Generators of some non-commutative stochastic processes

0.27 MB·English

by  Michael Anshelevich

GENERATORS OF SOME NON-COMMUTATIVE STOCHASTIC PROCESSES
MICHAELANSHELEVICH
3 ABSTRACT. AfundamentalresultofBiane(1998)statesthataprocesswithfreelyindependentin-
1 crementshastheMarkovproperty,butthattherearetwokindsoffreeLe´vyprocesses: thefirstkind
0
has stationary increments, while the second kind has stationarytransition operators. We show that
2
a processofthe first kind(with meanzeroand finite variance) hasthe same transitionoperatorsas

n
thefreeBrownianmotionwithappropriateinitialconditions,whileaprocessofthesecondkindhas
a
thesametransitionoperatorsasamonotoneLe´vyprocess. We computeanexplicitformulaforthe
J
generatorsofthesefamiliesoftransitionoperators,intermsofsingularintegraloperators,andprove
2
2 thatthisformulaholdsonafairlylargedomain. Wealsocomputethegeneratorsfortheq-Brownian
motion,andforthetwo-statefreeBrownianmotions.

]
A
O
.
h
1. INTRODUCTION
t
a
m
ALe´vyprocessisarandomprocess X(t) : t 0 whoseincrementsX(t) X(s)areindependent
[ { ≥ } −
and stationary,in thesensethatthedistributionµ ofX(t) X(s) dependsonlyon t s,
s,t − −
4
v µ = µ .
s,t t s
1 −
8 A Le´vy processis aMarkovprocess,and itstransitionoperators defined via
s,t
K
3
1 E[f(X(t)) s] = ( s,tf)(X(s))
| K
.
4 are alsostationary, = ; infact
0 Ks,t Kt−s
1
1 (f)(x) = f(x+y)dµ (y).
s,t t s
: K ZR −
v
i Here E[ s] is the conditional expectation onto time s. The maps : t 0 form a semigroup,
X t
·| {K ≥ }
which has a generator A. In fact A = ℓ( i∂ ), where ℓ is the cumulant generating function of the
r x
−
a process. It can also be expressed in terms of the Le´vy measure of the process. See Section 3 for
moredetails.
In a groundbreaking paper [Bia98], Biane showed that processes with freely independent incre-
ments,inthecontextoffreeprobability[VDN92,NS06],are alsoMarkovprocesses. Healsonoted
that there are two distinct classes of such processes which can be called (additive) free Le´vy pro-
cess (Biane also investigated multiplicative processes, which we will not study here). Free Le´vy
processesofthefirstkind(FL1)havestationaryincrements,inthesensethateachX(t) X(s)has
distribution µ . Then µ : t 0 form a semigroup with respect to free convolution−⊞. These
t s t
− { ≥ }
processesareMarkov,buttheirtransitionoperatorstypicallyarenotstationary. FreeLe´vyprocesses
ofthesecondkind(FL2)havestationarytransitionoperators ,whichformasemigroup,buttheir
t
K
Date:January23,2013.
2010MathematicsSubjectClassification. Primary46L54;Secondary60J25,47D06.
ThisworkwassupportedinpartbyNSFgrantsDMS-0900935andDMS-1160849.
1
2 MICHAELANSHELEVICH
incrementsare typicallynotstationary: if X(t) X(s) has distributionµ , then weonlyhavethe
s,t
property µ ⊞µ = µ (sothat themeasures−form afreeconvolutionhemigroup).
s,t t,u s,u
InthispaperwecomputethegeneratorsoffreeLe´vyprocesseswithfinitevariance. SinceforFL1,
thetransitionoperators do not form a semigroup,they havea family ofgenerators A : t 0 . In
t
{ ≥ }
thecaseofFL2,thereisagenuinegeneratorA. Ifthedistributionoftheprocesshasfinitemoments,
usingthefreeItoˆ formulafrom[Ans02],onecanexpressthegeneratorsintermsoftheR-transform,
the free analog of ℓ. However, it is unclear whethersuch an expression can be assigned a meaning
in theabsenceofmoments. But thereis an alternativedescription. Fora measure ν, denote
f(x) f(y)
L (f)(x) = − dν(y),
ν
Z x y
R
−
a singular integral operator. A free convolution semigroup with finite variance is characterized by
thefreecanonical pair(α,ρ),where α R,and (withappropriatenormalization) ρisaprobability
measure. Further,denotebyγ thesemi∈circulardistributionattimet,sothatρ⊞γ isthefreeanalog
t t
ofheatflowstartedatρ. ThenthegeneratorofthecorrespondingfreeLe´vyprocess ofthefirstkind
is
(1) α∂x +∂xLρ⊞γt.
