SEA s workshop- MIT - July 10-14

Transcription

1 Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA s workshop- MIT - July July 13, 2006

2 Outline Matrix-valued stochastic processes. 1- definition and examples. 2- Multivariate statistics and multivariate functions. 3- Eigenvalues process : non-colliding particles. 4- The Hermitian complex case : determinantal processes. Root systems 1- β-processes (β > 0). 2- Radial Dunkl processes. Free probability : free processes.

3 Matrix-valued processes : definitions Let (Ω, F, (F t ) t 0, P) be a filtered probability space. A matrix-valued process is defined by X : R Ω M m (C) (t, ω) X t (ω) where M m (C) is the set of square complex matrices. Let S m, H m denote the spaces of symmetric and Hermitian matrices.

4 Examples Dyson model : B 1 = (Bij 1) i,j, B 2 = (Bij 2) i,j : two independent m m real Brownian matrices. { B 1 ii (t) i = j X ij (t) = i < j B 1 ij (t)+ 1B 2 ij (t) 2

5 Examples Dyson model : B 1 = (Bij 1) i,j, B 2 = (Bij 2) i,j : two independent m m real Brownian matrices. { B 1 ii (t) i = j X ij (t) = i < j B 1 ij (t)+ 1B 2 ij (t) 2 Wishart process (Bru, Donati-Doumerc-Matsumoto-Yor,): X (t) = B T (t)b(t), B : n m real Brownian matrix. Complex version: Laguerre process. n is the dimension and m is the size.

6 Examples Dyson model : B 1 = (Bij 1) i,j, B 2 = (Bij 2) i,j : two independent m m real Brownian matrices. { B 1 ii (t) i = j X ij (t) = i < j B 1 ij (t)+ 1B 2 ij (t) 2 Wishart process (Bru, Donati-Doumerc-Matsumoto-Yor,): X (t) = B T (t)b(t), B : n m real Brownian matrix. Complex version: Laguerre process. n is the dimension and m is the size. Other models: matrix Jacobi process (Y. Doumerc), Hermitian model of Katori and Tanemura (Brownian bridges).

7 Some SDE When it makes sense, one has for : Wishart (W (n, m, X 0 )), Laguerre (L(n, m, X 0 )) processes : dx t = dnt Xt + X t dn t + βni m dt, (β = 1, 2) Real and complex matrix Jacobi processes J(p, q, m, X 0 ): dx t = X t dnt Im X t + I m X t dn t Xt + β(pi m (p + q)x t )dt, (β = 1, 2) (N t ) t 0 is a square real Brownian matrix of size m.

8 Multivariate statistics t = 1, X 0 = 0, Dyson model, symmetric BM GUE and GOE. t = 1, X 0 = 0, Wishart, Laguerre LOE, LUE. Non-central Wishart and complex Wishart distributions with parameters M = X 0, Σ = βti m (James, Muirhead, Chikuze). Stationnary Jacobi matrix MANOVA.

9 Key tools : multivariate functions Hypergeometric function of matrix argument : pf β q ((a i ) 1 i q, (b j ) 1 j q, X ) = (a 1 ) τ (a p ) τ Jτ β (X ) (b 1 ) τ (b q ) τ k! k=0 τ β = 1 : zonal polynomial (Muirhead). β = 2 : Schur function (Macdonald) : J 2 τ (x 1,, x m ) = det(x k j +m j i ) det(x m j i ) Hypergeometric function of two matrix arguments : pf β q ((a i ) 1 i q, (b j ) 1 j q, X, Y ) = (a 1 ) τ (a p ) τ Jτ β (X )Jτ β (Y ) (b 1 ) τ (b q ) τ Jτ β (I m )k! k=0 τ

10 Some expressions Wishart and Laguerre semi-groups : 0 F β 1, β = 1, 2. Generalized Hartman-Watson Law : 0 F β 1, β = 1, 2. Tail distribution of T 0 := inf{t, det(x t ) = 0} : 1 F β 1, β = 1, 2. Real symmetric case : quite complicated. More precise results in the complex Hermitian case : determinantal representations of multivariate functions (Gross and Richards, Demni, Lassalle for orthogonal polynomials).

