Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics

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106 5 Analysis on Groups of Diﬀeomorphisms

means that Y ∈ T

(M)isamapY : M → TM such that πY (m)=g(m)

where π : TM → M is the natural projection. For X ∈ T

(M)wehave

πX(m)=m. D

(M) also possesses the same features, i.e., T

(M)={X ∈

(M,TM)|π ◦X = g},etc.

Let N be a compact oriented Riemannian manifold without boundary.

Consider the mappings α

: H

(M,N) → H

(M,N)oftheformα

(f)=

f ◦ g,andω

: D

(M) → H

(M,N)oftheformω

(r)=h ◦ r.

Lemma 5.1 (α-lemma) α

is C

∞

-smooth and its derivative is also of the

form α

Lemma 5.2 (ω-lemma) ω

is continuous. If h ∈ H

s+k

, ω

: D

→

(M,N) is a C

-mapping with derivative of the form ω

. In particular, if

h ∈ C

∞

, ω

is C

∞

-smooth.

The right translation R

: D

(M) →D

(M), R

◦θ = θ ◦ f,whereθ, f ∈

(M), is C

∞

-smooth and thus one may consider right-invariant vector ﬁelds

on D

(M). The tangent to right translation takes the form TR

X = X ◦ f

for X ∈ T D

(M). For D

(M) we have analogous properties.

Convention 5.3 We shall consider TR

: T

(M) → T

η◦g

(M) for all

η, g ∈D

(M) as a right action of D

(M) on T D

(M).

Theorem 5.4 Let X ∈ T

(M) be a vector ﬁeld on M and

X be the cor-

responding right-invariant vector ﬁeld on D

(M),

= X ◦ g. The vector

ﬁeld

X on D

(M) is C

-smooth if and only if the vector ﬁeld X on M be-

longs to the class H

s+k

. In particular,

X is C

∞

-smooth if and only if X

is C

∞

-smooth. The same property holds for right-invariant vector ﬁelds on

(M).

This fact is a consequence of the ω-lemma 5.2 and is also true for more

complicated ﬁelds (for example, for right-invariant tensor ﬁelds).

The left translation L

f = g ◦f is continuous. If g ∈D

s+k

(M), L

is a C

mapping. In particular, in this case TL

(X)=Tg◦X where Tg : TM → TM

is the tangent mapping of g and X ∈ T D

(M). The mapping g → g

−1

continuous on D

(M). If g ∈D

s+k

(M), this is a C

-mapping from D

s+k

(M)

to D

(M). Thus, from the point of view of the standard ﬁnite-dimensional

deﬁnition, D

(M) is not a Lie group.

Theorem 5.5 Let s>

+1 and let the right-invariant vector ﬁeld

X on

(M) be C

-smooth. Then:

(i) for every g ∈D

(M) there exists a unique integral curve γ

(t) of this

ﬁeld, well-deﬁned for all t ∈ (−∞, ∞), such that γ

(0) = g;

(ii) γ

(t) is the ﬂow of the vector ﬁeld X =

on M, γ

(t)=γ

(t) ◦g;

(iii) if s>

+2, assertion (i) is valid for a continuous right-invariant

vector ﬁeld

X on D

(M).

The same results are true for right-invariant vector ﬁelds of D

(M).

5.1 General Concepts 107

For g ∈D

(M) consider the tangent space T

(M) (see above). Deﬁne

an inner product (·, ·)inT

(M) by the formula

(X, Y )



X(m),Y(m)

g(m)

μ(dm), (5.1)

where X, Y ∈ T

(M)andμ is the Riemannian volume form. Recall that

πX(m)=πY (m)=g(m) so that the vectors X(m)andY (m) belong to the

tangent space at g(m) and so the product is found with respect to ·, ·

g(m)

Since the metric tensor ·, · is C

∞

-smooth, from the ω-lemma 5.2 it follows

that (5.1)isC

∞

-smooth in g ∈D

(M).

Clearly this metric introduces the topology of the functional space L

in the tangent spaces, which is weaker than the initial topology on H

This is why (·, ·) is called a weak Riemannian metric.

One can easily check that the second tangent bundle TTD

(M) consists

of H

maps from M to TTM with the additional properties that they are

projected into maps from D

(M). Consider the connector K : TTM → TM

of the Levi-Civit´a connection H on M.

