On stochastic differential equations driven by the renormalized square of the Gaussian white noise

0.23 MB·English

by  Bilel Kacem Ben Ammou

On stochastic differential equations driven by the
renormalized square of the Gaussian white noise
5 Bilel Kacem Ben Ammou Alberto Lanconelli
∗ †
1
0
2

n
a
J
Abstract

4
1
We investigate the properties of the Wick square of Gaussian white noises

] through a new method to perform non linear operations on Hida distributions.
R
This method lays in between the Wick product interpretation and the usual def-
P
inition of nonlinear functions. We prove on Itˆo-type formula and solve stochastic
.
h differential equations driven by the renormalized square of the Gaussian white
t
a noise. Our approach works with standard assumptions on the coefficients of the
m
equations, Lipschitz continuity and linear growth condition, and produces exis-
[ tence and uniqueness results in the space where the noise lives. The linear case is

1 studied in details and positivity of the solution is proved.
v
0
Key words and phrases: Gaussian white noise, Hida distributions, stochastic differ-
6
3 ential equations, Wick product.
3
AMS 2000 classification: 60H40, 60H10.
0
.
1
0 1 Introduction
5
1
The Gaussian white noise W is a generalized process that can be formalized
: t t≥0
v { }
as the distributional time derivative of a standard one dimensional Brownian motion
i
X
B . One way to define its square is through the so-called Wick renormalization
t t≥0
r { }
a
W⋄2 := lim(Wε)2 E[(Wε)2] (1.1)
t ε→0 t − t
where Wε is a smooth approximation of the white noise W and the limit is interpreted
t t
in a suitable distributional sense. This object appears naturally in different contexts.
For instance, in the paper [26] the author considered gradient operators on the classical
Wiener space along directions that do not belong to the Cameron-Martin space and
obtained among other things the following integration by parts formula
1
E[D F(B)] = E F(B) W⋄2dt (1.2)
B h Z t i
0
∗DepartmentofMathematics,UniverstiyofTunis-ElManar,StreetMohamedAlayaKacem,Nabeul
- Tunisia. E-mail: [email protected]
†Dipartimento di Matematica, Universit´a degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125
Bari - Italia. E-mail: [email protected]
1
whereF(B)isafunctionaloftheBrownianpathbelongingtoacertainclassofregularity
andD denotesdifferentiationalongthedirectionB,i.e. D F(B) := lim F(B+δB)−F(B).
B B δ→0 δ
One encounters quantities like (1.1) also in connection with Itˆo-type formulas for solu-
tions to certain stochastic partial differential equations. More precisely, consider the
stochastic heat equation driven by an additive Gaussian space-time white noise (see
[24])
1
∂ u(t,x) = ∂ u(t,x)+W , t > 0, x [0,1], (1.3)
t xx tx
2 ∈
with initial condition u(0,x) = 0 and homogeneous Dirichlet boundary conditions . It
was proved in [25] (see also [17],[18]) that, if u(t,x) denotes the unique (weak)
t≥0,x∈[0,1]
{ }
solution to (1.3), then for any l C2(]0,1[) one has
∈ 0
1 t t
u(t, )2,l = u(s, )2,l′′ ds+2 u(s, ),ldW + E[u(t, )2],l
h · i 2 Z h · i Z h · si h · i
0 0
t
(∂ u(s, ))⋄2,l ds
−Z h x · i
0
where , denotes theinner product in 2([0,1]). The term(∂ u(s,x))⋄2 intheprevious
x
h· ·i L
equality is analogous to the one in (1.1) since ∂ u(t,x) is a generalized Gaussian field.
x
The existence of limits of the type (1.1) has also been considered in [20] where the
authors investigated the almost sure existence of the limit
+∞ X X ⋄k
lim t+ε − t g(t)dt
ε→0Z (cid:16) ε (cid:17)
0
where X is a Gaussian process with a covariance satisfying certain assumptions
t t≥0
{ }
(not satisfied by the standard Brownian motion), k 0 and g is a bounded and mea-
≥
surable function with compact support. Observe that for X = B , k = 2 and g = 1
t t [0,1]
the above limit would correspond to the integral in the right hand side of (1.2).
We also mention the paper [1] where the problem of defining powers of Gaussian white
noises is taken from the point of view of quantum probability.