In fact, we show that for α = 0, the full Markov structure of this process coincides with that of
thefree Brownian motion Y : t 0 with Y having distributionρ. Thisstatement clearly has no
t 0
{ ≥ }
classical analogue.
Inadditiontofreeprobabilitytheory,thereareonlytwoother“natural”non-commutativeprobabil-
ity theories [Mur03], the Boolean and the monotone. These theories do not, at least at this point,
approachthewealthofstructureoffreeorclassicalprobability. However,onereasontostudythem
is that they turn out to have unexpected connections to free probability. Indeed, we show that the
generator of a free Le´vy process of the second kind is α∂ + ∂ L , where ρ now is the monotone
x x ρ
canonical measure. In fact, Biane in [Bia98] already noted that each FL2 is associated to a semi-
group ofanalyticmaps,and Franz in[Fra09]observed thatexactly such semigroupsare associated
withmonotoneLe´vyprocesses: themeasures µ donotformafreesemigroup,buttheydoforma
0,t
semigroupundermonotoneconvolution. Inthemonotonecaseitselfthereisnodistinctionbetween
theLe´vyprocessesofthefirstandsecondkind(sothefreecaseisreallyspecialinthisrespect),and
the generatorof a monotoneLe´vy process is related to its monotoneLe´vy measure in theexpected
way [FM05].
We also compute the generators of the q-Brownian motion. This non-commutative process was
constructed in [BS91], and investigated in detail in [BKS97]. Building on the work of [DM03],
we prove a functional Itoˆ formula for it (for polynomial functions), from which the formulas for
generators easilyfollow.
We notethat the studyof “time-dependent generators”, ormoreusually theinverseproblem—how
to reconstruct from A —goes back to [Kat53]. This is typically formulated at the linear
s,t t
{K } { }
non-autonomous Cauchy problem, and a significant amount of general results on its solution is
known,seeforexampleSectionVI.9of[EN00],[NZ09],andtheirextensivereferences. Wedonot
usethesegeneral resultsinthepaper, butthismaybeamatterforfurtherstudy.
The paper is organized as follows. After the introduction and some general results in Section 2,
in Section 3 we give a short overview of the generators for classical processes. The next section,
covering free Le´vy processes, is the main part of the paper. We show that transition operators for
GENERATORSOFSOMENON-COMMUTATIVESTOCHASTICPROCESSES 3
suchaprocessformastronglycontinuouscontractivefamilyonLp(R, dx),andthattheirgenerators
are given by formula (1) on large domains in Lp(R, dx) and C (R). In Section 4.5 we find the
0
closures of these generators. We also compute the generators of FL2 processes. In section 4.6, we
showthattheoperatorL itselfisanisometrybetweencertainL2 spaces,andcomputethe“carre´ du
ν
champ” operator corresponding to ∂ L . In Section 5, we compute the Itoˆ formula and generators
x ν
for the q-Brownian motion, and in a short final section we apply similar analysis to the two-state
free Brownianmotionsfrom [Ans11b].
Acknowledgements. This work was prompted by a discussion with Włodek Bryc about his pa-
per [Bry10]; I am grateful to Włodek for showing me an early version of that paper. I have dis-
cussed various aspects of this article with a large number of people. Thanks to Dominique Bakry,
ToddKemp,MichelLedoux,Conni Liaw,AlexPoltoratski,andSergei Treilforhelpfulcomments.
Thanks especially to J.C. Wang and thereferee foracorrection in Remark 5. Finally,I am grateful
totheErwinSchro¨dingerInstitute,andtotheUniversite´ PaulSabatier, wherepart ofthisworkwas
completed.
