Unitary processes with independent increments and representations of Hilbert tensor algebras

0.33 MB·English

by  Lingaraj Sahu

Unitary processes with independent increments and
representations of Hilbert tensor algebras
Lingaraj Sahu 1, Michael Schu¨rmann 2
and
Kalyan B. Sinha 3 4
8
0
0 Abstract
2

n The aim of this article is to characterize unitary increment process by
a
a quantum stochastic integral representation on symmetric Fock space.
J

4 Under certain assumptions we have proved its unitary equivalence to a

Hudson-Parthasarathy flow.
]
A
F
.
h 1 Introduction
t
a
m
In the framework of the theory of quantum stochastic calculus developed by
[
pioneering work of Hudson andParthasarathy [6], quantum stochastic differential

2
v equations (qsde) of the form
6
9
8 dVt = VtLµνΛνµ(dt), V0 = 1h⊗Γ, (1.1)
1
µ,ν≥0
. X
2
1 (where the coefficients Lµ : µ, ν 0 are operators in the initial Hilbert
7 ν ≥
0 space h and Λνµ are fundamental processes in the symmetric Fock space Γ =
v: Γ (L2(R ,k)) with respect to a fixed orthonormal basis (in short ‘ONB’)
sym +
i
X E : j 1 of the noise Hilbert space k ) have been formulated and con-
j
{ ≥ }
r ditions for existence and uniqueness of a solution V are studied by Hudson
a t
{ }
and Parthasarathy and many other authors. In particular when the coefficients
Lµ : µ, ν 0 are bounded operators satisfying some conditions it is observed
ν ≥
that the solution V : t 0 is a unitary process.
t
{ ≥ }
In [4], using integral representation of regular quantum martingales in sym-
metric Fock space [15], the authorsshow that any covariant Fockadapted unitary
evolution (with norm-continuous expectation semigroup) V : 0 s t <
s,t
{ ≤ ≤ ∞}
1 Stat-Math Unit, Indian Statistical Institute, Bangalore Centre , 8th Mile, Mysore Road,
Bangalore-59,India. E-mail : [email protected]
2 Institutfu¨rMathematikundInformatik,F.-L.-Jahn-Strasse15a,D-17487Greifswald,Ger-
many. E-mail: [email protected]
3 JawaharlalNehru Centre for Advanced Scientific Research, Jakkur, Bangalore-64,India
4 Department of Mathematics, Indian Institute of Science, Bangalore-12,India.
E-mail: kbs [email protected]
1
satisfies a quantum stochastic differential equation (1.1) with constant coeffi-
cients Lµ (h). For situations where the expectation semigroup is not norm
ν ∈ B
continuous, the characterization problem is discussed in [5, 1]. In [9, 10], by ex-
tended semigroup methods, Lindsay and Wills have studied such problems for
Fock adapted contractive operator cocycles and completely positive cocyles.
In this article we are interested in the characterization of unitary evolutions
with stationary andindependent increments onh , where hand are separa-
⊗H H
ble Hilbert spaces. In [16, 17], by a co-algebraic treatment, the author has proved
that any weakly continuous unitary stationary independent increment process on
h ,h finite dimensional, is unitarily equivalent to a Hudson-Parthasarathy
⊗ H
flow with constant operator coefficients; see also [7, 8]. In this present paper we
treat the case of a unitary stationary independent increment process on h ,h
⊗H
not necessarily finite dimensional, with norm-continuous expectation semigroup.
By a GNS type construction we are able to get the noise space kand the bounded
operator coefficients Lµ such that the Hudson-Parthasarathy flow equation (1.1)
ν
admitsaunique unitarysolutionandisunitarilyequivalent totheunitaryprocess
we started with.
