QUASI-INVARIANCE ON ABSTRACT NILPOTENT LIE
GROUPS
TAI MELCHER∗
Abstract. We study heat kernel measures for non-degenerate Brownian motions on a general class of infinite-dimensional nilpotent Lie groups based on
an abstract Wiener space. We prove that a Cameron–Martin type quasiinvariance result holds, and give estimates on the associated Radon-Nikodym
derivative. We also prove that a log Sobolev estimate holds in this setting.
Contents
1. Introduction
1.1. Main results
1.2. Discussion
2. Abstract nilpotent Lie algebras and groups
2.1. Measurable group actions on G
2.2. Examples of abstract nilpotent Lie algebras
2.3. Properties of gCM
3. Brownian motion
3.1. Multiple Itô integrals
3.2. Brownian motion and finite-dimensional approximations
4. Heat kernel measure
References
1
2
3
4
5
7
9
11
11
14
20
22
1. Introduction
Quasi-invariance and other smoothness properties are fundamental subjects of
study for measures on infinite-dimensional spaces. The purpose of this paper is to
construct heat kernel measures on a general class of infinite-dimensional nilpotent
Lie groups, and to verify quasi-invariance for these measures. This class of groups is
quite rich, and the groups are well-structured for the study of hypoelliptic diffusions
in infinite dimensions. As nilpotent groups are standard first models for studying
hypoellipticity, examples of infinite-dimensional versions of such spaces are important for the more general study of hypoellipticity in infinite dimensions. This is an
area of great interest, and is of particular relevance in the study of stochastic PDEs
and their applications [1, 3, 6, 17, 22, 23]. The present paper studies heat kernel
2010 Mathematics Subject Classification. Primary 60J65 28D05; Secondary 58J65 22E65.
Key words and phrases. Heat kernel measure, infinite-dimensional Lie group, quasi-invariance,
logarithmic Sobolev inequality.
∗ This research was supported in part by NSF Grants DMS-0907293 and DMS-1255574.
1
2
MELCHER
measures for elliptic diffusions on these spaces, which is a necessary preceding step
to understanding the degenerate case.
1.1. Main results. Let (g, gCM , µ) denote an abstract Wiener space, where g is
a Banach space equipped with a centered non-degenerate Gaussian measure µ and
associated Cameron–Martin Hilbert space gCM . We will assume that gCM additionally carries a nilpotent Lie algebra structure [·, ·] : gCM × gCM → gCM , and we will
further assume that this Lie bracket is Hilbert-Schmidt. We will call such a space an
abstract nilpotent Lie algebra. Via the standard Baker-Campbell-Hausdorff-Dynkin
formula, we may then equip gCM with an explicit group operation under which
gCM becomes an infinite-dimensional group. When thought of as a group, we will
denote this space by GCM . We equip GCM with the left-invariant Riemannian
metric which agrees with the inner product on gCM ∼
= Te GCM , and we denote the
Riemannian distance by dCM . Note that, despite the use of the notation g, we do
not assume that the Lie bracket structure extends to g, and so this space is not
necessarily a Lie algebra or a Lie group. Still, when g is playing a role typically
played by a group, we will denote g by G.
If g were a Lie algebra (and thus carried an associated group structure), we could
construct a Brownian motion on G as the solution to the Stratonovich stochastic
differential equation
δgt = gt δBt := Lgt ∗ δBt , with g0 = e = (0, 0),
where Lx is left translation by x ∈ G and {Bt }t≥0 is a standard Brownian motion on
g (as a Banach space) with Law(B1 ) = µ. The solution to this stochastic differential
equation may be obtained explicitly as a formula involving the Lie bracket. In
particular, for t > 0 and n ∈ N, let ∆n (t) denote the simplex in Rn given by
{s = (s1 , · · · , sn ) ∈ Rn : 0 < s1 < s2 < · · · < sn < t}.
Let Sn denote the permutation group on (1, · · · , n), and, for each σ ∈ Sn , let
e(σ) denote the number of “errors” in the ordering (σ(1), σ(2), · · · , σ(n)), that is,
e(σ) = #{j < n : σ(j) > σ(j + 1)}. Then the Brownian motion on G could be
written as
(1.1)
Z
r−1 X X
e(σ)
2 n−1
n
gt =
(−1)
[[· · · [δBsσ(1) , δBsσ(2) ], · · · ], δBsσ(n) ],
e(σ)
∆n (t)
n=1 σ∈Sn
where this sum is finite under the assumed nilpotence. In Section 3.1, we show
that one may still make sense of the above expression when the Lie bracket on gCM
does not necessarily extend to g, and thus we are able to define a “group Brownian
motion” {gt }t≥0 on G via (1.1). We take νt := Law(gt ) to be the “heat kernel
measure” on G.
In particular, the stochastic integrals given above are defined via finitedimensional approximations. We consider finite-dimensional subgroups Gπ of GCM ,
and we show that these Gπ are nice in the sense that they may be used to approximate GCM and that there exists a uniform lower bound on their Ricci curvatures.
Additionally, we show that the (true) heat kernel measures on these spaces approximate νt .
Using these results, we are able to prove the following main theorem.
QUASI-INVARIANCE ON ABSTRACT NILPOTENT LIE GROUPS
3
Theorem 1.1. For h ∈ GCM , let Lh , Rh : GCM → GCM denote left and right
translation by h, respectively. Then Lh and Rh define measurable transformations
−1
on G, and for all T > 0, νt ◦L−1
h and νt ◦Rh are absolutely continuous with respect
to νt . Let
d(νt ◦ Rh−1 )
d(νt ◦ L−1
h )
and
Jtr (h, ·) :=
dνt
dνt
be the Radon-Nikodym derivatives, k be the uniform lower bound on the Ricci curvatures of the finite-dimensional approximation groups Gπ and
t
c(t) := t
,
for all t ∈ R,
e −1
with the convention that c(0) = 1. Then, for all p ∈ [1, ∞), Jtl (h, ·), Jtr (h, ·) ∈ Lp (νt )
and both satisfy the estimate
c(kt)(p − 1)
2
∗
d(e, h) ,
kJt (h, ·)kLp (νt ) ≤ exp
2t
Jtl (h, ·) :=
where ∗ = l or ∗ = r.
The fact that one may define a measurable left or right action on G by an element
of GCM is proved in Proposition 2.6. The lower bound on the Ricci curvature is
proved in Proposition 2.12.Fr
1.2. Discussion. The present paper builds on the previous work in [11] and [24],
significantly generalizing these previous works in several ways. In particular,
the paper [24] considered analogous results for “semi-infinite Lie groups”, which
are infinite-dimensional nilpotent Lie groups constructed as extensions of finitedimensional nilpotent Lie groups v by an infinite-dimensional abstract Wiener space
(see Example 2.3). At several points in the analysis there, the fact that dim(v) < ∞
was used in a critical way. In particular, it was used to show that the stochastic
integrals defining the Brownian motion on G as in equation (1.1) were well-defined.
In the present paper, we have removed this restriction, as well as removing the
“stratified” structure implicit in the construction as a Lie group extension.
Again, we note that, despite the use of the notation g, it is not assumed that the
Lie bracket structure on gCM extends to g, and so this space is not necessarily a
