Wick Calculus For Nonlinear Gaussian Functionals

0.2 MB·

by  Yaozhong Hu

Wick Calculus For Nonlinear Gaussian
Functionals
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2 Yaozhong Hu∗ and Jia-An Yan†
n Department of Mathematics, University of Kansas
a 405 Snow Hall, Lawrence, Kansas 66045-2142
J

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Abstract
]
R
ThispapersurveyssomeresultsonWickproductandWickrenormal-
P
ization. TheframeworkistheabstractWienerspace. Someknownresults
.
h on Wick product and Wick renormalization in the white noise analysis
t framework are presented for classical random variables. Some conditions
a
m are described for random variables whose Wick product or whose renor-
malization are integrable random variables. Relevant results on multiple
[
Wiener integrals, second quantization operator, Malliavin calculus and

1 their relations with the Wick product and Wick renormalization are also
v brieflypresented. AusefultoolforWickproductistheS-transformwhich
1 isalsodescribedwithouttheintroductionofgeneralizedrandomvariables.
1
9 Keyword: Malliavin calculus, Multiple integral, Chaos decomposition,
4 Wick product, Wick renormalization
. 2000 MR Subject Classification: 60G15, 60H05, 60H07, 60H40
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1 Introduction
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:
v
In the constructive Euclidean quantum field theory, such as the P(φ) theory or
i
X the : φ4 : theory, there have been encountered infinite quantities, which origi-
r nated (from mathematical point of view) from the product of generalized func-
a
tions (see [23], [4], or [9] andthe references therein for more details). To obtain
useful information out of these infinite quantities, Wick ([25]) first introduced
the now so-called Wick renormalization. According to [6], the Wick product in
stochastic analysis was first introduced by Hida and Ikeda [5]. Meyer and Yan
([21]) extended to cover the Wick products of Hida distributions. Now Wick
∗Y. Hu was supported in part by the National Science Foundation under Grant No.
DMS0504783, and the International Research Team on Complex Systems, Chines Academy
ofSciences.
†J.-A. Yan was supported by the National Natural Science Foundation of China (No.
10571167),theNationalBasicResearchProgramofChina(973Program)(No.2007CB814902),
andtheScienceFundforCreativeResearchGroups(No.10721101).
1
product is applied to stochastic differential equations ([13]), stochastic partial
differential equations, stochastic quantization ([9]) and many other fields.
In stochastic analysis, most of the research work on Wick product are on
Hida distribution spaces or other spaces of generalized random variables. In
this paper we survey some results that we have frequently used. To make the
concept of Wick product accessible to broader audience, we restrict ourselves
to the classical framework, namely, the classical random variables. In fact, a
motivation to write such a survey is from some students who need to know
some results relevant to Wick product and the way of how to use them in their
research. Afterwehavedonesomeresearchonthereferences,wefoundoutthat
manyresultsonWickproducthavebeenalreadyknownbythesecondauthorin
theframeworkofHidadistribution([26],[27]). Buttheyarerelativelyunknown
to the experts on the field.
In Section 2, we introduce the framework and some results which are useful
in Wick product. In particular, we introduce the multiple integrals and the
chaos expansion.
MalliavincalculusisveryusefulintheproblemswhereWickproductpresents.
InSection3,wegiveasimplestpresentationofsomeresultsinMalliavincalculus
which are relevant to wick product.
Wick product is introduced in Section 4. We present some basic results.
Some of them may be new.
InEuclideanquantumfieldtheory,the Wickrenormalizationis morewidely
used. In section 5, we present some results on Wick renomalization. In fact,
in Euclidean quantum field theory a very special abstract Wiener space and a
veryspecialrandomvariablesareneeded(see[9]). However,we willnotgointo
detail.
This paper is a condensed survey of some results on Wick product. We do
not intend to give a survey on the historical account. So in some cases, for a
concept or a result, probably not the original references are going to be cited.
2 Multiple integrals and chaos expansion
Let H be a separable Hilbert space with scalar product , . There is a
H
h· ·i
Banach space B (B is not unique) with the following properties.
(i) H is continuously embedded in B and H is dense in B. The dual B′
(the space of continuous linear functionals) of B is identified as a (dense)
subspace of H (B′ B).