11 determinantal formulae pf 2 q ((a) 1 i p, (b j ) 1 j q, X ) = (1) det(x m j i pf q ((a j + 1) 1 i p, (b l j + 1) 1 l q, x i ) det(x m j i ) (Gross et Richards) pfq 2 ((m + µ i ) 1 i p, (m + φ j ) 1 j q ; X, Y ) = Γ m (m) p π m(m 1) (p q 1) (Γ(µ i + 1)) m q Γ m (m + φ j ) 2 Γ m (m + µ i ) (Γ(φ j + 1)) m (2) i=1 j=1 det ( p F q ((µ i + 1) 1 i p, (1 + φ j ) 1 j q ; x l y f ) l,f h(x)h(y) for (µ i ), (φ j ) > 1 (Gross and Richards, Demni).

12 The complex case : determinantal processes A determinantal process : R m -valued process with semi group writable as determinant : q t (x, y) = det(k t (x i, y j )) i,j, x, y R m In the complex Hermitian case, the eigenvalue process is determinantal : Weyl integration formula + determinantal representation of the two matrix arguments functions. Probability technics (Doob h-transform). q t (x, y) = V (y) ( 1 V (x) det exp (y j x i ) 2 ) 2πt 2t Laguerre process : K t = the squared Bessel process semi-group. i,j

13 The eigenvalues process : SDE Let X be a matrix-valued process and (λ 1, λ 2,..., λ m ) its eigenvalues process with starting point (λ 1 (0) > > λ m (0)). Then dλ i (t) = d X i (t) + SD(t), 1 i m X i L = Xii, independent. SD : singular drift showing interaction between particles.

14 Examples Symmetric and Hermitian Brownian matrix : dλ i (t) = db i (t) + β 2 j i dt λ i (t) λ j (t)

15 Examples Symmetric and Hermitian Brownian matrix : dλ i (t) = db i (t) + β 2 j i dt λ i (t) λ j (t) Wishart and Laguerre processes δ > m 1, m. dλ i (t) = 2 λ i (t)db i (t) + βδdt + β j i λ i (t) + λ j (t) λ i (t) λ j (t) dt

16 Examples Symmetric and Hermitian Brownian matrix : dλ i (t) = db i (t) + β 2 j i dt λ i (t) λ j (t) Wishart and Laguerre processes δ > m 1, m. Jacobi dλ i (t) = 2 λ i (t)db i (t) + βδdt + β j i λ i (t) + λ j (t) λ i (t) λ j (t) dt dλ i (t) = 2 λ i (t)(1 λ i (t)db i (t) + (p (p + q)λ i (t))dt + β j i λ i (t)(1 λ j (t)) + λ j (t)(1 λ i (t)) dt λ i (t) λ j (t) for β = 1, 2. What happens for arbitrary β > 0?

17 Root systems Let α R m \ {0} and let σ α denotes the reflection with respect to the hyperplane H α orthogonal to α : σ α (x) = x 2 < x, α > < α, α > α. A root system R is a non-empty subset of non-null vectors of R m satisfying : R Rα = {±α}, A simple system is a basis of R such that each α R is either a positive or negative linear combination of vectors of. The first kind of roots constitute the positive subsystem, denoted by R +, and are called by the way positive roots.

18 Root systems Let α R m \ {0} and let σ α denotes the reflection with respect to the hyperplane H α orthogonal to α : σ α (x) = x 2 < x, α > < α, α > α. A root system R is a non-empty subset of non-null vectors of R m satisfying : R Rα = {±α}, σ α (R) = R, α R A simple system is a basis of R such that each α R is either a positive or negative linear combination of vectors of. The first kind of roots constitute the positive subsystem, denoted by R +, and are called by the way positive roots.