Lie algebra or Lie group. In [11] and [24], it was assumed that the Lie bracket was
a continuous map defined on g. However, it turns out that the group construction
on gCM is the only necessary structure for the subsequent analysis. As is usual for
the infinite-dimensional setting, while the heat kernel measure is itself supported on
the larger space g, its critical analysis depends more on the structure of gCM . Still,
as was originally done in [11] and then in [24], one may instead define an abstract
nilpotent Lie algebra starting with a continuous nilpotent bracket [·, ·] : g × g → g.
For example, in the event that g = W × v where v is a finite-dimensional Lie
algebra and [·, ·] : g × g → v, it is well-known that this implies that the restriction
of the bracket to gCM = H × v is Hilbert-Schmidt. (For any continuous bilinear
ω : W × W → K where K is a Hilbert space, one has that kωk(H ⊗2 )∗ ⊗K < ∞;
this follows for example from Corollary 4.4 of [21].) More generally, in order for
the subsequent theory to make sense, one would naturally need to require that
gCM be a Lie subalgebra of g, that is, for the restriction of the Lie bracket to gCM
to preserve gCM . As the proofs in the sequel rely strongly on the bracket being
Hilbert-Schmidt, it would be then necessary to add the Hilbert-Schmidt hypothesis
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as it does not follow immediately if one only assumes a continuous bracket on g
which preserves gCM .
Additionally, the spaces studied in the present paper are well-designed for the
study of infinite-dimensional hypoelliptic heat kernel measures, and there has already been progress on proving quasi-invariance and stronger smoothness properties for these measures in the simplest case of a step two Lie algebra with finitedimensional center; see [5] and [12].
Acknowledgements. The author thanks Bruce Driver and Nate Eldredge for
several helpful conversations during the writing of this paper.
2. Abstract nilpotent Lie algebras and groups
First we give a standard construction of abstract Wiener spaces. Suppose that
W is a real separable Banach space and BW is the Borel σ-algebra on W .
Definition 2.1. A measure µ on (W, BW ) is called a (mean zero, non-degenerate)
Gaussian measure provided that its characteristic functional is given by
Z
1
for all u ∈ W ∗ ,
µ̂(u) :=
eiu(x) dµ(x) = e− 2 q(u,u) ,
W
for q = qµ : W ∗ × W ∗ → R a symmetric, positive definite quadratic form. That is,
q is a real inner product on W ∗ .
Theorem 2.2. Let µ be a Gaussian measure on W . For w ∈ W , let
kwkH :=
|u(w)|
p
,
q(u, u)
u∈W ∗ \{0}
sup
and define the Cameron–Martin subspace H ⊂ W by
H := {h ∈ W : khkH < ∞}.
Then H is a dense subspace of W , and there exists a unique inner product h·, ·iH on
H such that khk2H = hh, hiH for all h ∈ H, and H is a separable Hilbert space with
respect to this inner product. For any h ∈ H, khkW ≤ CkhkH for some C < ∞.
Alternatively, given W a real separable Banach space and H a real separable
Hilbert space continuously embedded in W as a dense subspace, then for each
w∗ ∈ W ∗ there exists a unique hw∗ ∈ H such that hh, w∗ i = hh, hw∗ iH for all
h ∈ H. Then W ∗ 3 w∗ 7→ hw∗ ∈ H is continuous, linear, and one-to-one with a
dense range
(2.1)
H∗ := {hw∗ : w∗ ∈ W ∗ },
and W ∗ 3 w∗ 7→ hw∗ ∈ W is continuous. A Gaussian measure on W is a Borel
probability measure µ such that, for each w∗ ∈ W ∗ , the random variable w 7→
hw, w∗ i under µ is a centered Gaussian with variance khw∗ k2H .
Definition 2.3. Let (g, gCM , µ) be an abstract Wiener space such that gCM
is equipped with a nilpotent Hilbert-Schmidt Lie bracket. Then we will call
(g, gCM , µ) an abstract nilpotent Lie algebra.
The Baker-Campbell-Hausdorff-Dynkin formula implies that
log(eA eB ) = A + B +
r−1
X
X
k=1 (n,m)∈Ik
nk
mk
1
akn,m adnA1 adm
B · · · adA adB A,
QUASI-INVARIANCE ON ABSTRACT NILPOTENT LIE GROUPS
5
for all A, B ∈ gCM , where
akn,m :=
(−1)k
,
(k + 1)m!n!(|n| + 1)
Ik := {(n, m) ∈ Zk+ × Zk+ : ni + mi > 0 for all 1 ≤ i ≤ k}, and for each multi-index
n ∈ Zk+ ,
n! = n1 ! · · · nk ! and |n| = n1 + · · · + nk ;
see, for example, [15]. If gCM is nilpotent of step r, then
nk
mk
1
adnA1 adm
B · · · adA adB A = 0
if |n| + |m| ≥ r.
for A, B ∈ gCM . Since gCM is simply connected and nilpotent, the exponential
map is a global diffeomorphism (see, for example, Theorems 3.6.2 of [29] or 1.2.1
of [10]). In particular, we may view gCM as both a Lie algebra and Lie group, and
one may verify that
(2.2)
g·h=g+h+
r−1
X
X
mk
nk
1
akn,m adng 1 adm
h · · · adg adh g
k=1 (n,m)∈Ik
defines a group structure on gCM . Note that g −1 = −g and the identity e = (0, 0).
Definition 2.4. When we wish to emphasize the group structure on gCM , we will
denote gCM by GCM .
Lemma 2.5. The Banach space topology on gCM makes GCM into a topological
group.
Proof. Since gCM is a topological vector space, g 7→ g −1 = −g and (g1 , g2 ) 7→ g1 +g2
are continuous by definition. The map (g1 , g2 ) 7→ [g1 , g2 ] is continuous in the gCM
topology by the boundedness of the Lie bracket. It then follows from (2.2) that
(g1 , g2 ) 7→ g1 · g2 is continuous as well.
2.1. Measurable group actions on G. As discussed in the introduction, given
a Hilbert-Schmidt Lie bracket on gCM and a subsequently defined group operation
on GCM , one may define a measurable action on G by left or right multiplication
by an element of GCM .
In particular, let {en }∞
n=1 be an orthonormal basis of gCM . For now, fix n
and consider the mapping gCM → g∗CM given by h 7→ hadh ·, en i. Then this is a
continuous linear map on gCM and in the usual way we may make the identification
of g∗CM ∼
= gCM so that we define the operator An : gCM → gCM given by
hAn h, ki = hadh k, en i;
in particular, An h =
ad∗h en .
Note that, for any h, k ∈ gCM
hA∗n h, ki = hAn k, hi = had∗k en , hi = hen , adk hi = −hen , adh ki = −had∗h en , ki
and thus A∗n = −An . Now fix h ∈ GCM = gCM . Then for adh : gCM → gCM we
may write
X
X
adh k =
hadh k, en ien =
hAn h, kien .
n
n
Since each hAn h, ·i ∈ g∗CM has a measurable linear extension to G such that
khAn h, ·ikL2 (µ) = kAn hkgCM (see, for example, Theorem 2.10.11 of [8]), we may
6
MELCHER
extend adh to a measurable linear transformation from G = g to GCM = gCM (still
denoted by adh ) given by
X
adh g :=
hAn h, gien .
n
Note that here we are using the fact that
X
X
X
khAn h, ·ik2L2 (µ) =
kAn hk2H =
hAn h, em i2
n
n
n,m
=
X
hadh em , en i2 ≤ khk2 k[·, ·]k2HS
n,m
which implies that
X
hAn h, gi2 < ∞
g-a.s.
n
Similarly, we may define
adg h := −adh g = −
X
hAn h, gien .
n
In a similar way, note that we may write, for h, k ∈ GCM and m < r,
!
X X m−1
Y
m
···
hA`b h, e`b+1 i hA`m h, kie`1
adh k =
`1
`m
= (−1)
m
X
b=1
···
`1
X
m−1
Y
`m
b=1
!
hA`b e`b+1 , hi hA`m k, hie`1 .
and thus for h, k, x ∈ GCM and n + m < r
adnk adm
h x
n
= (−1)
X
···
j1
XX
jn
···
`1
X
n−1
Y
