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

.

Sn = X1 + . . . + Xn Xk = ±1

Brownian motion Scale by ε in time and √ε in space. √ε ↑ → ε

0.00 0.25 0.50 0.75 1.00 −3 3

.

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) = |πλϕ|2 πλ = orthogonal projection on eigenspace of λ P is supported on the spectrum of A.

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

If A1, . . . , An commute → diagonalized simultaneously Their joint distribution makes sense: Tr(ρf (A1, . . . , An)) =

• f (x1, . . . , xn)dµ

for µ proba on Rn

Basic example (Ω, F, P) probability space H = L2(Ω, F, P) x=real random variable Xx : H → H Xx(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 X = 1 1

• Y =

i −i

• Z =

1 −1

• In the state ϕ = e1,

X and Y are symmetric Bernoulli Z = 1 a.s. Combinations xX + yY + zZ, x2 + y 2 + z2 = 1 realize all possible Bernoulli distributions In the central state Tr(.1

2Id) all three are symmetric Bernoulli.

Quantization of head an tails game Xn =

n−1

• k=0

I ⊗k⊗x ⊗ I ∞ Yn =

n−1

• k=0

I ⊗k ⊗ y ⊗ I ∞ Zn =

n−1

• k=0

I ⊗k ⊗ z ⊗ I ∞ in M2(C)⊗∞. Xn, Yn, Zn define three simple random walks [Xn, Yn] = 2iZn

Let Rn =

• X 2

n + Y 2 n + Z 2 n + 1

Lemma [Rm, Rn] = 0; Rn is a Markov chain with probability transitions p(k, k + 1) = k + 1 2k p(k, k − 1) = k − 1 2k Proof: Rn 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 space ˆ SU(2)

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

• x2 + y 2 + z2 + 1

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

• W ξ(x)dµ(x)

state ω on A( ˆ W )

Ω = (W , µ)∞ M = ⊗∞(A( ˆ W ), ω) Yn = w1 + . . . + wn jn : A( ˆ W ) → M f → f (w1 + . . . + wn) jn+1 = (∆ ⊗ I ⊗(n+1)) ◦ I ⊗ jn Markov operator Φ(f )(x) =

• W f (x + y)dµ(y)

Φ(f ) = (I ⊗ ω) ◦ ∆

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. Φν : A(G) → A(G) Φν = (I ⊗ ν) ◦ ∆ is a completely positive map. It generates a semigroup Φn

ν; n ≥ 1.

(N, ω) = (A(G), ν)∞ jn : A(G) → N defined by jn(λg) = λ⊗n

g

⊗ I The morphisms (jn)n≥0, define a random walk on the noncommutative space dual to G, with Markov operator. Φν : A(G) → A(G) Φν = (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 jn : A → M2(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

• perator

P : A → A P = Id ⊗ Tr2(./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

k−1 (k−1)/2k (k+1)/2k k

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

Xn and Yn are symmetric Bernoulli Zn = n In the state Tr(.1

2Id)⊗∞

Xn Yn and Zn are symmetric Bernoulli

there is a basis εk, k = 0, 1, ... such that ε0 = e⊗∞

1

Znεk = n − 2k (Xn + iYn)εk =

• k(n − 2k + 2)εk−1

(Xn − iYn)εk =

• (k + 1)(n − 2k)εk+1

In the limit Zn/n, Xn/√n, Yn/√n converge to harmonic oscillator

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

Probabilistic interpretation a+ + a−=gaussian variable in state ε0 for any f one has [f (a+ + a−)ϕ, ϕ] =

• f (x)e−x2/2

√ 2π εk = Hk(a+ + a−)ε0 Hk =Hermite polynomial

Number operator a+a−εk = kεk is the number operator a+a− = lim(n − Zn) In the state ε0, a+a− is the zero random variable λ(a+ + a−) + a+a− has Poisson(λ2) distribution. εk = Ck(λ(a+ + a−) + a+a−)ε0 Ck =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 ψt(z, w) = exp t(iw − |z|2/2) form a multiplicative semigroup of positive type functions on sur H. ψt generate a semi-group Tt : C ∗(G) → C ∗(G); f → f ψt

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

Restrictions of Tt to subgroups (x, 0) and (iy, 0) give real Brownian motions Φt(ip) = Pt Φt(iq) = Qt Φt(iτ) = t Pt and Qt are Brownian motions and they satisfy [Ps, Pt] = 0, , [Qs, Qt] = 0, , PsQt − QtPs = is ∧ t Restriction to (0, w) gives a uniform translation.

Functions on H invariant by rotation (z, w) → (eiθz, w) form a commutative subalgebra C ∗

R(G) ⊂ C ∗(G)

They can be identified with functions of p2 + q2 and τ.

y=x y=2x y=−x y=3x y=−2x

Heisenberg fan

Let (Pt, Qt) be the quantum Brownian motion, then (P2

t + Q2 t , t)

is a stochastic process on the Heisenberg fan.

The semigroup Tt 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 = SL2(C) C ∗(G)= convolution algebra of the group Some important abelian subgroups: g(x) = ex e−x

• isomorphic to R.

Cartan decomposition: KAK g = k1ak2 (G, K) is a Gelfand pair: functions bi-invariant under K translations (depending only on a) form a commutative subalgebra

• f C ∗(G)

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

−1 +1

For ω ∈ iR ∪ [−1, 1] the spherical function is φω(g(x)) = sinh(ωx) ω sinh(x) φ0(g(x)) = x sinh(x) There exists a remarkable negative definite function ψ = lim

ω→1

1 − φω 1 − ω . ψ(g(x)) = 1 − x coth(x)

The semi-group exp(tψ) defines a Markov process on the spherical dual

On the dual of the diagonal subgroup it defines a L´ evy process Lt with E[exp(ixLt)] = exp(t(1 − x coth x) Starting from the center (0) of the spherical dual we obtain the positive definite function E[exp(ixLt)] = x sinh(x) exp(t(1 − x coth x) this formula is exactly Paul L´ evy’s area formula!! E[exp ix 1 X(s)dY (s) − Y (s)dX(s)|X1 = u, Y1 = v] = x sinh(x) exp((u2 + v 2)(1 − x coth x)/2

G automorphism group of a regular tree Tq K ⊂ G automorphism which fix e (G, K) is a Gelfand pair

• e has

[−(q1/2 + q−1/2), q1/2 + q−1/2] = [−2, 2] ∪ ([−(q1/2 + q−1/2), −2[∪]2, q1/2 + q−1/2]) principal series + complementary series ψ(g) = −d(e, g(e)) induces a semigroup in the complementary series it is a uniform translation followed by a jump process in the principal series governed by the Dirichlet form f (x) − f (y) x − y 2 mq(dx)mq(dy).