2. PRELIMINARIES
2.1. Generalities and definitions. Let ( ,E) be a tracial non-commutative probability space,
where is a von Neumann algebra andME is a tracial normal state on it. Possibly unbounded
M
randomvariablesare self-adjointelementsofthealgebra ofoperators affiliatedto .
M M
Aprocessisafamilyof(possiblynon-commutative)randomvariables X(t) : t 0 ina(possibly
f
non-commutative)probabilityspace ( ,E). { ≥ }
M
We will assume that X(0) = 0, and will denote by µ the distribution of X(t) X(s) with
s,t
respect to E (for s t), µ = µ the distribution X(t), and µ = µ . If ⋆ is a−convolution
t 0,t 1
≤
operation corresponding to some non-commutative independence, and the increments of X(t)
{ }
are independentinthatsense, then
µ ⋆µ = µ .
s,t t,u s,u
Foran unboundedoperator X,wewilldenoteby (X)itsdomain,and by(X, )itsrestrictionto
D D
asmallerdomain .
D
Definition 1. For a family of distributions µ , we say that the functional L is its generator at
t t
{ }
timetwithdomain (L ) if
t
D
∂ f(x)dµ (x) = L [f]
t t t
ZR
forf (L ). Frequently,
t
∈ D
L [f] = (A f)(x)dµ (x)
t t t
ZR
foran operator A . If X(t) is aprocesswithdistributions µ ,thisis equivalentto
t t
{ } { }
(2) ∂ E[f(X(t))] = E[(A f)(X(t))].
t t
Notehoweverthat thisproperty doesnot determine A ,evenon (L ).
t t
D
Foroperators ona Banach space , wewrite
s,t
{K } A
∂
= A
s,t s
∂t(cid:12) K
(cid:12)t=s
(cid:12)
(cid:12)
4 MICHAELANSHELEVICH
if
1
(3) lim ( f f) A f = 0.
s,s+h s,s s
h 0+(cid:13)h K −K − (cid:13)
→ (cid:13) (cid:13)
(cid:13) (cid:13)
In this case we say that A is the g(cid:13)enerator of the family (cid:13)at time s. Its domain (A )
s s,t s
{K } D ⊂ A
consistsofall f forwhichthelimit(3)holds.
∈ A
Now suppose that the process X(t) is a Markov process. That is, denoting E[ s] the E-
{ } ·| ≤
preserving conditional expectation onto the von Neumann algebra generated by X : u s , for
u
any f L (R, dx), E[f(X(t)) s] is in the von Neumann algebra generate{d by X(≤s).}(See
∞
∈ | ≤
the Introduction and Section 4 of [Bia98] for more details, and also for a weaker requirement,
sufficient for our purposes, that the classical version of X(t) is a Markov process.) In this case,
{ }
thecorrespondingtransitionoperators aredeterminedby
E[f(X(t)) s] = ( f)(X(s))
s,t
| ≤ K
WesaythattheoperatorA isthegeneratoroftheprocessattimesifitisthegeneratorofitsfamily
s
oftransitionoperators. Notethat if A exists,it hastheproperty inequation(2).
t
Proposition1. Let ( , ) beaBanachspace, a familyofcontractionson suchthat
s,t
A k·k {K } A
= , = I,
s,t t,v s,v s,s
K ◦K K K
and isstronglycontinuousint. Let A bethegeneratorsof inthesenseofequation(3),
s,t t s,t
K { } {K }
andconsiderasubspace (A )suchthatforanyf ,A f isacontinuousfunctionoft.
D ⊂ tD t ∈ D t
T
(a) EachA isdissipativeand closable.
t
(b) Let beanothersubspacesuchthat , and beanothernormon suchthat
B ⊂ A D ⊂ B k·k B
is -densein , f f ,and forf , B
D k·k B k k ≤ k k ∈ D
B B
(4) A f f .
t
k k ≤ k k
B
Then equation(3)holdsforf , sothat (A ) forallt.
t
∈ B B ⊂ D
(c) The closure k·kA of inthesup-graphnorm
D D
f = f +sup A f
k kA k k k t k
t
is in (A ) forallt.
t
D
Remark 1. Note that strong continuity of does not imply continuity of A . Indeed, already
s,t t
K { }
in onedimension,if = ef(t) f(s),then A = f (t).
