On a flow of transformations of a Wiener space

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by  J. Najnudel

ON A FLOW OF TRANSFORMATIONS OF A WIENER SPACE
J. NAJNUDEL, D. STROOCK, M. YOR
1 Abstract. In this paper, we define, via Fourier transform, an ergodic flow of transfor-
1 mations of a Wiener space which preserves the law of the Ornstein-Uhlenbeck process and
0 which interpolates the iterations of a transformation previously defined by Jeulin and Yor.
2
Then,wegiveamoreexplicitexpressionforthisflow,andweconstructfromitacontinuous

n gaussian process indexed by R2, such that all its restriction obtained by fixing the first
a
coordinate are Ornstein-Uhlenbeck processes.
J

5
2 1. Introduction

]
R An abstract Wiener space is a triple (H,E, ) consisting of a separable, real Hilbert
W
P space H, a separable real Banach space E in which H is continuously embedded as a dense
. subspace, and a Borel probability measure on E with the property that, for each x∗ E∗,
h
t the -distribution of the map x E Wx,x∗ R, from E to R, is a centered gau∈ssian
a W ∈ 7−→ h i ∈
m random distribution with variance khx∗k2H, where hx∗ is the element of H determined by
[ (h,hx∗)H = h,x∗ for all h H. See Chapter 8 of [5] for more information on this topic.
h i ∈
1 Because hx∗ : x∗ E∗ is dense in H and hx∗ H = ,x∗ L2(W), there is a unique
{ ∈ } k k kh· ik
v isometry, known as the Paley–Wiener map, : H L2( ) such that (h) = ,x∗
4 I 7−→ W I h· i
7 if h = hx∗. In fact, for each h ∈ H, I(h) under W is a centered Gaussian variable with
9 variance khk2H. Because, when h = hx∗, I(h) provides an extention of (·,h)H to E, for
4 intuitive purposes one can think of x [ (h)](x) as a giving meaning to the inner product
. I
1 x (x,h) , although for general h this will be defined only up to a set of -measure 0.
H
0 W
An important property of abstract Wiener spaces is that they are invariant under orthog-
1
1 onal transformations on H. To be precise, given an orthogonal transformation on H,
O
: there is a -almost surely unique T : E E with the property that, for each h H,
v O
W −→ ∈
i (h) T = ( ⊤h) -almost surely. Notice that this is the relation which one would
X O
I ◦ I O W
predict if one thinks of [ (h)](x) as the inner product of x with h. In general, T can be
r I O
a constructed by choosing {x∗m : m ≥ 1} ⊆ E∗ so the {hx∗m : m ≥ 1} is an orthonormal basis
in H and then taking
∞
TOx = hx,x∗miOhx∗m,
m=1
X
where the series converges in E for -almost every x as well as in Lp( ;E) for every
W W
p [1, ). See Theorem 8.3.14 in [5] for details. In the case when admits an extension
∈ ∞ O
as a continuous map on E into itself, T can be the taken equal to that extension. In any
O
case, it is an easy matter to check that the measure is preserved by T . Less obvious
O
W
is a theorem, originally formulated by I.M. Segal (cf. [6]), which says that T is ergodic if
O
and only admits no non-trivial, finite dimensional, invariant subspace. Equivalently, T
O
O
Date: January 27, 2011.
1
is ergodic if and only if the complexification has a continuous spectrum as a unitary
c
O
operator on the complexification H of H.
c
The classical Wiener space provides a rich source of examples to which the preceding
applies. Namely, take H = H1 to be the space of absolutely continuous h Θ whose
0 ∈
derivative h˙ is in L2([0,∞)), and set khkH01 = kh˙kL2([0,∞)). Then H01 with norm k · kH01 is
a separable Hilbert space. Next, take E = Θ, where Θ is the space of continuous paths
θ : [0, ) R such that θ(0) = 0 and
∞ −→
θ(t)
| | 0 as t > 0 tends to 0 or ,
t12 log(e+ logt ) −→ ∞
| |
and set
θ(t)
θ = sup | | .
k kΘ t>0 t12 log(e+ logt )
| |
Then Θ with norm is a separable Banach space in which H1 is continuously embedded
k · kΘ 0
as a dense subspace. Finally, the renowned theorem of Wiener combined with the Brownian
law of the iterated logarithm says that there is a Borel probability measure WH01 on Θ for
which (H01,Θ,WH01) is an abstract Wiener space. Indeed, it is the clasRsical Wiener space
on which the abstraction is modeled, and WH01 is the distribution of an -valued Brownian
motion.