2 Notation and Preliminaries
We assume that all the Hilbert spaces appearing in this article are complex sep-
arable with inner product anti-linear in the first variable. For any Hilbert spaces
, ( , ) and ( ) denote the Banach space of bounded linear operators
1
H K B H K B H
from to and trace class operators on respectively. For a linear map (not
H K H
necessarily bounded ) T we write its domain as (T). We denote the trace on
D
( ) by Tr or simply Tr. The von Neumann algebra of bounded linear oper-
1 H
B H
ators on is denoted by B( ). The Banach space ( , ) ρ ( , ) :
1
H H B H K ≡ { ∈ B H K
ρ := √ρ∗ρ ( ) with norm (Ref. Page no. 47 in [2])
1
| | ∈ B H }
ρ = ρ = sup φ ,ρψ : φ , ψ areONB of and resp.
k k1 k| |kB1(H) { |h k ki| { k} { k} K H }
k≥1
X
isthepredualof ( , ).Foranelementx ( , ), ( , ) ρ Tr (xρ)
1 H
B K H ∈ B K H B H K ∋ 7→
defines an element of the dual Banach space ( , )∗. For a linear map T
1
B H K
on the Banach space ( , ) the adjoint T∗ on the dual ( , ) is given by
1
B H K B K H
Tr (T∗(x)ρ) := Tr (xT(ρ)), x ( , ), ρ ( , ).
H H 1
∀ ∈ B K H ∈ B H K
For any ξ ,h the map
∈ H⊗K ∈ H
k ξ,h k
K ∋ 7→ h ⊗ i
2
defines a bounded linear functional on and thus by Riesz’s theorem there exists
K
a unique vector ξ,h in such that
hh ii K
ξ,h , k = ξ,h k , k . (2.1)
h hh ii i h ⊗ i ∀ ∈ K
In other words ξ,h = F∗ξ where F ( , ) is given by F k = h k.
hh ii h h ∈ B K H⊗K h ⊗
Let h and be two Hilbert spaces with some orthonormal bases e : j 1
j
H { ≥ }
and ζ : j 1 respectively. For A (h ) and u,v h we define a linear
j
{ ≥ } ∈ B ⊗H ∈
operator A(u,v) ( ) by
∈ B H
ξ ,A(u,v)ξ = u ξ ,A v ξ , ξ ,ξ
1 2 1 2 1 2
h i h ⊗ ⊗ i ∀ ∈ H
and read off the following properties:
Lemma 2.1. Let A,B (h ) then for any u,v,u and v ,i = 1,2 in h
i i
∈ B ⊗H
(i) A(u,v) ( ) with A(u,v) A u v and A(u,v)∗ = A∗(v,u).
∈ B H k k ≤ k k k k k k
(ii) h h A( , ) is 1 1, i.e. if A(u,v) = B(u,v), u,v h then A = B.
× 7→ · · − ∀ ∈
(iii) A(u ,v )B(u ,v ) = [A( v >< u 1 )B](u ,v )
1 1 2 2 1 2 H 1 2
| |⊗
(iv) AB(u,v) = A(u,e )B(e ,v) (strongly)
j≥1 j j
(v) 0 A(u,v)∗AP(u,v) u 2A∗A(v,v)
≤ ≤ k k
(vi) A(u,v)ξ ,B(p,w)ξ = p ζ ,[B( w >< v ξ >< ξ )A∗u ζ
h 1 2i j≥1h ⊗ j | |⊗| 2 1| ⊗ ji
= v ξ , [A∗( u >< p 1 )Bw ξ .
hP⊗ 1 | |⊗ H ⊗ 2i
Proof. We are omitting the proof of (i),(ii).
(iii) For any ξ,ζ we have
∈ H
ξ,A(u ,v )B(u ,v )ζ = u ξ,Av B(u ,v )ζ = A∗u ξ,v B(u ,v )ζ
1 1 2 2 1 1 2 2 1 1 2 2
h i h ⊗ ⊗ i h ⊗ ⊗ i
= A∗u ξ,v ζ ζ ,B(u ,v )ζ
1 1 n n 2 2
h ⊗ ⊗ ih i
n≥1
X
= A∗u ξ,v ζ u ζ ,Bv ζ
1 1 n 2 n 2
h ⊗ ⊗ ih ⊗ ⊗ i
n≥1
X
= A∗u ξ,( v >< u ζ >< ζ )Bv ζ
1 1 2 n n 2
h ⊗ | |⊗| | ⊗ i
n≥1
X
= u ξ,A( v >< u 1 )Bv ζ .
1 1 2 H 2
h ⊗ | |⊗ ⊗ i
Thus it follows that
A(u ,v )B(u ,v ) = [A( v >< u 1 )B](u ,v ).
