Matrix-valued Stochastic Processes- Eigenvalues Processes

Matrix-valued Stochastic Processes- EigenvaluesProcesses and Free Probability

SEA’s workshop- MIT - July 10 -14

Nizar Demni

July 13, 2006

Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14

Outline

I 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.

I Root systems

1- β-processes (β > 0).

2- Radial Dunkl processes.

I Free probability : free processes.


Matrix-valued processes : definitions

Let (Ω,F , (Ft)t≥0, P) be a filtered probability space. Amatrix-valued process is defined by

X : R× Ω −→ Mm(C)(t, ω) 7→ Xt(ω)

where Mm(C) is the set of square complex matrices. Let Sm,Hm

denote the spaces of symmetric and Hermitian matrices.


Examples

I Dyson model : B1 = (B1ij )i ,j ,B

2 = (B2ij )i ,j : two independent

m ×m real Brownian matrices.

Xij(t) =

B1

ii (t) i = jB1

ij (t)+√−1B2

ij (t)√2

i < j

I Wishart process (Bru, Donati-Doumerc-Matsumoto-Yor,):

X (t) = BT (t)B(t), B : n ×m real Brownian matrix.

Complex version: Laguerre process. n is the dimension and mis the size.

I Other models: matrix Jacobi process (Y. Doumerc),Hermitian model of Katori and Tanemura (Brownian bridges).


Examples




Xij(t) =

B1

ii (t) i = jB1

ij (t)+√−1B2

ij (t)√2

i < j






Examples




Xij(t) =

B1

ii (t) i = jB1

ij (t)+√−1B2

ij (t)√2

i < j






Some SDE

When it makes sense, one has for :

I Wishart (W (n,m,X0)), Laguerre (L(n,m,X0)) processes :

dXt = dN?t

√Xt +

√XtdNt + βnImdt, (β = 1, 2)

I Real and complex matrix Jacobi processes J(p, q,m,X0):

dXt =√

XtdN?t

√Im − Xt +

√Im − XtdNt

√Xt+

β(pIm − (p + q)Xt)dt, (β = 1, 2)

(Nt)t≥0 is a square real Brownian matrix of size m.


Multivariate statistics

I t = 1,X0 = 0, Dyson model, symmetric BM ⇒ GUE andGOE.

I t = 1,X0 = 0, Wishart, Laguerre ⇒ LOE, LUE.

I Non-central Wishart and complex Wishart distributions withparameters M = X0,Σ = βtIm (James, Muirhead, Chikuze).

I Stationnary Jacobi matrix ⇒ MANOVA.


Key tools : multivariate functions

Hypergeometric function of matrix argument :

pFβq ((ai )1≤i≤q, (bj)1≤j≤q,X ) =

∞∑k=0

∑τ

(a1)τ · · · (ap)τ

(b1)τ · · · (bq)τ

Jβτ (X )

k!

I β = 1 : zonal polynomial (Muirhead).

I β = 2 : Schur function (Macdonald) :

J2τ (x1, · · · , xm) =

det(xkj+m−ji )

det(xm−ji )

Hypergeometric function of two matrix arguments :

pFβq ((ai )1≤i≤q, (bj)1≤j≤q,X ,Y ) =

∞∑k=0

∑τ

(a1)τ · · · (ap)τ

(b1)τ · · · (bq)τ

Jβτ (X )Jβ

τ (Y )

Jβτ (Im)k!


Some expressions

I Wishart and Laguerre semi-groups : 0Fβ1 , β = 1, 2.

I Generalized Hartman-Watson Law : 0Fβ1 , β = 1, 2.

I Tail distribution of T0 := inft, det(Xt) = 0 : 1Fβ1 , β = 1, 2.

Real symmetric case : quite complicated.More precise results in the complex Hermitian case : determinantalrepresentations of multivariate functions (Gross and Richards,Demni, Lassalle for orthogonal polynomials).


determinantal formulae

pF2q ((a)1≤i≤p, (bj)1≤j≤q,X ) = (1)

det(xm−ji pFq((a− j + 1)1≤i≤p, (bl − j + 1)1≤l≤q, xi )

det(xm−ji )

(Gross et Richards)

pF2q ((m + µi )1≤i≤p, (m + φj)1≤j≤q;X ,Y ) = Γm(m)

πm(m−1)

2(p−q−1)

p∏i=1

(Γ(µi + 1))m

Γm(m + µi )

q∏j=1

Γm(m + φj)

(Γ(φj + 1))m(2)

det (pFq((µi + 1)1≤i≤p, (1 + φj)1≤j≤q; xlyf )l ,fh(x)h(y)

for (µi ), (φj) > −1 (Gross and Richards, Demni).


