1. Introduction
Let us consider a compact connected Lie group
G of dimension
d endowed with its normalized biinvariant Haar measure
. Let us consider the Laplacian
on it. It is equal to
where
is an orthonormal basis of the Lie algebra of
G. It generates a Markov semi-group
:
if
f is smooth. Moreover there is a
strictly positive heat kernel
when
Let us consider a Bilaplacian on
G, this means a power
. It generates still a semi-group
.
is not a Markovian semi-group. This means that the heat kernel
associated to
can
change sign. We have still when
In the first case, the heat semi-group is represented by the Brownian motion on
G. In the second case, there is until now no stochastic process associated to it. In the case of
, the path integral involved with the semi-group
is defined as a distribution in [
1].
We are motivated in this work by an extension in infinite dimension of these results, by considering the case of the path-group from continuous path from into G starting from e.
Let us recall that the Haar measure
on a topological group
exists as a full measure
if and only if the group is locally compact. Haar measure means that for all bounded measurable function
The difficult requirement to satisfy is the Lebesgue dominated convergence: Let
be a bounded increasing sequence of measurable functions tending almost surely to
. Then
Haar measures in infinite dimension were studied by Pickrell [
2] and Asada [
3]. We have defined the Haar distribution on a path group by using the Hida-Streit approach of path integrals as distribution [
4,
5,
6,
7]. We refer to the review of Albeverio for various rigorous approaches to path integrals [
8] and our review on geometrical path integrals [
4,
9].
In the case of a path group, we can consider the Wiener process on a path-group
starting from the unit path (See the work of Airault-Malliavin ([
10]), the work of Baxendale [
11] and the review paper of Léandre on that topic [
12]). We have shown that
when
where
is the Haar distribution on the path group and
F is a test functional of Hida type [
7].
Recently we are motivated by extending stochastic analysis tools in the non-Markovian context ([
13,
14,
15,
16,
17]). Especially in [
1], we are interested in constructing the sheet and martingales problem in distributional sense for a big-order differential operator on
. We consider for that the Connes test algebra.
Let us recall what is the main difference between the Hida test algebra and the Connes test algebra.
(1) Hida considers Fock spaces and tensor product of Hilbert spaces.
(2) Connes, motivated by his work on entire cyclic cohomology, considers Banach spaces. Tensor product of Banach spaces whose theory (mainly due to Grothendieck) is much more complicated than the theory of tensor product of Hilbert spaces.
In [
1], we are motivated by the generalization of martingale problems in the non-Markovian context. We consider Connes spaces in [
1]. In the present context, we are not motivated by that and we return in the original framework of [
7].
We consider the heat semi-group on a path group associated to a bilaplacian on the group in the manner of [
1]. In [
1], we look at the case of
. Here we consider the case of the compact Lie group
G. The analysis is similar because we have analog estimates of the heat-kernel [
18,
19,
20].
In order to resume, we consider an element of an Hida Fock space, we associate a functional on the path group. The heat-semi group (in the distributional sense) satisfies the three next properties:
- (1)
is still in the considered space
- (2)
- (3)
is not a Markovian semi-group on . Especially, is not represented by a stochastic process. However we expect to extend in this context (7).
2. A Brief Review on the Haar Distribution on a Path Group
Let us recall what is the Brownian motion
on
. We consider the set of continuous path
issued from 0 from
into
. We consider the sigma-algebra
spanned by
. The Brownian motion probability measure
is characterized as the solution of the following martingale problem: if
f is any bounded smooth function on
,
is a martingale associated to the filtration
. This means that
where
G is a bounded functional
measurable (
).
The Brownian motion is only continuous. However we can define stochastic integrals (as it was done by Itô). Let
be a bounded continuous process. We suppose that
is
measurable. Then the Itô integral is defined as follows:
Moreover we have the Itô isometry
Associated to the Brownian motion is classically associated the Bosonic Fock space.
Let
be the Hilbert space of
functions
from
into
. We consider the symmetric tensor product
of
. It can be realized as the set of symmetric maps
from
into
such that
The symmetric Fock space
coincides with the set of formal series
such that
. To each
we associate the
Wiener chaos
if
is the standard
-valued Brownian motion. The definition of the Wiener chaos
is a small improvement of the stochastic integral
. By using the symmetry of
, we have:
Moreover and and are orthogonal in . The of the Brownian motion can be realized as the symmetric Fock space through the isometry .
We introduce the Laplacian
on
and we consider the Sobolev space
associated to
. On the set of formal series
, we choose a slightly different Hilbert structure:
We get another symmetric Fock space denoted
. We remark that if
The Hida test function space
is the intersection of
,
endowed with the projective topology. A sequence
of the Hida Fock space converges to
for the topology of the Hida Fock space if
converges to
in all
. The map Wiener chaos
realized a map from
into the set of continuous Brownian functional dense in
. We refer to the books [
21] and [
22] for an extensive study between the Fock space and the
of the Wiener measure.
In infinite dimensional analysis, there are basically 3 objects:
(i) An algebraic model.
(ii) A mapping space and a map from the algebraic model into the space of functionals on this mapping space.
(iii) A path integral which is an element of the topological dual of the algebraic model.
In the standard case of the Brownian motion,
is the vacuum expectation:
A distribution on the Hida Fock space is a linear map
from
into
which satisfies the following requirement: there exists
such that for all
Getzler in his seminal paper [
23] is the first author who considered another map than the map Wiener chaos. Getzler is motivated by the heuristic considerations of Atiyah-Bismut-Witten relating the structure of the free loop space of a manifold and the Index theorem on a compact spin manifold. Getzler used as algebraic space a Connes space and as map
the map Chen iterated integrals.