Deﬁne the mapping

K : TTD

(M) → T D

(M) by the equality

K(Y )=K ◦ Y. (5.2)

Theorem 5.6

K is invariant with respect to right shifts on D

(M).

K is the connector of a connection

H that is proved to be the Levi-Civit´a

connection of the metric (5.1).

For vector ﬁelds X, Y on D

(M) and for a vector ﬁeld X(t) along a cer-

tain smooth curve g(t)inD

(M) deﬁne the covariant derivatives

∇

Y and

X(t), respectively, by the usual formulae (cf. Deﬁnitions 2.22 and 2.25)

∇

Y =

K ◦ TY(X)=K ◦TY(X),

X(t)=

K ◦

X(t)=K ◦

X(t). (5.3)

On D

(M) with connection

H a vector ﬁeld X along a curve g(t) is parallel

X(t) = 0. The curve g(t) is geodesic if it satisﬁes the equation

˙g(t)=0,

where ˙g(t)=

g(t).

Thegeodesicspray

Z of

H is described as follows:

Z(X)=Z◦X (5.4)

for X ∈ T D

(M), where Z is the geodesic spray of the connection H on M.

Since Z is C

∞

-smooth, from the ω-lemma 5.2 it follows that

Z is C

∞

-smooth

on T D

(M). From (5.4) it evidently follows that

Z is D

(M)-right-invariant.

Proposition 5.7 If g(t) is a geodesic of the connection

H on D

(M), then

for every f ∈D

(M) the curve R

g(t) is also a geodesic.

This follows from the fact that the geodesic spray

Z is right-invariant.

108 5 Analysis on Groups of Diﬀeomorphisms

Remark 5.8. The exponential mapping exp of the Levi-Civit´a connection

on D

(M) is well-deﬁned. It is a C

∞

-mapping of a neighborhood of the zero

cross-section in T D

(M)toD

. This follows from the existence, uniqueness

and smooth dependence on initial values of the local solution of the Cauchy

problem for integral curves of C

∞

-smooth vector ﬁelds. The fact that exp

sends a neighborhood in T

(M) onto a neighborhood of e in D

follows

from general properties of smooth exponential mappings.

In what follows the tangent space T

(M) will be called the algebra of

the group D

(M) in analogy with ﬁnite-dimensional Lie groups. In fact the

everywhere dense linear submanifold of C

∞

-vector ﬁelds in T

(M) really

is an inﬁnite-dimensional Lie algebra, on which the bracket coincides with

the ordinary Lie bracket of vector ﬁelds. However, the entire tangent space

(M) is not an inﬁnite-dimensional Lie algebra since the Lie bracket can

give values which are not in T

(M).

The restriction of (5.1)toT D

(M) is evidently right-invariant. It is a weak

Riemannian metric on D

(M).

Recall the Hodge decomposition for M [61]

(TM)=G

⊕ E

⊕ ker Δ = G

⊕ T

(M) (5.5)

where G

is the space of gradients of all H

s+1

functions on M, E

is the space

of all H

-co-gradients on M,kerΔ is the space of all harmonic (i.e., both

gradient and co-gradient) vector ﬁelds on M and ⊕ denotes the orthogonal

direct sum with respect to the L

-inner product (5.1)inT

(M). By a co-

gradient we mean a vector ﬁeld corresponding to a co-exact form on M with

respect to the Riemannian metric ·, ·.kerΔ is a ﬁnite-dimensional space

and consists of C

∞

-smooth vector ﬁelds.

Denote by P

: T

(M)=H

(TM) → E

⊕ker Δ = T

(M)the(·, ·)

orthogonal projection in (5.5). Consider the mapping

P : T D

(M)

→

T D

(M) determined for each η ∈D

(M) by the formula

= TR

◦ P

◦ TR

−1

. (5.6)

It is obvious that

P is D

(M)-right-invariant. There is an important and

rather complicated result (see [61]) that

P is C

∞

-smooth. Notice the conse-

quence of (5.5) and of the deﬁnition of P

: the relation

(Y )=Y − gradp (5.7)

holds for every Y ∈ T

(M)wherep is an H

s+1

-function on M, unique to

within an additive constant.