One of the simplest examples of an orthogonal transformation on H1 for which the as-
0
sociated transformation on Θ is ergodic is the Brownian scaling map S given by S θ(t) =
α α
1
α−2θ(αt) for α > 0. It is an easy matter to check that the restriction Oα of Sα to H01 is
orthogonal, and so, since S is continuous on Θ, we can take T = S . Furthermore, as long
α Oα α
as α = 1, an elementary computation shows that lim g, nh = 0, first for smooth
6 n→∞ Oα H
g, h H1 with compact support in (0, ) and thence for all g, h H1. Hence, when α = 1,
∈ 0 ∞ (cid:0) ∈(cid:1) 0 6
admits no non-trivial, finite dimensional subspace, and therefore S is ergodic; and so,
α α
O
by the Birkoff’s Individual Ergodic Theorem, for p ∈ [1,∞) and f ∈ Lp(WH01),
n−1
1
nl→im∞ n f ◦Sαn = f dWH01
m=0 Z
X
both WH01-almost surely and in Lp(WH01). Moreover, since {Sα : α ∈ (0,∞)} is a mul-
tiplicative semigroup in the sense that S = S S , one has the continuous parameter
αβ α β
◦
version
1 a dα
al→im∞ loga (f ◦Sα) α = f dWH01
Z1 Z
of the preceding result.
A more challenging ergodic transformation of the classical Wiener space was studied by
Jeulin and Yor (see [1], [2] and [4]), and, in the framework of this article, it is obtained by
considering the transformation on H1, defined by
O 0
t h(s)
[ h](t) = h(t) ds. (1.1)
O − s
2 Z0
An elementary calculation shows that is orthogonal. Moreover, admits a continuous
O O
extension to Θ given by replacing h H1 in (1.1) by θ Θ. That is
∈ 0 ∈
t θ(s)
[T θ] = θ(t) ds for θ Θ and t 0. (1.2)
O
− s ∈ ≥
Z0
In addition, one can check that lim g, nh = 0 for all g, h H1, which proves that
n→∞ O H01 ∈ 0
TO is ergodic for WH01. (cid:0) (cid:1)
In order to study the transformation T in greater detail, it will be convenient to refor-
O
mulate it in terms of the Ornstein–Uhlenbeck process. That is, take HU to be the space of
absolutely continuous functions h : R R such that
−→
h 1h(t)2 +h˙(t)2 dt < .
k kHU ≡ s R 4 ∞
Z
(cid:0) (cid:1)
Then HU becomes a separable Hilbert space with norm . Moreover, the map F :
HU
k · k
H1 HU given by
0 −→
[F(g)](t) = e−2tg(et), for g H1 and t R, (1.3)
∈ 0 ∈
is an isometric surjection which extends as an isometry from Θ onto Banach space of con-
U
tinuous ω : R satisfying lim |ω(t)| = 0 with norm ω = sup log(e+ t ) −1 ω(t) .
−→ |t|→∞ log|t| k kU t∈R | | | |
Thus, (HU,U,WHU) is an abstract Wiener space, where WHU = F∗W(cid:0) H01 is the(cid:1)image of
WH01 under the map F. In fact, WHU is the distribution of a standard, reverisible Ornstein–
Uhlenbeck process.
Notethatthe scaling transformationsfor theclassical Wiener space become translations in
the Ornstein–Uhlenbeck setting. Namely, for each α > 0, F S = τ F, where τ denotes
α logα s
◦ ◦
the time-translation map given by [τ ω](t) = ω(s+ t). Thus, for s = 0, the results proved
s
6
about the scaling maps say that τ is an ergodic transformation for . In particular, for
s HU
W
p [1, ) and f Lp( HU),
∈ ∞ ∈ W
1 n−1 1 T
lim f τ = lim f τ ds = f d
ns s HU
n→∞ n ◦ T→∞ T ◦ W
m=0 Z0 Z
X
both -almost surely and in Lp( ).
HU HU
W W
The main goal of this article is to show that the reformulation of transformation T
O
coming from the Jeulin–Yor transformation in terms of the Ornstein–Uhlenbeck process
allows us to embed T in a continuous-time flow of transformations on the space , each
O
U
of which is WH01-measure preserving and all but one of which is ergodic. In Section 2, this
flow is described via Fourier transforms. In Section 3, a direct and more explicit expression,
involving hypergeometric functions and principal values, is computed. In Section 4, we study
the two-parameter gaussian process which is induced by the flow introduced in Section 2.
