On bijections between monotone increment processes ...jgarzav/Slides_Uwe_Franz.pdf · Programme 1...

On bijections between monotone increment processes,classes of classical Markov processes

and Loewner Chains

Uwe Franzwith T. Hasebe (Sapporo) and S. Schleißinger (Würzburg)

to appear in Dissertationes Mathematicae (see also arXiv:1811.02873)

Université de Bourgogne Franche-Comté

UC Berkeley Probabilistic Operator Algebra Seminar11 mai 2020

Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 1 / 27

Programme

1 Noncommutative (or quantum) probability

2 Monotone independence

3 Monotone increment processes

4 Complex Analysis

5 The three 1-1 correspondences

6 Stationary case

7 ExamplesStationary caseNon-stationary case

8 Infinitesimal arrays1

1This work was supported by the French “Investissements d’Avenir” program, projectISITE-BFC (contract ANR-15-IDEX-03).


Main Theme: Three 1-1 Correspondences

Quantum Probability“SAIP” (Xt)t≥0 (q-stoch.

proc. with mon. indep.increments)↓↑

Complex AnalysisLöwner chains (evolutionfamilies) (Ft)t≥0 in the

upper half plane C+

↓↑

Classical Probability .-hom. Markov proc.(Mt)t≥0 in R = ∂C+


Noncommutative probability space (1)

DefinitionA nc probability space is a pair (A,Φ), where A is a unital ∗-algebra andΦ : A → C is a normalized positive linear functional, i.e.

1 Φ(1A) = 1,2 ∀a ∈ A,Φ(a∗a) ≥ 0.

Example

Every classical prob. space (Ω,F ,P) gives rise to a nc prob. space, take

A = L∞(Ω,F ,P) and Φ : L∞(Ω,F ,P) 3 f → Φ(f ) =

∫Ωf dP.


Noncommutative probability space (2)

ExampleIf H is a Hilbert space and ξ ∈ H a unit vector, then

A = B(H) and Φ : B(H) 3 X 7→ Φ(X ) = 〈ξ,X ξ〉

define a nc prob. space.

We will view a (possibly unbounded) self-adjoint operator X on a Hilbertspace H as a nc random variable. Its law (w.r.t. a unit vector ξ ∈ H) can bedefined by functional calculus: there exists a probability µ s.t.

Φ(f (X )

)= 〈ξ, f (X )ξ〉 =

∫Rf (x)µ(dx)

for f ∈ Cb(R). We write X ∼ µ.


For X an nc random variable with distribution µ, their Cauchy-Stieltjestransform is GX = Gµ : C+ → −C+,

Gµ(z) =

∫R

1z − x

µ(dx) =⟨ξ, (z − X )−1ξ

⟩and their reciprocal C-S transform is FX = Fµ : C+ → C+, Fµ(z) = 1

Gµ(z) .

Lemma

Let F : C+ → C+ be holomorphic. Then tfae:(1) There exists a probability measure µ on R such that F = Fµ.(2) limy→∞

F (iy)iy = 1.

(3) F has the Pick-Nevanlinna representation

F (z) = z + b +

∫R

1 + xz

x − zρ(dx),

where b ∈ R and ρ is a finite, non-negative measure on R.


Monotone independence (1)

Definition (Muraki)

*-subalgebras (Aι)ι∈I of B(H) indexed by a lin. ordered set I aremonotonically independent (w.r.t. ξ, with Φξ(X ) = 〈ξ,X ξ〉) if(1) For r , s ∈ N ∪ 0, i1, . . . , ir , j , k1 . . . , ks ∈ I with

i1 > · · · > ir > j < ks < · · · < k1

and for X1 ∈ Ai1 , . . . ,Xr ∈ Air , Y ∈ Aj , Z1 ∈ Ak1 , . . . ,Zs ∈ Aks , we have

Φξ(X1 · · ·XrYZs · · ·Z1) = Φξ(X1) · · ·Φξ(Xr )Φξ(Y )Φξ(Zs) · · ·Φξ(Z1).

(2) For i , j , k ∈ I with i < j > k and X ∈ Ai , Y ∈ Aj , Z ∈ Ak , we have

XYZ = Φξ(Y )XZ .


Monotone independence (2)

Definition(Xι)ι∈I normal random variables indexed by a lin. ordered set I aremonotonically independent if the family (Aι)ι∈I is monotonicallyindependent, where

Aι = f (Xι) | f ∈ Cb(C), f (0) = 0.


Monotone convolution

Let (X ,Y ) be mon. independent self-adjoint random variables (w.r.t. a cyclicξ), then X + Y is essentially self-adjoint.And if X ∼ µ and Y ∼ ν, i.e.

Φx((z − X )−1) = Gµ(z) =1

Fµ(z)

for Gµ(z) =∫R

1z−x dx the Cauchy-Stieltjes transformation of µ and similarly

for Y and ν, then the distribution λ of X + Y is given by

Fλ = Fµ Fν ;

This defines an assoc. binary operation µB ν := λ, the (additive) monotoneconvolution µ and ν.


