An equivalent representation of the Jacobi field of a Lévy process

0.13 MB·English

by  Eugene Lytvynov

An equivalent representation
of the Jacobi field of a L´evy process
5
0
E. Lytvynov
0
2
Department of Mathematics

n
University of Wales Swansea
a
J Singleton Park

5 Swansea SA2 8PP
2
U.K.

] E-mail: [email protected]
R
P
Dedicated to Professor Yuri Berezansky
.
h
on his 80th birhday
t
a
m
Abstract
[

In [8], the Jacobi field of a L´evy process was derived. This field consists
1
v of commuting self-adjoint operators acting in an extended (interacting) Fock
0 space. However, these operators have a quite complicated structure. In this
5
note, using ideas from [1, 17], we obtain a unitary equivalent representation
4
1 of the Jacobi field of a L´evy process. In this representation, the operators act
0 in a usual symmetric Fock space and have a much simpler structure.
5
0
AMS Mathematics Subject Classification: 60G20, 60G51, 60H40, 47B36
/
h
t
a
m 1 L´evy process and its Jacobi field
:
v
i ThenotionofaJacobifieldintheFockspace first appearedintheworks byBerezan-
X
sky and Koshmanenko [6, 7], devoted to the axiomatic quantum field theory, and
r
a
then was further developed by Bru¨ning (see e.g. [10]). These works, however, did
not contain any relations with probability measures. A detailed study of a general
commutative Jacobi field in the Fock space and a corresponding spectral measure
was carried out in a serious of works by Berezansky, see e.g. [3, 5] and the references
therein.
In [8] (see also [16, 18]), the Jacobi field of a L´evy process on a general manifold
X was studied. Let us shortly recall these results.
Let X be a complete, connected, oriented C∞ (non-compact) Riemannian man-
ifold and let B(X) be the Borel σ-algebra on X. Let σ be a Radon measure on
(X,B(X)) that is non-atomic and non-degenerate (i.e., σ(O) > 0 for any open set
O ⊂ X). As a typical example of measure σ, one can take the volume measure on
X.
1
We denote by D the space C∞(X) of all infinitely differentiable, real-valued
0
functions on X with compact support. It is known that D can be endowed with a
topology of a nuclear space. Thus, we can consider the standard nuclear triple
D ⊂ L2(X,σ) ⊂ D′,
where D′ is the dual space of D with respect to the zero space L2(X,σ). (Here and
below, all the linear spaces we deal with are real.) The dual pairing between ω ∈ D′
and ϕ ∈ D will be denoted by hω,ϕi. We denote the cylinder σ-algebra on D′ by
C(D′).
Let ν be a measure on (R,B(R)) whose support contains an infinite number of
points and assume ν({0}) = 0. Let
ν˜(ds) := s2ν(ds).
We further assume that ν˜ is a finite measure on (R,B(R)), and moreover, there
exists an ε > 0 such that
exp ε|s| ν˜(ds) < ∞. (1)
ZR
(cid:0) (cid:1)
We now define a centered L´evy process on X (without Gaussian part) as a
generalized process on D′ whose law is the probability measure µ on (D′,C(D′))
given by its Fourier transform
eihω,ϕiµ(dω) = exp (eisϕ(x) −1−isϕ(x))ν(ds)σ(dx) , ϕ ∈ D. (2)
ZD′ (cid:20)ZR×X (cid:21)
Thus, ν is the L´evy measure of the L´evy process µ. Without loss of generality,
we can suppose that ν˜ is a probability measure on R. (Indeed, if this is not the case,
define ν′ := c−1ν and σ′ := cσ, where c := ν˜(R).)
It follows from (1) that the measure ν˜ has all moments finite, and furthermore,
the set of all polynomials is dense in L2(R,ν˜). Therefore, by virtue of [2], there
exists a unique (infinite) Jacobi matrix
a b 0 0 0 ...
0 1
b a b 0 0 ...
1 1 2
J = 0 b2 a2 b3 0 ..., an ∈ R, bn > 0,
0 0 b a b ...
 3 3 4 
 ... ... ... ... ... ...
 