In particular, we compute its covariance and prove that it admits a version which is jointly
continuous in its parameters.
2. Preliminary description of the flow
Let and T be the transformations on H1 and Θ given by (1.1) and (1.2), and recall
O O 0
the unitary map F : H1 HU in (1.3) and its continuous extension as an isometry from
0 −→
3
Θ onto . Clearly, the inverse of F is given by
U
F−1(ω)(t) = √tω(logt) for t > 0.
Because F is unitary and is orthogonal on H1, F F−1 is an orthognal transfor-
O 0 − ◦O◦
mation on HU, and because
S := F T F−1
O
− ◦ ◦
is continuous extension of F F−1 to , we can identify S as T−F◦O◦F−1.
− ◦O◦ U
Another expression for action of S is
∞
[S(ω)](t) = ω(t)+ e−2sω(t s)ds for t R.
− − ∈
Z0
Equivalently,
S(ω) = ω µ,
∗
where µ is the finite, signed measure µ given by
µ := δ +e−2t1 dt.
0 t≥0
−
To confirm that ω µ is well-defined as a Lebesgue integral and that it maps continuously
into itself, note tha∗t, for any ω and t R, U
∈ U ∈
∞ ∞
s s
e−2 ω(t s) ds ω e−2 log e+ t +s)ds
U
| − | ≤ k k | |
Z0 Z0
(cid:0) ∞
s
ω log(e+ t ) e−2(1+s)ds 9 ω log(e+ t )
U U
≤ k k | | ≤ k k | |
Z0
The Fourier transform µ of µ is given by
∞ 1 1 2iλ
µ(λ) = e−iλxdµ(x) = 1+ e−x(1/2+iλ)dx = 1+ = − = e−2iArctg(2λ).
b
− − 1/2+iλ 1+2iλ
ZR Z0
Hbence, for all h HU and λ R,
∈ ∈
[
h µ(λ) = e−2iArctg(2λ)h(λ), (2.1)
∗
which, since
b
1
h 2 = h(λ) 2 1+4λ2 dλ,
k kHU 8π | |
R
Z
(cid:0) (cid:1)
provides another proof that S ↾ HU is isometbric.
The preceeding, and especially (2.1), suggests a natural way to embed S ↾ HU into a
continuous group of orthogonal transformations. Namely, for u R, let µ∗u to be the unique
∈
tempered distribution whose Fourier transform is given by
µ∗u(λ) = e−2iuArctg(2λ), (2.2)
and define uϕ = ϕ µ∗u for ϕ in the Schwartz test function class S of smooth functions
S ∗ c
which, together with all their derivatives, are rapidly decreasing. Because
uϕ(λ) = e−2iuArctg(2λ)ϕˆ(λ),
S
4
d
it is obvious that u has a unique extension as an orthogonal transformation on HU, which
we will again denoSte by u. Furthermore, it is clear that u+v = u v for all u,v R.
Finally, for all g,h HU,Su R, S S ◦ S ∈
∈ ∈
1
(g, uh)HU = g(λ)h(λ)e−2iuArctg(2λ)(1+4λ2)dλ
S 8π
R
Z
= 1 bπ/2 gb tan(τ) h tan(τ) 1+tan2(τ) 2e−2iuτ dτ,
16π 2 2
Z−π/2 (cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
where b
b
1 π/2 tan(τ) tan(τ) 1
g h 1+tan2(τ) 2 dτ = g(λ) h(λ) (1+4λ2)dλ
16π 2 2 8π | || |
Z−π/2(cid:12) (cid:18) (cid:19)(cid:12) (cid:12) (cid:18) (cid:19)(cid:12) ZR
(cid:12) (cid:12) (cid:12) (cid:12)(cid:0) (cid:1)
1 (cid:12)b (cid:12) (cid:12)b 1/2 (cid:12) 1/2 b b
(cid:12) g(λ) 2(1+(cid:12) (cid:12)4λ2)dλ (cid:12) h(λ) 2(1+4λ2)dλ = g HU h HU < .
≤ 8π | | | | || || || || ∞
R R
(cid:18)Z (cid:19) (cid:18)Z (cid:19)
Hence, by Riemann–Lebesgue lemma, shows that (g, uh) tends to zero when u goes to
b HU
b S | |
infinity.