`m
a=1
!
hAja eja+1 , ki hAjn e`1 , ki
!
m−1
Y
hA`b h, e`b+1 i hA`m h, xiej1 .
b=1
More generally for |n| + |m| < r
ns
ms
1
adnk 1 adm
h · · · adk adh k
= (−1)|n|
X
···
j11
×
s XY
nY
c −1
`sms
ac =1
c=1
!
hAjacc ejacc +1 , ki hAjnc c e`c1 , ki
mY
c −1
!
hA`cbc h, e`cbc +1 i hA`cmc h, ej1c i hA`sms h, kiej11 /hA`sms h, ej1s i.
bc =1
Thus, for h ∈ GCM and |n| + |m| < r, we may define a measurable action on G
given by
ns
ms
1
G 3 g 7→ adng 1 adm
h · · · adg adh g
|n|
:= (−1)
X
j11
···
s XY
nY
c −1
`sms c=1
ac =1
!
hAjacc ejacc +1 , gi hAjnc c e`c1 , gi
QUASI-INVARIANCE ON ABSTRACT NILPOTENT LIE GROUPS
×
mY
c −1
7
!
hA`cbc h, e`cbc +1 i hA`cmc h, ej1c i hA`sms h, giej11 /hA`sms h, ej1s i ∈ GCM .
bc =1
(For |n| + |m| ≥ r, we define this mapping to be 0, which is certainly measurable.)
Again, we are using that
( n −1
!
s
c
Y
X
X XY
···
hAjacc ejacc +1 , ·i hAjnc c e`c1 , ·i
1
s
1
`ms c=1
j2
j1
×
mY
c −1
bc =1
ac =1
2
hA`cbc h, e`cbc +1 i hA`cmc h, ej1c i hA`sms h, ·i/hA`sms h, ej1s i
2
!
)
< ∞.
L (µ)
This holds by straightforward but tedious computations
P — in fact, iterative applications of Cauchy-Schwarz combined with the fact that n,m kAn em k2 = k[·, ·]k2HS <
∞. Thus,
( n −1
!
s
c
Y
X X
XY
hAjacc ejacc +1 , gi hAjnc c e`c1 , gi
···
j11
j21
×
`sms c=1
mY
c −1
ac =1
!
)
!2
hA`cbc h, e`cbc +1 i hA`cmc h, ej1c i hA`sms h, gi/hA`sms h, ej1s i
< ∞,
bc =1
n1
1
adg adm
h
s
g-a.s. and
· · · adng s adm
h g as given above is defined a.s. Thus, we have
proved the following result.
Proposition 2.6. For h ∈ GCM , the mapping G 3 g 7→ g · h ∈ G defined analogously to (2.2) is a measurable right group action by GCM on G, and similarly for
the left action g 7→ h · g.
2.2. Examples of abstract nilpotent Lie algebras.
Example 2.1 (Free nilpotent Lie algebras). Starting with an abstract Wiener
space (W, H, µ), one may construct in the standard way the abstract free nilpotent
Lie algebra of step r with generators {hi }∞
i=1 an orthonormal basis of H. See for
example Section 0 of [16].
Example 2.2 (Heisenberg-like algebras). Let (Wi , Hi , µi ) for i = 1, 2 be abstract
Wiener spaces. Then for any ω : H1 × H1 → H2 a Hilbert-Schmidt map, we may
define a Lie bracket on gCM = H1 × H2 by
[(h1 , h2 ), (h01 , h02 )] := (0, ω(h1 , h01 )),
and g = W1 × W2 may be thought of as an abstract Heisenberg-like algebra as in
[11].
These abstract Heisenberg-like algebras are central extensions of one abstract
Wiener space by another abstract Wiener space. The next example generalizes this
construction.
Example 2.3 (Extensions of Lie algebras). Let v and h be Lie algebras, and let
Der(v) denote the set of derivations on v; that is, Der(v) consists of all linear maps
ρ : v → v satisfying Leibniz’s rule:
ρ([X, Y ]v ) = [ρ(X), Y ]v + [X, ρ(Y )]v .
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MELCHER
Now suppose there are a linear mapping α : h → Der(v) and a skew-symmetric
bilinear mapping ω : h × h → v, satisfying, for all X, Y, Z ∈ h,
[αX , αY ] − α[X,Y ]h = adω(X,Y )
(B1)
and
X
(B2)
(αX ω(Y, Z) − ω([X, Y ]h , Z)) = 0.
cyclic
Then, one may verify that, for X1 + V1 , X2 + V2 ∈ h ⊕ v,
[X1 + V1 , X2 + V2 ]g := [X1 , X2 ]h + ω(X1 , X2 ) + αX1 V2 − αX2 V1 + [V1 , V2 ]v
defines a Lie bracket on g := h ⊕ v, and we say g is an extension of h over v. That
is, g is the Lie algebra with ideal v and quotient algebra g/v = h. The associated
exact sequence is
ι1
π2
0 → v −→
g −→
h → 0,
where ι1 is inclusion and π2 is projection. In fact, these are the only extensions of
h over v (see, for example, [2]).
Now suppose that (W, H, µ) is a real abstract Wiener space, and (v, vCM , µ0 ) is
an abstract nilpotent Lie algebra. Motivated by the previous discussion, we may
consider H as an abelian Lie algebra and construct extensions of H over vCM . In
this case, we need a linear mapping α : H → Der(vCM ) and a skew-symmetric
bilinear mapping ω : H × H → vCM , such that ω and α : H × vCM → vCM are
both Hilbert-Schmidt and together ω and α satisfy (B1) and (B2), which in this
setting become
[αX , αY ] = adω(X,Y )
and
αX ω(Y, Z) + αY ω(Z, X) + αZ ω(X, Y ) = 0,
for all X, Y, Z ∈ H. Then we may define a Lie algebra structure on gCM = H ⊕vCM
via the Lie bracket
[(X1 , V1 ), (X2 , V2 )]gCM := (0, ω(X1 , X2 ) + αX1 V2 − αX2 V1 ).
Again, these are in fact the only extensions of H over vCM .
Combining Examples 2.2 and 2.3 shows that these constructions are iterative, in
that one may construct new abstract nilpotent Lie algebras as Lie algebra extensions
of another abstract nilpotent Lie algebra. The next example builds on the previous
one to give one precise way to construct some Lie algebra extensions.
Example 2.4. Let β : H → vCM be any Hilbert-Schmidt map, and for X, Y ∈ H
and V ∈ vCM define
ω(X, Y ) := [β(X), β(Y )]vCM
αX V := adβ(X) V = [β(X), V ]vCM .
Then the Lie bracket on gCM is given by
[(X, V ), (Y, U )]
= (0, [β(X), β(Y )]vCM + [β(X), U ]vCM − [β(Y ), V ]vCM + [V, U ]vCM ).
Note that, if vCM is nilpotent of step r, then gCM will automatically be nilpotent
of step r.
QUASI-INVARIANCE ON ABSTRACT NILPOTENT LIE GROUPS
9
The examples above demonstrate that the space of abstract nilpotent Lie algebras is quite rich, and there are many natural examples with a straightforward construction. This significantly improves the results of [24], which studied heat kernel
measures on nilpotent extensions of abstract Wiener spaces over finite-dimensional
nilpotent Lie algebras. For example, the restriction dim(vCM ) < ∞ trivializes
Example 2.4. We elaborate in the following remark.
Remark 2.7. Returning to Example 2.4, since β is linear and continuous, we have
the decomposition
H = Nul(β) ⊕ Nul(β)⊥ ,
where dim(Nul(β)⊥ ) ≤ dim(vCM ). Thus, for X, Y ∈ H we may write X = X1 +
X2 , Y = Y1 + Y2 ∈ Nul(β) ⊕ Nul(β)⊥ and
[(X1 + X2 , 0), (Y1 + Y2 , 0)] = [β(X1 + X2 ), β(Y1 + Y2 )] = [β(X2 ), β(Y2 )],
and ω is a map on Nul(β)⊥ × Nul(β)⊥ . Thus, [Nul(β), Nul(β)] = {0} and similarly
[Nul(β), v] = {0}. So
gCM = H ⊕ vCM = Nul(β) ⊕ Nul(β)⊥ ⊕ vCM ,
and, in particular, when dim(vCM ) < ∞, gCM is just an extension of the finitedimensional vector space Nul(β)⊥ by the finite-dimensional Lie algebra vCM , and
the construction is not truly infinite-dimensional.
2.3. Properties of gCM . This section collects some results for topological and
geometric properties of gCM that we’ll require for the sequel.
Proposition 2.8. For all m ≥ 2, [[[·, ·], . . .], ·] : g⊗m
CM → gCM is Hilbert-Schmidt.
Proof. For m = 2, this follows from the definition of G. Now assume the statement
holds for all m = `, and consider m = ` + 1. Writing [[hi1 , hi2 ], · · · , hi` +1 ] ∈ gCM
in terms of the orthonormal basis {ej }∞
j=1 and using Hölder’s inequality gives
k[[[·,·], . . .], ·]k22 = k[[[·, ·], . . .], ·]k(g∗CM )⊗`+1 ⊗gCM
∞
X
=
k[[[hi1 , hi2 ], · · · , hi` ], hi`+1 ]k2
i1 ,...,i`+1 =1
2
X
∞
=
[ej , hi`+1 ]hej , [[hi1 , hi2 ], · · · , hi` ]i
i1 ,...,i`+1 =1 j=1