1 1 2 2 1 2 H 1 2
| |⊗
3
(iv) By part(iii)
N
A(e ,u)ξ 2
j
k k
j=1
X
N
= ξ,A∗(u,e )A(e ,u)ξ
j j
h i
j=1
X
= ξ,[A∗(P 1 )A](u,u)ξ ,
N H
h ⊗ i
where P is the finite rank projection N e >< e on h. Since [A∗(P
N j=1| j j| { N ⊗
1 )A](u,u) is an increasing sequence of positive operators and 0 P 1
H } P ≤ N ⊗ H
converges strongly to 1h⊗H as N tends to , [A∗(PN 1H)A](u,u) converges
∞ ⊗
strongly to [A∗A](u,u) as N tends to . Thus
∞
N
lim A(e ,u)ξ 2 = ξ,[A∗A](u,u)ξ
j
N→∞ k k h i
j=1
X
and
N
A(e ,u)ξ 2 A u ξ 2 A 2 u 2 ξ 2, N 1.
j
k k ≤ k ⊗ k ≤ k k k k k k ∀ ≥
j=1
X
Now let us consider the following, for ξ,ζ
∈ H
N N
ξ, A(u,e )B(e ,v)ζ 2 = A∗(e ,u)ξ,B(e ,v)ζ 2
j j j j
|h i| | h i|
j=1 j=1
X X
N N
A∗(e ,u)ξ 2 B(e ,v)ζ 2
j j
≤ k k k k
j=1 j=1
X X
A 2 u 2 ξ 2 B 2 v 2 ζ 2.
≤ k k k k k k k k k k k k
So
N
ξ, A(u,e )B(e ,v)ζ A B u v ξ ζ
j j
|h i| ≤ k kk kk kk kk kk k
j=1
X
and strong convergence of A(u,e )B(e ,v) follows.
j≥1 j j
P
(v) We have
ξ,A(u,v)∗A(u,v)ξ = ξ,A∗(v,u)ζ ζ ,A(u,v)ξ
j j
h i h ih i
j≥1
X
= v ξ,A∗u ζ u ζ ,Av ξ
j j
h ⊗ ⊗ ih ⊗ ⊗ i
j≥1
X
= v ξ,A∗ u >< u ζ >< ζ Av ξ .
j j
h ⊗ {| |⊗ | |} ⊗ i
j≥1
X
4
Since ζ >< ζ converges strongly to the identity operator
j≥1| j j|
P ξ,A(u,v)∗A(u,v)ξ u 2 v ξ,A∗Av ξ
h i ≤ k k h ⊗ ⊗ i
and this proves the result.
(vi)We have
A(u,v)ξ ,B(p,w)ξ
1 2
h i
= A(u,v)ξ ,ζ ζ ,B(p,w)ξ
1 j j 2
h ih i
j≥1
X
= Av ξ ,u ζ p ζ ,Bw ξ
1 j j 2
h ⊗ ⊗ ih ⊗ ⊗ i
j≥1
X
= B∗p ζ ,w ξ v ξ ,A∗u ζ
j 2 1 j
h ⊗ ⊗ ih ⊗ ⊗ i
j≥1
X
= p ζ ,B( w >< v ξ >< ξ )A∗u ζ .
j 2 1 j
h ⊗ | |⊗| | ⊗ i
j≥1
X
This proves the first part of (vi), the other part follows from
p ζ ,B( w >< v ξ >< ξ )A∗u ζ
j 2 1 j
h ⊗ | |⊗| | ⊗ i
j≥1
X
= Trh⊗H[( u >< p 1H)B( w >< v ξ2 >< ξ1 )A∗]
| |⊗ | |⊗| |
= Trh⊗H[( w >< v ξ2 >< ξ1 )A∗( u >< p 1H)B]
| |⊗| | | |⊗
= v ξ , [A∗( u >< p 1 )Bw ξ
1 H 2
h ⊗ | |⊗ ⊗ i
2.1 Symmetric Fock Space and Quantum Stochastic Cal-
culus
Let us briefly recall the fundamental integrator processes of quantum stochastic
calculus and the flow equation, introduced by Hudson and Parthasarathy [6]. For
a Hilbert space k let us consider the symmetric Fock space Γ = Γ(L2(R ,k)).