The analogybetween (1.2) andthe usual integrationby parts formula fromthe Malliavin
calculus ([22])
1
E[D F(B)] = E F(B) g′(s)dB
g h Z si
0
where g is a deterministic absolutely continuous function, suggests that the quantity
1W⋄2dt or more generally the process t tW⋄2ds plays in the differential calculus
0 t 7→ 0 s
Rassociated to the gradient D (from (1.2)) Rthe role played by tW ds = B in the
B 0 s t
classical Malliavin calculus. There is however a major difference inRthese situations: the
fact that tW⋄2ds is a generalized process (more precisely, a Hida distribution) and
0 s
therefore tRhe question of finding Itˆo type formulas or studying stochastic differential
equations driven by that process is far from being obvious.
Consider for instance equations of the type
dY
t = b(Y )+σ(Y ) W⋄2 (1.4)
dt t t · t
2
for suitable measurable coefficients b and σ. First of all, note that for b = 0 and σ = 1
equation (1.4) is solved by Y = Y + tW⋄2ds which is a Hida distribution. Therefore,
t 0 0 s
one cannot expect the solution to (1.4R) to be more regular than that. This fact implies
the necessity to give a meaning to the nonlinear terms appearing in (1.4) and to the
multiplication between σ(Y ) and W⋄2. One possibility is to interpret all the nonlinear
t t
terms in the Wick-product sense (see [13] and the references quoted there), that means
to replace equation (1.4) with
dZ
t = b⋄(Z )+σ⋄(Z ) W⋄2 (1.5)
dt t t ⋄ t
(we refer to the next sections for precise definitions). However, this procedure has at
least two important drawbacks: firstly, to define b⋄(Z ) and σ⋄(Z ) one needs the ana-
t t
lyticity of the functions b and σ; secondly, the solution to (1.5) usually exists in spaces
that are much bigger than the one where the noise lives (such as the Kondratiev spaces).
Moreover, from a modeling point of view, solutions to equation (1.5) may exhibit behav-
iors that differs from what is expected to happen for solutions to (1.4) (see for instance
[3]).
Our aim in this paper is to introduce a new method to define nonlinear operation on
Hida distributions. This method lays in between the Wick product interpretation (1.5)
and the usual definition of nonlinear functions (1.4). Our approach requires standard
assumptions on the coefficients of the stochastic equations considered (Lipschitz conti-
nuity and linear growth condition) and produces existence and uniqueness results in the
space where the noise lives.
In the recent years, renormalization techniques for solving stochastic (partial) differen-
tial equations have attracted the attention of many authors (see [6], [8], [9], [10], [11] and
the references quoted there). The common basic idea in these references is to smooth
the noise, solve the corresponding equation and then try to compute the limit of the
solution as the degree of regularization of the noise decreases (the existence of a non
trivial limit usually requires a renormalization of the coefficients of the original equa-
tions). The way we treat nonlinear functions of distributions and solve related stochastic
differential equations follows the same principle: the only technical difference is that we
choose a specific regularizing procedure for the noise and then, once we have solved the
regularized equation, we apply to it the inverse of the regularizing map utilized before
(instead of letting a parameter tend to zero) (see Remark 5.6 below). Our procedure
deeply depends on the adopted smoothing map but this is intrinsically connected with
the construction of the Hida distribution space which is the natural accommodation of
the noise. We do not know whether the equations we consider can be handled with
the recent theory of regularity structures ([11]); certainly, the points of view of the two
approaches are different in the fact that our notion of distribution is related to the prob-
ability space where the noise is defined and not to the state space of the time parameter
describing the processes.
The paper is organized as follows: Section 2 is a quick review of the minimal back-
ground material needed to treat our problem; Section 3 introduces and describes our
new method of performing non linear operations on Hida distributions while in Section
4 and 5 we apply these concepts to deduce an Itˆo-type formula and a theorem on ex-
istence and uniqueness for solutions to stochastic differential equations driven by the
renormalized square of the Gaussian white noise, respectively; finally, in Section 6 we
propose few by-products of the introduced concepts.
3
2 Framework
The aim of this section is to briefly set up the framework utilized to prove our main
results. For more details we refer the interested reader to one of the books [12], [13],
[15] or to the paper [7] where many technical issues are usefully spelled out.