∞
∞
∞
X
X
X

k[ej , hi`+1 ]k2  
|hej , [[hi1 , hi2 ], · · · , hi` ]i|2 
≤
∞
X
i1 ,...,i`+1 =1

=
∞ X
∞
X
i`+1 =1 j=1
≤
j=1
j=1

k[ej , hi`+1 ]k2  
∞
X

k[[hi1 , hi2 ], · · · , hi` ]k2 
i1 ,...,i` =1
k[·, ·]k2(g∗ )⊗2 ⊗gCM k[[[·, ·], . . .], ·]k2(g∗ )⊗` ⊗gCM
CM
CM
and the last line is finite by the induction hypothesis.
Next we will recall that the flat and geometric topologies on gCM are equivalent.
First we set the following notation.
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MELCHER
Notation 2.9. For g ∈ GCM , let Lg : GCM → GCM and Rg : GCM → GCM
denote left and right multiplication by g, respectively. As GCM is a vector space,
to each g ∈ GCM we can associate the tangent space Tg GCM to GCM at g, which
is naturally isomorphic to G. For f : GCM → R a Frechét smooth function and
v, x ∈ GCM and h ∈ g, let
d
f 0 (x)h := ∂h f (x) = f (x + th),
dt 0
and let vx ∈ Tx G denote the tangent vector satisfying vx f = f 0 (x)v. If σ(t) is any
smooth curve in GCM such that σ(0) = x and σ̇(0) = v (for example, σ(t) = x+tv),
then
d Lg∗ vx = g · σ(t).
dt 0
Notation 2.10. Let C 1 ([0, 1], GCM ) denote the collection of C 1 -paths g : [0, 1] →
GCM . The length of g is defined as
Z T
`CM (g) :=
kLg−1 (s)∗ g 0 (s)kgCM ds.
0
The Riemannian distance between x, y ∈ GCM is then defined as
d(x, y) := inf{`CM (g) : g ∈ Ch1 ([0, 1], GCM ) such that g(0) = x and g(1) = y}.
The following proposition was proved as Corollary 4.13 in [24]. This was proved
under the conditions that gCM was a “semi-infinite Lie algebra”, that is, under the
assumption that gCM was a nilpotent Lie algebra extension (as in Example 2.3) of a
finite-dimensional nilpotent Lie algebra v by an abstract Wiener space. However, a
cursory inspection of the proofs there will show that they only depended on the fact
that the Lie bracket was Hilbert-Schmidt, and not on the “stratified” structure of
gCM = H × v or the fact that the image of the Lie bracket was a finite-dimensional
subspace.
Proposition 2.11. The topology on GCM induced by d is equivalent to the Hilbert
topology induced by k · kgCM .
We may find a uniform lower bound on the Ricci curvature of all finitedimensional subgroups of GCM . Finite-dimensional subgroups of GCM may be
obtained by taking the Lie algebra generated by any finite-dimensional subspace of
gCM (which is again necessarily finite-dimensional by the nilpotence of the bracket)
and endowing it with the standard group operation via the Baker-CampbellHausdorff-Dynkin formula. An analogue of the following proposition was proved as
Proposition 3.23 and Corollary 3.24 in [24]. It follows directly from the form of the
Ricci curvature on nilpotent groups (endowed with a left invariant metric) and the
assumption that the Lie bracket is Hilbert-Schmidt. This proof is essentially the
same as in [24], but it is quite brief and so is included for completeness.
Proposition 2.12. Let
n
o
1
k := − sup k[·, X]k2g∗CM ⊗gCM : kXkgCM = 1 .
2
Then k > −∞ and k is the largest constant such that
hRicπ X, Xigπ ≥ kkXk2gπ ,
for all X ∈ gπ ,
holds uniformly for all gπ finite-dimensional Lie subalgebras of gCM .
QUASI-INVARIANCE ON ABSTRACT NILPOTENT LIE GROUPS
11
Proof. For g any nilpotent Lie algebra with orthonormal basis Γ,
1 X
1 X
1 X
hRic X, Xi =
kad∗Y Xk2 −
kadY Xk2 ≥ −
k[Y, X]k2
4
2
2
Y ∈Γ
Y ∈Γ
Y ∈Γ
for all X ∈ g. Thus, for gπ any finite-dimensional Lie algebra
hRicπ X, Xigπ ≥ kπ kXk2gπ ,
for all X ∈ gπ ,
where
n
o
1
1
kπ := − sup k[·, X]k2g∗π ⊗gπ : kXkgπ = 1 ≥ − k[·, ·]k22 > −∞.
2
2
Taking the infimum of kπ over all gπ completes the proof.
(2.3)
3. Brownian motion
Suppose that Bt is a smooth curve in gCM with B0 = 0, and consider the
differential equation
ġt = gt Ḃt := Lgt ∗ Ḃt ,
with g0 = e.
The solution gt may be written as follows (see [27]): For t > 0, let ∆n (t) denote
the simplex in Rn given by
{s = (s1 , · · · , sn ) ∈ Rn : 0 < s1 < s2 < · · · < sn < t}.
Let Sn denote the permutation group on (1, · · · , n), and, for each σ ∈ Sn , let
e(σ) denote the number of “errors” in the ordering (σ(1), σ(2), · · · , σ(n)), that is,
e(σ) = #{j < n : σ(j) > σ(j + 1)}. Then
r X X
e(σ)
2 n−1
(−1)
n
×
(3.1) gt =
e(σ)
n=1 σ∈Sn
Z
[· · · [Ḃsσ(1) , Ḃsσ(2) ], . . . , ]Ḃsσ(n) ] ds.
∆n (t)
Using this as our motivation, we first explore stochastic integral analogues of equation (3.1) where the smooth curve B is replaced by Brownian motion on g.
3.1. Multiple Itô integrals. For this section, we return to the general setting
where (W, H, µ) is an abstract Wiener space. Let {Bt }t≥0 be a Brownian motion
on W with variance determined by
E [hBs , hiH hBt , kiH ] = hh, kiH min(s, t),
for all s, t ≥ 0 and h, k ∈ H∗ , where H∗ is as in (2.1).
Suppose that P : H → H is a finite rank orthogonal projection such that P H ⊂
H∗ . Let {hj }m
j=1 be an orthonormal basis for P H. Then we may extend P to a
(unique) continuous operator from W → H (still denoted by P ) by letting
(3.2)
P w :=
m
X
hw, hj iH hj
j=1
for all w ∈ W . Note that P B is a Brownian motion on P H.
Notation 3.1. Let Proj(W ) denote the collection of finite rank projections on W
such that P W ⊂ H∗ and P |H : H → H is an orthogonal projection, that is, P has
the form given in equation (3.2).
12
MELCHER
The following is Proposition 4.1 of [24]. Note that again this was stated in the
context where H = gCM was a semi-infinite Lie algebra, but a brief inspection
of the proof shows that this is a general statement about stochastic integrals on
Hilbert spaces.
Proposition 3.2. Let {Pm }∞
m=1 ⊂ Proj(W ) such that Pm |H ↑ IH . Then, for
ξ ∈ L2 (∆n (t), H ⊗n ) a continuous mapping, let
Z
m
⊗n
Jn (ξ)t :=
hPm
ξ(s), dBs1 ⊗ · · · ⊗ dBsn iH ⊗n
∆n (t)
Z
=
hξ(s), dPm Bs1 ⊗ · · · ⊗ dPm Bsn iH ⊗n .
∆n (t)
Then {Jnm (ξ)t }t≥0 is a continuous L2 -martingale, and there exists a continuous
L2 -martingale {Jn (ξ)t }t≥0 such that
m
2
(3.3)
lim E sup |Jn (ξ)τ − Jn (ξ)τ | = 0
m→∞
τ ≤t
and
E|Jn (ξ)t |2 ≤ kξk2L2 (∆n (t),H ⊗n )
(3.4)
for all t < ∞. The process Jn (ξ) is well-defined independent of the choice of
increasing orthogonal projections {Pm }∞
m=1 into H∗ , and so will be denoted by
Z
Jn (ξ)t =
hξ(s), dBs1 ⊗ · · · ⊗ dBsn iH ⊗n .
∆n (t)
Now we may use this result to define stochastic integrals taking values in another
Hilbert space K.
Proposition 3.3. Let K be a Hilbert space and F ∈ L2 (∆n (t), (H ⊗n )∗ ⊗ K) be a
continuous map. That is, F : ∆n (t) × H ⊗n → K is a map continuous in s and
linear on H ⊗n such that
Z
Z
∞
X
2
kF (s)k2 ds =
kF (s)(hj1 ⊗ · · · ⊗ hjn )k2K ds < ∞.
∆n (t)
∆n (t) j ,...,j =1
1
n
Then
Jnm (F )t :=
Z
F (dPm Bs1 ⊗ · · · ⊗ dPm Bsn )
∆n (t)
is a continuous K-valued L2 -martingale, and there exists a continuous K-valued
L2 -martingale {Jn (F )t }t≥0 such that
m
2
(3.5)
lim E sup kJn (F )τ − Jn (F )τ kK = 0,
m→∞
τ ≤t
for all t < ∞. The martingale Jn (F ) is well-defined independent of the choice of
orthogonal projections, and thus will be denoted by
Z
Jn (F )t =
F (dBs1 ⊗ · · · ⊗ dBsn ).
∆n (t)
QUASI-INVARIANCE ON ABSTRACT NILPOTENT LIE GROUPS
13
Proof. Let {ej }∞
j=1 be an orthonormal basis of K. Since hF (s)(·), ej i is linear on
H ⊗n , for each s there exists ξj (s) ∈ H ⊗n such that
hξj (s), k1 ⊗ · · · ⊗ kn i = hF (s)(k1 ⊗ · · · ⊗ kn ), ej i.
(3.6)
If ξj : ∆n (t) → H ⊗n is defined by equation (3.6), then clearly ξj ∈ L2 (∆n (t), H ⊗n )
and in particular
kF k2L2 (∆n (t)×H ⊗n ,K) =
∞
X
kξj kL2 (∆n (t),H ⊗n ) < ∞.
j=1
Thus, for Jn (ξj ) as defined in Proposition 3.2,