+
The exponential vector in the Fock space, associated with a vector f L2(R ,k)
+
∈
is given by
1
e(f) = f(n),
√n!
n≥0
M
where f(n) = f f f for n > 0 and by convention f(0) = 1. The expo-
⊗ ⊗···⊗
n−copies
nential vector e(0) is called the vacuum vector.
| {z }
5
LetusconsidertheHudson-Parthasarathy(HP)flowequationonh Γ(L2(R ,k)):
+
⊗
t
Vs,t = 1h⊗Γ + Vs,τLµνΛνµ(dτ). (2.2)
µ,ν≥0Zs
X
Here the coefficients Lµ : µ, ν 0 are operators in h and Λν are fundamental
ν ≥ µ
processes with respect to a fixed orthonormal basis E : j 1 of k :
j
{ ≥ }
t 1h⊗Γ for(µ,ν) = (0,0)
a(1 E ) for(µ,ν) = (j,0)
Λµ(t) =  [0,t] ⊗ j (2.3)
ν  a†(1[0,t] ⊗Ek) for(µ,ν) = (0,k)
Λ(1 E >< E ) for(µ,ν) = (j,k).
[0,t] k j
⊗| |


Theorem 2.2. [12, 14, 3] Let H (h) be self-adjoint, L , Wj : j,k 1 be a
∈ B { k k ≥ }
family of bounded linear operators in h such that W = Wj E >< E
j,k≥1 k ⊗| j k|
is an isometry (respectively co-isometry) operator in h k and for some constant
⊗P
c 0,
≥
L u 2 c u 2, u h.
k
k k ≤ k k ∀ ∈
k≥1
X
Let the coefficients Lµ be as follows,
ν
iH 1 L∗L for (µ,ν) = (0,0)
− 2 k≥1 k k
L for (µ,ν) = (j,0)
Lµ =  j P (2.4)
ν  L∗Wj for (µ,ν) = (0,k)
 − j≥1 j k
j j
W δ for (µ,ν) = (j,k).
kP− k



Then there exists a unique isometry ( respectively co-isometry) operator valued
process V satisfying (2.2) .
s,t
3 Hilbert tensor algebra
For a product vector u = u u u h⊗n we shall denote the product
1 2 n
⊗ ⊗···⊗ ∈
vector u u u by u. For the null vector in h⊗n we shall write 0. If
n n−1 1
⊗ ⊗···⊗
f ∞ is an ONB for h, then w←−e have a product ONB f = f f : j =
{ j}j=1 { j j1 ⊗···⊗ jn
(j ,j , ,j ),j 1 for the Hilbert space h⊗n.
1 2 n k
··· ≥ }
Consider Z = 0,1 , the finite field with addition modulo 2. For n 1,
2
{ } ≥
let Zn denotes the n-fold direct sum of Z and we write 0 = (0,0, ,0) and
2 2 ···
1 = (1,1, ,1). For ǫ = (ǫ ,ǫ , ,ǫ ),ǫ′ = (ǫ′,ǫ′, ,ǫ′ ) we put
··· 1 2 ··· n 1 2 ··· m
ǫ ǫ′ = (ǫ , ,ǫ ,ǫ′, ,ǫ′ ) Zn+m and we define ǫ∗ = 1+(ǫ ,ǫ , ,ǫ ) Zn.
⊕ 1 ··· n 1 ··· m ∈ 2 n n−1 ··· 1 ∈ 2
6
Let A (h ),ǫ Z = 0,1 . We define operators A(ǫ) (h ) by
2
∈ B ⊗H ∈ { } ∈ B ⊗H
A(ǫ) := A if ǫ = 0 and A(ǫ) := A∗ if ǫ = 1. For 1 k n, we define a unitary
≤ ≤
exchange map P : h⊗n h⊗n by putting
k,n
⊗H → ⊗H
P (u ξ) := u u u u (u ξ)
k,n 1 k−1 k+1 n k
⊗ ⊗···⊗ ⊗ ···⊗ ⊗ ⊗
on product vectors. Let ǫ = (ǫ ,ǫ , ,ǫ ) Zn. Consider the ampliation of the
1 2 ··· n ∈ 2
operator A(ǫk) in (h⊗n ) given by
B ⊗H
A(n,ǫk) := Pk∗,n(1h⊗n−1 ⊗A(ǫk))Pk,n.
