Mouvement brownien libre

Dans cette sous-section, on se place dans un C

∗

-espace de probabilit´e

(A,∗,k · k, φ) muni d’une filtration, c’est-`a-dire d’une famille (A

t

)

_t≥0

de

sous-alg`ebres ferm´ees pour la topologie faible-∗ sur A telles que pour tous

s≤t, on aA

s

⊂ A

t

.

D´efinition A.54. Unmouvement brownien libre est une famille (S

_t

)

_t≥0

de

variables non commutatives telle que :

– pour tout t≥0,S

_t

est un ´el´ement auto-adjoint de A

t

, dont la

distri-bution est la loi semi-circulaire de variance t, d´efinie par

dµ

_sc,t

(x) = ¹

2πt

p

4t−x

2

1

_[−2√ t,2√ t]

(x) dx;

– pour tous s≤t, l’´el´ement S

_t

−S

_s

est libre avec A

s

et sa distribution

est la loi semi-circulaire de variance t−s.

On remarquera que cette d´efinition est l’analogue libre de la d´efinition

du mouvement brownien

≪

classique

≫

.

Le mouvement brownien libre peut ˆetre interpr´et´e en termes de matrices

al´eatoires.

Proposition A.55(voir [Voi1]). Pour tout N ∈N

^∗

, on d´efinit le processus

matriciel (X

N

(t))

_t≥0

sur H

N

(C) par :

– pour tout j∈J1, NK, X

N

j,j

est un mouvement brownien r´eel standard ;

– pour tous j, k ∈J1, NK tels que j < k, √

2 ReX

_j,k^N

et √

2 ImX

_j,k^N

sont

deux mouvements browniens r´eels standards ind´ependants.

Le processus(X

N

(t)/√

N)

_t≥0

converge en distribution quandN →+∞vers

un mouvement brownien libre.

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