Monotone increment processes

Definition(Xt)t≥0 a family of ess. s.-a. r.v. on H with X0 = 0. (Xt) is a self-adjointadditive monotone increment process (SAIP) if

(a) The increment Xt − Xs with domain Dom(Xt) ∩ Dom(Xs) is ess. s.-a.for 0 ≤ s ≤ t.

(b) Dom(Xs) ∩ Dom(Xt) ∩ Dom(Xu) is dense and a core for Xu − Xs for0 ≤ s ≤ t ≤ u.

(c) For s ≥ 0, t 7→ µst is weakly continuous, where Xt − Xs ∼ µst .(d)

(Xt1 ,Xt2 − Xt1 , . . . ,Xtn − Xtn−1)

is mon. independent for n ∈ N and 0 ≤ t1 ≤ t2 ≤ · · · ≤ tn.

If Xt − Xs has the same distribution as Xt−s (stationary increments), then(Xt)t≥0 is a monotone Lévy process.


SAIP→ Löwner

Let (Xt)t≥0 be an SAIP.Let µst ∼ Xt − Xs (w.r.t. ξ) and Fst = Fµst for 0 ≤ s ≤ t.

Mon. independence of Xu − Xt and Xt − Xs implies

µsu = µtu . µst

andFsu = Ftu Fst

for 0 ≤ s ≤ t ≤ u, so (F0t)t≥0 is an additive (descreasing) Löwner chain.


Before we define Löwner chains, let us recall some background from complexanalysis.D = z ∈ C; |z | < 1,

Theorem (Denjoy-Wolff theorem, 1926)Let f : D → D be holomorphic. If f is not an automorphim (i.e. a Möbiustransformation), then there exists a unique z0 ∈ D such that

limn→∞

f n(z) = z0

uniformly on all compacts in D .


C+ = z ∈ C; Im(z) > 0.G(C+) = f : C+ → C+; f holomorphic.

Theorem (Berkson-Porta, 1978)

(i) Let D ⊆ C be open and (φt)t≥0 a one-parameter semigroup ofholomorphic mappings of D into D . Then there exists a holomorphicmapping G : D → C such that

∂

∂tφ(t, z) = G

(φ(t, z)

), t ≥ 0, z ∈ D.

(ii) Let D = C+ and z0 the Denjoy-Wolff point of (φt)t≥0.(a) If z0 =∞, then G ∈ G(C+);(b) if z0 = b ∈ R, then G (z) = −(z − b)2G0(z) for some G0 ∈ G(C+);(c) if z0 =∈ C+, then G (z) = (z − z0)(z + z0)G0(z) for some G0 ∈ G(C+).


Löwner chains

Definition1 Let (fst)0≤s≤t be a family of non-constant holomorphic self-mappings

fst : D → D satisfying1 fss(z) = z for all z ∈ D and s ≥ 0,2 fsu = fst ftu for all 0 ≤ s ≤ t ≤ u,3 fsu converges loc. uniformly to fst as u → t.

The family (ft)t≥0 := (f0t)t≥0 is called a (decreasing) Loewner chain onD . We will call the mappings fst the transition mappings of the Loewnerchain.

2 We call a Loewner chain (ft)t≥0 an additive Loewner chain if D = C+

and limy→∞ ft(iy)/(iy) = 1, or equivalently

ft = Fµt

for all t ≥ 0, where each µt is a probability measure on R.


SAIP→Markov

PropositionFor X ,Y mon. independent normal r.v., there exists a conditional expectation

EX : alg(X ,Y )→ alg(X ).

On functions of X + Y we have

EX

(1

z − (X + Y )

)=

1FY (z)− X

, z ∈ C+


SAIP & Löwner→Markov, cont’d

Corollary

Every SAIP (Xt)t≥0 has a classical version, i.e., there exists a class. stoch.proc. (Mst)0≤s≤t s.t.

Φξ

(f1(Xt1 − Xs1) · · · fn(Xtn − Xsn)

)= E

(f1(Ms1t1) · · · fn(Msntn)

)for all n ∈ N, 0 ≤ s1 ≤ t1 ≤ · · · ≤ sn ≤ tn, f1, . . . , fn ∈ Cb(R).

(Mst)0≤s≤t is the Markov process with Mss = 0 and transition kernels(kst)0≤s≤t given by∫

R

1z − y

kst(x , dy) =1

Fst(z)− x, z ∈ C+.


Markov→ SAIP

Let Ft = σ(Ms1s2 ; 0 ≤ s1 ≤ s2 ≤ t), Pt = E( · |Ft).

TheoremLet (Mst)0≤s≤t be a .-homogeneous Markov process on (Ω,F ,P), i.e., aMarkov process whose transition kernels satisfy

kst(x , dy) = δx . kst(0, dy), x ∈ R, 0 ≤ s ≤ t.

ThenXt = MtPt , t ≥ 0

defines an SAIP on L2(Ω,F ,P) (w.r.t. ξ = 1Ω).


The three 1-1 correspondences

TheoremOne-to-one correspondence between the following objects:

1 SAIPs (Xt)t≥0 (up to stoch. equivalence)2 additive Loewner chains (Ft)t≥0 in C+,3 real-valued B-homogeneous Markov processes (Mt)t≥0 with M0 = 0

(up to stoch. equivalence).