Now we define the operator A(ǫ) := n A(n,ǫk) := A(1,ǫ1) A(n,ǫn) in (h⊗n
k=1 ··· B ⊗
n
). Please note that as here, through out this article, the product symbol
H Q k=1
standsforproductwithorder1ton.Form n,weshallwriteǫ(m) = (ǫ ,ǫ , ,ǫ )
≤ 1 2 ·Q·· m
and consider the operator A(ǫ(m)) = m A(m,ǫi) in (h⊗m ) We have the fol-
i=1 B ⊗H
lowing preliminary observation.
Q
Lemma 3.1. (i) For product vectors u,v h⊗n
∈
m m n
A(n,ǫi)(u,v) = Aǫi(u ,v ) u ,v ( ).
i i i i
h i ∈ B H
i=1 i=1 i=m+1
Y Y Y
(ii) For ξ,ζ
∈ H
m
A(ǫi)(ξ,ζ) = A(ǫ(m))(ξ,ζ) 1h⊗n−m B(h⊗n).
⊗ ∈
i=1
Y
(iii) If A is an isometry (respectively unitary) then A(n,ǫk) and A(ǫ) are isometries
(respectively unitaries).
The proof is obvious and is omitted.
We note that part (i) of this Lemma in particular gives
n
A(ǫ)(u,v) = A(ǫi)(u ,v ) (3.1)
i i
i=1
Y
Let M := (u,v,ǫ) : u = n u , v = n v h⊗n,ǫ = (ǫ ,ǫ , ,ǫ )
0 { ⊗i=1 i ⊗i=1 i ∈ 1 2 ··· n ∈
Zn, n 1 . In M , we introduce an equivalence relation ‘ ’ : (u,v,ǫ)
2 ≥ } 0 ∼ ∼
(p,w,ǫ′) if ǫ = ǫ′ and u >< v = p >< w (h⊗n). Expanding the vectors
| | | | ∈ B
in term of the ONB e = e e : j = (j ,j , ,j ),j 1 , from
{ j j1 ⊗ ··· ⊗ jn 1 2 ··· n k ≥ }
7
u >< v = p >< w we get u v = p w for each multi-indices j,k. Thus in
j k j k
| | | |
particular when (u,v,0) (p,w,0), for any ξ ,ξ we have
1 2
∼ ∈ H
ξ ,A(u,v)ξ
1 2
h i
= u v e ξ ,Ae ξ
j k j 1 k 2
h ⊗ ⊗ i
j,k≥1
X
= p w e ξ ,Ae ξ
j k j 1 k 2
h ⊗ ⊗ i
j,k≥1
X
= ξ ,A(p,w)ξ .
1 2
h i
In fact A(u,v) = A(p,w) iff (u,v,0) (p,w,0) and more generally A(ǫ)(u,v) =
∼
A(ǫ′)(p,w)iff(u,v,ǫ) (p,w,ǫ′).Itiseasytoseethat(0,v,ǫ) (u,0,ǫ) (0,0,ǫ)
∼ ∼ ∼
and we call this class the 0 of the quotient set M .
0
Let us define multiplication and involution on M / by setting
0
∼
Vector multiplication: (u,v,ǫ).(p,w,ǫ′) = (u p,v w,ǫ ǫ′) and
⊗ ⊗ ⊕
Involution: (u,v,ǫ)∗ = (v, u,ǫ∗).
←− ←−
Since (u p) = p p u u = (p u) and (ǫ ǫ′)∗ = (ǫ′)∗ ǫ∗
m 1 n 1
⊗ ⊗···⊗ ⊗ ⊗···⊗ ⊗ ⊕ ⊕
←−−−− ←− ←−
[(u,v,ǫ).(p,w,ǫ′)]∗ = (u p,v w,ǫ ǫ′)∗
⊗ ⊗ ⊕
= (v w,u p,(ǫ ǫ′)∗)
⊗ ⊗ ⊕
= (←w−−−v←,−p−− u,(ǫ′)∗ ǫ∗)
⊗ ⊗ ⊕
= (←p−,w,ǫ←−′)∗.←(−u,v,←−ǫ)∗.