Z
∞
∞
n
X
X
t
E
|Jn (ξj )t |2  ≤ E 
|hξj (s), dBs1 ⊗ · · · ⊗ dBsn iH ⊗n |2 
n!
∆
(t)
n
j=1
j=1
∞
tn X
kξj k2L2 (∆n (t),H ⊗n ) < ∞,
n! j=1
=
and so we may write
∞
X
Jn (ξj )t ej =
∞ Z
X
j=1
j=1
hξj (s), dBs1 ⊗ · · · ⊗ dBsn iH ⊗n ej
∆n (t)
∞
X
hF (s)(dBs1 ⊗ · · · ⊗ dBsn ), ej iK ej
Z
=
∆n (t) j=1
Z
F (s)(dBs1 ⊗ · · · ⊗ dBsn ).
=
∆n (t)
Thus, taking Jn (F )t :=
P∞
Jn (ξj )t ej , we also have that


∞
X
EkJn (F )t − Jnm (F )t k2K = E 
|Jn (ξj )t − Jnm (ξj )t |2  → 0
j=1
j=1
as m → ∞ by (3.3) and dominated convergence since
E|Jn (ξj )t − Jnm (ξj )t |2 ≤ 4kξj k2L2 (∆n (t),H ⊗n )
by (3.4). Then equation (3.5) holds by Doob’s maximal inequality.
Note that the preceding results then imply that one may define the above stochastic integrals with respect to any increasing sequence of orthogonal projections
– that is, we need not require that the projections extend continuously to W .
Proposition 3.4. Let V be an arbitrary finite-dimensional subspace of H, and let
π : H → V denote orthogonal projection onto V . Then for any Hilbert space K and
F ∈ L2 (∆n (t), (H ⊗n )∗ ⊗ K) a continuous map, the stochastic integral
Z
π
Jn (F )t :=
F (dπBs1 ⊗ · · · ⊗ dπBsn )
∆n (t)
{Jnπ (F )t }t≥0
is well-defined, and
is a continuous K-valued L2 -martingale. Moreover, if Vm is an increasing sequence of finite-dimensional subspaces of H such that
14
MELCHER
the corresponding orthogonal projections πm ↑ IH , then
lim E sup kJnπm (F )τ − Jn (F )τ k2 = 0,
m→∞
τ ≤t
where Jn (F ) is as defined in Proposition 3.3.
Proof. First consider the case that K = R, and thus F (s) = hξ(s), ·i for a continuous
ξ ∈ L2 (∆n (t), H ⊗n ). Since π ⊗n ξ ∈ L2 (∆n (t), H ⊗n ), the definition of Jnπ (ξ) =
Jn (π ⊗n ξ) follows from Propostion 3.2. Moreover, by equation (3.4),
⊗n
⊗n
E|Jnπm (ξ)t − Jn (ξ)t |2 = E|Jn (πm
ξ)t − Jn (ξ)t |2 = E|Jn (πm
ξ − ξ)t |2
⊗n
≤ kπm
ξ − ξkL2 (∆n (t),H ⊗n ) → 0
as m → ∞. Now the proof for general F follows just as in Proposition 3.3.
3.2. Brownian motion and finite-dimensional approximations. We now return to the setting of an abstract Wiener space (g, gCM , µ) endowed with a nilpotent
Hilbert-Schmidt Lie bracket on gCM . Again, let Bt denote Brownian motion on g.
By equation (3.1), the solution to the Stratonovich stochastic differential equation
δgt = Lgt ∗ δBt ,
with g0 = e,
should be given by
(3.7)
gt =
r−1 X
X
cσn
Z
[[· · · [δBsσ(1) , δBsσ(2) ], · · · ], δBsσ(n) ],
∆n (t)
n=1 σ∈Sn
for coefficients cσn determined by equation (3.1).
To understand the integrals in (3.7), consider the following heuristic computation. Let {Mn (t)}t≥0 denote the process in g⊗n defined by
Z
Mn (t) :=
δBs1 ⊗ · · · ⊗ δBsn .
∆n (t)
By repeatedly applying the definition of the Stratonovich integral, the iterated
Stratonovich integral Mn (t) may be realized as a linear combination of iterated Itô
integrals:
n
X
X
1
Mn (t) =
Itn (α),
n−m
2
m
m=dn/2e
where
α∈Jn
(
Jnm
:=
m
(α1 , . . . , αm ) ∈ {1, 2}
:
m
X
)
αi = n ,
i=1
and, for α ∈ Jnm , Itn (α) is the iterated Itô integral
Z
Itn (α) =
dXs11 ⊗ · · · ⊗ dXsmm
∆m (t)
with
dXsi
=
if αi = 1
P∞ dBs
;
h
⊗
h
ds
if αi = 2
j
j=1 j
compare with Proposition 1 of [7]. This change from multiple Stratonovich integrals
to multiple Itô integrals may also be recognized as a specific case of the Hu-Meyer
formulas [18, 19], but we will compute more explicitly to verify that our integrals
are well-defined.
QUASI-INVARIANCE ON ABSTRACT NILPOTENT LIE GROUPS
15
For n ∈ {1, · · · , r}, define Fn : g⊗n
CM → gCM by
Fn (k1 ⊗ · · · ⊗ kn ) := [[[· · · [k1 , k2 ], k3 ], · · · ], kn ]
(3.8)
and, for σ ∈ Sn , define Fnσ : g⊗n
CM → gCM by
Fnσ (k1 ⊗ · · · ⊗ kn ) := Fn (kσ(1) ⊗ · · · ⊗ kσ(n) )
(3.9)
= [[· · · [kσ(1) , kσ(2) ], · · · ], kσ(n) ].
Then we may write
r−1 X
X
gt =
cσn Fnσ (Mn (t)) =
n=1 σ∈Sn
r−1 X
X
n
X
n=1 σ∈Sn m=dn/2e
cσn
n−m
2
X
Fnσ (Itn (α)),
α∈Jnm
presuming we can make sense of the integrals Fnσ (Itn (α)).