The complex case : determinantal processes

A determinantal process : Rm-valued process with semi groupwritable as determinant :

qt(x , y) = det(Kt(xi , yj))i ,j , x , y ∈ Rm

In the complex Hermitian case, the eigenvalue process isdeterminantal :

I Weyl integration formula + determinantal representation ofthe two matrix arguments functions.

I Probability technics (Doob h-transform).

qt(x , y) =V (y)

V (x)det

(1√2πt

exp−(yj − xi )

2

2t

)i ,j

Laguerre process : Kt = the squared Bessel process semi-group.


The eigenvalues process : SDE

Let X be a matrix-valued process and (λ1, λ2, . . . , λm) itseigenvalues process with starting point (λ1(0) > · · · > λm(0)).Then

dλi (t) = dXi (t) + SD(t), 1 ≤ i ≤ m

I XiL= Xii , independent.

I SD : singular drift showing interaction between particles.


Examples

I Symmetric and Hermitian Brownian matrix :

dλi (t) = dBi (t) +β

2

∑j 6=i

dt

λi (t)− λj(t)

I Wishart and Laguerre processes δ > m − 1, m.

dλi (t) = 2√

λi (t)dBi (t) + βδdt + β∑j 6=i

λi (t) + λj(t)

λi (t)− λj(t)dt

I Jacobi

dλi (t) = 2√

λi (t)(1− λi (t)dBi (t) + (p − (p + q)λi (t))dt

+ β∑j 6=i

λi (t)(1− λj(t)) + λj(t)(1− λi (t))

λi (t)− λj(t)dt

for β = 1, 2. What happens for arbitrary β > 0?


Examples



2

∑j 6=i

dt

λi (t)− λj(t)


dλi (t) = 2√


λi (t) + λj(t)

λi (t)− λj(t)dt

I Jacobi

dλi (t) = 2√


+ β∑j 6=i

λi (t)(1− λj(t)) + λj(t)(1− λi (t))

λi (t)− λj(t)dt



Examples



2

∑j 6=i

dt

λi (t)− λj(t)


dλi (t) = 2√


λi (t) + λj(t)

λi (t)− λj(t)dt

I Jacobi

dλi (t) = 2√


+ β∑j 6=i

λi (t)(1− λj(t)) + λj(t)(1− λi (t))

λi (t)− λj(t)dt



Root systems

Let α ∈ Rm \ 0 and let σα denotes the reflection with respect tothe hyperplane Hα orthogonal to α :

σα(x) = x − 2< x , α >

< α,α >α.

A root system R is a non-empty subset of non-null vectors of Rm

satisfying :

I R ∩ Rα = ±α,

I σα(R) = R, α ∈ R

A simple system ∆ is a basis of R such that each α ∈ R is eithera positive or negative linear combination of vectors of ∆. The firstkind of roots constitute the positive subsystem, denoted by R+,and are called by the way positive roots.


Root systems

Let α ∈ Rm \ 0 and let σα denotes the reflection with respect tothe hyperplane Hα orthogonal to α :

σα(x) = x − 2< x , α >

< α,α >α.

A root system R is a non-empty subset of non-null vectors of Rm

satisfying :

I R ∩ Rα = ±α,I σα(R) = R, α ∈ R

A simple system ∆ is a basis of R such that each α ∈ R is eithera positive or negative linear combination of vectors of ∆. The firstkind of roots constitute the positive subsystem, denoted by R+,and are called by the way positive roots.


Weyl Group

I Weyl group W :

W := spanσα, α ∈ R ⊂ O(Rm)

I Multiplicity function k :

k : orbits ofW → Rα 7→ k(α)

constant on each orbit.

I Positive Weyl chamber :

C := x ∈ Rm, < α, x >> 0∀α ∈ R+


Radial Dunkl process

Let C be the closure of C . The radial Dunkl process XW is apaths- continuous C -valued Markov process with extendedgenerator given by :

L u(x) =1

24u(x) +

∑α∈R+

k(α)< ∇u(x), α >

< x , α >,

where u ∈ C 20 (C ) such that < ∇u(x), α >= 0 for x ∈ Hα, α ∈ R+.


Am−1-type

I R = ±(ei ± ej), 1 ≤ i < j ≤ m.I R+ = ei − ej , 1 ≤ i < j ≤ m.I ∆ = ei − ei+1, 1 ≤ i ≤ m − 1.I C = x ∈ Rm, x1 > · · · > xm.I k = k0 > 0, (ei )1≤i≤m is the standard basis of Rm.

I Set k0 = β/2, β > 0.


2

∑j 6=i

dt

λi (t)− λj(t), 1 ≤ i ≤ m.

with λ1(0) > . . . λm(0) (Cepa and Lepingle).

I X0 = 0, t = 1 ⇒ β-Hermite ensemble (Edelman-Dumitriu).


Am−1-type


I Set k0 = β/2, β > 0.


