Stochastic processes arising from non commutative symmetries Final - PowerPoint PPT Presentation

Stochastic processes arising from non commutative symmetries Final conference of MADACA Domaine de Chal` es 23 juin 2016 Philippe Biane CNRS INSTITUT GASPARD MONGE UNIVERSIT´ E PARIS EST

The aim of this talk is to present some (interesting, exotic) stochastic processes arising from natural non commutative group theoretic constructions

Classical random walk . 0.0 7.5 15.0 22.5 30.0 S n = X 1 + . . . + X n X k = ± 1

Brownian motion Scale by ε in time and √ ε in space. √ ε ↑ → ε 3 . −3 0.00 0.25 0.50 0.75 1.00 Y

The Yule process A bacteria splits after an exponential time. Y ( t )=total number of bacteria at time t .

A crash course on quantum mechanics H = (complex) Hilbert space Observables=self-adjoint operators on H a unit vector ϕ ∈ H (state of the system) and an observable A give a probability measure P ( λ ) = | π λ ϕ | 2 π λ = orthogonal projection on eigenspace of λ P is supported on the spectrum of A .

Expectation of A is � A ϕ, ϕ � = Tr ( A π ϕ ) More generally: expectation of f ( A ) is � f ( A ) ϕ, ϕ � = Tr ( f ( A ) π ϕ ) One can convexify: replace π ϕ with a positive operator of trace 1. E [ f ( A )] = Tr ( ρ f ( A ))

If A 1 , . . . , A n commute → diagonalized simultaneously Their joint distribution makes sense: � Tr ( ρ f ( A 1 , . . . , A n )) = f ( x 1 , . . . , x n ) d µ for µ proba on R n

Basic example (Ω , F , P ) probability space H = L 2 (Ω , F , P ) x =real random variable X x : H → H X x ( z ) = xz is a self-adjoint operator Spectral theorem: any self-adjoint operator on a Hilbert space can be put in this form.

Spins dim( H )=2 The space of observables has dimension 4 Identity and Pauli matrices give a basis � 0 � 0 � � � 1 � 1 0 i X = Y = Z = − i − 1 1 0 0 0 In the state ϕ = e 1 , X and Y are symmetric Bernoulli Z = 1 a.s. Combinations xX + yY + zZ , x 2 + y 2 + z 2 = 1 realize all possible Bernoulli distributions In the central state Tr ( . 1 2 Id ) all three are symmetric Bernoulli.

Quantization of head an tails game n − 1 n − 1 I ⊗ k ⊗ y ⊗ I ∞ � � I ⊗ k ⊗ x ⊗ I ∞ X n = Y n = k =0 k =0 n − 1 I ⊗ k ⊗ z ⊗ I ∞ � Z n = k =0 in M 2 ( C ) ⊗∞ . X n , Y n , Z n define three simple random walks [ X n , Y n ] = 2 iZ n

� X 2 n + Y 2 n + Z 2 Let R n = n + 1 Lemma [ R m , R n ] = 0; R n is a Markov chain with probability transitions p ( k , k − 1) = k − 1 p ( k , k + 1) = k + 1 2 k 2 k Proof: R n corresponds to the Casimir operator. Clebsch-Gordan formula for representations of SU (2) [ k ] ⊗ [2] = [ k + 1] ⊕ [ k − 1]

We have defined a random walk with values in a noncommutative ˆ space SU (2)

A = group algebra of SU (2) ˆ x , y , z =generators of Lie( SU (2))=coordinates on the space SU (2) [ x , y ] = 2 iz In each direction of space the coordinates take integer values. x 2 + y 2 + z 2 + 1 � One can measure the distance to origin using

E = a set (e.g. Z d ) Ω a probability space A random variable with values in E : X : Ω → E this gives an algebra morphism: F ( E ) → F (Ω) f → f ◦ X We could drop the condition that the algebras are commutative

Random walks on groups ˆ W = abelian group W = dual group ξ ∈ ˆ W = character of W A ( ˆ W )=group algebra of ˆ F ( W )=algebra of functions on W W ∆ : A ( ˆ W ) → A ( ˆ W ) ⊗ A ( ˆ F ( W ) → F ( W × W ) W ) f ( x ) → f ( x + y ) ∆( ξ ) = ξ ⊗ ξ φ =positive definite function on ˆ µ : F ( W ) → C W � =probability measure on W φ ( ξ ) = W ξ ( x ) d µ ( x ) state ω on A ( ˆ W )

M = ⊗ ∞ ( A ( ˆ Ω = ( W , µ ) ∞ W ) , ω ) j n : A ( ˆ Y n = w 1 + . . . + w n W ) → M j n +1 = (∆ ⊗ I ⊗ ( n +1) ) ◦ I ⊗ j n f → f ( w 1 + . . . + w n ) Markov operator � Φ( f ) = ( I ⊗ ω ) ◦ ∆ Φ( f )( x ) = W f ( x + y ) d µ ( y )

Random walks on duals of compact groups Replace ˆ W by a compact group G . φ =continuous positive definite functions on G , with φ ( e ) = 1. =state ν on A ( G ). ν = distribution of the increments. Φ ν : A ( G ) → A ( G ) Φ ν = ( I ⊗ ν ) ◦ ∆ is a completely positive map. It generates a semigroup Φ n ν ; n ≥ 1.

