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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).
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 2 / 27
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+
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 3 / 27
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+
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 3 / 27
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+
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 3 / 27
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.
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 4 / 27
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.
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 5 / 27
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.
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 6 / 27
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 .
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 7 / 27
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.
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 8 / 27
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.
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 9 / 27
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.
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 10 / 27
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.
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 11 / 27
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 .
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 12 / 27
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+).
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 13 / 27
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.
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 14 / 27
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.
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 14 / 27
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+
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 15 / 27
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+.
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 16 / 27
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).
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 17 / 27
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.
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 18 / 27
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.
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 18 / 27
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.
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 19 / 27
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.
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 20 / 27
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.
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 21 / 27
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).
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 22 / 27
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).
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 22 / 27
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).
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 22 / 27
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).
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 22 / 27
The simplest vector fields
drift to the left no drift drift to the right
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 23 / 27
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
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 24 / 27
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.
Uwe Franz (UBFC) On Monotone Increment Processes Berkeley, 11/05/2020 25 / 27
Another visualization for the Wigner case
F (z) = 12
(z +√z2 − 4
)
-4 -2 0 2 4
0
1
2
3
4
5
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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).
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