Let (Ω, , ) be a complete probability space endowed with a standard one dimensional
F P
Brownian motion B and denote by its augmented natural filtration. Write
t t≥0 t t≥0
{ } {F }
to denote the smallest sigma-algebra containing all the ’s for t 0. According to
t
G F ≥
the Wiener-Itoˆ chaos decomposition theorem any X 2(Ω, , ) ((L2) for short) can
∈ L G P
be uniquely represented as
X = I (h ) (2.1)
n n
X
n≥0
where h 2(Rn)is asymmetric functionandI (h )stands forthen-thordermultiple
n ∈ L + n n
Itˆo integral of h with respect to the Brownian motion B . Observe that from (2.1)
n t t≥0
{ }
one gets
E[X2] = n! h 2
X | n|L2(Rn+)
n≥0
(here E[ ] denotes the expectation on the probability space (Ω, , )). For r [1,+ ]
· G P ∈ ∞
we also set (Lr) := r(Ω, , ), the classic Lebesgue spaces over the measure space
L G P
(Ω, , ).
G P
We are now going to introduce the Schwartz space over R and an analogous class of
+
smooth random variables. Consider the differential operator
d d 1
A := t + t+1
−dt dt 4
acting on a subset of 2(R ) and recall that for any k 0 one has
+
L ≥
3
Aξ = k + ξ (2.2)
k k
(cid:16) 2(cid:17)
where ξ is the complete orthonormal system in 2(R ) formed by the Laguerre
k k≥0 +
{ } L
functions. For p 0 define S to be the Hilbert space of functions in 2(R ) such that
p +
≥ L
f := Apf < +
| |p | |L2(R+) ∞
(for consistency we will denote from now on the norm with the symbol ).
|·|L2(R+) |·|0
It is clear from (2.2) that for p < q one has S S ; one can then prove that
q p
⊂
S := S
p
\
p≥0
endowed with the projective limit topology, coincides with the Schwartz space over R
+
of infinitely differentiable functions that vanish, together with all their derivatives, at
+ faster than any inverse power (see [7]). The dual space of S, denoted by S′, is
∞
the space of tempered distributions over R and it can be represented as the union of
+
the spaces S for p 0. One of its most representative elements is the Dirac delta
−p
≥
4
distribution δ , t 0. The dual pairing between S′ and S will be denoted by , .
t
≥ h· ·i
We now lift this construction to the space (L2); more precisely, for X = I (h )
n≥0 n n
and p 0 let P
≥
Γ(Ap)X := I ((Ap)⊗nh )
n n
X
n≥0
and define (S ) to be the Hilbert space of those X’s such that
p
n! (Ap)⊗nh 2 < + .
X | n|L2(Rn+) ∞
n≥0
The space
(S) := (S )
p
\
p≥0
endowed with the projective limit topology is called Hida test function space. Its dual
(S)∗, the Hida distribution space, accommodates the white noise W := dBt which can
t dt
be represented as W := I (δ ). We will write , for the dual pairing between (S)∗
t 1 t
hh· ·ii
and (S). For f 2(R ) set
+
∈ L
f 2
(f) := exp I (f) | |0 .
1
E n − 2 o
It is easy to prove that (f) (S) for f S and that
E ∈ ∈
X, (f) = Y, (f) , for all f S
hh E ii hh E ii ∈
implies X = Y in (S)∗. The map
f (SX)(f) := X, (f)
7→ hh E ii
is called S-transform of X (S)∗. The celebrated characterization theorem ([12]) pro-
∈
vides a necessary and sufficient condition for the invertibility of the S-transform: let
f S F(f) R be a measurable function such that
∈ 7→ ∈
for all f,g S the function x R F(f +xg) has an entire analytic extension
• ∈ ∈ →
to the complex plane (denoted with the same symbol);
there exist positive constants K,p such that for any f S and z C one has
• F(zf) Kexp |z|2 f 2 . ∈ ∈
| | ≤ { 2 | |p}
Then there exists X (S)∗ such that (SX)(f) = F(f). If these conditions are met for
∈
the function
b
f S(X )(f)dt
7→ Z t
a
we will say that the process X is Pettis integrable in (S)∗ over the interval [a,b];
t t≥0
{ }
b
the value of the integral is denoted by X dt and it verifies
a t
R
b b
X dt,ϕ = X ,ϕ dt
DDZ t EE Z hh t ii
a a
5
for all ϕ (S). Let τ 0; a Hida distribution X will said to be -measurable if
τ
∈ ≥ F
(SX)(f +g) = (SX)(f) (2.3)
for all f,g S such that the support of g in contained in [0,τ]c ([7]). Finally, for
∈
X,Y (S)∗ we write X Y for the unique element in (S)∗ such that
∈ ⋄
X Y, (f) = X, (f) Y, (f) ,
hh ⋄ E ii hh E ii·hh E ii
for all f S. The quantity X Y is named Wick product of X and Y.