s,t − t ′
K
Proof. For part (a), recall from Section X.8 of [RS75] that for f , a normalized tangent func-
∈ A
tionalϕ is an elementof suchthat ϕ = f and ϕ [f] = f 2. Forany suchfunctional,
f ∗ f f
A k k k k k k
1 1
ϕ [A f] = lim ϕ [ f f] lim ϕ [ f] f 2
f s f s,s+h f s,s+h
ℜ h 0+ hℜ K − ≤ h 0+ h | K |−k k
→ → (cid:0) (cid:1)
1
lim f f f 2 0
s,s+h
≤ h 0+ h k k·kK k−k k ≤
→ (cid:0) (cid:1)
since isacontraction,soA isdissipative. CombiningthiswithTheoremII.3.23andPropo-
s,s+h s
K
sitionII.3.14(iv)of[EN00], itfollowsthatA isclosable.
s
GENERATORSOFSOMENON-COMMUTATIVESTOCHASTICPROCESSES 5
Forpart (b), wefirst notethat for s t, since isacontraction,for f ,
s,t
≤ K ∈ D
1 1
lim ( f f) (A f) = lim ( f f) A f
s,t+h s,t s,t t s,t t,t+h t
h 0+(cid:13)h K −K −K (cid:13) h 0+(cid:13)K (cid:18)h K − − (cid:19)(cid:13)
→ (cid:13) (cid:13) → (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13)1 (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
lim ( f f) A f = 0,
t,t+h t
≤ h 0+(cid:13)h K − − (cid:13)
→ (cid:13) (cid:13)
so (cid:13) (cid:13)
(cid:13) (cid:13)
∂ (f) = (A f).
t s,t s,t t
K K
Also, since is a contraction, A f is continuous in v. Therefore we have the Riemann
s,v s,v v
K K
integralidentity
t
(f) = f + (A f)dv.
s,t s,v v
K Z K
s
Since for f , (4) holds, A has a continuous extension ( , ) satisfying the same
t
∈ D B k·k → A
property. Wewillshowthatthiscontinuousextension A˜ coincideswBithA .
t t
Fix g and a time t. For each ε > 0, we can find a f such that f g < ε, so that
∈ B ∈ D k − k
f g < ε, B
k − k
f g < ε,
s,t s,t
kK −K k
and
˜ ˜ ˜ ˜
(A f) (A g) A f A g < ε
s,v v s,v v v v
(cid:13)K −K (cid:13) ≤ (cid:13) − (cid:13)
forall s v t. Then (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
≤ ≤
t
g g (A˜ g)dv < 2ε+(t s)ε.
s,t s,v v
(cid:13)K − −Z K (cid:13) −
(cid:13) s (cid:13)
(cid:13) (cid:13)
So (cid:13) (cid:13)
t
(5) (g) = g + (A˜ g)dv
s,t s,v v
K Z K
s
(in particular,theintegraliswell defined), and
∂
A g = g = A˜ g.
t s,t t
∂t(cid:12) K
(cid:12)t=s
(cid:12)
(cid:12)
Finally,forpart (c)wetake = k·kA and = ,and applypart (b). (cid:3)
B D k·k k·kA
B
Remark 2. Undertheassumptionsoftheprecedingproposition,from equation(5)wealsoget
∂
g = A g,
s,t s
∂s(cid:12) K −
(cid:12)s=t
(cid:12)
A (cid:12) (g) = ∂ (g),
s s,t s s,t
K − K
which inturnimplies
t
(g) = g + A (g)dv.
s,t v v,t
K Z K
s
6 MICHAELANSHELEVICH
Lemma2. Let X(t) beanon-commutativeMarkovprocess,withtransitionoperators and
s,t
distributions µ{ . L}et ( , ) be either (C (R), ) or Lp(R, dx), and suppose th{aKt } ,
t 0 s,t
{ } A k·k k·k {K }
theirgenerators A , and satisfythehypothes∞esof Proposition1. Then forf ,
s
{ } D ⊂ A ∈ D
t
(6) f(X(t)) (A f)(X(v))dv
v
−Z
0
isamartingale. Conversely,suppose B isanotherfamilyofoperatorsstronglycontinuouson
s
{ } D
suchthat (6)isa martingale. Then forf , A f = B f in therestrictionof tosupp(µ ).