19 Weyl Group Weyl group W : Multiplicity function k : constant on each orbit. W := span{σ α, α R} O(R m ) Positive Weyl chamber : k : {orbits of W } R α k(α) C := {x R m, < α, x >> 0 α R + }

20 Radial Dunkl process Let C be the closure of C. The radial Dunkl process X W is a paths- continuous C-valued Markov process with extended generator given by : L u(x) = 1 2 u(x) + k(α) < u(x), α > < x, α >, α R + where u C 2 0 (C) such that < u(x), α >= 0 for x H α, α R +.

21 A m 1 -type R = {±(e i ± e j ), 1 i < j m}. R + = {e i e j, 1 i < j m}. = {e i e i+1, 1 i m 1}. C = {x R m, x 1 > > x m }. k = k 0 > 0, (e i ) 1 i m is the standard basis of R m.

22 A m 1 -type R = {±(e i ± e j ), 1 i < j m}. R + = {e i e j, 1 i < j m}. = {e i e i+1, 1 i m 1}. C = {x R m, x 1 > > x m }. k = k 0 > 0, (e i ) 1 i m is the standard basis of R m. Set k 0 = β/2, β > 0. dλ i (t) = db i (t) + β 2 j i dt λ i (t) λ j (t), 1 i m. with λ 1 (0) >... λ m (0) (Cépa and Lépingle).

23 A m 1 -type R = {±(e i ± e j ), 1 i < j m}. R + = {e i e j, 1 i < j m}. = {e i e i+1, 1 i m 1}. C = {x R m, x 1 > > x m }. k = k 0 > 0, (e i ) 1 i m is the standard basis of R m. Set k 0 = β/2, β > 0. dλ i (t) = db i (t) + β 2 j i dt λ i (t) λ j (t), 1 i m. with λ 1 (0) >... λ m (0) (Cépa and Lépingle). X 0 = 0, t = 1 β-hermite ensemble (Edelman-Dumitriu).

24 The B m -type R = {±e i, 1 i m, ±(e i ± e j ), 1 i < j m}. R + = {e i, 1 i m, e i ± e j, 1 i < j m}. = {e i e i+1, 1 i m 1, e m }. C = {x R m, x 1 > x 2... x m > 0}. Two conjugacy classes k = (k 0, k 1 ).

25 β-laguerre processes (Demni) Let β, δ > 0. A β-laguerre process (λ(t)) t 0 starting at (λ 1 (0) > > λ m (0)) is a solution when it exists of dλ i (t) = 2 λ i (t)dν i (t)+β δ + λ i (t) + λ j (t) dt, λ i (t) λ j (t) i j 1 i m for t < τ, the first collision time, where (ν i ) are independent BM. Let R 0 := inf{t, λ m (t) = 0 for some i} then T 0 = τ R 0, moreover :

26 A B m - Radial Dunkl process (r(t)) t T0 = ( λ(t)) t T0 satisfies : dr i (t) = dν i (t)+ β 2 + [ j i 1 r i (t) r j (t) + 1 r i (t) + r j (t) β(δ m + 1) 1 dt, 2r i (t) 1 i m (r t ) t T0 is a B m -radial Dunkl process with multiplicity function given by 2k 0 = β(δ m + 1) 1 > 0 and 2k 1 = β > 0. ] dt

27 A B m - Radial Dunkl process (r(t)) t T0 = ( λ(t)) t T0 satisfies : dr i (t) = dν i (t)+ β 2 + [ j i 1 r i (t) r j (t) + 1 r i (t) + r j (t) β(δ m + 1) 1 dt, 2r i (t) 1 i m (r t ) t T0 is a B m -radial Dunkl process with multiplicity function given by 2k 0 = β(δ m + 1) 1 > 0 and 2k 1 = β > 0. ] dt X 0 = 0, t = 1 β-laguerre ensemble (Edelman-Dumitriu).