RemarkThere exist similar bijections between

1 MUIP, i.e. multiplicative unitary increment processes,2 “multiplicative” Löwner chains in the unit disk,3 and class of Markov chains with values in the unit circle.


TheoremLet µ be a probability measure on R. The following statements are equivalent.

(1) Fµ is univalent.(2) There exists a SAIP (Xt)t≥0 such that the distribution of X1 is µ.(3) There exists an additive Löwner chain (Ft)t≥0 such that F1 = Fµ.

Note that

Univ(R) = µ probability on R s.t. Fµ is univalent

is strictly bigger than

ID(.) = µ probabilty on R and . -inf. divisible.


Mon. Lévy-Khintchine formula

Theorem (Muraki 2000)1 Let µtt≥0 be a weakly cont. B-conv. semigroup of prob. s.t. µ0 = δ0

and let Ft = Fµt . Then the right derivative A(z) = ddt

∣∣t=0Ft(z) exists

and Ft satisfies

ddt

Ft(z) = A(Ft(z)), F0(z) = z , z ∈ C+, t ≥ 0. (1)

Moreover, A is of the form

−A(z) = γ +

∫R

1 + zx

z − xρ(dx), z ∈ C+,

where γ ∈ R, ρ a finite non-neg. measure on R. The pair (γ, ρ) is unique.2 Given a pair (γ, ρ) as above, define A as above. Then (1) defines a flowFtt≥0 on C+, and there exists a weakly cont. B-conv. semigroupµtt≥0 s.t. Ft = Fµt for all t ≥ 0.


Mon. Hunt formula

TheoremLet (Mt)t≥0 be the stationary B-homogeneous Markov process withtransition kernels (kt)t≥0, associated to a mon. Lévy proc. LetTt : Bb(R)→ Bb(R) be its transition semigroup,

(Tt f )(x) =

∫Rf (y) kt(x , dy), f ∈ Bb(R).

Its generator is then given by

(Gf )(x) :=ddt

∣∣∣∣t=0

(Tt f )(x)

= γf ′(x) +

∫R

(1 + y2)(

f (y)− f (x)− (y − x)f ′(x)

(y − x)2 + yf ′(x)

dρ(y)

for f ∈ Cb(R) ∩ C 2(R), x ∈ R, where (γ, ρ) is the pair associated to the mon.conv. semigroup kt(0, ·)t≥0.


Azéma and Belton martingales

For ρ = 0, we get A(z) = c (with c ∈ R), Ft(z) = z + ct, µt = ct, andMt is deterministic.

For ρ = αδx0 , we have two cases.

1 A(z) = αz−x0

(no drift): We get an arcsince distribution

µt =1

π√

2t − (x − x0)21]−√

2t−x0,√

2t−x0[(x)dx

and (Mt)t≥0 is an Azéma martingale.

2 A(z) = γ + αz−x0

(γ 6= 0, i.e. non-vanishing drift): We get Belton’smartingales (plus a drift).


Azéma and Belton martingales

For ρ = 0, we get A(z) = c (with c ∈ R), Ft(z) = z + ct, µt = ct, andMt is deterministic.For ρ = αδx0 , we have two cases.

1 A(z) = αz−x0

(no drift): We get an arcsince distribution

µt =1

π√

2t − (x − x0)21]−√

2t−x0,√

2t−x0[(x)dx

and (Mt)t≥0 is an Azéma martingale.

2 A(z) = γ + αz−x0

(γ 6= 0, i.e. non-vanishing drift): We get Belton’smartingales (plus a drift).


The simplest vector fields

drift to the left no drift drift to the right


Visualisation of holomorphic function

The F-transform of the arcsine distribution “folds” the upper half plane aroundthe segment from 0 to i

√t:

Complex plane Image of H under F (z) =√z2 − 2


Visualisation of holomorphic function, cont’d

The F-transform of the Wigner distribution (= inverse of the Joukowski map)occors for a non-stationary SAIP:

Complex plane F (z) = 12

(z +√z2 − 4

)F−1(w) = w + 4

w

It “eats” a semi-disk of radius 2√t.


Another visualization for the Wigner case

F (z) = 12

(z +√z2 − 4

)

-4 -2 0 2 4

0

1

2

3

4

5


Infinitesimal triangular arrays (and an open problem)

Theorem(1) If µ is a probability measure such that Fµ is univalent, then there exists

an infinitesimal triangular array µn,j1≤j≤kn,1≤n such that

µn,1 B µn,2 B · · ·B µn,kn

converges weakly to µ as n→∞.(2) If an infinitesimal triangular array µn,j1≤j≤kn,1≤n satisfies the variance

conditionsup

1≤j≤knσ2(µn,j)→ 0 as n→∞

and if µn,1 B µn,2 B · · ·B µn,kn converges weakly to a probabilitymeasure µ, then Fµ is univalent.

Conjecture: (2) holds without the variance conditon and we haveUniv(R) = InfArr(B).


On bijections between monotone increment processes ...jgarzav/Slides_Uwe_Franz.pdf · Programme 1...

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