s s s
∈ D A
Proof. AsshowninProposition1, undertheseassumptions,for f
∈ D
t
(f) = f + (A f)dv.
s,t s,v v
K Z K
s
It followsthat
t
E f(X(t)) (A f)(X(v))dv s
v
(cid:20) −Z (cid:12) ≤ (cid:21)
0 (cid:12)
s (cid:12) t
(cid:12)
= ( (f))(X(s)) (A f)(X(v))dv (A f)(X(s))dv
s,t v s,v v
K −Z −Z K
0 s
s
= f(X(s)) (A f)(X(v))dv
v
−Z
0
and theprocessis amartingale.
t
Conversely, suppose that f(X(t)) (B f)(X(s))dv is a martingale. The last equality then
− 0 v
impliesthat R
t
(f)(X(s)) = f(X(s))+ (B f)(X(s))dv.
s,t s,v v
K Z K
s
It followsthat inC (supp(µ ))orLp(supp(µ ), dx),
0 s s
t
(f) = f + (B f)dv,
s,t s,v v
K Z K
s
and thereforein thisspace
∂
B f = (f) = A f.
v v,t v
∂t(cid:12) K
(cid:12)t=v
(cid:12) (cid:3)
(cid:12)
2.2. Cumulants. Let µ beaconvolutionsemigroupwithrespecttosomeconvolutionoperation
t
{ }
⋆. Inallcaseswewillconsider,µ = δ ,µ [x] = t µ[x],and µ isweaklycontinuous. Almostby
0 0 t t
· { }
definition (see Property (K1’) in Section 3 of [HS11]), thecumulantfunctional of µ corresponding
to theconvolutionoperation ⋆ is
∂
(7) C [f] = µ [f].
µ t
∂t(cid:12)
(cid:12)t=0
(cid:12)
This approach works for all the convolutions asso(cid:12)ciated to natural types of independence (tensor,
free, Boolean, monotone),butalso forotheroperationssuch as theq-convolutionfrom [Ans01].
GENERATORSOFSOMENON-COMMUTATIVESTOCHASTICPROCESSES 7
Proposition 3. Assume that µ has finite moments of all orders. Then, at least on the space of
P
polynomials,
f(y) f(0) yf (0)
′
(8) C [f] = αf (0)+ − − dρ(y)
µ ′ ZR y2
fora finitemeasureρ.
Proof. First note that C [1] = 0 and C [x] = µ[x] = α for some α. Since each µ is positive, it
µ µ t
follows that C is a conditionally positive function on polynomials, so it has the canonical repre-
µ
sentation C [xn] = ρ[xn 2] for n 2, where ρ is a finite measure, the canonical measure of the
µ −
≥
semigroup. Wecompute,for f(x) = xn
f(x) f(0) xf (0) ∂ f(x) f(y)
C [xn] = ρ[xn 2] = αf (0)+ρ − − ′ = αf (0)+ρ −
µ − ′ ′
(cid:20) x2 (cid:21) (cid:20)∂x(cid:12) x y (cid:21)
(cid:12)x=0 −
and thisformulaisalso validfor f(x) = 1 andx. (cid:12)(cid:12) (cid:3)
3. CLASSICAL LE´VY PROCESSES.
See Section 1.2of[Ber96](exceptfora smallmisprint)forthefollowingresults.
Theorem. Let X(t) beaLe´vyprocesscorrespondingtotheconvolutionsemigroup µ . Denote
t
{ } { }
ℓ(θ) = logE eiθX = log eiθxdµ(x)
ZR
(cid:2) (cid:3)
the cumulant generating function of the process. Then the generator of the process is the pseudo-
differentialoperatorℓ( i∂ )with densedomain
x
−
2
f L2(R,dx) : ℓ(θ) 2 fˆ(θ) dθ < .
(cid:26) ∈ ZR| | (cid:12) (cid:12) ∞(cid:27)
(cid:12) (cid:12)
In otherwords, iftheprocesshastheLe´vy-Khintchine(cid:12)repre(cid:12)sentation
1
ℓ(θ) = iαθ Vθ2 + (eiyθ 1 iyθ1 )Π(dy),
y<1
− 2 ZR 0 − − | |
\{ }
then
1
Af(x) = αf (x)+ Vf (x)+ f(x+y) f(x) 1 yf (x) Π(dy).
