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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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).
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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).
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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).
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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!
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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).
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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).
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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?
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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?
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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?
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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+
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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+.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
dλi (t) = dBi (t) +β
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).
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
dλi (t) = dBi (t) +β
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).
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
dλi (t) = dBi (t) +β
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).
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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).
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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 :
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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).
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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).
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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+.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
A β-matrix model (Demni)
I Q : Is there a matrix-valued process corresponding to :
dλi (t) = dBi (t) +β
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
A β-matrix model (Demni)
I Q : Is there a matrix-valued process corresponding to :
dλi (t) = dBi (t) +β
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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).
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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).
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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).
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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.
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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
are free and stationnary.
Biane showed that this process is the limit in distribution of them ×m unitary Brownian motion Y (m).
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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
are free and stationnary.
Biane showed that this process is the limit in distribution of them ×m unitary Brownian motion Y (m).
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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
are free and stationnary.
Biane showed that this process is the limit in distribution of them ×m unitary Brownian motion Y (m).
Nizar Demni Matrix-valued Stochastic Processes- Eigenvalues Processes and Free Probability SEA’s workshop- MIT - July 10 -14
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
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
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
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