LEANDRE, REMI

Osaka City University Repository

Leandre, R.
Osaka J. Math.
37(2000),139-145
LOGARITHMIC SOBOLEV INEQUALITIES
FOR DIFFERENTIABLE PATHS
REMI LEANDRE
(Received March 20, 1998)
1. Introduction
Measures over differentiable paths can be interesting in two ways:
-)The spin representation of a loop groop is very well understood for Cl loops
([25]).
-)There is a Morse theory theory for Cl loops, which is involved with the energy
functional (See [11] and the references therein).
For the first of these two reasons, there exist measures which are introduced for
differentiable paths or differentiable loops of a Lie group. In the case of a loop group,
they are introduced in [20] or in [21], in order to understand what is a string structure
over the Brownian bridge. Let us recall that string structures over the Brownian bridge
are very important in order to define the Dirac operator over the loop space (See [26],
[29], [14]).
In the case of an analytical approach of the Morse theory related to the symplectic
action, we can work with the measure defined over the loop space by the classical
Brownian bridge (See [19], [10]). But if the function over the loop space is the energy
functional (See [11]), we need to introduce a new measure, which is involved with
differentiable paths. It is the subject of [22].
In [22], we introduced a measure over the path space. We give integration by parts
formulas for differentiable paths, and we establish a Sobolev Calculus, which is in the
spirit of [13] and [17], [18]. The integration by parts formulas are very badly written in
terms of the tangent vector fields which are considered in [22]. In [17] or in [6], there
is the remark that if we follow Bismut's indication to writte the shape of the tangent
vector fields ([2]), the integration by parts formulas are better written.
In the first part, we consider the measure of [22] over the space of differentiable
paths, and we establish a Bismut Calculus associated to it. The main result is the
following: the transformation which gives a vector field in the manner of [13] or in the
manner of [22] into a vector into a vector field of Bismut is path by path bounded, but
not uniformly bounded. In order to control this transformation, we have to assume that
a quantity of the form exp[I(^)} (/(T) is the square of the supremum of the modulus
of the speed of the path) is in I/2. It is possible that this assumption is satisfied if
140 R. LEANDRE
we introduce a small parameter, which leads to a formalism analogous to small time
asymptotics of heat kernels. (The reader who is interested by short time asymptotics
expansions can see the surveys of Kusuoka ([15]), Leandre ([16]) or Watanabe ([27])).
If this integrability condition is checked, or in other words, if the parameter of the
equation which gives the measure is small enough, the cylindrical functionals belong
to Bismut's Sobolev spaces with one derivative in L2.
In the second part of this work, we show that there is a Clark-Ocone formula in
this context, associated to Bismut's derivatives of a functional, when we have fixed the
speed of the path in the starting time. The proof follows the ideas of [6].
This allows us, by following the method of [3] to deduce a Logarithmic Sobolev
inequality over the differentiable path space, if we suppose given the speed of the path
at the starting time. For that, we take the tangent space of [22]. But this gives a
weighted Logarithmic Sobolev inequality, because the transformation which gives a
tangent vector of [22] into a Bismut tangent vector is not uniformly bounded.
We thank S. Fang for helpfull discussions.
2. Bismut Calculus for Differentiable Paths
Let M be a compact Riemannian manifold. We endow it with the Levi-Civita
connection. Let B
s
 be a Brownian motion over TX(M) starting from x. We consider
the stochastic differential equation:
70 = x
e is a small parameter. C follows a Gaussian law of covariance Id and average 0 over
TX(M), independent of Bs. rs is the parallel transport over the path 7^, starting from
Id.
[22] has introduced the following tangent space of a path 7.: it is constituted of
the path τ
s
H
s
, where H
s
 has two derivatives with HQ = 0. [22] considers the Hubert
structure H-ffolP+Ji)1 \\d2/ds'2H\\2ds. Let us recall one of the main theorem of [22]. Let
F be a cylindrical functional over PX(M), the based path space of Cl differentiables