Now define the associated transformations Su := TSu on for each u R. By the
U ∈
general theory summarized in the introduction and the preceding discussion, we know that
Su : u R is a flow of HU-measure preserving transformations and that, for each u = 0,
{ ∈ } W 6
Su is ergodic.
3. A more explicit expression
So far we know very little about the transformations Su for general u R. By getting a
∈
handle on the tempered distributions µ∗u, in this section we will attempt to find out a little
more.
We begin with the case when u is an integer n Z. Recalling that µ = δ0 +e−2t1t≥0dt,
∈ −
one can use induction to check that, for n 0,
≥
µ∗n = ( 1)n δ +e−2tL′ (t)1 dt ,
− 0 n t≥0
where L is the nth Laguerre polynomial. Indeed, the Laguerre polynomials satisfy the
n (cid:0) (cid:1)
following relations: for all n 0,
≥
L (0) = 1
n
and for all n 0, t R,
≥ ∈
L′ (t) = L′ (t) L (t).
n+1 n − n
Similarly, starting from µ∗−1 = δ0 +e2t1t≥0dt, one finds that
−
µ∗n = ( 1)n δ +e2tL′ ( t)1 dt
− 0 n − t≤0
for n 0. In particular, µ∗n is a finite, signed measure for n Z and Snω can be identified
(cid:0) (cid:1)
as µ∗n≤ ω for all ω and n Z. ∈
As th∗e next resul∈t sUhows, wh∈en u / Z, µ∗u is more singular tempered distribution than a
∈
finite, signed measure.
Proposition 3.1. For each u / Z, the distribution µ∗u is given by the following formula:
∈
sin(πu)
µ∗u = cos(πu)δ (x)+ pv(1/x)+Φ (x), (3.1)
0 u
π
5
where pv denotes the principal value, and Φ L2(R) is the function for which Φ (x) equals
u u
∈
u sin(πu) ∞ (1 usgn(x)) x k Γ′ Γ′
e−|x|/2 − k| | (1+k usgn(x)) (1+k)
− π k!(k +1)! Γ − − Γ

k=0 (cid:20)
X
Γ′ sin(πu) sinπu
(2+k)+log( x ) + ,
−Γ | | πx − πx
(cid:21) (cid:19)
Γ′/Γ being the logarithmic derivative of the Euler gamma function and ( ) being the
k
Pochhammer symbol.
Proof. Define the functions ψu and θu from R∗ = R 0 to R so that θu(x) = e−x2ψu(x) and
\{ }
ψ (x) equals
u
u sin(πu) ∞ (1 usgn(x)) x k Γ′ Γ′
k
− | | (1+k usgn(x)) (1+k)
− π k!(k +1)! Γ − − Γ
k=0 (cid:20)
X
Γ′ sin(πu)
(2+k)+log( x ) + .
−Γ | | πx
(cid:21)
From Lebedev [3], p. 264, equation (9.10.6), with the parameters α = 1 u or α = 1+u,
n = 1, z = x or z = x, the function ψ satisfies, for all x R∗, the differ−ential equation:
u
− ∈
xψ′′(x)+(2 x )ψ′(x)+(u sgn(x))ψ (x) = 0,
u −| | u − u
and grows at most polynomially at infinity. One then deduces that θ decreases as least
u
exponentially at infinity, and satisfies (for x = 0) the following equation:
6
x
xθ′′(x)+2θ′(x)+ u θ (x) = 0. (3.2)
u u − 4 u
(cid:16) (cid:17)
At the same time, by writing
e−|x|/2 = (e−|x|/2 1)+1
−
and expanding θ (x) accordingly, we obtain:
u
sin(πu) usin(πu) Γ′ Γ′ Γ′
θ (x) = (1 usgn(x)) (1) (2)+log( x )
u
πx − π Γ − − Γ − Γ | |
(cid:20) (cid:21)
sin(πu)
sgn(x)+η (x),
u
− 2π
for
η (x) = xη(1)(x)+ x η(2)(x)+xlog( x )η(3)(x)+ x log( x )η(4)(x),
u u | | u | | u | | | | u
(1) (2) (3) (4)
where η , η , η , η are all smooth functions. The derivatives of the functions x, x ,
u u u u
| |
xlog x , x log x inthesense ofthedistributions areobtainedbyinterpreting their ordinary
| | | | | |
derivatives as distributions. Similarly, the product by x of their second distributional deriva-
tives are obtained by multiplying their ordinary second derivatives by x. Hence, both η′(x)
u
and xη′′(x) as distributions can be obtained by computing η′(x) and xη′′(x) as functions on
u u u
R∗.
