Stochastic and Deterministic Bundles
We consider a bundle determined by a classifying map with skeleton smooth in the Chen — Souriau sense. We show that the stochastic classifying map is homotopic to a deterministic classifying map on the Holder loop space.
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irk-123456789-1658342020-02-17T01:27:21Z Stochastic and Deterministic Bundles Leandre, R. Статті We consider a bundle determined by a classifying map with skeleton smooth in the Chen — Souriau sense. We show that the stochastic classifying map is homotopic to a deterministic classifying map on the Holder loop space. Розглядаються розшарування, що визначаються класифікуючим відображенням із скелетом, гладким у сенсі Chen - Souriau. Показано, що стохастичне класифікуюче відображення гомотопне детерміністичному класифікуючому відображенню на просторах гьольдерових петель. 2005 Article Stochastic and Deterministic Bundles / R. Leandre // Український математичний журнал. — 2005. — Т. 57, № 9. — С. 1284–1288. — Бібліогр.: 23 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165834 519.21 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Leandre, R. Stochastic and Deterministic Bundles Український математичний журнал |
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We consider a bundle determined by a classifying map with skeleton smooth in the Chen — Souriau sense. We show that the stochastic classifying map is homotopic to a deterministic classifying map on the Holder loop space. |
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Article |
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Leandre, R. |
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Leandre, R. |
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Leandre, R. |
| title |
Stochastic and Deterministic Bundles |
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Stochastic and Deterministic Bundles |
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Stochastic and Deterministic Bundles |
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Stochastic and Deterministic Bundles |
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Stochastic and Deterministic Bundles |
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stochastic and deterministic bundles |
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Інститут математики НАН України |
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2005 |
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Stochastic and Deterministic Bundles / R. Leandre // Український математичний журнал. — 2005. — Т. 57, № 9. — С. 1284–1288. — Бібліогр.: 23 назв. — англ. |
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Український математичний журнал |
| work_keys_str_mv |
AT leandrer stochasticanddeterministicbundles |
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2025-07-14T20:05:45Z |
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2025-07-14T20:05:45Z |
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1837654131882328064 |
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UDC 519.21
R. Léandre (Inst. Math. Univ. Bourgogne, France)
STOCHASTIC AND DETERMINISTIC BUNDLES
STOXASTYÇNI TA DETERMINISTYÇNI ROZÍARUVANNQ
In honour of Professor A. Skorokhod for his 75 birthday
We consider a bundle determined by a classifying map with skelettum smooth in Chen – Souriau sense. We
show that the stochastic classifying map is homotopic to a deterministic classifying map on the Hölder loop
space.
Rozhlqdagt\sq rozßaruvannq, wo vyznaçagt\sq klasyfikugçym vidobraΩennqm iz skeletom, hladkym
u sensi Chen – Souriau. Pokazano, wo stoxastyçne klasyfikugçe vidobraΩennq homotopne determinis-
tyçnomu klasyfikugçomu vidobraΩenng na prostorax h\ol\derovyx petel\.
1. Introduction. Let Lx(M) be the based loop space of a compact Riemannian mani-
fold, endowed with the Brownian bridge measure.
There are two stochastic de Rham cohomology theories associated to it, for forms
almost surely defined:
The first one is the de Rham cohomology in Nualart – Pardoux sense, which is a refin-
ment of Malliavin Calculus [1 – 3]. If the manifold is simply connected, it is equal to the
de Rham cohomology of the finite energy based loop space.
The second one is the de Rham cohomology in Chen – Souriau sense, which uses a
wide variety of stochastic diffeologies on the based loop space [4 – 7]. The main theorem
is that the stochastic cohomology in Chen – Souriau sense is equal to the deterministic de
Rham cohomology of the Hölder loop space.
On a compact manifold, associated to a complex bundle, we can study character-
istic classes. In particular, Chern – Weil isomorphism states that the complex K-theory
tensorized by C of a compact manifold is equal to the complex de Rham cohomology
associated to the manifold.
There are a big variety of definition of stochastic bundle (with fiber almost surly de-
fined!) on the loop space:
Either, in Nualart – Pardoux sense, [8 – 10] consider stochastic K-theories, by consid-
ering finite dimensional random projectors. Let us recall that one of the main originality of
Malliavin Calculus with respects of its preliminary versions (see works of Hida, Elworthy,
Fomin, Albeverio, Berezanskii ...) is that the space of test functionals is an algebra. This
allows to Léandre [8, 11] to define a stochastic Chern – Character in Nualart – Pardoux
sense.
Or in the Chen – Souriau sense, various stochastic Z-valued Stiefel – Whitney classes
were defined [8]. Léandre [8] uses a suitable classifying map in Chen – Souriau sense in
the classifying space (see the book of Milnor – Stasheff [12] for the study of such objects
in the deterministic context).
Moreover, for these two Calculi, in order to understand characteristic classes, various
algebraic de Rham complexes were studied in [13] (see the book of Loday [14] for an
extensive study of such objects associated to differential algebras).
