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

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326 13 Some Problems on Lorentz Manifolds

a charge is given, i.e., a map e: F→g

∗

that is constant on the orbits of G.

The space g

∗

is a coalgebra, i.e., g

∗

is dual to g.

Note that the electromagnetic ﬁeld is an example of a gauge ﬁeld with

G = U(1) (the group of unitary operators in the complex plane C

), F = C

and h(X, Y )=X

Y , where the bar indicates complex conjugation. Below we

show how in this case Maxwell’s equations (13.4) can be derived from (13.15)

(see formula (13.22)).

Consider the bundle π : Q→M

associated to E with standard ﬁber F (see

Section 1.3). Recall that Q is obtained by the factorization λ : E×F →Q with

respect to the right action of G on E×Fdetermined as (¯e,

f)g =(bg, g

−1

for g ∈ G, b ∈Eand f ∈F. In particular, the tangent map Tλ sends the

spaces of the connection H from the tangent spaces to E into the tangent

spaces to Q (see Section 2.7). The connection on Q obtained from H (recall

that Tλ is one-to-one on H) is denoted by H

.SinceQ is a bundle with

connection, the notation V and H will be also used to denote the projections

onto vertical and horizontal, respectively, subspaces in tangent spaces to Q,

as was mentioned above. To avoid any confusion we shall denote the vertical

distribution on Q by V

Consider a point q =(m, c) ∈Q,wherem = πq so that c belongs to the

ﬁber Q

of Q at m. Note that by deﬁnition V

(m,c)

, the vertical subspace in

(m,c)

Q, is the tangent space to Q

and so, since the latter is a linear space,

these spaces are naturally isomorphic to each other. Denote by p

: V

(m,c)

→

the natural linear isomorphism as in formulae (1.2)and(2.11). Recall

that the connector (connection map) of H

is the map K

: T Q→Qdeﬁned

as the composition K = p

◦ V.

The restriction of Tπ : H

→ TM

to H

is a linear isomorphism of H

onto T

for m = πq. Its inverse map Tπ

−1

sends the tangent space T

into the connection H

⊂ T

Q at a speciﬁed point q ∈Qsuch that πq = m.

Let U be a chart on M

. The bundle Q over U is represented as the

cartesian product U ×F. The subspace

in T

Q, q =(m, c), m = πq ∈ U,

corresponding to the tangent space to U in the above product, is obviously

complementary to V

. It is called the Euclidean connection of the coordinate

system in U (see Sections 2.1 and 2.2). Using the presentation T

Q =

⊕

we obtain the description of any vector Y ∈ T

Q in local coordinates as

a quadruple Y =(m, c, Y

), Y

∈

and Y

∈ V

Since

is complementary to V

=KerTπ,themapTπ :

→

is one-to-one (as for H

, see above). For a vector X ∈ T

consider

the vector Tπ

−1

X − Tπ

−1

X. Since this vector belongs to V

,wecan

apply p

to it. As in Sections 2.2 and 2.3, the resulting vector Γ

(c, X)=

(Tπ

−1

X −Tπ

−1

X) ∈Q

is called the local coeﬃcient of the connection

(or local connector of H

)inthechartU. Recall that the connector K

has the following description in local coordinates:

(m, c, Y

)=(m, Y

+ Γ

(c, Y

)).

13.3 A Classical Particle in a Classical Gauge Field 327

Consider the tangent bundle τ : TM

→ M

(here, to avoid any confu-

sion, we denote the natural projection by τ)andletH

be the Levi-Civit´a

connection of the metric g(·, ·). This means that H

is a distribution on TM

complementary to the vertical distribution V

. Special features of H

can be

found in Section 2.6. For a point (m, X) ∈ TM

,wherem = τ (m, X)and

X ∈ T

, the subspace V

(m,X)

in the second tangent space T

(m,X)

is tangent to the ﬁber T

. Denote by p

: V

(m,X)

