ornstein-Uhlenbeck processes just asthe one considered in [Bru91]. We thus can write for all t ≥ M t = M e − γt + κ Z t e γ ( s − t ) dW s . The idea is to use a suitable rectangular free convolution, see [BG09]. More precisely, the matrices A n,t = M e − γt and B n,t = κ R t e γ ( s − t ) dW s are independent and we may use a version of Voiculescu asymptoticfreeness theorem, see [HP00]. The precise result to use is Theorem 4.4 (see [BG09, Theorem 3.13]). Indeed, B n,t is a matrix filled with i.i.d Gaussian random variables of variance σ t = E "(cid:12)(cid:12)(cid:12)(cid:12) κ Z t e − γ ( s − t ) dW s (cid:12)(cid:12)(cid:12)(cid:12) = κ Z t e γ ( s − t ) ds = κ γ (1 − e − γt ) if γ = 0 κ t if γ = 0by Ito’s isometry, and is thus bi-unitary invariant. The Marcenko-Pastur theorem (see for instance [BS10,Theorem 3.10]) tells us that in the sense of convergence in probability, µ m B n,t B ∗ n,t = 1 n n X i =1 δ λ i ( m B n,t B ∗ n,t ) −→ n →∞ µ MP α,σt weakly, and thus we have : µ √ m B n,t B ∗ n,t = 1 n n X i =1 δ λ i ( √ m B n,t B ∗ n,t ) −→ n →∞ √ .♯µ MP α,σt which is a non random measure.An application of Theorem 4.4 to A n,t and B n,t for all t > γ = 0. According to [BG09, Theorem 2.12], the binary operation ⊞ α is continuous (withrespect to the weak convergence) on the set of symmetric probability measures on the real line, and so doesthe the rectangular R-transform C . Thus, aslim t → + ∞ sym( e − γt µ ) = δ and lim t → + ∞ q µ MP α,σt = p µ MP α,σ ∞ ,
12e have lim t → + ∞ sym( µ t ) = δ ⊞ α p µ MP α,σ ∞ . Moreover, the formula in Subsection 4.2 allows to compute for all z ∈ C : M ( δ ) ( z ) = 0 ,H δ ( z ) = z,C δ ( z ) = U (0) = 0 . Consequently, applying Theorem 4.5, C δ ⊞ α √ µ MP α,σ ∞ = C δ + C √ µ MP α,σ ∞ = C √ µ MP α,σ ∞ which allows to conclude, applying Theorem 4.6, that δ ⊞ α p µ MP α,σ ∞ = p µ MP α,σ ∞ which ends the proof. Proof of Theorem 1.2.
Let us first show that if ( λ ,n , . . . , λ n,n ) is a random vector distributed according tothe distribution with density with respect to the Lebesgue measure:( λ , . . . , λ n ) Z n Y i =1 ( λ i ) m − n +12 − e − mγκ λ i Y j = i | λ j − λ i | / ≤ λ ≤···≤ λ n (9)where Z is a normalizing constant, and if we define the empirical measure ν n = 1 n n X i =1 δ λ i,n and note σ ∞ = q κ γ , then, in the sense of weak convergence, ν n −→ n → + ∞ n/m → α µ MP α,σ ∞ , which is a peculiar case of (5).Let us consider the n × m random matrix M whose coordinates are independent identically distributedcentered real Gaussian random variables of variance σ ∞ = κ γ . An application of [PS11, Proposition 7.4.1]shows that of the matrix m M M ∗ follow the density (9). Thus, we have µ m M M ∗ = ν n in the sense of equality in law.Moreover, an application of the Marcenko-Pastur theorem (see for instance [BS10, Theorem 3.10]) showsthat in the sense of convergence in probability µ m M M ∗ −→ n → + ∞ n/m → α µ MP α,σ ∞ n ∈ N ∗ , an application of [JK20, Lemma 3.1 and Proposition 2.8] shows that in the sense of weakconvergence, ν nt −→ t → + ∞ ν n . We proved earlier that we have in the sense of weak convergence ν n −→ n → + ∞ n/m → α µ MP α,σ ∞ . Theorem 1.1 gives the two other limits and concludes the proof.
Proof of Theorem 2.1.
The proof of this Theorem mimics the proof of Theorem 1.1.
Proof of Theorem 3.1.