′ ′′ y<1 ′
2 ZR(cid:16) − − | | (cid:17)
Ifµ has meanα and finitevariance,wealso havetheKolmogorovrepresentation,
ℓ(θ) = iαθ + (eiyθ 1 iyθ)y 2dρ(y),
−
ZR − −
where ρ isthecanonicalmeasure. In thiscasethegeneratoris
f(x+y) f(x) yf (x)
′
(9) Af(x) = αf (x)+ − − dρ(y).
′
ZR y2
Iftheprocess has(say) finiteexponentialmoments,wehavemoveover
∞ 1
ℓ(θ) = c (iθ)n,
n
n!
Xn=1
8 MICHAELANSHELEVICH
where c are the cumulants [Shi96] of (the distribution of) the process. So the generator of the
n
{ }
process is
∞ 1
c ∂n.
n! n x
Xn=1
Notealsothatc = α and
1
c = xn 2dρ(x)
n −
ZR
forn 2.
≥
4. FREE LE´VY PROCESSES
4.1. Background. Let µ be a probability measure on R. Its Cauchy transform is the analytic
functionG : C+ C defined by
µ −
→
1
G (z) = dµ(x).
µ
ZR z x
−
Wewillalso denoteF (z) = 1 , sothatF : C+ C+.
µ Gµ(z) µ →
G isinvertiblein aStolzanglenearinfinity,and Voiculescu’sR-transform isdefined by
µ
1
R (z) = G 1(z) .
µ −µ − z
Afreeconvolutionµ ⊞µ oftwoprobabilitymeasures µ ,µ ischaracterized bythepropertythat
1 2 1 2
G 1 (z) = G 1(z)+R (z).
−µ1⊞µ2 −µ1 µ2
µ is ⊞-infinitely divisibleifand only it can be included as µ = µ in a free convolutionsemigroup
1
µ : t 0 , µ ⊞ µ = µ . This is the case if and only if R extends to an analytic function
t s t s+t µ
R{ : C+≥ }C+ R. In this case, wehave thefree Le´vy-Khintchinerepresentation (Theorem 5.10
µ
→ ∪
of[BV93])
z +x
R (z) = α+ dν(x).
µ
Z 1 xz
R
−
Moreover,if µhas finitevariance,wealso havethefree Kolmogorovrepresentation,
z
R (z) = α+ dρ(x).
µ
Z 1 xz
R
−
Here α is the mean of µ, ρ is a finite measure, the free canonical measure for the semigroup µ ,
t
{ }
and (α,ρ)isthefreecanonical pair. Forconvenience,throughoutmostofthepaperwewillrescale
timeso thatthevariance
(10) Var[µ = µ ] = 1
1
in whichcaseρ isaprobabilitymeasure.
We will also encounter two other convolution operations, the monotone convolution ⊲ and the
Boolean convolution , determinedby
⊎
F (z) = F (F (z))
µ1⊲µ2 µ1 µ2
and
F (z) = F (z)+F (z) z.
µ1⊎µ2 µ1 µ2 −
GENERATORSOFSOMENON-COMMUTATIVESTOCHASTICPROCESSES 9
Wewilldenote
1
dγ (x) = √4t x21 (x)dx
t 2πt − [ 2√t,2√t]
−
thesemicirculardistributions,theanalogsofthenormaldistributionsinfreeprobability. Theyform
afree convolutionsemigroupwith thefree canonical pair (0,δ ).
0
Remark 3. If ν is a probability measure, there exists a probability measure µ = Φ [ν] with mean
t
zero andvariance tsuch that
F (z) = z tG (z).
Φt[ν] − ν
In particular, forthemap Φ = Φ , see[BN09]andProposition2.2of[Maa92], and
1
Φ [ν] = Φ[ν] t.
t ⊎
Conversely,ifµisaprobabilitymeasurewithmean αandvarianceβ > 0,thereexistsaprobability
measure ν = [µ]such that
J
(11) F (z) = z α βG (z).