→ T

the natu-

ral linear isomorphism as in formulae (1.2)and(2.11). Now we can consider

the connector K

: T

→ TM

of H

deﬁned by the natural formula

= p

◦ V

(here T

is the second tangent bundle TTM

The general construction of the Euclidean connection, the presentation of

vectors as quadruples and the deﬁnition of the local coeﬃcient of a connection

in a chart described above for the bundle Q are valid for the tangent bundle

with H

. Denote the local coeﬃcient of H

by Γ

(Y,X), X, Y ∈ T

(Here T

plays the role of Q

and so c ∈Q

is replaced by Y ∈ T

We shall use the connection H

on the manifold Q (the total space of

the bundle Q) constructed from the connections H

and H

in Section 2.8.

Denote by K : T

Q→T Q its connector.

Recall that K = K

⊕ K

where K

: T

Q→H

and K

: T

Q→V

is determined by the formula K

= Tπ

−1

◦K

◦T

π, T

π : T

Q→T

is the tangent map to Tπ : T Q→TM

and the latter is the tangent map to

π. Note that the image K

◦T

π(T

(q,C)

T Q) belongs to T

where m = πq

so that Tπ

−1

sends it into H

⊂ T

Q as is required in the deﬁnition of a

connector.

On the other hand K

=(p

)

−1

◦ K

◦ TK

. Note that the image K

◦

(q,C)

T Q) belongs to the ﬁber Q

where m = πq and (p

)

−1

sends it

into V

as required.

The covariant derivative of H

on Q is deﬁned by the usual formula

K ◦

. By construction we have

,where

= K

◦

and

= K

◦

Deﬁne a Riemannian metric g

(·, ·)onQ so that it coincides with h(·, ·)

in vertical subspaces V

, with the inverse image Tπ

∗

g in the horizontal sub-

spaces H

and so that the spaces H

and V

at any point q ∈ M

are

orthogonal to each other.

In this particular case a vector ¯α and a 1-form ˜α on Q are said to be

physically equivalent to each other with respect to the metric g

if for any

vector Y we have ˜α(Y )=g

(¯α, Y ) (cf. Section 1.4). We shall denote them

by the same symbol, using a bar over vectors and a tilde over 1-forms.

328 13 Some Problems on Lorentz Manifolds

13.3.2 The equation of motion

Since λ is a one-to-one map from the standard ﬁber F onto each ﬁber of

Q, the charge e is well-deﬁned on the whole manifold Q.AlsoTλ is a one-

to-one map of the connections and Φ is equivariant (see [26]). Hence we

can set up the form

Φ with values in g on the manifold Q by the equality

(X, Y )=Φ

(Tλ

−1

HX, T λ

−1

HY )forq = λ(b, f ), where b ∈E, f ∈F,and

for X,Y ∈ T

Q. Unlike

(·, ·), the 2-form e(q) •

(·, ·)onT

Q takes values

in R and so it is an ordinary scalar-valued 2-form. The symbol • denotes the

coupling of the elements e(q) ∈ g

∗

and

(·, ·) ∈ g.

Thus, on the total space Q we may consider the following equation, an

analog of Newton’s second law (11.2) and of the Lorentz equation (13.5), in

the form

˙q =

e(q) •

(·, ˙q). (13.16)

We interpret these equations as equations of motion for a classical particle

with charge in a classical gauge ﬁeld.

Proposition 13.35 The vector

e(q) •

(·, ˙q) is horizontal, i.e., at any point

q ∈Qit belongs to H

Proof. Since Φ is a horizontal form by construction, so too is

Φ(·, ·), i.e.,

(Y, ˙q) = 0 for all Y ∈ V

.Thuse(q) •

(Y, ˙q) = 0 for the same Y . Hence

(e(q) •

(·, ˙q),Y)=0. 

From Proposition 13.35 it follows that equation (13.16) is equivalent to

the system

˙q = e(q) •

Φ(·, ˙q), (13.17)

˙q =0. (13.18)

Theorem 13.36 If q(t) is a solution of (13.16) with horizontal initial con-

dition ˙q(0) = ˙q

∈ H

, then q(t) is a horizontal curve (i.e., ˙q ∈ H

for all t,

for which it exists) and e(q(t)) is constant.