Now, let ν be the distribution given by the expression:
u
sin(πu) sin(πu)
ν (x) = cos(πu)δ (x)+ pv(1/x)+ θ (x) . (3.3)
u 0 u
π − πx
(cid:20) (cid:21)
6
Note that the term in brackets, in the definition of ν , is a locally integrable function, and
u
that ν coincides with the function θ in the complement of the neighborhood of zero. Let us
u u
now prove that ν satisfies the analog of the equation (3.2), in the sense of the distributions.
u
One has:
sin(πu) usin(πu) Γ′
ν (x) = cos(πu)δ (x)+ pv(1/x) (1 usgn(x))
u 0
π − π Γ −
(cid:20)
Γ′ Γ′ sin(πu)
(1) (2)+log( x ) sgn(x)+η (x).
u
−Γ − Γ | | − 2π
(cid:21)
Since
Γ′ Γ′ d (Γ(1+u)Γ(1 u)) d (πu/sin(πu)) 1
(1+u) (1 u) = du − = du = πcot(πu),
Γ − Γ − Γ(1+u)Γ(1 u) πu/sin(πu) u −
−
one obtains, after straightforward computation,
sin(πu) ucos(πu) usin(πu)
ν (x) = cos(π)δ (x)+ pv(1/x) sgn(x) log( x )+c(u)+η (x),
u 0 u
π − 2 − π | |
where c(u) does not depend on x. One deduces that
sin(πu)
ν (x) = cos(πu)δ (x)+ pv(1/x)+χ (x),
u 0 u,1
π
where χ denotes a locally integrable function. Moreover,
u,1
sin(πu) usin(πu)
ν′(x) = cos(πu)δ′(x) fp(1/x2) ucos(πu)δ (x) pv(1/x)+η′(x),
u 0 − π − 0 − π u
where fp(1/x2) denotes the finite part of 1/x2, and then
sin(πu) usin(πu)
xν′(x) = cos(πu)δ (x) pv(1/x) +xη′(x).
u − 0 − π − π u
By differentiating again, one obtains:
sin(πu)
ν′(x)+xν′′(x) = cos(πu)δ′(x)+ fp(1/x2)+η′(x)+xη′′(x).
u u − 0 π u u
Therefore,
x sin(πu)
xν′′(x)+2ν′(x)+ u ν (x) = χ (x)+ cos(πu)δ′(x)+ fp(1/x2)
u u − 4 u u,2 − 0 π
(cid:18) (cid:19)
(cid:16) (cid:17)
sin(πu) usin(πu)
+ cos(πu)δ′(x) fp(1/x2) ucos(πu)δ (x) pv(1/x)
0 − π − 0 − π
(cid:18) (cid:19)
sin(πu)
+u cos(πu)δ (x)+ pv(1/x) = χ (x),
0 u,2
π
(cid:18) (cid:19)
where χ is a locally integrable function. Since θ satisfies (3.2), χ is identically zero.
u,2 u u,2
Hence, ν is a tempered distribution solving the differential equation:
u
x
xν′′(x)+2ν′(x)+ u ν (x) = 0,
u u − 4 u
(cid:16) (cid:17)
or equivalently,
x d2
ν (x) (xν (x)) uν (x) = 0.
4 u − d2x u − u
7
Multiplying by 4i and taking the Fourier transform (in the sense of the distributions), one
−
deduces:
ν ′(λ)(1+4λ2) = 4iuν (λ).
u u
−
This linear equation admits a unique solution, up to a multiplicative factor c:
b b
λ 4iu
ν (λ) = cexp − dt = cexp( 2iuArctg(2λ)).
u
1+4t2 −
(cid:18)Z0 (cid:19)
Hence, ν is proportional to µ∗u. In order to determine the constant c, let us observe that
u b
the distribution ν given by
u,0
csin(πu)
ν (x) = ν (x) ccos(πu)δ (x) pv(1/x)
u,0 u 0
− − π
admits the Fourier transform:
ν (λ) = ce−2iuArctg(2λ) ce−πiusgn(λ).
u,0
−
One deduces that ν is a function in L2, which implies that ν is also a function in L2,
u,0 u,0
and then locally integrable.dSince the last term in (3.3) is also a locally integrable function,
one deduces that c = 1, and then
d
µ∗u = ν ,
u
which proves Proposition 3.1. (cid:3)
The reasonably explicit expression for µ∗u found in Proposition 3.1 yields a reaonably
explicit expression for the action of u. Indeed, only the term pv(1/x) is a source of concern.