On the other hand, if the based loop space is simply connected, a line bundle is deter-
mined by its curvature. This justifies the fact [7] that a stochastic line bundle (with fiber
c© R. LÉANDRE, 2005
1284 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
STOCHASTIC AND DETERMINISTIC BUNDLES 1285
almost surely defined!) in Chen – Souriau sense is isomorphic to a true line bundle over
the Hölder loop space, if this one is simply connected.
This motivates the question: for what theories is a stochastic bundle (with fiber almost-
surely defined!) isomorphic to a deterministic bundle?
Let us recall that study of bundles on the loop spaces are motivated by the works of
Witten [15, 16] relating the K-theory of the free loop space with elliptic cohomology.
Moreover, Jaffe – Lesniewski – Osterwalder [17] have considered some K-theory on the
loop space associated to the Wess – Zumino model on the loop space.
The goal of this work is to answer to this question.
Let us recall that we have defined in [18, 19] the notion of stochastic forms smooth in
Chen – Souriau sense with skelettum.
We consider in this paper a bundle determined by a classifying map with skelettum
smooth in Chen – Souriau sense. We show that the stochastic classifying map is homo-
topic to a deterministic classifying map on the Hölder loop space. We get:
Main theorem. A stochastic bundle, smooth in Chen – Souriau sense (or in the
Froelicher sense), with skelettum, over the based loop space is isomorphic to a deter-
ministic bundle on the strong Hölder loop space.
By strong Hölder loop space, we mean the set of loops s → γ(s) such that
lim
s→t
d(γ(s), γ(t))
|s− t|1/2−ε
= 0
for some small ε where d is the Riemannian distance on the manifold.
We refer to the survey of Albeverio [20], and to the 3 surveys of Léandre for the
relation between stochastic analysis and mathematical physics [11, 21, 22]. We thank
D. Arnal for helpfull comments.
2. The model. We consider the injective limit M∞(C) of the linear space Mn(C)
of linear applications from Cn into Cn. We consider the Hilbert norm ‖A‖2 =
=
∑
|Aei|2 where ei is the canonical basis of Cn. This norm is compatible with
the canonical injection from Mn(C) into Mn+1(C). We get a structure of Prehilbert
space M∞(C) (it is not complete). In order to take derivatives, we consider its natural
completion M∞(C).
B∞(U(n)) is the set of orthogonal projectors of rank n belonging to M∞(C). It
is the classifying space [12] of n-dimensional complex Hermitian bundles endowed with
an Hermitian structure. There is on B∞(U(n)) a canonical universal U(n) bundle.
Let M ⊆ Rn a Riemannian manifold of dimension d isometrically imbedded in
Rn. We consider the based loop space Lx(M) of strongs Hölder loops s → γ(s) of
loops such that
lim
s→t
d(γ(s), γ(t))
|s− t|1/2−ε
= 0.
It is a Banach manifold. It is endowed with the Brownian bridge measure.
Definition 2.1. A stochastic plot of dimension m, φst = (U, φi,Ωi)i∈N is given by
the following data:
a open subset U of Rm;
a countable partition Ωi of Lx(M);
a family of smooth applications (u, s, y) → Fi(u, s, y) from U × S1 ×M bounded
with bounded derivatives of all orders, with values in Rn;
over Ωi, φi(u) = {s → Fi(u, s, γ(s))} belongs to Lx(M).
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1286 R. LÉANDRE
Let Lx(M)N be the set of loops such that sup|s−t|<1/N d(γ(s), γ(t)) < r where
r is small enough. If γ belongs to Lx(M)N , we can define its polygonal approxima-
tion γN .
We call Lx,∞(M) the finite energy based loop space.
Theorem 2.1. A map Ψ from Lx,∞(M) into B∞(Un) is called smooth in the
Frechet sense if it is Frechet smooth considered as map with values in M∞(C).
Analoguous definition works for a Frechet smooth map from the strong Hölder loop
space in the classifying space.
Definition 2.2. A Frechet smooth map Ψ from Lx,∞(M) in the classifying space
is called the skelettum of a map smooth in the Chen – Souriau sense (or in the Froelicher
sense) Ψst with values in the classifying space if for all stochastic plot u → Ψ(φN
st)
tends almost surely smoothly in N (for a suitable compactification of R+ ) for the
smooth topology to a smooth random map u → Ψst(φst(u)) from U into the classi-
fying space (almost surely defined!).
The application Ψst defines a map smooth in the stochastic Chen – Souriau sense (or
in the stochastic Froelicher sense) from Lx(M) endowed with the Brownian bridge mea-
sure according to our diffeology and the definitions of [8] with values in the classifying
space. In particular, the pullback by Ψst of the universal bundle E∞(U(n)) on the clas-
sifying space defines a stochastic bundle in the stochastic Chen – Souriau sense (or in the
stochastic Froelicher sense) over Lx(M). We denote Ψ∗
stE∞(Un) it. It is a stochastic
bundle in Chen – Souriau sense with skelettum Ψ∗E∞(Un).
3. Proof of the main theorem. Let us recall what is a stochastic bundle in Chen –
Souriau sense (or in Froelicher sense) over the strong Hölder loop space [8].