Proof. To prove this we consider local coordinates so that q(t)=(m(t),c(t)),

where m(t) ∈ M

,andc(t) belongs to the ﬁber Q

m(t)

. Hence ˙q =(m, c, ˙m, ˙c)

and K

˙q = K

(m, c, ˙m, ˙c)=(m, ˙c + Γ

(c, ˙m)). So, the condition ˙q

∈ H

the local coordinates means that ˙c

+ Γ

, ˙m

)=0.Wehave,

˙q =

(m, ˙c + Γ

(c, ˙m))



m, ˙c + Γ

(c, ˙m), ˙m,

(˙c + Γ

(c, ˙m))



13.3 A Classical Particle in a Classical Gauge Field 329

By (13.18)

˙q =(p

)

−1

◦K

◦TK

˙q =0sothatTK

˙q belongs to the

connection. This means that

(m, ˙c + Γ

(c, ˙m)) = (m, ˙c + Γ

(c, ˙m), ˙m, −Γ

(˙c + Γ

(c, ˙m), ˙m)) .

Thus

(˙c + Γ

(c, ˙m)) = −Γ

(˙c + Γ

(c, ˙m), ˙m)) . (13.19)

Equality (13.19) is an ordinary linear diﬀerential equation with respect to

˙c + Γ

(c, ˙m). For zero initial conditions (see above) it has a unique solution

˙c + Γ

(c, ˙m) = 0. This proves the ﬁrst statement.

By construction the curve q(t)=(m(t),c(t)) is horizontal and so it can

be represented as (m(t),b

(f)), where b

is a horizontal lift of m(t)intoE

and f is some vector in F. Hence b

(f) belongs to an orbit of G in F and so

e(q(t)) = e(b

(f)) is constant. 

Theorem 13.37 If m(t)=πq(t) is the projection of a solution of (13.16),

then m(t) satisﬁes

˙m(t)=Tπ(e(q) •

Φ(·, ˙q)) where

is the covariant

derivative of the Levi-Civit´a connection on M

Proof. Denote q =(m, c) as above. Describing vectors in local coordinates

by quadruples, we obtain ˙q =(m, c, ˙m, ˙q)and

˙q =(m, c, ˙m, ˙c, ˙m, ˙c, ¨m, ¨c).

For m = πq the vector ˙m clearly belongs to T

and so we represent it

in the form (m, ˙m) as a point of TM

.Then

˙m =(m, ˙m, ˙m, ¨m).

By the deﬁnition of covariant derivative

˙m = K

(m, ˙m, ˙m, ¨m).

On the other hand, from the construction of K

Tπ

˙q = TπK

˙q = K

◦ T

π(m, c, ˙m, ˙c, ˙m, ˙c, ¨m, ¨c)=K

(m, ˙m, ˙m, ¨m).

Thus,

˙m = Tπ

˙q. So, applying Tπ to both sides of (13.17) we obtain

the theorem. 

The most important cases are those where G is U (1), or SU(2), or SU(3)

and F is a complex space with appropriate dimension. They correspond to

the well-known gauge ﬁelds. As an example we consider the particular case

of an electromagnetic ﬁeld with G = U(1), F = C

, h(X, Y )=X

Y ,where

the bar denotes complex conjugation.

Here the algebra U(1) = R

is one dimensional and hence commutative.

Thus ϕ and Φ are ordinary diﬀerential forms with values in R. The charge

takes values in real numbers and it is constant on the circles in C

centered at

zero. Recall the structure equation (2.40): dϕ = −

[ϕ, ϕ]+Φ.Fromthecom-

mutativity of U(1) it follows that [ϕ, ϕ](X, Y )=ϕ(X)ϕ(Y )−ϕ(Y )ϕ(X)=0,

330 13 Some Problems on Lorentz Manifolds

for each pair X, Y ∈ T

E, b ∈E.ThusΦ =Dϕ =dϕ and

DΦ =Ddϕ =ddϕ(H·, H·, H·)=0

D ∗Φ =D∗ dϕ =d∗ dϕ(H·, H·, H·). (13.20)