For f a twice continuously differentiable real test function, we have thanks to theSDE (3) : d h ν n,Wt , f i = 2 κn √ m n X j =1 f ′ ( λ j,n,Wt ) q λ j,n,Wt dB jt + κ nm n X j =1 λ j,n,Wt f ′′ ( λ j,n,Wt ) + 1 n n X j =1 ( β κ − γλ j,n,Wt ) f ′ ( λ j,n,Wt ) dt + β β κ nm n X j =1 f ′ ( λ j,n,Wt ) X k = j λ j,n,Wt + λ k,n,Wt λ j,n,Wt − λ k,n,Wt dt We have :1 nm n X j =1 f ′ ( λ j,n,Wt ) X k = j λ j,n,Wt + λ k,n,Wt λ j,n,Wt − λ k,n,Wt = 12 nm n X j =1 X k = j ( f ′ ( λ j,n,Wt ) − f ′ ( λ k,n,Wt )) λ j,n,Wt + λ k,n,Wt λ j,n,Wt − λ k,n,Wt = n m Z Z { x = y } ( f ′ ( x ) − f ′ ( y )) x + yx − y ν n,Wt ( dx ) ν n,Wt ( dy )= n m Z Z ( x + y ) f ′ ( x ) − f ′ ( y ) x − y ν n,Wt ( dx ) ν n,Wt ( dy ) − m Z xf ′′ ( x ) ν n,Wt ( dx ) . Finally we have d h ν n,Wt , f i = 2 κn √ m n X j =1 f ′ ( λ j,n,Wt ) q λ j,n,Wt dB jt + κ nm n X j =1 λ j,n,Wt f ′′ ( λ j,n,Wt ) + 1 n n X j =1 ( β κ − γλ j,n,Wt ) f ′ ( λ j,n,Wt ) dt + β β κ (cid:18) n m Z Z ( x + y ) f ′ ( x ) − f ′ ( y ) x − y ν n,Wt ( dx ) ν n,Wt ( dy ) − m Z xf ′′ ( x ) ν n,Wt ( dx ) (cid:19) dt = 2 κn √ m n X j =1 f ′ ( λ j,n,Wt ) q λ j,n,Wt dB jt + h ν n,Wt , κ (2 − β β )Φ m f ′′ + ( β κ − γ Φ) f ′ i dt + β β κ (cid:18) n m Z Z ( x + y ) f ′ ( x ) − f ′ ( y ) x − y ν n,Wt ( dx ) ν n,Wt ( dy ) (cid:19) dt = dM ( n,f ) t + h ν n,Wt , κ (2 − β β )Φ m f ′′ + ( β κ − γ Φ) f ′ i dt + β β κ (cid:18) n m Z Z ( x + y ) f ′ ( x ) − f ′ ( y ) x − y ν n,Wt ( dx ) ν n,Wt ( dy ) (cid:19) dt (10)14ith Φ : x → x defined on R and M ( n,f ) a continuous martingale verifying d h M ( n,f ) i t = 4 κ n m n X i =1 | λ i,n,Wt f ′ ( λ i,n,Wt ) | dt. The reader will find in [MP19] a proof of the tightness of the family { ( ν n,Wt ) t ≥ ; n ≥ } , which allows toconclude with the previous computations that any accumulation point of this family satisfies the evolutionequation (6). The reader will also find in [MP19] the proof of the uniqueness of the solution to the equation(6) in the case where ν admits a characteristic function and this function is analytic on a neighbourhood ofthe origin, this proof being based on the considerations made in Remark 3.2.Let us now prove the second part of the Theorem. Applying (6) with f ( v ) = 1 z − v we get that G t ( z ) obeys G t ( z ) = G ( z ) + Z t β κ z − x ) − γ x ( z − x ) ν s ( dx ) ds + αβ β κ Z Z ( x + y ) f ′ ( x ) − f ′ ( y ) x − y ν t ( dx ) ν t ( dy ) . We have12