∈ ⋄
3 A renormalized product
The following definition introduces the main tool utilized in the investigation pre-
sented in this paper.
Definition 3.1 Let ϕ : R R be a bounded function and let X belong to (S ) (S)∗
−p
→ ⊂
for some p 0. We define
≥
ϕ˜ (X) := Γ(Ap)(ϕ(Γ(A−p)X)). (3.1)
p
The idea behind this definition is clear: we take a Hida distribution X; by construction
there exists a p 0 such that X (S ) that means Γ(A−p)X (L2); now we apply
−p
≥ ∈ ∈
the nonlinear function ϕ to Γ(A−p)X, viewed as a smooth approximation of X and then
we ”remove” the regularization by applying Γ(Ap). Observe that here we cannot remove
the regularization by taking the limit as p tends to zero of ϕ(Γ(A−p)X) since it wouldn’t
exists without some other renormalization.
The definition of ϕ˜ (X) does indeed depend on p, as it is stressed in the notation.
p
Since (S ) (S ) for p < q we could have chosen a bigger value of the parameter
−p −q
⊂
and obtain a different renormalized object. The point is however to prefer the smallest
possible exponent in Γ(A−p)X in order to preserve X as much as possible. If for instance
X (L2), we do not need to perform any approximation and we can simply take p = 0
∈
and get the usual ϕ(X).
The assumption of boundedness on ϕ in Definition 3.1 can be clearly relaxed for specific
choices of X. In general, the quantity ϕ˜ (X) belongs to (S ), the same space of X.
p −p
Definition 3.2 Let X,Y (S ) (S)∗ for some q 0: then for any p > q we set
−q
∈ ⊂ ≥
X Y := Γ(Ap)(Γ(A−p)X Γ(A−p)Y). (3.2)
p
∗ ·
Remark 3.3 If X (S ) then Γ(A−p)X belongs to (Lr) for some r > 2. In fact
−q
∈
Γ(A−p)X = Γ(A−p+q)Γ(A−q)X;
by definition Γ(A−q)X (L2); moreover, be the Nelson hyper-contractive estimate ([21])
∈
the operator Γ(A−p+q) maps (L2) into a smaller (Lr) for some r > 2. This is necessary
to guarantee that the product Γ(A−p)X Γ(A−p)Y belongs to (Lu) for some u > 1.
·
The product defined in (3.2) is commutative, associative and distributive with respect
to the sum. It was introduced in a slightly different form in [4] in connection to Wong-
Zakai approximation theorems and utilized subsequently in [5] in the study of certain
generalizations of the Poincar´e inequality. It is instructive to observe (see [4]) that
lim X Y = X Y in (S)∗.
p
p→+∞ ∗ ⋄
Therefore the product ⋆ is collocated between the ordinary product (when p = 0) and
p
the Wick product (when p = + ).
∞
6
4 Itoˆ-type formula for the quadratic white noise process
In this section we are going to prove a chain-rule formula for the object of our
investigation, i.e. tW⋄2ds. We begin with the following technical lemma.
0 s
R
Lemma 4.1 For p > 1 the function t A−pδ 2 is continuous. For p > 1 the process
2 7→ | t|0
Γ(A−p)W is almost surely continuous.
t t≥0
{ }
Proof. Let t 0; then δ S′ and we can write
t
≥ ∈
δ = ξ (t)ξ .
t k k
X
k≥0
Then
3 −p
A−pδ = k + ξ (t)ξ
t k k
X(cid:16) 2(cid:17)
k≥0
and hence
3 −2p
A−pδ 2 = k + ξ2(t).
| t|0 X(cid:16) 2(cid:17) k
k≥0
Since sup ξ (t) = 1 ([7]) we get that the previous series is uniformly convergent for
t∈R+ | k |
p > 1 entailing the continuity of the function t A−pδ 2.