From (12.6)and(13.20) we obtain

dΦ =0, d ∗ Φ = ∗J. (13.21)

Recall that Φ is horizontal and equivariant (see [26]). Since the group

U(1) is commutative, the latter means that Φ is invariant with respect to the

natural right action of G on E: Φ

(TR

X, T R

Y )=Φ

(X, Y )whereR

the right action of g ∈ G on E, b ∈Eand X, Y ∈ T

E. Thus the equality

(X, Y )=Φ

(TΠ

−1

X, T Π

−1

Y ),

for any m ∈ M

, b ∈E, Πb = m and X, Y ∈ T

, deﬁnes a 2-form Ψ (·, ·)

on M that is well-deﬁned (i.e., does not depend on the choice of b over m).

For Ψ equations (13.21) are reduced to

dΨ =0, d ∗ Ψ = ∗

J. (13.22)

Taking into account Deﬁnition 1.71 of the codiﬀerential δ = ∗

−1

d∗, one can

easily see that (13.22) are the ordinary Maxwell equations (13.4)onthe

Lorentz manifold M

where

(X)=J(TΠ

−1

X), m = Πb and X ∈ T

are well-deﬁned. Thus we may consider Ψ as the strength of the electromag-

netic ﬁeld on M

Clearly Tπ(

e(q) •

Φ(·, ˙q)) = e(q)Ψ(·, ˙m). Hence, from Theorem 13.37 it

follows that a particle with charge e(q) is governed by the Lorentz equation

(13.5):

˙m = e(q)

Ψ(·, ˙m)

on M

and the charge e is constant.

Chapter 14

Mechanical Systems with Random

Perturbations

14.1 Setting Up the Problem

It is a well-known fact that a second order diﬀerential equation ¨x(t)=

¯α(t, x(t), ˙x(t)) expressing Newton’s law in R

may be represented as a ﬁrst

order system on the space of dimension 2n:



˙x(t)=v(t)

˙v(t)=¯α(t, x(t),v(t)).

(14.1)

We call the ﬁrst equation of the above system horizontal and the second

one vertical. This is consistent with the terminology in the general case of a

mechanical system on a non-linear conﬁguration space M, where Newton’s

law is presented by means of covariant derivatives in the form (11.2), equiv-

alent to equation (11.3) with a special vector ﬁeld (second order diﬀerential

equation) on the phase space TM. Recall that the ﬁeld (11.3)isthesum

of the Levi-Civit´a geodesic spray Z (which is horizontal, i.e. belongs to the

connection) and the vertical lift of the vector force ﬁeld (which belongs to

the vertical subspace).

Thus a random perturbation of Newton’s law can arise in the horizontal

equation and in the vertical equation (possibly both). Note that a vertical

perturbation is a perturbation of the force ﬁeld while a horizontal perturba-

tion is a perturbation of velocity. All the cases are physically reasonable but

they require diﬀerent methods of investigation. It should also be pointed out

that under random perturbations, Newton’s law becomes a random diﬀeren-

tial equation, which we shall present in terms of mean derivatives. Taking

into account various possible constructions of second order mean derivatives

(forward, backward, mixed, etc.), this yields several equations of motion from

diﬀerent parts of physics.

In this chapter we deal with Newton’s law in terms of forward mean deriva-

tives. Its physical interpretation is the description of the motion of an ordi-

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

to Mathematical Physics, Theoretical and Mathematical Physics,

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

14, © Springer-Verlag London Limited 2011

331

332 14 Mechanical Systems with Random Perturbations

nary mechanical system with random perturbations. We consider both per-

turbations of forces and of velocities.

The versions of Newton’s law in terms of backward and mixed mean deriva-

tives will be considered in forthcoming chapters. Newton’s law in special

mixed derivatives (the so-called Newton-Nelson equation) describes the mo-

tion of a quantum particle (see Chapter 15). Newton’s law in backward mean

derivatives on groups of diﬀeomorphisms describes the motion of a viscous

incompressible ﬂuid (Chapter 16).