Z Z ( x + y ) f ′ ( x ) − f ′ ( y ) x − y ν t ( dx ) ν t ( dy ) = 12 Z Z (cid:18) z − x ) − z − y ) (cid:19) x + yx − y ν t ( dx ) ν t ( dy )= 12 Z Z (2 z − x − y )( x + y )( z − x ) ( z − y ) ν t ( dx ) ν t ( dy )= Z Z zx − xy − x + z − z ( z − x ) ( z − y ) ν t ( dx ) ν t ( dy )= Z Z − z − y ) − xy ( z − x ) ( z − y ) + z ( z − x ) ( z − y ) ν t ( dx ) ν t ( dy )= Z Z − z − y ) + z ( z − x ) ( z − y ) ν t ( dx ) ν t ( dy ) − (cid:18)Z x ( z − x ) ν t ( dx ) (cid:19) = Z Z − z − y ) + z ( z − x ) ( z − y ) ν t ( dx ) ν t ( dy ) − (cid:18)Z x − z + z ( z − x ) ν t ( dx ) (cid:19) = − Z z − x ) ν t ( dx ) + z (cid:18)Z z − x ) ν t ( dx ) (cid:19) − (cid:18)Z z − x ν t ( dx ) (cid:19) + 2 z (cid:18)Z z − x ν t ( dx ) (cid:19) (cid:18)Z z − x ) ν t ( dx ) (cid:19) − z (cid:18)Z z − x ) ν t ( dx ) (cid:19) , so that G t ( z ) obeys G t ( z ) = G ( z ) + Z t β κ z − x ) − γ (cid:18) x − z + z ( z − x ) (cid:19) ν s ( dx ) ds − αβ β κ "Z z − x ) ν t ( dx ) + (cid:18)Z z − x ν t ( dx ) (cid:19) − z (cid:18)Z z − x ν t ( dx ) (cid:19) (cid:18)Z z − x ) ν t ( dx ) (cid:19) . G . Proof of Proposition 3.4. (i) We first recall the Cauchy-Stieltjes transform of a Marcenko-Pastur law (seefor instance [BS10, Lemma 3.11] : G µ MPρ,σ ( z ) = Z µ MP ρ,σ ( dv ) z − v = − σ (1 − ρ ) + z − p ( z − σ − ρσ ) − ρσ ρzσ for all z ∈ C + .We now want to find conditions on the functions t → σ ( t ) ∈ R + and t → ρ ( t ) ∈ R + with σ (0) = σ and ρ (0) = ρ such that ( t, z ) −→ G µ MPρ ( t ) ,σ ( t ) ( z ) is solution to the PDE (7): ∂∂t G t ( z ) = ( αβ β κ − β κ + 2 γz ) ∂∂z G t ( z ) − αβ β κ zG t ( z ) ∂∂z G t ( z ) − αβ β κ G t ( z ) + 2 γG t ( z ) ,G ( z ) = Z ν W ( dv ) z − v = φ ( z ) . We have ∂∂t G µ MPρ ( t ) ,σ ( t ) ( z ) = − σ ( z − σ ) ˙ ρ + 2 ˙ σρz ρ zσ − σ ((1 − ρ ) σ − z ( ρ + 2) σ + z ) ˙ ρ + 2( z − σ (1 + ρ )) z ˙ σρ ρ zσ p ( z − σ − ρσ ) − ρσ ∂∂z G µ MPρ ( t ) ,σ ( t ) ( z ) = 1 − ρ ρz − z (1 + ρ ) − σ (1 − ρ ) ρz p ( z − σ − ρσ ) − ρσ ( G µ MPρ ( t ) ,σ ( t ) ( z )) = σ (1 − ρ ) − zσ + z + [ z − σ (1 − ρ )] p ( z − σ − ρσ ) − ρσ ρ z σ G µ MPρ ( t ) ,σ ( t ) ( z ) ∂∂z G µ MPρ ( t ) ,σ ( t ) ( z ) = − (1 − ρ ) σ + z ρ z σ + z + zσ [ ρ ρ −
2] + σ (1 − ρ ) ρ z σ p ( z − σ − ρσ ) − ρσ . The equation solved by G can be written for z ∈ C + , t ≥ H ( z, t ) = 0 (11)with H ( z, t ) = zσ ( z − σ ) ˙ ρ + 2 σρ ˙ σz + β κ (1 − ρ )( β α − ρ ) σ + 2 σ ργz − β β ακ z ρ σ z − z ((1 − ρ ) σ − z ( ρ + 2) σ + z ) σ ˙ ρ + 2 z ( z − (1 + ρ ) σ ) ρσ ˙ σ + β κ ( ρ − ( β α − ρ ) σ ρ σ z p ( z − σ − ρσ ) − ρσ + − z ((2 γz + β κ ) ρ − β β ακ )( ρ + 1) σ + z ((2 γz + β β ακ ) ρ + β β ακ ) σ − β β ακ z ρ σ z p ( z − σ − ρσ ) − ρσ . Using the fact that(( z − σ − ρσ ) − ρσ ) − = 1 z + σ (1 + ρ ) z + σ ρ + 4 ρ + 1 z + o | z |→ + ∞ (cid:18) | z | (cid:19) ,