28 A unique strong solution for all t 0 Theorem 1 (Demni): Let B be a m-dimensional BM. Then, the radial Dunkl process (X W t ) t 0 is the unique strong solution of the SDE dy t = db t Φ(Y t )dt, Y 0 C where Φ(x) = α R + k(α) ln(< α, x >) for k(α) > 0 α R +.

29 A β-matrix model (Demni) Q : Is there a matrix-valued process corresponding to : dλ i (t) = db i (t) + β 2 j i dt λ i (t) λ j (t) 1 i m

30 A β-matrix model (Demni) Q : Is there a matrix-valued process corresponding to : dλ i (t) = db i (t) + β 2 j i A : β-hermitian model, 0 < β 2 : Bii 1(t) X ij (t) = β 2 dt λ i (t) λ j (t) ( ) Bij 1(t)+B2 ij (t) 2 1 i m i = j i < j where B 1 ii, B2 jj t = (1 β/2)t and B 1, B 2 are two ind. m m Brownian matrices.

31 Free probability : free processes Non-commutative probability space : unital algebra A + state A C, Φ(1) = 1. Examples A m = p>0 L p (Ω, F, (F t ) t 0, P) M m (C) the set of m m random matrices with finite moments, and the normalized trace expectation : Φ m := 1 m E(tr) := E(tr m) B(H) : the set of bounded linear operators on a Hilbert space H with the pure state Φ(a) =< ax, x >, a B(H), where x H is a unit element.

32 Other properties - algebra C or W - non commutative probability space. involutive Banach algebra : norm s.t a = a, a A + completion. C -algebra involutive Banach algebra + aa = a 2 for all a A. Φ can be : 1. tracial : Φ(ab) = Φ(ba). In the matrix example, involution has to be the usual adjonction and conditions are obviously fulfilled. As in classical probability, we endow our space with a family (A t ) t 0 of increasing C -subalgebras called filtration conditional expectation Φ(a t /A s ), s t.

33 Other properties - algebra C or W - non commutative probability space. involutive Banach algebra : norm s.t a = a, a A + completion. C -algebra involutive Banach algebra + aa = a 2 for all a A. Φ can be : 1. tracial : Φ(ab) = Φ(ba). 2. faithful : Φ(aa ) = 0 a = 0. In the matrix example, involution has to be the usual adjonction and conditions are obviously fulfilled. As in classical probability, we endow our space with a family (A t ) t 0 of increasing C -subalgebras called filtration conditional expectation Φ(a t /A s ), s t.

34 Other properties - algebra C or W - non commutative probability space. involutive Banach algebra : norm s.t a = a, a A + completion. C -algebra involutive Banach algebra + aa = a 2 for all a A. Φ can be : 1. tracial : Φ(ab) = Φ(ba). 2. faithful : Φ(aa ) = 0 a = normal In the matrix example, involution has to be the usual adjonction and conditions are obviously fulfilled. As in classical probability, we endow our space with a family (A t ) t 0 of increasing C -subalgebras called filtration conditional expectation Φ(a t /A s ), s t.

35 Examples A free process is a family (a t ) t 0 of free variables. It is said to be adapted if a t A t for all t 0. The free additive and free multiplicative Brownian motion.

36 Examples A free process is a family (a t ) t 0 of free variables. It is said to be adapted if a t A t for all t 0. The free additive and free multiplicative Brownian motion. The free Wishart process (Donati-Capitaine).

37 Examples A free process is a family (a t ) t 0 of free variables. It is said to be adapted if a t A t for all t 0. The free additive and free multiplicative Brownian motion. The free Wishart process (Donati-Capitaine). The free Jacobi process (Demni).

38 Asymptotic freeness and random matrices Definitions: A family (U i (m)) i I of random matrices converges in distribution to (U i ) i I in (A, φ) if lim m E(tr m(u i1 (m)... U ip (m))) = φ(u i1... U ip ) for any collection i 1,..., i p I. A family is said to be asymptotically free if it cv in distribution to free variables. Connection with random matrices : Voiculescu result on GUE.