S
However, convolution with respect of pv(1/x) is, apart from a multiplicative constant, just
the Hilbert transform, whose properties are well-known. In particular, it is a translation
invariant, bounded map on L2(R), and as such it is also a bounded map on HU. Thus,
we can unambiguously write u(h) = h µ∗u for all h HU. On the other hand, the
S ∗ ∈
interpretation of ω µ∗u for ω needs some thought. No doubt, ω µ∗u is well-defined as
∗ ∈ U ∗
an element of S ′, the space tempered distributions, but it is not immediately obvious that it
is can be represented by an element of or, if it can, that the element of which represents
U U
it can be identified as Suω. In fact, the best that we should expect is that such statements
will be true of -almost every ω . The following result justifies that expectation.
HU
W ∈ U
Proposition 3.2. For -almost every ω , the tempered distribution ω µ∗u is repre-
HU
W ∈ U ∗
sented by an element of which can be can be identified as Suω.
U
Proof. Recall that, for ϕ S, ϕ µ∗−u is the element of S whose Fourier transform is given
∈ ∗
by
ϕ\µ∗−u(λ) = ϕ(λ)e2iuArctg(2λ) for all λ R.
∗ ∈
Also, if T S ′, then T µ∗u is the tempered distribution whose action on ϕ S is given
∈ ∗ ∈
by b
S ϕ,T µ∗u S′ = S ϕ µ∗−u,T S′.
h ∗ i h ∗ i
Now choose an orthonormal basis h : n 1 for HU all of whose members are elements
n
{ ≥ }
of S, and, for each n 1, set g = 1h +h′′. Next, think of g as the element of ∗ whose
≥ n 4 n n n U
action on ω is given by
∈ U
U ω,gn U∗ = S gn,ω S′.
h i h i
8
It is then an easy matter to check that, in the notation of the introduction, h = h . Hence,
n gn
if B is the subset of ω for which
∈ U
n n
ω = lim S gn,ω S′hn and Suω = lim S gn,ω S′hn µ∗u,
n→∞ h i n→∞ h i ∗
m=1 m=1
X X
where the convergence is in , then (B) = 1.
HU
U W
Now let ω B. Then, for each ϕ S,
∈ ∈
n
S ϕ,ω µ∗u S′ = S ϕ µ∗−u,ω S′ = lim S gn,ω S′S ϕ,hn µ∗u S′
h ∗ i h ∗ i n→∞ h i h ∗ i
m=1
X
n
= lim S gn,ω S′S ϕ, uhn S′ = S ϕ,Suω S′.
n→∞ h i h S i h i
m=1
X
Thus, for ω B, ω µ∗u S ′ is represented by Suω . (cid:3)
∈ ∗ ∈ ∈ U
4. A two parameter gaussian process
By construction, Suω(t) : (u,t) R2 is a gaussian family in L2( ). In this conclud-
HU
{ ∈ } W
ing section, we will show that this family admits a modification which is jointly continuous
in (u,t).
Let ϕ,ψ S and u,v R2 be given. Then, by Proposition 3.2, for HU-almost every
∈ ∈ W
ω ,
∈ U
ϕ(s)ψ(t)(Su(ω))(s)(Sv(ω))(t)dsdt = S ϕ,ω µ∗u S′S ψ,ω µ∗v S′,
h ∗ i h ∗ i
ZZ
R2
where the integral in the left-hand side is absolutely convergent. Because E Suω(t)2 is
WHU
finite and independent of (u,t) ∈ R2, by taking the expectation with respect t(cid:2)o WHU a(cid:3)nd
using (2.2), one can pass from this to
ϕ(s)ψ(t)EWHU (Su(ω))(s)(Sv(ω))(t) dsdt = EWHU Shϕ,ω ∗µ∗uiS′Shψ,ω ∗µ∗viS′
ZZ
R2 h i
(cid:2) (cid:3)
2 ∞ e2i(u−v)Arctg(2λ) 2 ei[(t−s)λ+2(u−v)Arctg(2λ)]
= ϕ(λ)ψ(λ)dλ = ϕ(s)ψ(t)dsdtdλ.