For each α, let pα = #{i : αi = 1} and qα = #{i : αi = 2} (so that pα + qα = m
when α ∈ Jnm ), and let
n
[
Jn :=
Jnm .
m=dn/2e
Then, for each σ ∈ Sn and α ∈ Jn ,
Z
Fnσ (Itn (α)) =
fα (s, t)F̂nσ,α (dBs1 ⊗ · · · ⊗ dBspα ),
∆pα (t)
where F̂nσ,α and fα are as follows.
The map F̂nσ,α : g⊗pα → g is defined by
(3.10) F̂nσ,α (k1 ⊗ · · · ⊗ kpα )
:=
∞
X
0
Fnσ (k1 ⊗ · · · ⊗ kpα ⊗ hj1 ⊗ hj1 ⊗ · · · ⊗ hjqα ⊗ hjqα ),
j1 ,...,jqα =1
{hj }∞
j=1
for
an orthonormal basis of gCM and σ 0 = σ 0 (α) ∈ Sn given by σ 0 = σ◦τ −1 ,
for any τ ∈ Sn such that
τ (dXs11 ⊗ · · · ⊗ dXsmm )
=
∞
X
dBs1 ⊗ · · · ⊗ dBspα ⊗ hj1 ⊗ hj1 ⊗ · · · ⊗ hjqα ⊗ hjqα ds1 · · · dsqα .
j1 ,··· ,jqα =1
To define fα , first consider the polynomial of order qα , in the variables si with i
such that αi = 1 and in the variable t, given by evaluating the integral
Z
Y
(3.11)
fα0 ((si : αi = 1), t) =
dsi ,
∆0qα (t) i:α =2
i
where ∆0qα (t) = {si−1 < si < si+1 : αi = 2} with s0 = 0 and sm+1 = t. Then fα
is fα0 with the variables reindexed by the bijection {i : αi = 1} → {1, . . . , pα } that
maintains the natural ordering of these sets. (For example, for α = (1, 2, 1, 2) ∈ J64 ,
Z
0
fα (s1 , s3 , t) =
ds2 ds4 = (t − s3 )(s3 − s1 ),
{s1 <s2 <s3 ,s3 <s4 <t}
so that fα (s1 , s2 , t) = (t − s2 )(s2 − s1 ).)
16
MELCHER
This explicit realization of fα is not critical to the sequel. It is really only
necessary to know that fα is a polynomial of order qα in s = (s1 , . . . , spα ) and t,
and thus may be written as
qα
X
fα (s, t) =
baα ta f˜α,a (s),
a=0
for some coefficients
∈ R and polynomials f˜α,a of degree qα − a in s. Now, if
α
σ,α
F̂n is Hilbert-Schmidt on g⊗p
CM , then
Z
2
σ,α 2
˜
F̂n < ∞,
fα,a (s)F̂nσ,α ds = f˜α,a 2
baα
L (∆pα (t))
2
∆pα (t)
2
and
Fnσ (Itn (α)) =
(3.12)
qα
X
baα ta Jn (f˜α,a F̂nσ,α )t
a=0
may be understood in the sense of the limit integrals in Proposition 3.3. (In particular, if αm = 1, then fα = fα (s) does not depend on t, and Proposition 3.3 implies
that Fnσ (Itn (α)) is a v-valued L2 -martingale.)
The above computations show that, if for all n F̂nσ,α is Hilbert-Schmidt for all
σ ∈ Sn and α ∈ Jn , then we may rewrite (3.7) as
gt =
r−1 X
X
n
X
n=1 σ∈Sn m=dn/2e
cσn
2n−m
qα
X X
baα ta Jn (f˜α,a F̂nσ,α )t ,
α∈Jnm a=0
where Jn is as defined in Proposition 3.3. The next proposition shows that F̂nσ,α is
Hilbert-Schmidt as desired, and thus gt in (3.7) is well-defined.
⊗pα
→
Proposition 3.5. Let n ∈ {2, . . . , r}, σ ∈ Sn , and α ∈ Jn . Then F̂nσ,α : gCM
gCM is Hilbert-Schmidt.
Proof. For the whole of this proof, all sums will be taken over an orthonormal basis
of gCM .
Now, for Fn and Fnσ as defined in equations (3.8) and (3.9), we may write
Fnσ (k1 ⊗ · · · ⊗ kpα ⊗ h1 ⊗ h1 ⊗ · · · ⊗ hqα ⊗ hqα ) = Fn (Aσ(1) ⊗ . . . ⊗ Aσ(n) )
where
kb
if b = 1, . . . , pα
.
hd(b−pα )/2e if b = pα + 1, . . . , n
If qα = 0, then pα = n, each Aσ(j) = ki for some i = 1, . . . , n, and
X
kF̂nσ,α k22 =
kFnσ (k1 ⊗ · · · ⊗ kn )k2gCM
Ab :=
k1 ,...,kn
=
X
kFn (k1 ⊗ · · · ⊗ kn )k2gCM = k[[[. . . [·, ·], . . .], ·]k2(g∗
⊗n ⊗g
CM
CM )
k1 ,...,kn
which is finite by Proposition 2.8.
Now, if qα = 1, let N = N (σ) denote the second j such that σ(j) ∈ {n − 1, n};
that is,
Fnσ (k1 ⊗ · · · ⊗kn−1 ⊗ h1 ⊗ h1 )
= Fn (Aσ(1) ⊗ · · · ⊗ Aσ(N −1) ⊗ h1 ⊗ Aσ(N +1) ⊗ . . . ⊗ Aσ(n) )
QUASI-INVARIANCE ON ABSTRACT NILPOTENT LIE GROUPS
17
= [[[[[. . . [Aσ(1) , Aσ(2) ], . . .], Aσ(N −1) ], h1 ], Aσ(N +1) ], . . .], Aσ(n) ]
where exactly one of Aσ(1) , . . . , Aσ(N −1) is h1 and, for all j > N , Aσ(j) = ki for
some
i ∈ I := I(σ) := {σ(j) : j = N + 1, . . . , n} ⊆ {1, . . . , pα } = {1, . . . , n − 2}.
Thus, writing
A(h1 , ki : i ∈ I c ) := [[. . . [Aσ(1) , Aσ(2) ], . . .], Aσ(N −1) ],
we have that
Fnσ (k1 ⊗ · · · ⊗ kn−1 ⊗ h1 ⊗ h1 )
X
=
hA(h1 , ki : i ∈ I c ), e1 igCM [[. . . [e1 , h1 ], Aσ(N +1) ], . . . , Aσ(n) ],
e1
and so
2
X
kF̂nσ,α k22 =
Fnσ (k1 ⊗ · · · ⊗ kn−1 ⊗ h1 ⊗ h1 )
k1 ,...,kn−1
h1
g