whose spectral measure is ν˜.
Next, we denote by P(D′) the set of continuous polynomials onD′, i.e., functions
on D′ of the form F(ω) = n hω⊗i,f i, ω⊗0:=1, f ∈ D⊗ˆi, i = 0,...,n, n ∈ Z .
i=0 i i +
P
2
Here, ⊗ˆ stands for symmetric tensor product. The greatest number i for which
f(i) 6= 0 is called the power of a polynomial. We denote by P (D′) the set of
n
continuous polynomials of power ≤ n.
By (1), (2), and [19, Sect. 11], P(D′) is a dense subset of L2(D′,µ). Let P∼(D′)
n
denote the closure of P (D′) in L2(D′,µ), let P (D′), n ∈ N, denote the orthogonal
n n
difference P∼(D′)⊖P∼ (D′), and let P (D′):=P∼(D′). Then, we evidently have:
n n−1 0 0
∞
L2(D′,µ) = P (D′). (3)
n
Mn=0
The set of all projections :h·⊗n,f i: of continuous monomials h·⊗n,f i, f ∈ D⊗ˆn,
n n n
onto P (D′) is dense in P (D′). For each n ∈ N, we define a Hilbert space F as
n n n
the closure of the set D⊗ˆn in the norm generated by the scalar product
1
(f ,g ) := :hω⊗n,f i::hω⊗n,g i:µ(dω), f ,g ∈ D⊗ˆn. (4)
n n Fn n! Z n n n n
D′
Denote
∞
F:= F n!, (5)
n
Mn=0
where F :=R. By (3)–(5), we get the unitary operator
0
U : F → L2(D′,µ)
that is defined through Uf :=:h·⊗n,f i:, f ∈ D⊗ˆn, n ∈ Z , and then extended by
n n n +
linearity and continuity to the whole space F.
An explicit formula for the scalar product (·,·) looks as follows. We denote by
Fn
Z∞ the set of all sequences α of the form
+,0
α = (α ,α ,...,α ,0,0,...), α ∈ Z , n ∈ N.
1 2 n i +
Let |α|:= ∞ α . For each α ∈ Z∞ , 1α + 2α + ··· = n, n ∈ N, and for any
i=1 i +,0 1 2
function fP: Xn → R we define a function D f : X|α| → R by setting
n α n
(D f )(x ,...,x ):=f(x ,...,x ,x ,x ,x ,x ,...,x ,x ,
α n 1 |α| 1 α1 α1+1 α1+1 α1+2 α1+2 α1+α2 α1+α2
2times 2times 2times
x ,x| {z,x } | ,.{.z.). } | {z }
α1+α2+1 α1+α2+1 α1+α2+1
3times
| {z }
We have (cf. [16]):
3
Theorem 1 For any f(n),g(n) ∈ D⊗ˆn, we have:
(f(n),g(n)) = K (D f )(x ,...,x )
Fn αZ α n 1 |α|
α∈Z∞+,0:1αX1+2α2+···=n X|α|
×(D g )(x ,...,x )σ⊗|α|(dx ,...,dx ),
α n 1 |α| 1 |α|
where
(1α +2α +···)! k−1b 2αk
K = 1 2 i=1 i . (6)
α
α !α !··· (cid:18)Qk! (cid:19)
1 2 kY≥2
Next, we find the elements which belong to the space F after the completion of
n
D⊗ˆn. To this end, we define, for each α ∈ Z∞ , the Hilbert space
+,0
L2(X|α|,σ⊗|α|):=L2(X,σ)⊗ˆα1 ⊗L2(X,σ)⊗ˆα2 ⊗··· .
α
Define a mapping
U(n) : D⊗ˆn → L2(X|α|,σ⊗|α|)K
α α
α∈Z∞+,0:1Mα1+2α2+···=n
by setting, for each f(n) ∈ D⊗ˆn, the L2(X|α|,σ⊗|α|)K -coordinate of U(n)f(n) to be
α α
D f(n). By virtue of Theorem 1, U(n) may be extended by continuity to anisometric
α
mapping of F into
n
L2(X|α|,σ⊗|α|)K .
α α
α∈Z∞+,0:1Mα1+2α2+···=n
Furthermore, we have (cf. [9, 16]):
Theorem 2 The mapping
U(n) : F → L2(X|α|,σ⊗|α|)K
n α α
α∈Z∞+,0:1Mα1+2α2+···=n
is a unitary opertator.
By virtue of Theorem 2 and (5), we can identify F with the space
n
L2(X|α|,σ⊗|α|)K
α α
α∈Z∞+,0:1Mα1+2α2+···=n
and the space F with
L2(X|α|,σ⊗|α|)K (1α +2α +···)!.
α α 1 2
α∈MZ∞
+,0
4
For a vector f ∈ F, we will denote its α-coordinate by f .
α
Note that, for for α = (n,0,0,...), we have