( N , ω ) = ( A ( G ) , ν ) ∞ j n : A ( G ) → N defined by j n ( λ g ) = λ ⊗ n ⊗ I g The morphisms ( j n ) n ≥ 0 , define a random walk on the noncommutative space dual to G , with Markov operator. Φ ν : A ( G ) → A ( G ) Φ ν = ( I ⊗ ν ) ◦ ∆ The quantum Bernoulli random walk is obtained for G = SU (2), and ν the tracial state associated with the 2-dimensional representation.

A = group algebra of SU (2)= Hopf algebra with coproduct ∆ : A → A ⊗ A ∆( x ) = x ⊗ I + I ⊗ x j n : A → M 2 ( C ) ⊗∞ = n -fold tensor product of 2-dimensional representations for n = 1 , 2 , ... form a quantum Bernoulli random walk the quantum Bernoulli walk is a Markov chain with Markov operator P : A → A P = Id ⊗ Tr 2 ( ./ 2) o ∆

RESTRICTIONS We can restrict the Markov operator P to commutative subalgebras: One parameter subgroup: Bernoulli random walk 1/2 1/2 � � � � � � � � � � � � � � � � � � � � � � � � � � Center: ”discrete Bessel process” (k−1)/2k (k+1)/2k � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � k−1 k k+1

This scheme can be extended using other semi-simple Lie groups. This leads to representation theoretic interpretation of familiar random walks in cones, non-intersecting paths, etc...

Quantum central limit theorem In the state e ⊗∞ 1 X n and Y n are symmetric Bernoulli Z n = n In the state Tr ( . 1 2 Id ) ⊗∞ X n Y n and Z n are symmetric Bernoulli

there is a basis ε k , k = 0 , 1 , ... such that ε 0 = e ⊗∞ 1 Z n ε k = n − 2 k � ( X n + iY n ) ε k = k ( n − 2 k + 2) ε k − 1 � ( X n − iY n ) ε k = ( k + 1)( n − 2 k ) ε k +1 In the limit Z n / n , X n / √ n , Y n / √ n converge to harmonic oscillator

Harmonic oscillator H Hilbert space, ε k , k = 0 , 1 , . . . orthonormal basis a + , a − creation and annihilation operators a + = ( a − ) ∗ [ a − , a + ] = I √ a + ε k = k + 1 ε k +1 √ a − ε k = k ε k − 1 ”Heisenberg representation”

Probabilistic interpretation a + + a − =gaussian variable in state ε 0 for any f one has f ( x ) e − x 2 / 2 � [ f ( a + + a − ) ϕ, ϕ ] = √ 2 π ε k = H k ( a + + a − ) ε 0 H k =Hermite polynomial

Number operator a + a − ε k = k ε k is the number operator a + a − = lim( n − Z n ) In the state ε 0 , a + a − is the zero random variable λ ( a + + a − ) + a + a − has Poisson( λ 2 ) distribution. ε k = C k ( λ ( a + + a − ) + a + a − ) ε 0 C k =Charlier polynomial cf Poisson as limit of binomial + recurrence relation for Charlier polynomials.

Dual of Heisenberg group H = Heisenberg group ( z , w ) ∗ ( z ′ , w ′ ) = ( z + z ′ , w + w ′ + ℑ ( z ¯ z ′ )); w ∈ R , z ∈ C C ∗ ( H )= convolution algebra of the group There a three important abelian subgroups: ( x , 0) , ( iy , 0) , (0 , t ) all isomorphic to R . The generators of these three one-parameter subgroups are like three ”coordinates” The generators p , q , τ of these three subgroups generate the Lie algebra, they satisfy: [ p , q ] = τ

QUANTUM BROWNIAN MOTION The functions ψ t ( z , w ) = exp t ( iw − | z | 2 / 2) form a multiplicative semigroup of positive type functions on sur H . ψ t generate a semi-group T t : C ∗ ( G ) → C ∗ ( G ); f → f ψ t

One can construct morphisms: Φ t : C ∗ ( G ) → B ( H ) which correspond to a Markov process with semi-group T t .

Restrictions of T t to subgroups ( x , 0) and ( iy , 0) give real Brownian motions Φ t ( ip ) = P t Φ t ( iq ) = Q t Φ t ( i τ ) = t P t and Q t are Brownian motions and they satisfy [ P s , P t ] = 0 , , [ Q s , Q t ] = 0 , , P s Q t − Q t P s = is ∧ t Restriction to (0 , w ) gives a uniform translation.

Functions on H invariant by rotation ( z , w ) → ( e i θ z , w ) form a commutative subalgebra C ∗ R ( G ) ⊂ C ∗ ( G ) They can be identified with functions of p 2 + q 2 and τ . y=3x y=2x y=−2x y=x y=−x Heisenberg fan

Let ( P t , Q t ) be the quantum Brownian motion, then ( P 2 t + Q 2 t , t ) is a stochastic process on the Heisenberg fan.

The semigroup T t restricted to functions invariant by rotation is that of space-time Yule process (for τ < 0) for τ > 0 it is the dual death process. The Heisenberg fan gives a realization of the space-time Martin boundary of the Yule process.

Dual of special linear group G = SL 2 ( C ) C ∗ ( G )= convolution algebra of the group Some important abelian subgroups: � e x � 0 g ( x ) = isomorphic to R . e − x 0 Cartan decomposition: KAK g = k 1 ak 2 ( G , K ) is a Gelfand pair: functions bi-invariant under K translations (depending only on a ) form a commutative subalgebra of C ∗ ( G )

Spherical dual: Ω = i R ∪ [ − 1 , 1] i R =principal series and [ − 1 , 1]=complementary series. +1 −1