π 1+4λ2 π 1+4λ2
Z−∞ ZZZ
R3
b
b
Hence,
2 ∞ ei[(t−s)λ+2(u−v)Arctg(2λ)]
E [Su(ω))(s)(Sv(ω))(t)] = dλ, (4.1)
WHU π 1+4λ2
Z−∞
first for almost every and then, by continuity, for all (s,t) R2. In particular, we now know
that the -distribution of Su(ω))(t) : (u,t) R2 is s∈tationary.
HU
W { ∈ }
To show that there is a continuous version of this process, we will use Kolmogorov’s
continuity criterion, which, because it is stationary and gaussian, comes down to showing
that
1 E [(Su(ω))(s)(Sv(ω))(t)] C (u,s) (v,t) α
− WHU ≤ −
9
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
for some C < and α > 0. But
∞
2 ∞ dλ
1 E [(Su(ω))(s)(Sv(ω))(t)] ei[(t−s)λ+2(u−v)Arctg(2λ)] 1
− WHU ≤ π 1+4λ2 −
Z−∞
(cid:12) 2 ∞ dλ (cid:12) 2 ∞ dλ (cid:12) (cid:12)
(cid:12) ei(t−s)λ 1 +(cid:12) e2i(cid:12)(u−v)Arctg(2λ) 1 (cid:12)
≤ π 1+4λ2 − π 1+4λ2 −
Z−∞ Z−∞
2 ∞ dλ (cid:12) (cid:12) 4 ∞ dλ (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
( t s λ 2)+ (u v)Arctg(2λ) ,
≤ π 1+4λ2 | − || |∧ π 1+4λ2| − |
Z−∞ Z−∞
and, after simple estimation, this shows that
1
1 E[(Su(ω))(s)(Sv(ω))(t)] C u v + t s 1+log 1+ ,
| − | ≤ | − | | − | (t s)2
(cid:20) (cid:18) (cid:18) − (cid:19)(cid:19)(cid:21)
where C < . Clearly, the desired conclusion follows.
∞
Remark 4.1. A question about filtrations comes naturally when one considers the group of
transformations (Su)u∈R on the space U. Indeed, for all t,u ∈ R, let Ftu be the σ-algebra
generatedby the -negligiblesubsets of ofandthevariables(Su(ω))(s), fors ( ,t]
HU
W U ∈ −∞
(these variables are well-defined up to a negligible set). From the results of Jeulin and Yor,
oune Rqu:ite easily deduces the following properties of the filtrations of the form (Ftu)t∈R for
∈
For all t,u R, u is generated by u+1 and (Su(ω))(t).
• For all t,u ∈ R, Ftu+1 and (Su(ω))(tF) tare independent under .
• Forall t,u ∈R, tFhet decreasing intersection of u+n for n Z iWs tHriUvial (i.e. it satisfies
• ∈ Ft ∈
the zero-one law).
If u R is fixed, the σ-algebra generated by u+n for t R does not depend on
• n Z∈. Ft ∈
∈
All these statements concern the sequence of filtrations ( u+n)n∈Z for fixed u R. A natural
F ∈
question arises: how can these results be extended to the continuous family of filtrations
( u)u∈R? Unfortunately, for the moment, we have no answer to this question (in particular
F
the family does not seem to be decreasing with u).
References
[1] T. Jeulin, M. Yor, Filtration des ponts browniens et ´equations diff´erentielles stochastiques lin´eaires,
S´eminaire de Probabilit´es, XXIV, Lecture Notes in Math., vol. 1426, Springer-Verlag, 1990, pp. 227–
265.
[2] P.-A. Meyer, Sur une transformation du mouvement brownien due `a Jeulin et Yor, S´eminaire de Proba-
bilit´es, XXVIII, Lecture Notes in Math., vol. 1583, Springer-Verlag,1994, pp. 98–101.
[3] N.-N. Lebedev, Special functions and their applications (Translated from Russian), Dover, New York,
1972.
[4] M. Yor, Some aspects of Brownian motion. Part I: Some Special Functionals, Lectures in Mathematics,
Ed. ETH Zu¨rich, Birkhau¨ser, 1992.
[5] D. Stroock, Probability Theory, an Analytic View, 2nd edition, Cambridge University Press, 2011.
[6] D. Stroock, Some thoughts about Segal’s ergodic theorem, Colloq. Math. vol. 118 #1, pp. 89-105,2010.
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