We begin this Chapter with an investigation of the so-called Langevin

equation.

The Langevin equation describes mechanical systems with both determin-

istic and random forces which have comparable magnitudes (i.e., neither the

deterministic nor random part can be neglected) where the random force

is a transformed white noise. Examples of such processes are well-known in

physics. One can easily see that in this case the trajectories of the process

are a.s. C

-smooth. This makes the analysis technically much simpler than

that for more general systems. In particular we can apply the machinery

of ordinary integral operators with Riemannian parallel translation of Sec-

tion 3.2 and study the Langevin equations arising on non-linear conﬁguration

spaces.

In Section 14.2, we introduce the Langevin equation on a Riemannian

manifold and reduce it to the velocity hodograph equation, which is an equa-

tion in a single tangent (i.e., vector) space. This enables us to apply some

standard results and carry out a detailed analysis. We also study an impor-

tant particular case of the Langevin equation: the equation describing the

so-called Ornstein-Uhlenbeck processes arising, for example, in the mathe-

matical model of physical Brownian motion [188]. Sometimes only the latter

equation is called the Langevin equation, whereas the equation applicable in

a more general context is said to be the generalized Langevin equation.

In Section 14.3 we study the case where the force ﬁeld in the Langevin

equation is set-valued (i.e., it is constructed from an essentially discontinuous

force or where a force with feedback control is under consideration).

Throughout Sections 14.2 and 14.3, all Riemannian manifolds are assumed

to be complete, but not necessarily uniformly or stochastically complete.

In Section 14.4 we investigate mechanical systems with random perturba-

tions of velocity motivated by the motion of a particle, subjected to a deter-

ministic force, that in addition moves in an enveloping medium with random

inﬂuence. First we consider systems in R

with single-valued and set-valued

forces. The systems on manifolds are investigated under some more restric-

tive assumptions. In particular, we assume the manifold to be stochastically

complete and use the machinery of stochastic parallel translation to reduce

the system to the corresponding velocity hodograph equation.

14.2 The Langevin Equation and Ornstein-Uhlenbeck Processes on Manifolds 333

14.2 The Langevin Equation and Ornstein-Uhlenbeck

Processes on Manifolds

Everywhere in this section we deal with processes given on a ﬁnite time

interval [0,l] ⊂ R.

Consider a mechanical system in the sense of Section 11.1, i.e., a Rieman-

nian manifold M together with a force ﬁeld on M. As just mentioned, M is

assumed to be complete, i.e., a free particle on M cannot escape to inﬁnity in

ﬁnite time. The Riemannian metric enables us to identify diﬀerential forms

and vector ﬁelds on M , so henceforth we regard the force ﬁeld as a vector

ﬁeld (see Section 11.1). In addition to the Deﬁnition 11.2 of a vector force

ﬁeld we give the following:

Deﬁnition 14.1. AmapA from R × TM to the bundle of (1, 1)-tensors

over M (either single or set-valued), such that π

A(t, m, X)=π(m, X)=m

(where π

is the projection of the bundle of (1, 1)-tensors onto M ) will be

called a tensor force ﬁeld.

Recall that a (1, 1)-tensor at m ∈ M is a linear operator in T

M.Thusfor

a tensor force ﬁeld A(t, m, X) and a vector ﬁeld Y (m)onM the composition

A(t, m, X)Y (m) is a vector force ﬁeld.

Let ¯α(t, m, X) be a vector force ﬁeld and A(t, m, X)a(1, 1)-tensor ﬁeld

on M.Inotherwords,foreveryt ∈ [0,l], m ∈ M,andX ∈ T

M,we

have a vector ¯α(t, m, X) ∈ T

M and a linear operator A(t, m, X): T

M →

M. Making use of the construction given in Section 7.7.2,specifyaWiener

process w(t) on the tangent spaces to M and denote by ˙w the white noise of

w(t) (see Section 6.2.1).