39 The A t -free additive Brownian motion X It is a adapted and selfadjoint process such that : X 0 = 0.

40 The A t -free additive Brownian motion X It is a adapted and selfadjoint process such that : X 0 = 0. X t has the semi-circle law of mean 0 and variance t given by : σ t (dy) = 1 4t y 2πt 2 1 { 2 t,2 t} (y)dy

41 The A t -free additive Brownian motion X It is a adapted and selfadjoint process such that : X 0 = 0. X t has the semi-circle law of mean 0 and variance t given by : σ t (dy) = 1 4t y 2πt 2 1 { 2 t,2 t} (y)dy For any collection t 0 < t 1... < t k, X t0, X t1 X t0,..., X tk X tk 1 are free and stationnary.

42 Matrix-valued and free processes Let X m be a normalized Hermitian Brownian matrix : Xij m (t) = B m ii (t) m if i = j B m ij (t)+ 1 B m ij (t) 2m if i < j where B m, B m are two independent m m real Brownian matrices. Voiculescu result : X m additive free Brownian motion X. Corollary: Let Z m = (B m + 1B m )/ 2m be a non-selfadjoint process. Z m complex free Brownian motion Z defined by Z = (X 1 + 1X 2 )/ 2 where X 1, X 2 are free A t -free Brownian motions.

43 The A t -free multiplicative Brownian motion Y It is a adapted unitary process such that : Y 0 = I Biane showed that this process is the limit in distribution of the m m unitary Brownian motion Y (m).

44 The A t -free multiplicative Brownian motion Y It is a adapted unitary process such that : Y 0 = I ν t, the law of Y t is supported in the unit circle and is given by its Σ-transform (Bercovici et Voiculescu) : Σ νt (z) = e t 2 1+z 1 z ν t+s = ν t ν s Biane showed that this process is the limit in distribution of the m m unitary Brownian motion Y (m).

45 The A t -free multiplicative Brownian motion Y It is a adapted unitary process such that : Y 0 = I ν t, the law of Y t is supported in the unit circle and is given by its Σ-transform (Bercovici et Voiculescu) : Σ νt (z) = e t 2 1+z 1 z ν t+s = ν t ν s For any collection t 0 < t 1... < t k, Y t0, Y t1 Yt 1 0,..., Y tk Yt 1 k 1 are free and stationnary. Biane showed that this process is the limit in distribution of the m m unitary Brownian motion Y (m).

46 Free SDE For suitable parameters : Free Wishart process (Capitaine-Donati) : dw t = W t dz t + dzt Wt + λpdt Thanks.

47 Free SDE For suitable parameters : Free Wishart process (Capitaine-Donati) : dw t = W t dz t + dz t Wt + λpdt Free multiplicative BM (Kummerer-Speicher) Let X be a free additive BM. Then dy t = i dx t Y t 1 2 Y tdt, Y 0 = 1 Thanks.

48 Free SDE For suitable parameters : Free Wishart process (Capitaine-Donati) : dw t = W t dz t + dz t Wt + λpdt Free multiplicative BM (Kummerer-Speicher) Let X be a free additive BM. Then dy t = i dx t Y t 1 2 Y tdt, Y 0 = 1 Free Jacobi process (Demni) : dj t = λθ P J t dz t Jt + λθ J t dz t P Jt +(θp J t ) dt where Z is a complex free BM. Thanks.

49 Free SDE For suitable parameters : Free Wishart process (Capitaine-Donati) : dw t = W t dz t + dz t Wt + λpdt Free multiplicative BM (Kummerer-Speicher) Let X be a free additive BM. Then dy t = i dx t Y t 1 2 Y tdt, Y 0 = 1 Free Jacobi process (Demni) : dj t = λθ P J t dz t Jt + λθ J t dz t P Jt +(θp J t ) dt where Z is a complex free BM. condition : Injectivity. Thanks.