L2(X|α|,σ⊗|α|) = L2(X,σ)⊗ˆn, K = 1, (1α +2α +···)! = n!.
α α 1 2
Hence, the space F contains the symmetric Fock space
∞
F(L2(X,σ)) = L2(X,σ)⊗ˆnn!
Mn=0
as a proper subspace. Therefore, we call F an extended Fock space. We also note
that the space F satisfies the axioms of an interacting Fock space, see [11].
In the space L2(D′,µ), we consider, for each ϕ ∈ D, the operator M(ϕ) of
multiplication by the function h·,ϕi. Let J(ϕ) := UM(ϕ)U−1. Denote by F (D)
fin
the set of all vectors of the form (f ,f ,...,f ,0,0,...), f ∈ D⊗ˆi, i = 0,...,n,
0 1 n i
n ∈ Z . Evidently, F (D) is a dense subset of F. We have the following theorem,
+ fin
see [8].
Theorem 3 For any ϕ ∈ D, we have:
F (D) ⊂ Dom(J(ϕ)), J(ϕ) ↾ F (D) = J+(ϕ)+J0(ϕ)+J−(ϕ). (7)
fin fin
Here, J+(ϕ) is the usual creation operator:
J+(ϕ)f = ϕ⊗ˆf , f ∈ D⊗ˆn, n ∈ Z . (8)
n n n +
Next, for each f(n) ∈ D⊗ˆn, J0(ϕ)f(n) ∈ F and
n
(J0(ξ)f(n)) (x ,...,x )
α 1 |α|
∞
= α a S ξ(x )(D f(n))(x ,...,x )
k k−1 α α1+···+αk α 1 |α|
Xk=1 (cid:0) (cid:1)
σ⊗|α|-a.e., α ∈ Z∞ , 1α +2α +··· = n, (9)
+,0 1 2
J−(ξ)f(n) = 0 if n = 0, J−(ξ)f(n) ∈ F if n ∈ N and
n−1
(J−(ξ)f(n)) (x ,...,x )
α 1 |α|
= nS ξ(x)(D f(n))(x,x ,...,x )σ(dx)
α(cid:18)Z α+11 1 |α| (cid:19)
X
n
+ α b2 S ξ(x )(D f(n))(x ,...,x )
k k−1 k−1 α α1+···+αk α−1k−1+1k 1 |α|
Xk≥2 (cid:0) (cid:1)
σ⊗|α|-a.e., α ∈ Z∞ , 1α +2α +··· = n−1. (10)
+,0 1 2
5
In formulas (9) and (10), we denoted by S the orthogonal projection of
α
L2(X|α|,σ⊗|α|) onto L2(X|α|,σ⊗|α|),
α
α±1 :=(α ,...,α ,α ±1,α ,...), α ∈ Z∞ , n ∈ N.
n 1 n−1 n n+1 +,0
Finally, each operator J(ϕ), ϕ ∈ D, is essentially self-adjoint on F (D).
fin
By (7), theoperatorJ(ϕ) ↾ F (D)isasumofcreation, neutral, andannihilation
fin
operators, and hence J(ϕ) ↾ F (D) has a Jacobi operator’s structure. The family
fin
of operators (J(ϕ)) is called the Jacobi field corresponding to the L´evy process
ϕ∈D
µ.
2 An equivalent representation
As shown in [4, 13, 15, 16], in some cases, the formulas describing the operators
J0(ϕ) and J−(ϕ) can be significantly simplified. However, in the case of a general
L´evy process this is not possible, see [8]. We will now present a unitarily equivalent
description of the Jacobi field (J(ϕ)) , which will have a simpler form.
ϕ∈D
Let us consider the Hilbert space ℓ spanned by the orthonormal basis (e )∞
2 n n=0
with
e = (0,...,0, 1 ,0,0...).
n
(n+1)-stplace
|{z}
Consider the tensor product ℓ ⊗L2(X,σ), and let
2
∞
F(ℓ ⊗L2(X,σ)) = (ℓ ⊗L2(X,σ))⊗ˆnn!
2 2
Mn=0
be the (usual) symmetric Fock space over ℓ ⊗L2(X,σ).
2
Denote by ℓ the dense subset of ℓ consisting of all finite vectors, i.e.,
2,0 2
ℓ :={(f(n))∞ : ∃N ∈ Z such that f(n) = 0 for all n ≥ N}.
2,0 n=0 +
The Jacobi matrix J determines a linear symmetric operator in ℓ with domain ℓ
2 2,0
by the following formula:
Je = b e +a e +b e , n ∈ Z , e :=0. (11)
n n+1 n+1 n n n n−1 + −1
DenotebyJ+,J0,J− thecorrespondingcreation, neutral,andannihilationoperators
in ℓ , so that J = J+ +J0 +J−.
2,0
Denote by Φ the linear subspace of F(ℓ ⊗L2(X,σ)) that is the linear span of