It is clear that ǫ = ǫ′ = ǫ∗ = (ǫ′)∗ and u >< v = p >< w implies
⇒ | | | |
v >< u = w >< p . Thus (u,v,ǫ) (p,w,ǫ′) implies (u,v,ǫ)∗ (p,w,ǫ′)∗.
| | | | ∼ ∼
M←−oreove←r−, (u,v←,−ǫ) (←u−′,v′,ǫ′) and (p,w,α) (p′,w′,α′) implies ǫ α = ǫ′ α′
∼ ∼ ⊕ ⊕
and u p >< v w = u >< v p >< w = u′ >< v′ p′ >< w′ =
| ⊗ ⊗ | | | ⊗ | | | | ⊗ | |
u′ p′ >< v′ w′ . So that the involution and multiplication respect .
| ⊗ ⊗ | ∼
Let M be the complex vector space spanned by M / . The elements of
0
∼
M are formal finite linear combinations of elements of M / . With the above
0
∼
multiplication and involution M is a -algebra.
∗
4 Unitary processes with stationary and inde-
pendent increment
Let U : 0 s t < be a family of unitary operators in (h ) and
s,t
{ ≤ ≤ ∞} B ⊗ H
Ω be a fixed unit vector in . We shall also set U := U for simplicity. As we
t 0,t
H
8
(ǫ)
discussed in the previous section, let us consider the family of operators U in
s,t
{ }
(h ) for ǫ Z given by U(ǫ) = U if ǫ = 0,U(ǫ) = U∗ if ǫ = 1. Furthermore
B ⊗H ∈ 2 s,t s,t s,t s,t
for n 1,ǫ Zn fixed, 1 k n, we consider the families of operators U(ǫk)
≥ ∈ 2 ≤ ≤ { s,t }
and U(ǫ) in (h⊗n ). By Lemma 3.1 we observe that
s,t
{ } B ⊗H
n
U(ǫ)(u,v) = U(ǫi)(u ,v ).
s,t s,t i i
i=1
Y
For ǫ = 0 Zn and 1 k n, we shall write U(n,k) for the unitary operator
∈ 2 ≤ ≤ s,t
U(n,ǫk) andU(n) fortheunitaryU(0) onh⊗n .Forn 1,s = (s ,s , ,s ),t =
s,t s,t s,t 1 2 n
⊗H ≥ ···
(k) (k) (k)
(t ,t , ,t ): 0 s t s ... s t < ,ǫ = (α ,α , ,α )
1 2 ··· n ≤ 1 ≤ 1 ≤ 2 ≤ ≤ n ≤ n ∞ k 1 2 ··· mk ∈
Zmk : 1 k n,m = m +m + +m ǫ = ǫ ǫ ǫ Zm, we define
2 ≤ ≤ 1 2 ··· n 1 ⊕ 2⊕···⊕ n ∈ 2
U(ǫ) (h⊗m ) by setting
s,t ∈ B ⊗H
n
(ǫ) (ǫ )
U := U k . (4.1)
s,t sk,tk
k=1
Y
Here U(ǫk) is looked upon as an operator in (h⊗m ) by ampliation and
sk,tk B ⊗ H
appropriate tensor flip. So for u = n u ,v = n v h⊗m we have
⊗k=1 k ⊗k=1 k ∈
n
(ǫ) (ǫ )
U (u,v) = U k (u ,v ).
s,t sk,tk k k
k=1
Y
(ǫ)
When there can be no confusion, for ǫ = 0 we write U for U . For a,b 0,s =
s,t s,t ≥
(s ,s , ,s ),t = (t ,t , ,t ) we write a s,t b if a s t s
1 2 n 1 2 n 1 1 2
··· ··· ≤ ≤ ≤ ≤ ≤ ≤
... s t b.
n n
≤ ≤ ≤
Let us assume the following properties on the unitary family U for further dis-
s,t
cussion to prove unitary equivalence of U with an HP flow.