µ [µ]
− − J
Note that Φ = Id, while Φ [ [µ]] = µ if µ has mean zero and variance t. If ν has finite
t t
J ◦ J
momentsofall orders, itsCauchytransform hasa continuedfraction expansion
1
G (z) = .
µ
β
1
z α
0
− − β
2
z α
1
− − β
3
z α
2
− − z ...
−
Here β = 1, α is the mean of µ, β is the variance of µ, and in general α ,β are its Jacobi
0 0 1 n n
{ }
parameters. Then for ν = Φ [µ],
t
1
G (z) = ,
ν
t
z
− β
1
z α
0
− − β
2
z α
1
− − β
3
z α
2
− − z ...
−
while is the inverse map, namely coefficient stripping[DKS10]. Note that there are also related
J
maps which involve finite rather than only probability measures, but because of the normaliza-
tion(10), wewillnotneed to considerthem.
4.2. Transitionoperators. Thefollowingis areformulationofTheorem 3.1of[Bia98].
Theorem. Let X andY befreelyindependent. Then thetransitionoperator defined via
K
E[f(X +Y) X] = ( f)(X)
| K
is amap onC (R) (which extendsto a mapon L (R, dx))suchthatforanyz C R
0 ∞
∈ \
1 1 1
= = .
K(cid:20)z x(cid:21) F(z) x F (z) x
ν
− − −
10 MICHAELANSHELEVICH
Here F(z) = F (z) forsomeprobabilitymeasureν, andF isuniquelydeterminedby
ν
G (z) = G (F(z)).
X+Y X
Proposition 4. For processes with freely independent increments, the transition operators have
K
theform
[f](x) = f(y)d(δ ⊲ν)(y).
x
K ZR
ForthefreeLe´vyprocessesofthesecondkind, for we haveν = µ . ForthefreeLe´vy processes
t t t
K
of thefirstkind, for , ν is determinedby
s,t s,t
K
G (z) = G (F (z)).
µt µs νs,t
In otherwords, ν = ν = µ µ , thesubordinationdistribution,see[Len07, Nic09].
s,t t s s
− ⊢
Proof. AccordingtoTheorem 3.1of[Bia98],
1 1 1 1 1
= = = = G (z) = d(δ ⊲ν)(y),
K(cid:20)z x(cid:21) F(z) x F (z) x F (z) δx⊲ν Z z y x
− − ν − δx⊲ν R −
and, still according to [Bia98], this property entirely determines . For FL2, F = F and
s,t t s
K −
F = F F , so fors = 0,
µt µs ◦ t−s
F = F F = F = F .
µt δ0 ◦ t t νt
ForFL1, ν = µ µ by definition. (cid:3)
s,t t s s
− ⊢
Remark 4. Notethat
(x,dy) = (δ ⊲ν)(y) = (δ ν)(y) = (ν δ )(y).
x x x
K ⊎ ⊎
In fact, measures δ ⊲ ν are well-known in classical spectral theory. Indeed, if X is an operator
x
with cyclic vector ξ and corresponding distribution ν, then δ ⊲ ν is the distribution with respect
x
to ξ of the rank-one perturbation X x ,ξ ξ. Finally, note that for the classical processes and
− h· i
convolution,wecan also write
[f](x) = f(y)dµ(y x) = f(y)d(δ µ)(y)
x
K ZR − ZR ∗
Proposition5. is acontractionon eachLp(R,dx) for1 p .
K ≤ ≤ ∞
Proof. Forf L (R,dx),
∞
∈
[f] = esssup f(y) d(δ ⊲ν)(y) esssup f = f ,
x
kK k∞ x R ZR| | ≤ x R k k∞ k k∞
∈ ∈
so is a contraction on L (R,dx). On the other hand, Alexandrov’s averaging theorem (Theo-
∞
remK11.8from [Sim05])states thatfor f L1(R, dx)
∈
[f](x)dx = f(y)d(δ ⊲ν)(y) dx = f(x)dx,
x
Z K Z (cid:18)Z (cid:19) Z
R R R R
so that f = f for f 0, and is a contraction on L1(R, dx). For 1 < p < , theresult
nowfolkloKwskb1y Rkieskz1-Thorin≥interpolaKtion,seeSectionIX.4 from [RS75]. ∞ (cid:3)