16e have for z ∈ C + , t ≥ H ( z, t ) = σ ˙ ρ + 2 σρ ˙ σ − β β ακ + 2 γρσ σ ρ − ˙ ρρ z − β κ ( ρ − β α − ρ ) − σρ ˙ σ + ( β κ − γσ ) ρ + β κ (1 + β α ) ρ − β β ακ ρ z + o | z |→ + ∞ (cid:18) | z | (cid:19) which gives in particular by identity (11) : σ ˙ ρ + 2 σρ ˙ σ − β β ακ + 2 γρσ = 0 , ˙ ρ = 0 ,β κ ( ρ − β α − ρ ) − σρ ˙ σ + ( β κ − γσ ) ρ + β κ (1 + β α ) ρ − β β ακ = 0 . We thus have ρ ( t ) = β α, ˙ σ = β κ − γσ and σ ( t ) = (cid:18) σ − β κ γ (cid:19) e − γt + β κ γ if γ = 0 σ + β κ t if γ = 0for all t ∈ R + .Reciprocally, we verify with this definition of σ that G MP β α,σ ( t ) µ is a solution to the PDE (7).(ii) Let us consider the stationary version of the PDE (7) :( β β ακ − β κ + 2 γz ) ddz G ( z ) − β β ακ zG ( z ) ddz G ( z ) − β β ακ G ( z ) + 2 γG ( z ) = 0 . We can integrate it − β β ακ zG ( z ) + (2 γz + ( β α − β κ ) G ( z ) = C where C ∈ C is an integration constant. It gives G ( z ) = 2 γz + ( β α − β κ ± p (2 γz + ( β α − β κ ) − Cβ β ακ z β β ακ z . For this function to be the Cauchy-Stieltjes transform of a probability measure on R , we need it toverify the condition given in assertion (3) of Theorem 4.1 :lim sup y →∞ y | G (i y ) | = 1 , necessarily, the sign ± must be replaced by a minus sign and C = 2 γ . Thus, the only stationary solutionto the PDE (7) corresponding to the Cauchy-Stieltjes transform of a probability measure on R is G ( z ) = 2 γz + ( β α − β κ − p (2 γz + ( β α − β κ ) − γβ β ακ z β β ακ z = − σ (1 − ρ ) + z − p ( z − σ − ρσ ) − ρσ ρzσ ρ = β α and σ = β κ γ . We recognize the Cauchy-Stieltjes transform of the Marcenko-Pasturdistribution with parameters ρ and σ , and verify that G is solution to the stationary version of thePDE (7). Under the assumptions of Theorem 3.1, and applying Theorem 3.1 with ν = µ MP β α,σ ,then µ MP β α,σ is the unique solution to the equation (6). Proof of Theorem 3.5.
By Theorem 3.1, the integro-differential equation (6) is verified by any accumulationpoint of the family { ( ν n,Wt ) t ≥ , n ≥ } .Let n ≤ m with nm −→ n → + ∞ β α . Let us define the stochastic process ( M t ) t taking its values in the spaceof n × m matrices with real entries verifying the following SDE dM t = p β κdW t − γM t dt. The eigenvalues of m M t M ∗ t verify the SDE (2) with κ replaced by √ β κ . By Theorem 3.1 (see more preciselythe computation (10) in the proof), the integro-differential equation (6) is also verified by any accumulationpoint of the family of the empirical spectral measures of (cid:0) m M t M ∗ t (cid:1) ≤ n ≤ m with nm −→ n → + ∞ β α , which endsthe proof.By Theorem 3.1, we have uniqueness of the solution to the integro-differential equation (6), and byTheorem 1.1 we have the expression of the limit of µ n,Wt = q ν n,Wt for all t ≥ Proof of Theorem 3.6.
For all n ∈ N ∗ , an application of [JK20, Lemma 3.1 and Proposition 2.8] shows thatin convergence in law ν n,Wt −→ t → + ∞ ν n,W where ν n,W is defined such as in equation (5). By equation (5), we have ν n,W −→ n → + ∞ n/m → α µ MP α,σ ∞ . Theorem 3.5 gives the two other limits and concludes the proof.
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