2
the vacuum vector (1,0,0,...) and vectors of the form (ξ ⊗ ϕ)⊗n, where ξ ∈ ℓ ,
2,0
ϕ ∈ D, and n ∈ N. The set Φ is evidently a dense subset of F(ℓ ⊗L2(X,σ)).
2
6
Now, for each ϕ,ψ ∈ D and ξ ∈ ℓ , we set
2,0
A+(ϕ)(ξ ⊗ψ)⊗n:=(e ⊗ϕ)⊗ˆ(ξ ⊗ψ)⊗n
0
+n((J+ξ)⊗(ϕψ))⊗ˆ(ξ ⊗ϕ)⊗(n−1),
A0(ϕ)(ξ ⊗ψ)⊗n:=n((J0ξ)⊗(ϕψ))⊗ˆ(ξ ⊗ϕ)⊗(n−1),
A−(ϕ)(ξ ⊗ψ)⊗n:=nhξ,e ihϕ,ψi(ξ ⊗ϕ)⊗(n−1)
0
+n((J−ξ)⊗(ϕψ))⊗ˆ(ξ ⊗ϕ)⊗(n−1), (12)
and then extend these operators by linearity to the whole Φ. Thus,
A+(ϕ) = a+(e ⊗ϕ)+a0(J+ ⊗ϕ),
0
A0(ϕ) = a0(J0 ⊗ϕ),
A−(ϕ) = a−(e ⊗ϕ)+a0(J− ⊗ϕ),
0
where a+(·), a0(·), a−(·) are the usual creation, neutral, and annihilation operators
in F(ℓ ⊗L2(X,σ)). (In fact, under, e.g., a0(J+⊗ϕ) we understand the differential
2
second quantization of the operator J+ ⊗ ϕ in ℓ ⊗ L2(X,σ), which, in turn, is
2
the tensor product of the operator J+ in ℓ defined on ℓ and the operator of
2 2,0
multiplication by ϕ in L2(X,σ) defined on D.) Note also that
A(ϕ):=A+(ϕ)+A0(ϕ)+A−(ϕ)
=a+(e ⊗ϕ)+a0((J+ +J0 +J−)⊗ϕ)+a−(e ⊗ϕ)
0 0
=a+(e ⊗ϕ)+a0(J ⊗ϕ)+a−(e ⊗ϕ).
0 0
In the following theorem, for a linear operator A in a Hilbert space H, we denote
by A its closure (if it esxists).
Theorem 4 There exists a unitary operator
I : F → F(ℓ ⊗L2(X,σ))
2
for which the following assertions hold. Let J+(ϕ), J0(ϕ), and J−(ϕ), ϕ ∈ D, be
linear operators in F with domain F (D) as in Theorem 3. Then, for all ϕ ∈ D,
fin
IJ+(ϕ)I−1 = A+(ϕ),
IJ0(ϕ)I−1 = A0(ϕ),
IJ−(ϕ)I−1 = A−(ϕ),
and
IJ(ϕ)I−1 = A(ϕ).
7
Remark 1 Note, however, that the image of F under I does not coincide with the
n
subspace (ℓ ⊗L2(X,σ))⊗ˆnn! of the Fock space F(ℓ ⊗L2(X,σ)).
2 2
The proof of Theorem 4 is a straightforward generalization of the proof of The-
orem 1 in [17], so we only outline it.
First, we recall the classical unitary isomorphism between the usual Fock space
over L2(R×X,ν ⊗σ) and L2(D′,µ):
U : F(L2(R×X,ν ⊗σ)) → L2(D′,µ).
1
This isomorphism was broved by Itˆo [12] and extended in [16] to a general L´evy
process on X. Note also that
L2(R×X,ν ⊗σ) = L2(R,ν)⊗L2(X,σ).
Next, we have the unitary operator
U : L2(R,ν˜) → L2(R,ν)
2
defined by
1
(U f)(s) := f(s).
2 s
Let
U : ℓ → L2(R,ν˜)
3 2
be the Fourier transform in generalized joint eigenvectors of the Jacobi matrix J, see
[2]. The U can be characterized as a unique unitary operator which maps the vector
3
(1,0,0,...) into the function identically equal to 1, and which maps the closure J
of the symmetric operator J in ℓ into the multiplication operator by the variable s.
2
Let
U : ℓ ⊗L2(X,σ) → L2(R×X,ν ⊗σ)
4 2
be given by
U := (U U )⊗id,
4 2 3
where id denotes the identity operator. Using U , we naturally construct the unitary
4
operator
U : F(ℓ ⊗L2(X,σ)) → F(L2(R×X,ν ⊗σ)).
5 2
We now define a unitary operator
I := UU−1U−1 : F → F(ℓ ⊗L2(X,σ)).
1 5 2
Then, the assertions of Theorem 4 about the unitary operator I follow from Theo-
rem 3, the construction of the unitary operator I (see in particular Theorem 3.1 in
[16]), (12), and a limiting procedure.
8
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