⊂
(ii) There is a Borel measure µ on (B, ), where is the Borel σ-algebra of
B B
B such that
exp i l,x dµ(x)=exp l 2 /2 , l Φ′, (2.1)
{ h i} {−k kH } ∈
ZB
where , means the pairing between B and B′ (namely, l,x =l(x)).
h· ·i h i
2
Thetriple(B,H,µ)isanabstractWienerspace. WedenoteE(f)= f(x)dµ(x)
B
and Lp =Lp(B, ,µ).
For any l BB′, l, :B R is a mean zeroGaussianrandomvRariable with
variance l 2∈. Byha ·liimitin→g argument, for any l H, l, : B R can be
k kH ∈ h ·i →
defined as a Gaussian random variable, denoted by ˜l.
FixedannandintroducethesymmetrictensorproductH⊗ˆnbythefollowing
procedure.
Let e ,e , beanorthonormalbasisofH andlet ˆ denotethesymmetric
1 2
{ ···} ⊗
tensor product. Then
f = f e ˆ ˆe , f R (2.2)
n i1,···,in i1⊗···⊗ in i1,···,in ∈
finite
X
is an element of H⊗ˆn with the Hilbert norm
f 2 = f 2, (2.3)
k nkH⊗ˆn | i1,···,in|
finite
X
H⊗ˆn is the completionofallthe elements of aboveformunder the abovenorm.
Todefinethemultipleintegral,weneedtousetheHermitepolynomials. Let
Hn(x)=(−1)nex22 ddxnne−x22 = 2kk(!−(n1)kn2!k)!xn−2k, x∈R
k≤Xn/2 −
be the n-th Hermite polynomial (n=0,1,2, ). Its generating function is
···
etx−t22 = ∞ tnHn(x).
n!
n=0
X
Any element f in H⊗ˆn of the form (2.2) can be rewritten as
n
f = f e⊗ˆk1ˆ ˆe⊗ˆkm, f R, , (2.4)
n j1,···,jm j1 ⊗···⊗ jm j1,···,jm ∈
finite
X
where j , ,j are different. For this type of integrands the multiple integral
1 m
···
is defined as
I (f ):= f H (e˜ ) H (e˜ ) . (2.5)
n n j1,···,jm j1 j1 ··· jm jm
finite
X
In particular, we have for f H
∈
I (f⊗n)= f nH ( f −1f˜). (2.6)
n k kH n k kH
Forgeneralelementf inH⊗ˆn wecandefinethemultipleintegralI (f )bythe
n n n
L2 convergence. It is straightforwardto obtain the following isometry equality
EI (f )2 =n! f 2 . (2.7)
| n n | k nkH⊗ˆn
3
One can also construct the Fock space Φ(H) on H as follows.
∞
Φ(H)= H⊗ˆn.
n=0
M
The scalar product of two elements f = (f ,f ,f , ) and g = (g ,g ,g , )
0 1 2 0 1 2
··· ···
in Φ(H) is defined as
∞
f,g = n! f ,g . (2.8)
h iΦ(H) h n niH⊗ˆn
n=0
X
Thechaosexpansiontheoremstatesthatanysquareintegrablerandomvariable
F on (B, ,µ) can be written as
B
∞
F = I (f ), f =(f ,f ,f , ) Φ(H) (2.9)
n n 0 1 2
··· ∈
n=0
X
and
∞
E(F2)= f 2 = n! f 2 . (2.10)
k kΦ(H) k n||H⊗ˆn
n=0
X
We refer to [20] and the references therein for further details.
Example 2.1 Let B = H = Rd and let µ be the standard Gaussian measure
on B. Then (B, ,µ) is the d-dimensional standard Gaussian measure space.
B
Example 2.2 Let
H = f :[0,T] R; f(0)=0 f is absolutely continuous on [0,T] .
{ → }
It is a Hilbert space under the norm f, g = T f′(t)g′(t)dt. Let
h i 0
Ω= f :[0,T] R; f(0)=0 f isRcontinuous on [0,T]
{ → }
with the sup norm. Then (Ω, ,µ) is a canonical Wiener space, where is the
F F
Borel σ-algebra on Ω (with respect to the sup norm).
Example 2.3 Consider a domain D of Rd (d dimensional Euclidean space).