Since D

(M) is a submanifold in D

(M), according to the standard con-

structions of diﬀerential geometry there is a corresponding connection

H on

(M) whose connector

K and covariant derivatives

∇ and

are described

by the formulae

5.1 General Concepts 109

K =

P ◦K,

∇

Y =

P ◦

∇

Y =

P ◦K ◦TY(X)=

K ◦ TY(X),

X(t)=

P ◦

X(t)=

P ◦K

X(t)=

X(t) (5.8)

where X, Y are vector ﬁelds on D

(M)andX(t) is a vector ﬁeld along a

smooth curve g(t)inD

(M). So, a vector ﬁeld X along a curve g(t)in

(M) is parallel if

X(t) = 0 and the curve g(t) is geodesic if

˙g(t)=0.

Theorem 5.9 The geodesic spray S of the Levi-Civit´a connection

H of

the metric (5.1) on D

(M) is a C

∞

-smooth right-invariant vector ﬁeld on

T D

(M) of the form S = T

P (

Z) where

Z is the geodesic spray (5.4) on

T D

(M).

Indeed,

P and

Z are D

(M)-right-invariant and C

∞

-smooth on T D

(M),

hence so is S. Denote by exp the corresponding exponential map of a neigh-

borhood of the zero section in T D

(M)ontoD

(M). Clearly the map exp is

∞

-smooth and D

(M)-right-invariant.

Theorem 5.10 There exists a neighborhood W of the unit e in D

(M) that

is covered by the image of T

(M) under the exponential mapping of the

Levi-Civit´a connection on D

(M).

This follows from the smoothness of S.

The case of M with boundary

Let, as above, s>

+1 and let M be a compact oriented manifold with

boundary ∂M. Denote by D

(M)thesetofC

-diﬀeomorphisms of M belong-

ing to the Sobolev class H

and by

◦

(M)thesetinD

(M) consisting of

diﬀeomorphisms coinciding with the identity on ∂M. In this case we cannot

use H

(M,M) to introduce the smooth manifold structure on D

(M)andon

◦

(M)sinceH

(M,M) has inﬁnite-dimensional corners.

Consider an arbitrary compact oriented manifold N without boundary

that has the same dimension n as M and is such that M is embedded into

N. We can take, say, the double of M as N (see Section 1.1) with Riemannian

metric smoothly expanded over the boundary. Consider the Hilbert manifold

(M,N).

Theorem 5.11 ([61]) D

(M) and

◦

(M) are smooth sub-manifolds in

(M,N).Fore =id∈D

(M) the tangent space T

(M) is the space

of H

-vector ﬁelds on M tangent to the boundary ∂M,andT

◦

(M) is the

space of H

-vector ﬁelds on M equaltozeroon∂M.

110 5 Analysis on Groups of Diﬀeomorphisms

The description of tangent bundles, group structures as well as of smooth

properties of right and left shifts are completely analogous to those described

above. For convenience of reference we summarize the properties of right-

invariant vector ﬁelds in the following Remark.

Remark 5.12. If X ∈ T

(M)isanH

s+k

-vector ﬁeld on M , the corre-

sponding right-invariant vector ﬁeld

X on D

(M)isC

-smooth. Neverthe-

less, generally speaking, the converse is not true. If

X is C

on D

(M), X is

s+k

in the interior of M and in the directions tangent to the boundary ∂M,

but it may not be H

s+k

-smooth in the directions normal to the boundary.

On the manifold H

(M,N) we introduce a weakly Riemannian metric in

the same way as (5.1). For this metric the analogs of the above-mentioned

theorems are obviously fulﬁlled. It is important to mention that this metric

can be considered at the points of the manifold D

(M) ⊂ H

(M,N). It is

also possible to deﬁne a geodesic spray Z◦X at X ∈ T D

(m) but in this

case the geodesics may not exist (

exp is not well-deﬁned) since the boundary

∂M, generally speaking, is not a completely geodesic manifold in N.

Deﬁnition 5.13. We say that a k-form α on M is tangent (normal)tothe

boundary ∂M if the restriction to ∂M of the form ∗α (form α, respectively)

is the identically zero form.