Getzler’s idea was developed by Léandre ([
9]) to study various path integrals in the Hida-Streit approach with a geometrical meaning. Especially Léandre ([
5,
6]) succeeded to define the Haar measure
as a distribution on a current group. Let us recall quickly the definition on it. We consider a compact Riemannian manifold
M (
) and a compact Lie group
G (
). We consider the current group
of continuous maps
from
M into
G. We consider the cylindrical functional
on the current group. We have
We would like to close this operation consistently. It is the object of [
5,
6].
(1)
Construction of the algebraic model. We consider the positive self-adjoint Laplacian on
. We consider the Sobolev space
of maps from
h into
such that
We consider the tensor product
associated to it and we consider the natural Hilbert norm on it (
and
are normalized Riemannian measures on
M and
G respectively).
is the set of formal series
such that
The Hida test functional space is the space
endowed with the projective topology.
(2)
Construction of the map Ψ. To
we associate
We put if
The map
realizes a continuous map from
into the set of continuous functional on
.
(3)
Construction of the path integral. We put if
belongs to all the Sobolev Hilbert spaces
This map can be extended into a linear continuous application from
(We say it is a Hida distribution) into
. This realizes our definition ([
5,
6,
7]) of the Haar distribution on the current group
.
Let
. We consider the normalized Lebesgue measure
on
. Let
be the
partial Laplacian on
. We consider the total operator
which operates on function
on
and we consider its power
. Let
be a function on
. We put
(
is the normalized Haar measure on
and
the normalized Lebesgue measure on
).
We put
and we consider the Hilbert norm
Definition 1: The Hida Fock space is the space constituted of the defined above such that for all ,
If
belongs to
, we associate
where
belongs to
.
Theorem 2: If , is a continuous bounded function on .
We put
Let us recall three of the main theorems of [
7]:
Theorem 3: can be extended as a distribution on the Hida Fock space. This means that there exists such that for all Theorem 4: If , .
Theorem 5: If , .
3. A Non-Markovian Semi-group on a Path Group
In the sequel, we will suppose that
. In such a case ([
20]), we have
where
.
is the heat-kernel associated to the heat semi-group
and
d is the biinvariant Riemannian distance on
G.
Moreover, since
is biinvariant
Since it is an heat kernel associated to a semi-group, it satisfies the Kolmogorov equation:
This shows that if
that
and that
Remark: We could get in the sequel more general convolution semi-groups [
20] with generators of degree
whose associated heat-kernels satisfied still (33).
Let us divide the interval time
into in time intervals
of length
. Let
F be a cylindrical functional
. Let us introduce
(
). This defines a semi-group on
. Let us show this statement. We remark
We do the change of variable
. We recognize in the last expression
But
We have used the semi-group property (36) of
and the fact that
is biinvariant (35).
We would like to extend by continuity this formula for functionals which depend on an infinite number of variables
of the previous type. We put for
:
. We order
without to loose generality.
We extend by linearity.
Theorem 6: is a Hida distribution. Moreover if , .
Proof: By the property of the cylindrical semi-group listed in the beginning of this part, we have
where
is the uniform norm of
. This uniform norm can be estimated by Sobolev imbedding theorem by
for some big
k and
C independent of
n. It follows clearly from that
is an Hida distribution.
Let us give some details in order to estimate
. We introduce the ordered set of eigenvalues
of
. Let
. Let
be the normalized eigenvectors associated to
. We consider
-valued functions to do that. We introduce
. We get
Therefore
By Garding and Sobolev inequality, the right-hand side of the previous inequality is smaller than
for some big
k, some big
C and some big
l.
Let us recall that
and that
for some
m ([
24]). We apply Cauchy-Schwartz inequality in (47). We deduce that
But
Moreover, by [
8],
for some
i. Therefore if
l is big enough,
is finite bounded independently of
n.
Let us consider the polygonal approximation of mesh
of
. If
, we get
. But
is a cylindrical functional which depends only of
. We use the properties listed in the beginning of this part. We get
by the property listed of the beginning of the cylindrical semi-group
. It remains to show that when
that
is very close from
. This follows from the next consideration. Let
be an elementary tensor product. We get clearly
where
denotes the supremum of the time of the subdivision smaller to
s and
denotes the infimum of the time of the subdivision larger to
s.
is the uniform
norm of
. This norm can be estimated by the Sobolev imbedding theorem by
for
and
independent of
n as in (50).
It remains to use the inequality (35) to conclude.◊
Definition 7: is called the Wiener distribution issued from the unit path associated to .
Let
be a smooth function from
into
. We suppose that
in order to simplify the exposition. We put
is the cylindrical semi-group on cylindrical functional associated to
.
lemma 8: There exist a bounded whentis bounded and which depend not ofn,
a which depend only ofkand not onnsuch that Proof: If we take derivative in , the result comes by taking derivative under the sign integral in (43). The result arises then from (37). Let us take first of all derivative in . Either we take derivative of and the result goes by the same way. Or we take derivative in or of the heat kernel . We represent in the way (43) the integral, we remark that the heat kernel satisfies the heat-equation and we integrate by parts in order to conclude.◊
Let us suppose that the time subdivision is fixed. Clearly
Let
be a function from
into
. We put
Theorem 9: can be extended by linearity as a continuous linear operator on the Hida Fock space. If ,
and we get the semi-group propertyif belong to .
Proof: The fact that
can be extended by linearity follows from the previous lemma.
if
holds exactly as in the proof of Theorem 6. For a simple element
of the Hida Fock space, we have clearly:
This result can be extended by continuity.◊