 CM
X
X

|hA(h1 , ki : i ∈ I c ), e1 igCM |2 
≤
X
k1 ,...,kn−1
h1 ,e1

×

X
k[[[e1 , h1 ], Aσ(N +1) ], . . . , Aσ(n) ]k2gCM 
h1 ,e1

=

X
c
|hA(h1 , ki : i ∈ I ), e1 igCM |
2
ki :i∈I c ,h1 ,e1

×

X
k[[. . . [[e1 , h1 ], Aσ(N +1) ], . . .], Aσ(n) ]k2gCM 
ki :i∈I,h1 ,e1
≤
k[[. . . [·, ·], . . .], ·]k2(g∗ )⊗N −1 ⊗gCM k[[. . . [·, ·], . . .], ·]k2(g∗ )⊗n−N +1 ⊗gCM .
CM
CM
Now more generally when qα ≥ 2, we may similarly “separate” the pairs of hj ’s
as above. More precisely, define
b
if b = 1, . . . , pα
.
Φ(b) := Φα (b) :=
α
e
+
p
if
b = pα + 1, . . . , n
d b−p
α
2
Let N0 = 1, and set
Ωj1 := {Φ(σ(`)) : ` = N0 , . . . , j − 1}
and
N1 := min{j > N0 : Φ(σ(j)) ∈ Ωj1 },
Ωj2 := {Φ(σ(`)) : ` = N1 , . . . , j − 1}
Similarly, we define
and
N2 := min{j > N1 : Φ(σ(j)) ∈ Ωj2 }.
N2m+1
Ωj2m+1 := {Φ(σ(`)) : ` = N2m , . . . , j − 1},
(
)
m−1
[ N
j
2i+1
:= min j > N2m : Φ(σ(j)) ∈
Ω2i+1 ∪ Ω2m+1 ,
i=0
Ωj2m
:= {Φ(σ(`)) : ` = N2m−1 , . . . , j − 1}, and
18
MELCHER
(
N2m := min j > N2m−1 : Φ(σ(j)) ∈
m−1
[
)
2m
ΩN
2m
∪
Ωj2m
.
i=1
−1
Then there is an M < qα such that the sets {Aσ(Ni ) , . . . , Aσ(Ni+1 −1) }M
i=0 and
{Aσ(NM ) , . . . , Aσ(n) } separate the hj ’s in the sense that no hj is repeated inside any
one of these sets, and moreover the union of “even” sets contains exactly one copy
of each hj and similarly with the “odd” sets. We can write
Fnσ (k1 ⊗ · · · ⊗ kpα ⊗ h1 ⊗ h1 ⊗ · · · ⊗ hqα ⊗ hqα )
X =
he1 , [[[Aσ(1) , Aσ(2) ], . . .], Aσ(N1 −1) ]i
e1 ,...,eM
×
M
−1
Y
!
hei+1 , [[[ei , Aσ(Ni ) ], . . .], Aσ(Ni+1 −1) ]i [[[eM , Aσ(NM ) ], . . .], Aσ(n) ] .
i=1
If M is even, then
A = he1 , [[[Aσ(1) , Aσ(2) ], . . .], Aσ(N1 −1) ]i


M/2−1
Y
×
he2i+1 , [[[e2i , Aσ(N2i ) ], . . .], Aσ(N2i+1 )−1 ]i [[[eM , Aσ(NM ) ], . . .], Aσ(n) ]
i=1
is a function of h1 , . . . , hqα , e1 , . . . , eM and ki for i ∈ I some subset of {1, . . . , n},
and
M/2
Y
B=
he2i , [[[e2i−1 , Aσ(N2i−1 ) ], . . .], Aσ(N2i )−1 ]i
i=1
is a function of h1 , . . . , hqα , e1 , . . . , eM and ki for i ∈ I c . Thus,


X
X
kAk2g  
kF̂nσ,α k22 ≤ 

|B|2 
CM
ki :i∈I c ,h1 ,...,hqα ,e1 ,...,eM
ki :i∈I,h1 ,...,hqα ,e1 ,...,eM
which is finite again by Proposition 2.8. Similarly, if M is odd,
A = he1 , [[[Aσ(1) , Aσ(2) ], . . .], Aσ(N1 −1) ]i
(M −1)/2
×
Y
he2i+1 , [[[e2i , Aσ(N2i ) ], . . .], Aσ(N2i+1 )−1 ]i
i=1
and


(M −1)/2
B=
Y
he2i , [[[e2i−1 , Aσ(N2i−1 ) ], . . .], Aσ(N2i )−1 ]i [[[eM , Aσ(NM ) ], . . .], Aσ(n) ]
i=1
are both functions of h1 , . . . , hqα , e1 , . . . , eM and some ki , and