Together with some nice boundary conditions (if D is not the whole space Rd)
we can prove (see [4]) that there is a kernel K(x,y) such that the following
equation (with mass m=1) holds
( ∆+1)K(x,y)=δ(x y),
− −
where ∆ is the Dirichlet Laplacian on D. A Hilbert space of (generalized) func-
tions can be determined by by
f,g = K(x,y)f(x)f(y)dxdy,
h i
ZD
where f and g are two (generalized) real-valued functions. The Gaussian mea-
sure associated with this Hilbert space is useful in the Euclidean quantum field
theory (see also [9] and the references therein).
4
Definition 2.4 Let f H⊗ˆn. f is called negative definite on H⊗ˆn if
n n
∈
f ,h⊗ˆn 0 h H
n
H⊗ˆn ≤ ∀ ∈
D E
Example 2.5 The following are proved in [9].
(i) If f H⊗ˆ2, then there is an α > 0 such that Eexp[αI (f )] < . And
2 2 2
Eexp[∈αI (f )]< is true for all α>0 iff f is negative definite∞(see [9],
2 2 2
∞
Theorem 5.1).
(ii) Foranynonzerof inH⊗ˆ2n+1(n 1)andanyλ R,E(exp λI (f ) )=
2n+1 2n+1 2n+1
≥ ∈ { }
. (see [9], Theorem 5.2).
∞
It is conjectured (see [9]) that
(iii) If n is even, then Eexp[I (f )]< iff f is negative definite on H⊗ˆn.
n n n
∞
Definition 2.6 If F has a chaos expansion F = ∞ F and α R, then the
n=0 n ∈
second quantization operator of α acting on F is defined as
P
∞
Γ(α)F = αnF . (2.11)
n
n=0
X
For this operator we have the following famous theorem.
Theorem 2.7 (Nelson’s hypercontractivity) Let 1 p<q < . The following
≤ ∞
inequality holds
Γ(α)F F , F Lp (2.12)
q p
k k ≤k k ∀ ∈
if and only if α p−1.
| |≤ q−1
q
This inequality was first obtained by Nelson and appears in many places. See
[2], [7] and the references therein for further detail.
The following theorem is due to U¨stu¨nel and Zakai [24] and see [16] for a
simpler proof.
Theorem 2.8 Let f H⊗ˆn and g H⊗ˆm. Then I (f) and I (g) are inde-
n m
∈ ∈
pendent if and only if
f,g =0, (2.13)
H
h i
where f,g H⊗ˆn+m−2 defined by
H
h i ∈
∞
f,g = f,e ˆ g,e .
H n H n H
h i h i ⊗h i
n=1
X
For an f H⊗ˆn satisfying some more conditions on the existence of trace
∈
of f, the multiple Stratonovich integral S (f) can also be introduced in the
n
following way:
5
Let e ,e ,... be an orthonormal basis of H. Let f H⊗ˆn and consider
1 2
{ } ∈
the following random variable:
N
SN(f)= f,e ˆ ˆe e˜ e˜ . (2.14)
n h k1⊗···⊗ kniH⊗ˆn k1··· kn
k1,·X··,kn=1
Definition 2.9 If as N , SN(f) converges in L2, then we say themultiple
→∞ n
Stratonovich integral of f exists. The limit is called the multiple Stratonovich
integral of f and is denoted by S (f).
n
Definition 2.10 Denote
N
Trk,Nf = f,e ˆe ˆ ˆe ˆe
n h i1⊗ i1⊗···⊗ ik⊗ ikiH⊗ˆ2k
i1,·X··,ik=1
which is considered as an element in H⊗ˆ(n−2k). If as N , Trk,Nf con-
n
verges in H⊗ˆ(n−2k), then we say the trace of order k exists→an∞d denote it by
∞
Trkf = f,e ˆe ˆ ˆe ˆe .
h i1⊗ i1⊗···⊗ ik⊗ ikiH⊗ˆ2k
i1,·X··,ik=1
Ifthetracesoforderkoff existforallk n/2,thenthemultipleStratonovich
≤
integral of f, namely S (f), exists and the following Hu-Meyer formula holds
n
n!
S (f) = I ( Trkf)
n 2kk!(n 2k)! n−2k
k≤Xn/2 −
( 1)kn!
I (f) = − S ( Trkf) (2.15)
n 2kk!(n 2k)! n−2k
k≤Xn/2 −
For this result and other results see [1], [12], [15] and the references therein.