There are several versions of the Hodge decomposition on a manifold with

boundary. We shall mainly deal with the following one (see [61]):

(∧

)=dH

s+1

(∧

k−1

) ⊕E

(∧

), (5.9)

where ⊕ is the orthogonal direct sum with respect to the H

-inner prod-

uct (·, ·)

(5.1)andE

(∧

) denotes the co-closed H

-ﬁelds tangent to the

boundary ∂M.

From (5.9) we obtain the following important statement:

Theorem 5.14 ([61, 170]) For every H

-vector ﬁeld X, s ≥ 0,onM with

boundary ∂M there exists a unique divergence-free H

-vector ﬁeld Y tangent

to the boundary ∂M and unique to within an additive constant.

For M with boundary, D

(M) is a smooth submanifold in D

(M)and

consequently in H

(M,N). The space T

(M)=E

(∧

)isthespaceof

all divergence-free H

-vector ﬁelds on M tangent to the boundary ∂M.Let

: T

(M,N) → T

(M) be the orthogonal projector in (5.9). The

corresponding morphism

P , deﬁned by analogy with (5.6), is C

∞

-smooth

and right-invariant. Thus we can introduce the covariant derivatives

∇ and

by formulae (5.8), the geodesic spray S as in Theorem 5.9,etc.From

Theorem 5.14 it follows that

Y = P

X = X −grad p. (5.10)

5.2 The Group of Diﬀeomorphisms of a Flat Torus 111

Remark 5.15. In this case the following modiﬁcation of Remark 5.12 holds.

If a divergence-free vector X tangent to the boundary belongs to the class

s+k

, the corresponding right-invariant vector ﬁeld

X on D

(M)isC

smooth. However, if a right-invariant vector ﬁeld

X on D

(M)isC

-smooth,

the ﬁeld X =

on M belongs to H

s+k

in the interior of M andinthe

directions tangent to boundary, but it may not belong to this class in the

directions normal to the boundary.

Note that in the case of a manifold M with boundary direct analogs of

Theorems 5.9 and 5.10 remain true.

Strong Riemannian metrics

Let M be a manifold without boundary as above. Consider g ∈D

(M)and

also X

and Y

∈ T

(M) with X

= X ◦ g and Y

= Y ◦ g where X, Y ∈

(M) (see the deﬁnition of T

(M) above). Introduce on T

(M)a

“strong” inner product (·, ·)

(s)

by the formula

)

(s)



(X

(m),Y

(m)

g(m)

(5.11)

+ (d + δ)

X ◦ g(m), (d + δ)

Y ◦ g(m)

g(m)

)μ(dm),

where d is the diﬀerential, δ is the co-diﬀerential and (d+δ)

=(dδ+δd) = Δ

is the Laplace-de Rham operator. Since the Riemannian metric is given on

M, we do not distinguish between 1-forms and vector ﬁelds.

We shall also use another strong right-invariant Riemannian metric:

((X

))

(s)

=(TR

−1

,TR

−1

)

(s)

. (5.12)

5.2 The Group of Diﬀeomorphisms of a Flat Torus

Consider the constructions, introduced above, in the particular case where

M is a ﬂat n-dimensional torus T

, i.e., T

= R

and the Riemannian

metric on T

is inherited from the Euclidean space R

via factorization with

respect to the integral lattice Z

(see Remark 1.18).

Recall that the tangent bundle T T

is trivial, i.e., there is a canonical

identiﬁcation of TT

with T

×R

that is also inherited from T R

= R

×R

(see Remark 1.40). Note that here the connection

H is generated by the ﬂat

connection on T

inherited from the ordinary ﬂat connection on R

(the

Levi-Civit´a connection of the Euclidean inner product).

Deﬁnition 5.16. We introduce the operators:

(i) B : T T

→ R

, the projection onto the second factor in T

× R

;

112 5 Analysis on Groups of Diﬀeomorphisms

(ii) A(m):R

→ T

,theinversetoB (see (i)) sending R

onto the

tangent space T

to T

at m ∈T

;

(iii) Q

= A(g(m)) ◦ B, the linear isomorphism Q

: T

→ T

g(m)

where g ∈D

and m ∈T

Note that if we specify a vector X ∈ R

, A(X):T

→ T T

is a vector

ﬁeld on T

with constant coordinates with respect to the standard basis

∂

∂q

,...,

∂

∂q

in tangent spaces. In particular, this vector ﬁeld is C

∞

-smooth

and divergence-free with respect to the above-mentioned metric on T

By construction, for every f ∈D

)andX ∈ T

) the vector

X lies in T

). In particular, Q

X ∈ T

). Note that even for

f ∈D

), the operator Q

does not send T

)toT

), but

)) = T

We describe the action of operator Q

on tangent vectors to D

)as

on mappings from T

to T T

and compare this action with the right shift.

