A Quasi-Invariance Theorem for Measures on Banach Spaces
Author(s): Denis Bell
Source: Transactions of the American Mathematical Society, Vol. 290, No. 2 (Aug., 1985), pp.
851-855
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2000320
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TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 290 Number 2} August 1985
A QUASI-INVARIANCETHEOREM
FOR MEASURES ON BANACH SPACES
BY
DENIS BELL
ABSTRACT. We show that for a measure ? on a Banach space directional
differentiabilityimplies quasi-translationinvariance. This result is shown to
imply the Cameron-Martintheorem. A second application is given in which ty
is the image of a Gaussian measure under a suitably regularmap.
1. Introduction. A (Borel)measureon a vectorspaceis said to be quasiinvariantundera vectorh if the classof its nullsets is preserved
by h-translation.
It is knownthatfora measureon Rn quasi-invariance
underalltranslations
implies
equivalence
to the Lebesguemeasure(see §4.3). This resulthas no analoguein
infinitedimensions.Howeverknowledgeof the quasi-invariance
propertiesof a
measureon an infinitedimensional
vectorspaceprovidesimportantinformation
aboutthe structureof the measure.
In the workdescribedherewe showthat for a measureon a Banachspace
differentiability
in a directionh impliesquasi-invariance
underh. Twoapplications
of this theoremaregiven.In the firsteyis assumedto be a Gaussianmeasureand
the Cameron-Martin
theoremis obtained.In the secondapplicationwe consider
the projectionof a Gaussianmeasureundera map satisfyingcertainregularity
conditions.
Thisworkwasmotivatedby the recentdevelopment
of the MalliavinCalculus.
MalliavinXs
theoryprovidesa methodof establishing
the directional
differentiability
of the measureinducedon Rn by the solutionof a stochasticdifferential
equation
(see [4],forexample).The corresponding
resultobtainedby an extensionof this
programto an infinitedimensional
settingwouldproducemeasureson a Banach
spaceE whichare differentiable
in a denselinearsubspace.In conjunction
with
the presenttheorem,this wouldthenestablishthe quasi-invariance
of the induced
measureson E.
Thereare manydifferentnotionsof measuredifferentiability.
The definition
adoptedhereis closelyrelatedto Skorohod's
Logarithmic
Differentiability
(see [5,
p. 121])andwasmotivatedby Malliavin's
result.
Supposethen that ? is a finiteBorelmeasureon a BanachspaceE, and h is
an elementof E. Wewill say that a functionX definedon E is h- C1 if + has
a Gateauxderivativein the directionh, denotedby DhX, andforalmostall x the
function+(x + *)is continuous
alongthe linespannedby h. Themeasurea(h + )
willbe denotedby Ah
Received by the editors October 15, 1983 and, in revised form, June 8, 1984 and November
10, 1984.
1980 MdhematicsSuUectC- ifiaXn. Primary 46G12.
t1985 American Mathematical Society
0002-9947/85 $1.00 + $.25 per page
851
As N oo this converges to
DENIS BELL
852
DEFINITION.eyis h-differentiableif there exists an L2 functionX such that the
relation
|
(1)
E
d?(X)= |
Dh(z)
E
(x)X(x) d?(x)
holds for all h- C1 functionsX with X and DhX bounded.
2. In this section we provethe following
THEOREM. S?lpposethat ? is h-differentiable,X(x + ) is continuousin the
directionh for almost all x and that
rt
SUp lx(X +
tE [O,1]
sup exp -4
th) 14 and
s,tE [O,1]
X(x + uh) du
j
8
?lnderh with
are locallyintegrable.Theneyis q?lasi-invariant
dah = exp- J: X(x + uh) du a.s.
The proofof the theoremrequiresthe following
LEMMA. If ? is h-differentiable,then (1) holdsfor everyh - C1 functionX such
that X and SUpt[O
]
are in L2 for some E > O.
+ th)l
gDh(x
PROOF. We first assumethat X is bounded. For each N let
N(X)
Then XN
Now
=
1/N
X(x + uh) du.