Example 2.11 If f , ,f H, then
1 n
··· ∈
S (f ˆf ˆ ˆf )=f˜f˜ f˜ .
n 1 2 n 1 2 n
⊗ ⊗···⊗ ···
3 Malliavin calculus
Iff H⊗ˆn andg H,then f ,g isanelementinH⊗ˆ(n−1). Forf H⊗ˆn,
n n H n
∈ ∈ h i ∈
we define
D I (f )=nI ( f ,g ) .
g n n n−1 n H
h i
If for almost every x B, the above right hand is a continuous functional of g
∈
on H, then
DI (f )=nI (f )
n n n−1 n
6
is a random variable with values in H. We can extend D and D to general
g
random variable F by linearity and limiting argument.
It is easy to check that D(FG)=FDG+GDF.
InthesamewaywecanintroducehigherorderderivativesDkF :B H⊗ˆk.
→
The space Dk,p is defined as
k
Dk,p = F :B R; F p := E DiF p < .
( → k kk,p k kH⊗ˆi ∞)
i=0
X
To describe this space, one may introduce the Ornstein-Uhlenbeck operator L
defined by
∞ ∞
LF = nF , if F has the chaos expansion F = F
n n
n=1 n=1
X X
(which is the generator of the semigroup P F =Γ(e−t)F).
t
The following result is called the Meyer’s inequality (see [19] and also [22]
for a simpler analytic proof).
Theorem 3.1 There is a constant c and C such that
k,p k,p
c (L+1)k/2F F C (L+1)k/2F .
k,p p k,p k,p p
k k ≤k k ≤ k k
Meyer’s inequality can be used to give a detailed description of Dk,p even for
non integer k.
Example 3.2 Let f H. Then
∈
∞
1 1
ε(f):= I (f⊗n)=exp f˜ f 2
n! n − 2k kH
n=0 (cid:18) (cid:19)
X
is called an exponential vector (in L2(B, ,µ)).
B
For an exponential vector ε(f) and a g H, we have
∈
D ε(f)=ε(f) f,g , Dε(f)=ε(f)f.
g
h i
Let
= F =h(f˜ , ,f˜ ), f , ,f H and h is smooth
1 n 1 n
S ( ··· ··· ∈
function on Rn, n 1 .
≥ )
Example 3.3 If F =h(f˜ , ,f˜ ) is in with h being of polynomial growth,
1 n
··· S
then F D for all k 0 and p 1 and
k,p
∈ ≥ ≥
n
∂h
D F = (f˜ , ,f˜ ) f ,g
g 1 n i
∂x ··· h i
i
i=1
X
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and
n
∂h
DF = (f˜ , ,f˜ )f .
1 n i
∂x ···
i
i=1
X
Definition 3.4 If F :B H and there is a random variable Z such that
→
E( F ,DG )=E(ZG) G , (3.1)
h i ∀ ∈S
Then we say the divergence of F exists and we denote it by Z =δ(F).
This means that the divergence operator δ is the adjoint operator of the
derivative operator D.
Example 3.5 If g H, then δ(g)=g˜.
∈
4 Wick product
Definition 4.1 If f H⊗ˆn and g H⊗ˆm, then the Wick product of I (f )
n m n n
∈ ∈
and I (g ) is defined as
m m
I (f ) I (g )=I (f ˆg ),
n n m m n+m n m
⋄ ⊗
where f ˆg denotes the symmetric tensor product of f and g . If F =
n m n m
N1 I (⊗f ) and G= N2 I (g ), then we define
n=0 n n m=0 m m
P P N1 N2
F G= I (f ˆg ),
n+m n m
⋄ ⊗
n=0m=0
X X
By a limiting argument, we can extend the Wick product to general random
variables (see [20] and [9] for example).
Remark 4.2 Of course, when we use “by limiting argument” the definition
depends on the topology that we use. We can approximate F and G by finite
combination of multiple integrals, F and G , and define F G as the limit
N N
⋄
of F G . Different choices of the topology (for example, in probability, Lp,
N N
⋄
almost surely etc) will lead to different definitions of the Wick product. In this
paper, we shall use the L2 limit.
It is clear that F,G L2 does not imply that F G is a well-defined object
∈ ⋄
in L2. Now we presenta sufficient condition on F and G suchthat F G L2.
⋄ ∈
To this end, we need to introduce some new norms.