2

∑j 6=i

dt

λi (t)− λj(t), 1 ≤ i ≤ m.




Am−1-type


I Set k0 = β/2, β > 0.


2

∑j 6=i

dt

λi (t)− λj(t), 1 ≤ i ≤ m.




The Bm-type

I R = ±ei , 1 ≤ i ≤ m, ±(ei ± ej), 1 ≤ i < j ≤ m.I R+ = ei , 1 ≤ i ≤ m, ei ± ej , 1 ≤ i < j ≤ m.I ∆ = ei − ei+1, 1 ≤ i ≤ m − 1, em.I C = x ∈ Rm, x1 > x2 . . . xm > 0.I Two conjugacy classes ⇒ k = (k0, k1).


β-Laguerre processes (Demni)

Let β, δ > 0. A β-Laguerre process (λ(t))t≥0 starting at(λ1(0) > · · · > λm(0)) is a solution when it exists of

dλi (t) = 2√

λi (t)dνi (t)+β

δ +∑i 6=j

λi (t) + λj(t)

λi (t)− λj(t)

dt, 1 ≤ i ≤ m

for t < τ , the first collision time, where (νi ) are independent BM.Let

R0 := inft, λm(t) = 0 for some i

then T0 = τ ∧ R0, moreover :


A Bm- Radial Dunkl process

I (r(t))t≤T0 = (√

λ(t))t≤T0 satisfies :

dri (t) = dνi (t)+β

2

∑j 6=i

[1

ri (t)− rj(t)+

1

ri (t) + rj(t)

]dt

+β(δ −m + 1)− 1

2ri (t)dt, 1 ≤ i ≤ m

⇒ (rt)t≤T0 is a Bm-radial Dunkl process with multiplicityfunction given by 2k0 = β(δ −m + 1)− 1 > 0 and2k1 = β > 0.

I X0 = 0, t = 1 ⇒ β-Laguerre ensemble (Edelman-Dumitriu).


A Bm- Radial Dunkl process

I (r(t))t≤T0 = (√

λ(t))t≤T0 satisfies :

dri (t) = dνi (t)+β

2

∑j 6=i

[1

ri (t)− rj(t)+

1

ri (t) + rj(t)

]dt

+β(δ −m + 1)− 1

2ri (t)dt, 1 ≤ i ≤ m

⇒ (rt)t≤T0 is a Bm-radial Dunkl process with multiplicityfunction given by 2k0 = β(δ −m + 1)− 1 > 0 and2k1 = β > 0.

I X0 = 0, t = 1 ⇒ β-Laguerre ensemble (Edelman-Dumitriu).


A unique strong solution for all t ≥ 0

Theorem 1 (Demni):Let B be a m-dimensional BM. Then, the radial Dunkl process(XW

t )t≥0 is the unique strong solution of the SDE

dYt = dBt −∇Φ(Yt)dt, Y0 ∈ C

where Φ(x) = −∑

α∈R+k(α) ln(< α, x >) for k(α) > 0∀α ∈ R+.


A β-matrix model (Demni)

I Q : Is there a matrix-valued process corresponding to :


2

∑j 6=i

dt

λi (t)− λj(t)1 ≤ i ≤ m

I A : β-Hermitian model, 0 < β ≤ 2 :

Xij(t) =

B1ii (t) i = j√β2

(B1

ij (t)+B2ij (t)√

2

)i < j

where 〈B1ii ,B

2jj〉t = (1− β/2)t and B1,B2 are two ind. m×m

Brownian matrices.


A β-matrix model (Demni)

I Q : Is there a matrix-valued process corresponding to :


2

∑j 6=i

dt

λi (t)− λj(t)1 ≤ i ≤ m

I A : β-Hermitian model, 0 < β ≤ 2 :

Xij(t) =

B1ii (t) i = j√β2

(B1

ij (t)+B2ij (t)√

2

)i < j

where 〈B1ii ,B

2jj〉t = (1− β/2)t and B1,B2 are two ind. m×m

Brownian matrices.


Free probability : free processes

Non-commutative probability space : unital algebra A + stateA 7→ C, Φ(1) = 1.Examples

I

Am =⋂p>0

Lp(Ω,F , (Ft)t≥0, P)×Mm(C)

the set of m ×m random matrices with finite moments, andthe normalized trace expectation :

Φm :=1

mE(tr) := E(trm)

I B(H) : the set of bounded linear operators on a Hilbert spaceH with the pure state Φ(a) =< ax , x >, a ∈ B(H), wherex ∈ H is a unit element.


Other properties

I ?- algebra C ? or W ?- non commutative probability space.

I involutive Banach algebra : norm || · || s.t ||a?|| = ||a||, a ∈ A+ completion.

I C ?-algebra involutive Banach algebra + ||aa?|| = ||a||2 for alla ∈ A .