NJ
o
is bounded,h - C1 and DhXN is also bounded. Hence (1) holds for XN
IE
DhXN(z) d?(x) = |
N
X
(x+-N)
-Q(x)
d?(x).
dt | (z+th)deS(z)lt=o
(2)
providedthis exists. For t E [O,s]
1
J
t
p
[X(x+ th) - X(x)]dol(x)=
/
(3)
j
Dh+(x + uh) dud(x)
Et °
= |
O E
|
Dh(x+uh)d(x)du.
Herewe have used Fubini'stheoremto interchangeordersof integration.Differentiating with respect to t at t = Oin (3) gives (2) = J8E Dh+(x) d(x). Hence we
have shownthat as N ) oo
DhXN(z) d?(x) ) | Dh+(x) d?(x)
E
E
N in (1), appliedto each s)N
over
limit
the
) X in L2 so taking
Now XN
that (1) also holds for X.
|
shows
E* H*
1
H E.
A QUASI-INVARIANCE THEOREM
853
Wewill nowremovethe boundedness
assumption
on X. SupposeX satisfiesthe
assumption
of the lemma.Let BM: R > [0,1] be a sequenceof smoothfunctions
suchthat
(i) BM(X)= 1 if Ixl < M.
(ii) BM(X)= Oif Ixl > M + 1.
(iii) DBM(x)is uniformly
boundedin M andx.
ThenforeachM, FM = XBMo X is a boundedfunctionwhichalsosatisfiesthe
hypothesisof the lemma.So (1) holdsforFM. Nowtakingthe limitoverM and
usingthe factthat X, DhX andX arein L2 gives(1) forX. z
PROOF OF THE THEOREM.It sufficesto showthat, for everyboundedreal
valuedcontinuous
function0 withboundedsupport,
(4)
y
E
0(x) d?(x) = y
E
0(x + h) exp- y
O
X(x + uh)dud7(x).
Wewill showthis underthe additionalassumptions
that 0 is h- C1 and has
DhObounded.(4) will thenfollowin the requiredgeneralityby approximating
0
by a sequenceof functionsONdefinedas in the proofof the lemma.
Foreaseof notation,let us denoteexp- t0 X(x+uh) du by E(t). Wenowdefine
a functiong from[0,1]intoR by
(5)
g(t) = X 0(x + th)E(t) da(x).
E
It is clearthat g is continuous.Wewill provethat g is constantby showingthat
g'--O on (0,1). First note that the integrandin (5) is differentiable
in t with
derivative
(6)
G(X,t) = [Dh0(X+ th) - 0(X+ th)X(x + th)]E(t).
As G(x, t) is continuous
in t andsupt[O1]IG(x,t)l E L1it followsbytheDominated
Convergence
Theoremthat SE G(x, t) da(x) is alsocontinuous
in t. Hencewe may
differentiate
underthe integralsignin (5) to obtaing'(t) = SE G(x, t) da(x).
Nowobservethat foralmosteveryx, E(t) is h-differentiable
and
DhE(t) = E(t) [X(X)-X(X + th)]
Substituting
this into (6) gives
(7)
G(x, t) = Dh0(x+ th)E(t)+ 0(x + th)DhE(t)- 0(x + th)E(t)X(x).
NowthefunctionX(x)-- 0(x+th)E(t) satisfiestheconditions
on X in thelemma,
so (1) holds.Sincethe h-derivative
of X consistsof the firsttwotermson the righthandsidein (7), integrating
withrespectto eyin (7) givesSE G(x, t) da(x) = Oas
required.
3. Wenowgivetwoapplications
of this result.
(1) Supposethat (i, H, E) is anabstractWienerspaceandeyis Gaussianmeasure
on E. Let E* be embeddedin E in the usualway,via the inclusions
(See [2]for detailsof abstractWienerspaces.) It is shownin [3]that if h E H,
then eyis h-differentiable
and X(x) = (h,x), where(h, ) is an extensionof the
DENIS BELL
854
innerproducton H to an L2 randomvariableon E. In this case, for almost all x,
X(x + th) = (h, x) + tlhlH. This implies the continuityrequiredby the theorem
togetherwith the inequalities
(8)
sup IX(x+th)l4
<
[|(h,x)l + IhlH]4,
tE[O,l]
6
(9)
pt
sup exp g -4
s,tE [O, 1]
8
X(x + uh) du 0
j
t
s
<
exp{41(h, x) | + 2ghlH}.