If F = ∞ I (f ), we define
n=0 n n
P ∞
F 2 = Γ(r)F 2 = n!r2n f 2.
k k(r) k k k nk2
n=0
X
The following proposition can be found in [27](Theorem 3.1).
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1 1 1
Proposition 4.3 Let + = , p,q,r >0. Then
p2 q2 r2
F G F G
(r) (p) (q)
k ⋄ k ≤k k k k
Proof LetF = ∞ I (f )andG= ∞ I (g ). Denoteh = f ˆg .
n=0 n n n=0 n n n k+j=n k⊗ j
Let a=r−2q2 1. Then 1+a−1 =r−2p2. We have
−P P P
n 1/2
√n! h √n! f g = k!j! f g
n k j k j
k k ≤ k kk k k k kk k
k+Xj=n k+Xj=n(cid:16) (cid:17) p
1/2 1/2
n
ak a−kk!j! f 2 g 2
k j
≤  k   k k k k 
k+Xj=n(cid:16) (cid:17) k+Xj=n
   1/2 
= (1+a)na−kk!j! f 2 g 2
k j
 k k k k 
k+j=n
X
  1/2
= (1+a−1)kk! f 2(1+a)jj! g 2 ,
k j
 k k k k 
k+j=n
X
 
which implies the result.
As a direct consequence of the above proposition we obtain a condition on
F and G such that F G is in L2.
⋄
Theorem 4.4 If Γ(p)F ,Γ(q)G are in L2 with 1 + 1 = 1, then F G exists
p2 q2 ⋄
as an element in L2.
AusefultoolinstudyingtheWickproductistheso-calledS-transformation.
Definition 4.5 Let F Lp for some p > 1. Then for any ξ H F( +ξ) :
B R is well-defined∈integrable random variable. The follow∈ing fun·ctional
from→H to R
S(F)(ξ)=E[F( +ξ)] , ξ H. (4.1)
· ∀ ∈
is called the S-transformation of F.
By Cameron-Martintheorem we have
S(F)(ξ)=E[Fε(ξ)], ξ H. (4.2)
∀ ∈
Consequently, if F = ∞ I (f ), then
n=0 n n
P ∞
S(F)(ξ)= n! f ,ξ⊗n , ξ H. (4.3)
n
h i ∀ ∈
n=0
X
This implies that
S(F G)(ξ)=S(F)(ξ)S(G)(ξ) (4.4)
⋄
9
for suitable F and G. For f H, since ε(f)= ∞ 1I (f⊗n), we have
∈ n=0 n! n
Sε(f)(ξ)=exp f,Pξ ,
{h i}
and consequently
ε(f) ε(g)=ε(f +g), f,g H. (4.5)
⋄ ∀ ∈
We refer to [14] [17], [18] and the references therein for more details in the
framework of white noise analysis.
Proposition 4.6 If f ,f H are two unit vectors which are orthogonal, then
1 2
∈
H (f˜) H (f˜)=H (f˜)H (f˜).
n 1 m 2 n 1 m 2
⋄
Under some suitable condition on F and G, we have
D (F G) = D F G+F D G (4.6)
g g g
⋄ ⋄ ⋄
D(F G) = DF G+F DG (4.7)
⋄ ⋄ ⋄
The following proposition can be found in [27] (Theorem 5.5).
Proposition 4.7 If g H, F L2(B, ,µ) and if D F exists and is in
g
∈ ∈ B
L2(B, ,µ), then F g˜ exists in L1(B, ,µ) and
B ⋄ B
F g˜=Fg˜ D F =Fg˜ DF,g . (4.8)
g H
⋄ − −h i
Proof Let
= a ef˜i, where a R, f H .
E  i i ∈ i ∈ 
fXinite 
Then is dense in L2. It is easy to see that (4.8) is true for all elements in
E E
by (4.5). A density argument shows the proposition.
Proposition 4.8 If g H, F L2(B, ,µ) and if D F exists and is in
g
∈ ∈ B
L2(B, ,µ), then F g˜ exists in L1(B, ,µ) and
B ⋄ B
F g˜=δ(Fg). (4.9)
⋄
Proof Let G . Then
∈S
E((F g)G) = E(FGg˜ G DF,g )
H
⋄ − h i
= E( D(FG),g GDF,g )
H H
h i −h i
= E( FDG,g )=E( DG,Fg ) .
H H
h i h i
Since G is arbitrary, we show the proposition.
The following proposition is used in [3] and [8].
10