paths over M.
We get:
(1.2) E[< dF, X>} = E[FdίvX]
if Xt = τtHt where Ήt is deterministic. The expression of dίvX is given in [22], and
is slightly complicated:
DlFFERENTIABLE PATHS 141
dίvX = -/W1JO
•L
(I 3}V
 ' ' Γ
l
 <d2/ds2H
s
,δB
s
>
R is the curvature tensor of the manifold. In order to simplify the expression of
the divergence, we give the notion of Bismut's tangent vector field.
Let d/dsH0 and d
2/ds2K
s
 be deterministic given. A Bismut's vector field Xf is
given by XB — τtHf where HB is the solution of the differential equation:
Hg = 0
(1.4) d/dsHg = cH0
d2/ds2Ht
B
 = ed2/ds2Kt + τt-
lR(d/ds%, XtB)d/ds^t
By using the Gronwall lemma, we get that
( / \\d2/ds2HtB\\2dt)ϊ < C((f \\d2/ds2K\\2ds)ϊ
(1 5)
2
 + l)exp[Csup \\d/ds
Ίt\\
2}
We can choose e small enough such that for a vector field d/dsHo -\-f^d2/ds2K
u
du
adapted in L2, XB is still adapted in L2.
We define a tangent vector as XB(d/dsHQ,d2/ds2K) associated to the solution
of (1.4).
We get:
Theorem I. 1. Let e be small enough. Let d/dsHQ and let d
2/ds2K be determin-
istic. We get for all cylindrical functionals:
dF,XB(d/dsH0,d2/ds2K.) >} = E[< d/dsH0,(λ f\\
ί
l
 j2fJ 2π
-f / < τ
s
d2/ds K,
Jo
is the formal acceleration of 7S given by eτsδBs and δ is the Itό inte-
gral.
This integration by parts formula leads to a first order Sobolev Calculus, called
Bismut's Sobolev Calculus, which is not equivalent to the Sobolev Calculus developped
142 R. LEANDRE
in [22]. Namely the transformation (d/dsHo,d2/ds2K,) -> XB(d/dsH0,d2/ds2K.)
is not bounded in 7.
For a cylindrical functional, we put:
< dF, XB(d/dsH0, d2/ds2K.) > = < A, d/dsH0 >
+ [ <C,d2/ds2K
s
 >ds
Jo
in a unique way. We call A = dF°'B and C
s
 = dFB such that:
< dF,XB(d/dsHQ,d2/ds2K) > =
(1.8) /•i
/ <dFB,d2/ds2K
s
 >ds
Jo
We get:
DEFINITION I. 2: The first order Bismut Sobolev norm W\$ are given by:
(1.9) ||F||1|2 = ||F||La + || ||dF° *|| \\L, + ||( ΓJo
From the estimate (1.5), we get:
Theorem I. 3. If e is small enough, J lFHi^ is finite for any cylindrical functional
F.
3. Clark-Ocone Formula
We fix C in the equation (1.2). We get a Sobolev Calculus as in the previous part
with C fixed, or in other words, if d/ds^Q fixed. This implies that in the new Sobolev
Calculus that HQ = 0. We suppose e small enough in order that the exponential which
appears in (1.5) is in L2. We do in the sequel as if e — 1, in order to simplify the
formulas.
Let F be a cylindrical functional. Since C is supposed fixed, we have for a suitable
vector fields Z
a
(F) adapted to the filtration generated by the process #.,
Γ
1
(2.1) F - E[F] + / < τ
s
Z
s
(F),
Jo
Let Z
s
 be another adapted vector fields. We have:
DlFFERENTIABLE PATHS 143
r
l
 r
l
E( \ < dF
s
B
, Z
s
 > ds] = E[F < τ
s
Z
s
, δVd/ds
Ίs
 >]
Jo Jo
= E[ί <τ
s
Z
s
(F),δVd/ds
Ίs
>
(2.2) 7o
/ <τ
s
Z
s
,δVd/ds% >}
Jo
= E[ί Z
s
(F)Z
s
ds\
Jo
Moreover, we get the following Clark-Ocone formula (See [24], [6]):
Theorem II. 1.
Γ
l
(2.3) F = E(F] + / < τ
s
EG* [dF
s
B], δVd/ds
Ίs
 >
Jo
where G
s
 is the filtration spanned by B
s
.
4. Logarithmic Sobolev Inequalities
We follow exactly the method of Capitaine-Hsu-Ledoux ([3]), with C fixed. We
do as if e was small enough.
We underline that there are the Bismut tangent space and the tangent space of [22].
This leads to an H-derivative in the sense of Bismut:
/•i
(3.1) < dF,XB(d2/ds2K) >= \ < dFB,d2/ds2K
s
 > ds
Jo
and to an H-derivative in the sense of [22]:
Γ
l
(3.2) < dF,X(d2/ds2K,) >= < dFJ'L,d2/ds2K
s
 > ds
Jo
where X(d2/ds2K.) = rt(/0* /Qs d2/ds2Kududv).
Moreover by the estimates (1.5), we get:
r
l
 r
l
(3.3) / \\dF
s
B\\2ds<Cexp[\sup\\B
s
\\2} \ \\dFJ
s
'
L\\2ds
Jo Jo
for λ small enough, by the assumption which is done.
144 R. LEANDRE
The method of Capitaine-Hsu-Ledoux shows that:
£[F2LogF2] < E[F2}LogE(F2} + 2E[ f \\dF
s
B\\2ds]
Jo
(3.4)
Let be 1(7) = sup \\d/ds^t\\2. Since exp[λ sup \\BS\\2} < Cexp[λ7(7)], we de-
duce the following Logarithmic Sobolev inequality:
Theorem HI. 1.
(3.5) E[F2LogF2} < E[F2}LogE[F2} + C£[exp[λ7(7)] / \\dFsJ'L\\2ds]
Jo
where the exponential is in L2 for e small enough.
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DlFFERENTIABLE PATHS 145
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departement de Mathematiques
Institut Elie Cartan
Faculte des Sciences
Universite de Nancy I
54000. Vandoeuvre-les-Nancy
FRANCE



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