X
X
kF̂nσ,α k22 ≤ 
|A|2gCM  
ki :i∈I,h1 ,...,hqα ,e1 ,...,eM
ki
:i∈I c ,h

kBk2gCM 
1 ,...,hqα ,e1 ,...,eM
is again finite by Proposition 2.8 and this completes the proof.
Propositions 3.3 and 3.5 now allow us to make the following definition.
QUASI-INVARIANCE ON ABSTRACT NILPOTENT LIE GROUPS
19
Definition 3.6. A Brownian motion on G is the continuous G-valued process
defined by
r X
n
X
X
X Z
cσn
gt =
fα (s, t)F̂nσ,α (dBs1 ⊗ · · · ⊗ dBspα ),
n−m
2
∆
(t)
m
pα
n=1
σ∈Sn m=dn/2e
α∈Jn
where
cσn = (−1)e(σ)
n2
n−1
,
e(σ)
F̂nσ,α is as defined in equation (3.10), and fα is as defined below equation (3.11).
For t > 0, let νt = Law(gt ) be the heat kernel measure at time t, a probability
measure on G.
Proposition 3.7 (Finite-dimensional approximations). For Gπ a finite-dimensional
Lie subgroup of GCM , let π denote orthogonal projection of GCM onto Gπ and let
gtπ be the continuous process on Gπ defined by
r X
n
X Z
X
X
cσn
π
fα (s, t)F̂nσ,α (dπBs1 ⊗ · · · ⊗ dπBspα ),
gt =
2n−m
∆pα (t)
m
n=1
σ∈Sn m=dn/2e
α∈Jn
where the stochastic integrals are defined as in Proposition 3.4. Then gtπ is Brownian motion on Gπ . In particular, for G` = Gπ` an increasing sequence of finitedimensional Lie subgroups such that the associated orthogonal projections π` are
increasing to IgCM , let gt` = gtπ` . Then, for all t < ∞,
2
(3.13)
lim E gt` − gt g = 0.
`→∞
gtπ
Proof. First note that
solves the Stratonovich equation δgtπ = Lgtπ ∗ δπBt with
π
g0 = e, see [7, 9, 4] where hπBit is a standard gπ -valued Brownian motion. Thus,
gtπ is a Gπ -valued Brownian motion.
By equation (3.12) and its preceding discussion,
gt` =
r X
X
n
X
n=1 σ∈Sn m=dn/2e
cσn
2n−m
qα
X X
baα ta Jn` (f˜α F̂nσ,α )t ,
α∈Jnm a=0
and thus, to verify (3.13), it suffices to show that
lim Ekπ` Bt − Bt k2g = 0
`→∞
and
2
lim E Jn` (f˜α F̂nσ,α )t − Jn (f˜α F̂nσ,α )t = 0,
`→∞
for all n ∈ {2, . . . , r}, σ ∈ Sn and α ∈ Jn .
So let µt = Law(Bt ). Then it is known that, if V is a finite-dimensional subspace
of gCM and πV is the orthogonal projection from gCM to V , then πV admits a
µt -a.s. unique extension to g. Moreover, if Vn is an increasing sequence of finitedimensional subspaces, then
lim EkπVn Bt − Bt k2g = 0;
n→∞
see for example Section 8.3.3 of [28].
By Proposition 3.5, F̂nσ,α is Hilbert-Schmidt, and recall that f˜α is a deterministic polynomial function in s. Thus Jn` (f˜α F̂nσ,α ) and Jn (f˜α F̂nσ,α ) are gCM -valued
20
MELCHER
martingales as defined in Proposition 3.4, and Proposition 3.4 gives the desired
convergence as well (in gCM and thus in g).
Remark 3.8. In fact, for each of the stochastic integrals Jn (f˜α F̂nσ,α ), it is possible
to prove the stronger convergence that, for all p ∈ [1, ∞),
p lim E sup Jn` (f˜α F̂nσ,α )τ − Jn (f˜α F̂nσ,α )τ = 0,
`→∞
τ ≤t
for all n ∈ {2, . . . , r}, σ ∈ Sn and α ∈ Jn . Again, Proposition 3.4 gives the limit
for p = 2 and thus for p ∈ [1, 2]. For p > 2, Doob’s maximal inequality implies it
suffices to show that
p
lim E J ` (f˜α F̂ σ,α )t − Jn (f˜α F̂ σ,α )t = 0.
`→∞
n
n
n
Since each Jn` (f˜α F̂nσ,α ) and Jn (f˜α F̂nσ,α ) has chaos expansion terminating at degree
n, a theorem of Nelson (see Lemma 2 of [26] and pp. 216-217 of [25]) implies that,
for each j ∈ N, there exists cj < ∞ such that
2j
2 j
E Jn` (f˜α F̂nσ,α )t − Jn (f˜α F̂nσ,α )t ≤ cj E Jn` (f˜α F̂nσ,α )t − Jn (f˜α F̂nσ,α )t .
In a similar way, one may prove the following convergence for the Brownian
motions under right translations by elements of GCM .
Proposition 3.9. For any y ∈ GCM ,
lim Ekgt` y − gt yk2g = 0.
`→∞
where gt y is the measurable right group action of y ∈ GCM on gt ∈ G, as in
Proposition 2.6.
4. Heat kernel measure
We are now able to prove Theorem 1.1, which states that the heat kernel measure
νt = Law(gt ) is quasi-invariant under left and right translation by elements of
GCM and gives estimates for the Radon-Nikodym derivatives of the “translated”
measures.
Proof of Theorem 1.1. Fix t > 0 and π0 an orthogonal projection onto a finitedimensional subspace G0 of gCM . Let h ∈ G0 , and {πn }∞
n=1 be an increasing
sequence of projections such that G0 ⊂ πn GCM for all n and πn |GCM ↑ IGCM . Let
Jtn,r (h, ·) denote the Radon-Nikodym derivative of νtn ◦ Rh−1 with respect to νtn .
Then for each n and for any q ∈ [1, ∞)
Z
1/q
(q − 1)k
n,r q
2
dn (e, h)
(Jt ) pt (x) dx
≤ exp
2(ekt − 1)
G
where k is the uniform lower bound on the Ricci curvature as in Proposition 2.12
and dn is Riemannian distance on Gn ; see for example Theorem 1.5 of [13].
By Proposition 3.9, we have that for any f ∈ Cb (G), the class of bounded
continuous functions on G
Z
f (gh)dνt (g) = E[f (gt h)]
G
Z
(4.1)
= lim E[f (gtn h)] = lim
f ◦ in (gh) dνtn (g),
n→∞
n→∞
Gn
QUASI-INVARIANCE ON ABSTRACT NILPOTENT LIE GROUPS
21
where in : Gn → G denotes the inclusion map. Note that
Z
Z
|(f ◦ in )(gh)| dνtn (g) =
Jtn,r (h, g)|(f ◦ in )(g)|dνtn (g)
Gn
Gn
k(q − 1)
2
0
≤ kf ◦ in kLq (Gn ,νtn ) exp
dn (e, h) ,
2(ekt − 1)
where q 0 is the conjugate exponent to q. Allowing n → ∞ in this last inequality
yields
Z
k(q − 1)
2
(4.2)
d(e,
h)
,
|f (gh)| dνt (g) ≤ kf kLq0 (G,νt ) exp
2(ekt − 1)
G
by equation (4.1) and the fact that the length of a path in GCM can be approximated
by the lengths of paths in the finite-dimensional projections. That is, for any π0 and
ϕ ∈ C 1 ([0, 1], GCM ) with ϕ(0) = e, there exists an increasing sequence {πn }∞
n=1 of
orthogonal projections such that π0 ⊂ πn , πn |gCM ↑ IgCM , and
`CM (ϕ) = lim `Gπn (πn ◦ ϕ).
n→∞
To see this, let ϕ be a path in GCM . Then one may show that
Z 1
r−1
X
`
0
0
`Gπn (πn ◦ ϕ) =
c` adπn ϕ(s) πn ϕ (s)
πn ϕ (s) +
0 `=1
ds
gCM
for appropriate coefficients c` ; see for example Section 3 of [24]. Thus, we have
proved that (4.2) holds for f ∈ Cb (G) and h ∈ ∪π Gπ . As this union is dense in G
by Proposition 2.11, dominated convergence along with the continuity of d(e, h) in
h implies that (4.2) holds for all h ∈ GCM .
0
Since the bounded continuous functions are dense in Lq (G, νt ) (see for example
Theorem A.1 of [20]), the inequality in (4.2) implies that the linear functional
ϕh : Cb (G) → R defined by
Z
ϕh (f ) =
f (gh) dνt (g)
G
0
has a unique extension to an element, still denoted by ϕh , of Lq (G, νt )∗ which
satisfies the bound
k(q − 1)
2
d(e,
h)
|ϕh (f )| ≤ kf kLq0 (G,νt ) exp
2(ekt − 1)
0
0
for all f ∈ Lq (G, νt ). Since Lq (G, νt )∗ ∼
= Lq (G, νt ), there then exists a function
r
q
Jt (h, ·) ∈ L (G, νt ) such that
Z
(4.3)
ϕh (f ) =
f (g)Jtr (h, g) dνt (g),
G
0
for all f ∈ Lq (G, νt ), and
kJtr (h, ·)kLq (G,νt ) ≤ exp
k(q − 1)
2
d(e,
h)
.
2(ekt − 1)
Now restricting (4.3) to f ∈ Cb (G), we may rewrite this equation as
Z
Z
−1
(4.4)
f (g) dνt (gh ) =
f (g)Jtr (h, g) dνt (g).
G
G
22
MELCHER
Then a monotone class argument (again use Theorem A.1 of [20]) shows that (4.4)
is valid for all bounded measurable functions f on G. Thus, d(νt ◦ Rh−1 )/dνt exists
and is given by Jtr (h, ·), which is in Lq for all q ∈ (1, ∞) and satisfies the desired
bound.
A parallel argument gives the analogous result for d(νt ◦L−1
h )/dνt . Alternatively,
one could use the right translation invariance just proved along with the facts that
νt inherits invariance under the inversion map g 7→ g −1 from its finite-dimensional
projections and that d(e, h−1 ) = d(e, h).
The following also records the straightforward fact that the heat kernel measure
does not charge GCM .
Proposition 4.1. For all t > 0, νt (GCM ) = 0.
Proof. This follows trivially from the fact that gt is the sum of a Brownian motion
Bt on g with a finite sequence of stochastic integrals taking values in gCM .
Thus, GCM maintains its role as a dense subspace of G of measure 0 with respect
to the distribution of the “group Brownian motion”.
Definition 4.2. A function f : G → R is said to be a (smooth) cylinder function
if f = F ◦ π for some finite-dimensional projection π and some (smooth) function
F : Gπ → R. Also, f is a cylinder polynomial if f = F ◦ π for F a polynomial
function on Gπ .
Theorem 4.3. Given a cylinder polynomial f on G, let ∇f : G → gCM be the
gradient of f , the unique element of gCM such that
h∇f (g), higCM = h̃f (g) := f 0 (g)(Lg∗ he ),
for all h ∈ gCM . Then for k as in Proposition 2.12,
Z
Z
Z
Z
1 − e−kt
k∇f k2gCM dνt .
(f 2 ln f 2 ) dνt −
f 2 dνt · ln
f 2 dνt ≤ 2
k
G
G
G
G
Proof. Following the method of Bakry and Ledoux applied to GP (see Theorem 2.9
of [14] for the case needed here) shows that
E
1 − e−kπ t
2
f 2 ln f 2 (gtπ ) − E f 2 (gtπ ) ln E f 2 (gtπ ) ≤ 2
E k(∇π f ) (gtπ )kgπ ,
kπ
for kπ as in equation (2.3). Since the function x 7→ (1 − e−x )/x is decreasing and
k ≤ kπ for all finite-dimensional projections π, this estimate also holds with kπ
replaced with k. Now applying Proposition 3.7 to pass to the limit as π ↑ I gives
the desired result.
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Department of Mathematics, University of Virginia, Charlottesville, VA 22903 USA
E-mail address: melcher@virginia.edu