J
Since (h,x) has a normal distribution,both (8) and (9) are integrable. So the
theoremimpliesthat eyis quasi-invariantunder h with
dah = exp -{(h, x) + 1/21hlH} a.s.
This is the Cameron-Martin
theorem.
(2) Assumingthe same notationas in example(1), supposethat T is an injective
map from E to a second Banach space K. Suppose that for some h E K the
followingconditionsare satisfied:
(i) For any y E RangeT and t E R, y + th E RangeT and T-1(y + th) is
continuousin t.
(ii) T is C1 and thereexists a C1 map S fromE into E* suchthat DT(x)S(x)h.
(iii) The maps T, T-1, S and x ) lYaceHDS(x) are boundedon boundedsets.l
Let us denote the inducedmeasureon K, T(a) by v. The followingcalculation
showsthat v is h-differentiablewith directionalderivative
(10)
X(x)= (S(T-lx),T-1x)-lYaceHDS(T-lx),
v-a.s.,
where (, ) denotesthe pairingbetweenE* and E. Since underconditions(i), (ii)
and (iii) X satisfies the requiredhypothesesthe theoremimplies that M is quasiinvariantunderh.
To establish (10) we proceedas follows:
r
/
Dh(x) dv(x) =
/
K
,.
Dh(TX) da(X)=
j
E
=
X
D( o T)(z)S(z) d?(z)
j
E
X o T(x){(S(x),
x) - AaceHDS(x)} d?(z).
E
The last equalityis obtainedby applyingthe divergencetheorem (see [1]). Hence
(10) follows.
4. Concluding remarks. 1. The conclusionof the theoremmay be stated in
the form: eyis quasi-invariantunderth, for any t E [0,1], with
dath = exp- J: X(X + Uh)dU a.s.
2. Skorohodhas studiedthe relationshipbetweendirectionaldifferentiabilityand
quasi-translationfor a measureon a Hilbert space [5, hh21-23].In this setting he
lBy way of explanation here we remarkthat as S is a C1 map from E to E*, then, for each
x, DS(x) restricts to give a trace class operator on H.
A QUASI-INVARIANCE THEOREM
855
the techniques
provesa theoremsimilarto oursunderweakerhypotheses.SIowever
workdependuponthe Hilbertstructureof the measurespace
usedin Skorohod's
andhencedo not applyin the presentsituationunlessE is Hilbertizable.
if (1) holds
theneycanbe shownto be h-differentiable
3. IfE is finitedimensional,
for all C1 functionswith compactsupport.Supposethat, for someorthonormal
Then the
basis el,...,en, tyhas continuousdirectionalderivativesX1,...,X.
underall translations.An elementary
theoremimpliesthat tyis quasi-invariant
to the Lebesguemeasurewith
argumentthen givesthe resultthat eyis equivalent
densityfunction
p1
n
F(x) = C exp -E
i=l
Xi(ux)du
xqy
a.s.,
°
whereC is a constant.
REFERENCES
1. K. D. Elworthy, Ga?>n measureson Banarhsparesandmanifolds,Global Analysis and Applications, Vol. 2, InternationalAtomic Energy Agency, Vienna, 1974, pp. 151-166.
2. L. Gross, Abst Wienerspace,Proc. Fifth BerkeleySympos. Math. Statist. and Probability,
part 1, University of California,1965, pp. 31-42.
on Hibertspace,J. Funct. Anal. 1 (1967), 123-181.
3.
, Potenil tt
Proc. Internat. Conf. on
anzlhypoelliptic
operaZors,
4. P. Malliavin, Stochasticcalculusof variti
Stochastic DifferentialEquations of Kyoto, 1976, pp. 195-263.
in Hibertspge, Springer-Verlag,New York, 1974.
5. A. V. Skorohod,Integr
THE UNIVERSITYOF TEXAS)AUSTIN,TEXAS78712
OF MATHEMATICS,
DEPARTMENT
Cuw7erd
add7ess:Department of Mathematics, SuSolk University,Boston, Massachusetts02114