s,t
Assumption A
A1 (Evolution) For any 0 r s t < , U U = U .
r,s s,t r,t
≤ ≤ ≤ ∞
A2 (Independence of increments) For any 0 s t < : i = 1,2 such
i i
≤ ≤ ∞
that [s ,t ) [s ,t ) = ∅
1 1 2 2
∩
(a) U (u ,v ) commutes with U (u ,v ) and U∗ (u ,v ) for every
s1,t1 1 1 s2,t2 2 2 s2,t2 2 2
u ,v h.
i i
∈
(b) For s a,b t , s q,r t and u,v h⊗n, p,w h⊗m,ǫ
1 1 2 2
≤ ≤ ≤ ≤ ∈ ∈ ∈
Zn,ǫ′ Zm
2 ∈ 2
(ǫ) (ǫ′) (ǫ) (ǫ′)
hΩ,Ua,b(u,v)Uq,r(p,w)Ωi = hΩ,Ua,b(u,v)ΩihΩ,Uq,r(p,w)Ωi.
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A3 (Stationarity) For any 0 s t < and u,v h⊗n,ǫ Zn
≤ ≤ ∞ ∈ ∈ 2
(ǫ) (ǫ)
Ω,U (u,v)Ω = Ω,U (u,v)Ω .
h s,t i h t−s i
Assumption B (Uniform continuity)
lim sup Ω,(U 1)(u,v)Ω : u , v = 1 = 0.
t→0 t
{|h − i| k k k k }
Assumption C (Gaussian Condition) For any u ,v h,ǫ Z : i = 1,2,3
i i i 2
∈ ∈
1
lim Ω, (U(ǫ1) 1)(u ,v )(U(ǫ2) 1)(u ,v )(U(ǫ3) 1)(u ,v )Ω = 0. (4.2)
t→0 th t − 1 1 t − 2 2 t − 3 3 i
Assumption D (Minimality)
Theset = U (u,v)Ω := U (u ,v ) U (u ,v )Ω : s = (s ,s , ,s ),
S { s,t s1,t1 1 1 ··· sn,tn n n 1 2 ··· n
t = (t ,t , ,t ) : 0 s t s ... s t < ,n 1,u =
1 2 n 1 1 2 n n
··· ≤ ≤ ≤ ≤ ≤ ≤ ∞ ≥
n u ,v = n v h⊗n is total in .
⊗i=1 i ⊗i=1 i ∈ } H
Remark 4.1. (a) The hypothesis A, B and C hold in many situations, for ex-
ample for unitary solutions of the Hudson-Parthasarathy flow (2.2) with bounded
operator coefficients and having no Poisson terms.
(b) The assumption D is not really a restriction, one can as well work with re-
placing by span closure of S. Taking 0 s t s ... s t <
1 1 2 n n
H ≤ ≤ ≤ ≤ ≤ ≤ ∞
in the definition of S is enough for totality of the set S because : for
⊆ H
0 r s t , we have U (p,w)) = U (p,e )U (e ,w). So if there are
≤ ≤ ≤ ≤ ∞ r,t j r,s j s,t j
overlapping intervals [s ,t ) [s ,t ) = ∅ then the vector ξ = U (u,v)Ω in
k k ∩ k+1 k+1 P6 s,t
can be obtained as a vector in the closure of the linear span of S.
H
For any n 1 we have the following useful observations.
≥
Lemma 4.2. (i) For any 0 r s t < ,
≤ ≤ ≤ ∞
U(n,k) = U(n,k)U(n,k). (4.3)
r,t r,s s,t
(ii) For any 1 k ,k , ,k n : k = k for i = j and 0 s t < :
1 2 m i j i i
≤ ··· ≤ 6 6 ≤ ≤ ∞
i = 1,2, ,n
···
m m
U(n,ǫki)(u,v) = U(n,ǫki)(u ,v ) u ,v (4.4)
si,ti si,ti ki ki h j ji
Yi=1 Yi=1 jY6=ki
for every u = n u ,v = n v h⊗n and ǫ Zn.
⊗i=1 i ⊗i=1 i ∈ ∈ 2
(iii)
U(n) = U(n)U(n). (4.5)
r,t r,s s,t
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