Definition 3.1. Let us consider a map Ψst smooth in Chen – Souriau sense with
values in the classifying space B∞(Un). This means, that for all stochastic plots φst
U → Lx(M), we can define Ψst(φst(u)) as a random smooth map from U into the
classifying space. Moreover the set of these random smooth maps have to satisfied the
two following requirements:
If j : U1 → U2 is a smooth deterministic map from U1 into U2, if φst,2 is a
stochastic plot from U2 into Lx(M), we can consider the composite plot φst,1 =
= φst,2 ◦ j. Therefore, the following statement has to be satisfied: Ψst(φst(u1)) =
= Ψst(φst,2(j(u1)) has smooth random applications from U1 into the classifying space.
If two stochastic plots φst,1 and φst,2 are deduced one from the other by a measur-
able transformation ψ of the strong Hölder based loop space on a set of probability not
equal to 0, we have
Ψst(φst,1) = Ψst(φst,2) ◦ ψ
almost surely.
All stochastic bundle in the Chen – Souriau sense (or in the Froelicher sense) can be
seen as isomorphic to a pullback bundle Ψ∗
stE∞(Un) [8]. This generalizes this very well
known fact in algebraic topology [12] in the deterministic context.
Let us recall the following theorem:
Theorem 3.1. Ψ∗
st,1E∞(Un) and Ψ∗
st,2E∞(Un) are isomorphic as soon as there
exists a smooth homotopy Ψst,t between Ψst,1 and Ψst,2. This means that for all
stochastic plot φst, (u, t) → Ψst,t(φ(u)) is almost surely smooth in t in u with val-
ues in the classifying space and satisfies consistency requirements analoguous to Defini-
tion 3.1.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
STOCHASTIC AND DETERMINISTIC BUNDLES 1287
Lemma 3.1. Let L0(Rn) be the strong
1
2
− ε-Hölder based loop space of Rn.
There exists θ and p ∈ 2N such that the map which to γ in the strong Hölder based
loop space of Rm associates
F (γ) =
∫
S1
|γ(s)|pds+
∫
S1×S1
|γ(s) − γ(t)|p
|s− t|1+θp
ds dt
is Frechet smooth.
Proof. Let us perturb γ by h in the strong Hölder based loop space:∣∣|γ(s) − γ(t)|p − |γ(s) + h(s) − γ(t) − h(t)|p
∣∣ ≤
≤ |h(s) − h(t)|t
∑
|γ(s) − γ(t)|k|γ(s) + h(s) − γ(t) − h(t)|p−k−1.
If θ is small enough,∫
S1×S1
|γ(s) − γ(t)|k|γ(s) + h(s) − γ(t) − h(t)|p−k−1
|s− t|1+θp−1/2+ε
ds dt
is finite. Therefore the result for the Frechet differentiability of F. Higher Frechet deriv-
abilities are shown in the same way.
The lemma is proved.
F 1/p defines a norm. Some Hölder loop space is continuously imbedded in the Ba-
nach space of loops such the norm F 1/p is finite [23]. This shows us that if F (γ) <
< Ck for some real number k, we can define γC′
for C ′ ≥ C if γ is a loop on
the strong
1
2
− ε-Hölder loop space of the manifold, imbedded in L0(Rn). Moreover,
‖γC′ − γ‖∞ < r for r small enough where ‖.‖∞ is the supremum norm on the based
loop space of Rn. Let π be the projection from a small tubular neighborhood from M
in Rn into M. Let us consider γ̃ a loop in this tubular neighborhood. Let us introduce
π∞ — the Nemystki map:
π∞(γ̃) = {s → π(γ̃)(s)},
π∞ is Frechet smooth for the strong Hölder topology (see [7] for a proof of this statement
in a similar situation).
Proof of the main theorem. Let h be a smooth decreasing function from R+ into
[0, 1] equal to 0 outside a small neighborhood of 0 and equal to 1 in a small neighborhood
of 0. Let Ψ be the skelettum of Ψst. We put
Ψst,t = Ψ
(
π∞
( ∑
n≥0
γnC exp[C/t](−h(n−kF (γ)) + h((n+ 1)−kF (γ)
))
.
We remark that
∑
n≥0
−h(n−kF (γ)) + h((n + 1)−kF (γ)) = 1 and is a series of
positives numbers almost all equal to 0. Moreover,∥∥∥∑
γnC exp[C/t](h((n+ 1)−kF (γ)) − h(n−kF (γ)) − γ
∥∥∥
∞
≤ r.
So we can define π∞
(∑
γnC exp[C/t]
(
h((n+ 1)−kF (γ)) − h(n−kF (γ)
))
. Since F
is Frechet smooth for the strong Hölder topology, and since Ψ is the skelettum of Ψst,
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1288 R. LÉANDRE
we deduce that that t → Ψst,t is a smooth homotopy between Ψst,0 = Ψst and Ψst,1
with values in the classifying space. But Ψst,1 is a Frechet smooth map from the strong
Hölder based loop space of M into the classifying space. Therefore the results.
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Received 17.06.2005
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