Φ can be :

1. tracial : Φ(ab) = Φ(ba).

2. faithful : Φ(aa?) = 0 ⇒ a = 0.

3. normal

In the matrix example, involution has to be the usual adjonctionand conditions are obviously fulfilled.As in classical probability, we endow our space with a family(At)t≥0 of increasing C ?-subalgebras called filtration ⇒conditional expectation Φ(at/As), s ≤ t.


Other properties




Φ can be :


2. faithful : Φ(aa?) = 0 ⇒ a = 0.

3. normal



Other properties




Φ can be :


2. faithful : Φ(aa?) = 0 ⇒ a = 0.

3. normal



Examples

A free process is a family (at)t≥0 of free variables. It is said to beadapted if at ∈ At for all t ≥ 0.

I The free additive and free multiplicative Brownian motion.

I The free Wishart process (Donati-Capitaine).

I The free Jacobi process (Demni).


Examples






Examples






Asymptotic freeness and random matrices

Definitions: A family (Ui (m))i∈I of random matrices converges indistribution to (Ui )i∈I in (A , φ) if

limm→∞

E(trm(Ui1(m) . . .Uip(m))) = φ(Ui1 . . .Uip)

for any collection i1, . . . , ip ∈ I . A family is said to beasymptotically free if it cv in distribution to free variables.Connection with random matrices : Voiculescu result on GUE.


The At-free additive Brownian motion X

It is a adapted and selfadjoint process such that :

I X0 = 0.

I Xt has the semi-circle law of mean 0 and variance t given by :

σt(dy) =1

2πt

√4t − y21−2

√t,2√

t(y)dy

I For any collection t0 < t1 . . . < tk ,Xt0 ,Xt1 − Xt0 , . . . ,Xtk − Xtk−1

are free and stationnary.




I X0 = 0.


σt(dy) =1

2πt

√4t − y21−2

√t,2√

t(y)dy






I X0 = 0.


σt(dy) =1

2πt

√4t − y21−2

√t,2√

t(y)dy




Matrix-valued and free processes

Let Xm be a normalized Hermitian Brownian matrix :

Xmij (t) =

Bm

ii (t)√m

if i = jBm

ij (t)+√−1Bm

ij (t)√

2mif i < j

where Bm, Bm are two independent m×m real Brownian matrices.Voiculescu result ⇒: Xm → additive free Brownian motion X .Corollary: Let Zm = (Bm +

√−1Bm)/

√2m be a non-selfadjoint

process.Zm → complex free Brownian motion Z defined by

Z = (X 1 +√−1X 2)/

√2

where X 1,X 2 are free At-free Brownian motions.


The At-free multiplicative Brownian motion Y

It is a adapted unitary process such that :

I Y0 = I

I νt , the law of Yt is supported in the unit circle and is given byits Σ-transform (Bercovici et Voiculescu) :

Σνt (z) = et2

1+z1−z νt+s = νt νs

I For any collection t0 < t1 . . . < tk , Yt0 ,Yt1Y−1t0 , . . . ,Ytk Y

−1tk−1


Biane showed that this process is the limit in distribution of them ×m unitary Brownian motion Y (m).




I Y0 = I


Σνt (z) = et2



−1tk−1






I Y0 = I


Σνt (z) = et2



−1tk−1




Free SDE

For suitable parameters :

I Free Wishart process (Capitaine-Donati) :

dWt =√

WtdZt + dZ ?t

√Wt + λPdt

I Free multiplicative BM (Kummerer-Speicher) Let X be a freeadditive BM. Then

dYt = i dXt Yt −1

2Ytdt, Y0 = 1

I Free Jacobi process (Demni) :

dJt =√

λθ√

P − JtdZt

√Jt+

√λθ

√JtdZ ?

t

√P − Jt+(θP − Jt) dt

where Z is a complex free BM.

I condition : Injectivity.

Thanks.Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14

Free SDE



dWt =√

WtdZt + dZ ?t

√Wt + λPdt


dYt = i dXt Yt −1

2Ytdt, Y0 = 1


dJt =√

λθ√

P − JtdZt

√Jt+

√λθ

√JtdZ ?

t





Free SDE



dWt =√

WtdZt + dZ ?t

√Wt + λPdt


dYt = i dXt Yt −1

2Ytdt, Y0 = 1


dJt =√

λθ√

P − JtdZt

√Jt+

√λθ

√JtdZ ?

t





Free SDE



dWt =√

WtdZt + dZ ?t

√Wt + λPdt


dYt = i dXt Yt −1

2Ytdt, Y0 = 1


dJt =√

λθ√

P − JtdZt

√Jt+

√λθ

√JtdZ ?

t





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Matrix-valued Stochastic Processes- Eigenvalues Processes