2 7→ | t|0
Now consider
W = ξ (t)I (ξ ).
t k 1 k
X
k≥0
Since
3 −p
Γ(A−p)W = k + ξ (t)I (ξ )
t k 1 k
X(cid:16) 2(cid:17)
k≥0
and the I (ξ )’s are independent and identical distributed random variables, we deduce
1 k
that for p > 1 the last series is almost surely uniformly convergent and hence the process
Γ(A−p)W is almost surely continuous.
t t≥0
{ }
Observe that for t 0 and p > 1 we have
≥ 2
t t
W⋄2 ds = (Γ(A−p)W )⋄2 ds
Z k s k−p Z k s k0
0 0
t
= I (A−pδ )⋄2 ds
Z k 1 s k0
0
t
= I ((A−pδ )⊗2) ds
Z k 2 s k0
0
t
= Z (2(|A−pδs|20)2)21ds
0
t
= √2 A−pδ 2ds
Z | s|0
0
7
t 3 −2p
= √2 k + ξ (s)2ds
Z0 Xk≥0(cid:16) 2(cid:17) k
3 −2p t
= √2 k + ξ (s)2ds
Xn≥0(cid:16) 2(cid:17) Z0 k
3 −2p
√2 k +
≤ X(cid:16) 2(cid:17)
k≥0
where the last series is convergent.
Definition 4.2 For each t 0 the process
≥
t
t X := W⋄2ds
7→ t Z s
0
is an element of (S)∗. We will refer to it as the quadratic white noise process.
Note that, since
t
Γ(A−p)X = (Γ(A−p)W )⋄2ds
t Z s
0
t
= (Γ(A−p)W )2 A−pδ 2ds,
Z s −| s|0
0
we deduce from the previous lemma that the function t Γ(A−p)X is almost surely
t
→
differentiable for p > 1.
The next theorem is the main result of the present section. Its proof makes use of several
formulas relating the Malliavin derivative D, translation operator T and Wick product
; the books [13], [14] and [15] are excellent references for the definition and properties
⋄
of those operators as well as for the formulas just mentioned.
Theorem 4.3 Let ϕ C3(R) be such that for each i 0,1,2,3 , ϕ(i) has at most
∈ ∈ { }
polynomial growth at infinity. Then for any p > 1 one has
t s ⋆p2 t s
ϕ˜ (X ) = ϕ(0)+4 ϕ′′′(X )⋆ K (r,s)dB ds+2 ϕ′′(X ) K2(r,s)dr ds
p t Z p s p (cid:16)Z p r(cid:17) Z p s ·(cid:16)Z p (cid:17)
0 0 0 0
t f s t f dX
s
+4 ϕ′′(X )⋆ K (r,s)dB W ds+ ϕ′(X ) ds,
Z (cid:16) p s p Z p r(cid:17)⋄ s Z p s ⋄ ds
0 0 0
f f
where K (r,s) := δp,δp and δp := A−pδ for any u R. Moreover, ( sK (r,s)dB )⋆p2
p h r si s u u s ∈ 0 p r
is a shorthand notation for K (r,s)dB ⋆ K (r,s)dB . R
0 p r p 0 p r
R R
Proof. We start applying the S-transform; for f S one has
∈
ϕ (X ), (f) = Γ(Ap)ϕ(Γ(A−p)X ), (f)
p t t
hh E ii hh E ii
= E[ϕ(Γ(A−p)X ) (Apf)]
f t E
= E[T ϕ(Γ(A−p)X )]
Apf t
= E[ϕ(T Γ(A−p)X )]
Apf t
8
= E[ϕ(Γ(A−p)T X )].
f t
Here we utilized the Girsanov theorem and a commutation relation between the trans-
lation operator T and the operator Γ(A−p). Now observe that
t t t t
T X = (W +f(s))⋄2ds = W⋄2ds+2 W f(s)ds+ f2(s)ds
f t Z s Z s Z s Z
0 0 0 0
and
t t t
Γ(A−p)T X = (Γ(A−p)W )⋄2ds+2 Γ(A−p)W f(s)ds+ f2(s)ds.
f t Z s Z s · Z
0 0 0
To ease the notation, write Wp for Γ(A−p)W ; therefore
t t
ϕ˜ (X ), (f) = E[ϕ(Γ(A−p)T X )]
p t f t
hh E ii
t t t
= E ϕ (Wp)⋄2ds+2 Wp f(s)ds+ f2(s)ds .