Recall that a vector X ∈ T

), i.e., a vector ﬁeld on T

, sends a point

m ∈T

to the vector (m, X(m)). The right shift TR

on T

) sends

the latter vector to (g(m),X(g(m)), and Q

(m, X(m)) = (g(m),X(m)).

Lemma 5.17 The following relations hold:

−1

X)=Q

(TR

−1

X); (5.13)

−1

X)=Q

(TR

X). (5.14)

Proof. According to the above formulae, Q

(TR

−1

X) sends a point m ∈

to (m, X(g

−1

(m))). On the other hand, Q

X =(g(m),X(m)) and

−1

X)=(m, X(g

−1

(m))). From this (5.13) follows. Formula (5.14)

is obtained from (5.13) by replacing g with g

−1

. 

Theorem 5.18 Q

: T

) → T

) is the parallel translation in

) with respect to

Indeed, since (roughly speaking) the connectors on T

and on D

)

coincide, the parallel translations coincide as well.

For a speciﬁed x ∈ R

we introduce the diﬀeomorphism L

: T

→T

by the formula L

(m)=m + x modulo factorization with respect to the

integral lattice. Evidently L

is C

∞

-smooth and preserves the volume. Note

that L

g(m)=g(m)+x is in fact the left shift of g ∈D

)byL

.One

can easily see that TL

= Q

Theorem 5.19 Let g(t) be a geodesic of the ﬂat connection

H on D

)

(i.e.,

˙g(t)=0). Then L

g(t) is also a geodesic.

Proof. Note that the derivative ˙g(t) sends the point m ∈T

to the vec-

tor (g(t)(m), ˙g(t)(m)). Since g(t) is a geodesic of a ﬂat connection, it satis-

ﬁes the relation (g(t)(m), ¨g(t)(m)) = (g(t)(m), 0). Then, since x is constant,

g(t)) sends m to (g(t)(m)+x, ˙g(t)(m)) and for the covariant derivative

we obtain (g(t)(m)+x, ¨g(t)(m)) = (g(t)(m)+x, 0). 

Part II

Stochastic Analysis

Chapter 6

Essentials from Stochastic Analysis in

Linear Spaces

6.1 Some Deﬁnitions from Probability Theory and the

Theory of Stochastic Processes

In this section we describe some facts and constructions from probability

theory and the theory of stochastic processes, some of which are not generally

included in standard university courses on these subjects. This is done mainly

for convenience of reference, but if necessary this material can be used as an

introduction to the subject. Nevertheless the reader is assumed to be familiar

with the main notions of probability theory including the notions of a σ-

algebra (in particular, a Borel σ-algebra), measure and independent random

variables.

We consider random variables (measurable mappings) given on a complete

probability space (Ω,F, P) and taking values in a ﬁnite-dimensional linear

space R

equipped with the Borel σ-algebra. For random variables with values

in non-linear manifolds and in inﬁnite-dimensional spaces there are analogous

constructions.

A detailed exposition of this material can be found, e.g., in [175, 176, 194,

204, 208].

We say that a σ-subalgebra B

in F is generated by a random variable

ξ : Ω → R

if B

is the minimal σ-algebra containing the pre-images of all

Borel sets in R

under the mapping ξ or, equivalently, B

is the minimal

σ-algebra with respect to which ξ is measurable.

6.1.1 Stochastic processes. Cylinder sets

A stochastic process is a random variable, given on a probability space and

taking values in R

, that depends on time. A process η(t), t ∈ [0, ∞), has

almost surely (a.s.) continuous sample paths (or trajectories) P-a.s. if for

Y.E. Gliklikh, Global and Stochastic Analysis with Applications

to Mathematical Physics, Theoretical and Mathematical Physics,

DOI 10.1007/978-0-85729-163-9

115