h (cid:16)Z s Z s · Z (cid:17)i
0 0 0
To further facilitate the writing of the next calculation we also set
t t t
Y := (Wp)⋄2ds+2 Wp f(s)ds+ f2(s)ds.
t Z s Z s · Z
0 0 0
Wenowapplytheusualchainrule(thediscussionprecedingthestatementofthetheorem
guarantees the differentiability of Y while the assumptions on ϕ allow us to differentiate
t
inside the expected value) to obtain
d d
ϕ˜ (X ), (f) = E[ϕ(Y )]
p t t
dthh E ii dt
= E[ϕ′(Y )((Wp)⋄2 +2Wp f(t)+f2(t))]
t t t
·
= E[ϕ′(Y )(Wp)⋄2]+2E[ϕ′(Y )Wp] f(t)+E[ϕ′(Y )]f2(t)
t t t t t
·
= + + ,
A B C
where
:= E[ϕ′(Y )(Wp)⋄2]
t t
A
:= 2E[ϕ′(Y )Wp] f(t)
t t
B ·
:= E[ϕ′(Y )] f2(t).
t
C ·
Recalling that Y = Γ(A−p)T X we can write
t f t
= E[ϕ′(Y )(Wp)⋄2]
t t
A
= E[ϕ′(Γ(A−p)T X )(Wp)⋄2]
f t t
= E[T (ϕ′(Γ(A−p)X )) (Wp)⋄2]
Apf t t
·
= E[ϕ′(Γ(A−p)X ) ((Wp)⋄2 (Apf))].
t t
· ⋄E
We now integrate twice by parts to obtain
= E[ϕ′(Γ(A−p)X ) ((Wp)⋄2 (Apf))]
t t
A · ⋄E
9
= E[D2 (ϕ′(Γ(A−p)X )) (Apf)]
δtp t ·E
= Γ(Ap)D2 (ϕ′(Γ(A−p)X )), (f)
hh δtp t E ii
that means
= (SΓ(Ap)D2 (ϕ′(Γ(A−p)X )))(f).
A δtp t
Moreover, by simple direct calculations we get
Dδ2tp(ϕ′(Γ(A−p)Xt)) = Dδtp(Dδtp(ϕ′(Γ(A−p)Xt)))
= Dδp(ϕ′′(Γ(A−p)Xt) DδpΓ(A−p)Xt)
t · t
= ϕ′′′(Γ(A−p)Xt)·(DδtpΓ(A−p)Xt)2 +ϕ′′(Γ(A−p)Xt)·Dδ2tpΓ(A−p)Xt,
as well as
t
DδtpΓ(A−p)Xt = Dδtp(cid:16)Z (Wsp)⋄2ds(cid:17)
0
t
= 2Z Wsp ⋄DδtpWspds
0
t
= 2 Wp δp,δp ds
Z s ·h s ti
0
t
= 2Γ(A−p) K (s,t)dB ,
(cid:16)Z p s(cid:17)
0
and
t
Dδ2tpΓ(A−p)Xt = Dδtp(cid:16)2Z WspKp(s,t)ds(cid:17)
0
t
= 2 K2(s,t)ds.
Z p
0
Plugging all the preceding quantities together we deduce
t 2
Γ(Ap)D2 (ϕ′(Γ(A−p)X )) = Γ(Ap) ϕ′′′(Γ(A−p)X ) 2Γ(A−p) K (s,t)dB
δtp t (cid:16) t ·(cid:16) Z p s(cid:17) (cid:17)
0
t
+Γ(Ap) ϕ′′(Γ(A−p)X ) 2 K2(s,t)ds
(cid:16) t · Z p (cid:17)
0
t ⋆p2 t
= 4ϕ′′′(X )⋆ K (s,t)dB +2ϕ′′(X ) K2(s,t)ds.
p t p (cid:16)Z p s(cid:17) p t ·Z p
0 0
f f
Similarly for the second term,
= 2E[ϕ′(Y )Wp] f(t)
t t
B ·
= 2E[T (ϕ′(Γ(A−p)X ))Wp] f(t)
Apf t t
·
= 2E[ϕ′(Γ(A−p)X ) (Wp (Apf))] f(t)
t t
· ⋄E ·
= 2E[Dδp(ϕ′(Γ(A−p)Xt)) (Apf)] f(t)
t ·E ·
= 2 Γ(Ap)Dδp(ϕ′(Γ(A−p)Xt)), (f) f(t).
hh t E ii·
10