The Langevin equation describes the evolution of a system with the force

ﬁeld

¯α(t, m, X)+A(t, m, X)˙w. (14.2)

More formally, the equation of motion must read

ξ(t)=¯α



t, ξ(t),

ξ(t)



+ A



t, ξ(t),

ξ(t)



˙w(t), (14.3)

however this expression is meaningful only in the sense of distributions.

Our ﬁrst goal is to give a rigorous meaning to (14.3) without using distri-

butions. We do this in terms of forward mean derivatives, the construction of

which can be simpliﬁed in the case under consideration. Then we transform

the obtained equation into an equivalent integral form employing the inte-

grals with Riemannian parallel translation from Section 3.2 by analogy with

that given in Section 11.8. In particular we use the transition to the corre-

sponding velocity hodograph equation which is much easier to study since it

is an equation in a single tangent (i.e., linear) space. Note that previously,

in [94, 118, 119], the Langevin equation was considered only in this integral

334 14 Mechanical Systems with Random Perturbations

form. A local coordinate version of the equation was independently given in

[180].

We assume that ¯α(t, m, X)andA(t, m, X) are jointly continuous in all

variables and that these ﬁelds have linear growth in X. In other words, there

exists a constant K>0 such that

¯α(t, m, X) + A(t, m, X) <K(1 + X) (14.4)

for all t ∈ [0,l], m ∈ M,andX ∈ T

M, where the norm is given by the

Riemannian metric. The estimate (14.4) is a version of the Itˆo condition.

One can show, appealing to physical reasoning, that a process subjected to

the force (14.2) a.s. has continuous velocities and as a consequence it a.s. has

-smooth sample paths. Below we shall show that solutions of the Langevin

equation do indeed exist in the class of processes with C

-smooth sample

paths. This is why we start by examining some features of such processes.

Let ξ(t) be a stochastic process on M with a.s. C

-smooth sample paths

given on a probability space (Ω,F, P), and let a vector ﬁeld Y be given on

M.Asabove,byΓ

t.s

we denote the operator of parallel translation along

a C

-smooth curve x(·)fromx(s)tox(t). Since the sample paths of ξ(t)

are C

-smooth, the parallel translation in the deﬁnition of a forward mean

derivative, by formula (9.15), is the ordinary parallel translation (i.e., we

needn’t deal with the general construction of a stochastic parallel translation

from Section 7.6). Thus we have obtained:

Lemma 14.2 The covariant forward mean derivative of the vector ﬁeld Y

along the process ξ(t) on M with a.s. C

-smooth sample paths at time t is

the L

random element of the form

DY (t, ξ(t)) = lim

t↓0



t,t+t

Y (t + t, ξ(t + t)) − Y (t, ξ(t))

t



, (14.5)

where Γ

t,s

is the ordinary parallel translation along C

-smooth curves.

Consider a probability space (Ω,F, P) and a non-decreasing family of com-

plete σ-subalgebras B

of F. In a given tangent space T

M introduce a

Wiener process w(t) adapted to B

,andanItˆo diﬀusion type process v(t)of

the form v(t)=



b(s)ds +



B(s)dw(s) with b(t)andB(t) a.s. having con-

tinuous sample paths. In particular this means that v(t) is non-anticipative

with respect to B

and a.s. has continuous sample paths. Thus we can apply

the operator S introduced in Section 3.2 to the sample paths of v(t). Then

we obtain the process ξ(t)=Sv(t)havingC

-smooth sample paths. Recall

that S: C

([0,l],T

M) → C

([0,l],M) is continuous. Since in addition

v(t) is non-anticipative with respect to B

, this proves the following:

Lemma 14.3 The process Sv(t) is non-anticipative with respect to B

Consider a special case of the probability space (

Ω,

P)where

Ω =

([0,l],T

M),

F is the σ-algebra generated by cylinder sets and the mea-

14.2 The Langevin Equation and Ornstein-Uhlenbeck Processes on Manifolds 335

sure

P is the measure generated by a certain stochastic process in T

M.In

this case we shall deal with the family

of σ-subalgebras of

F where for

some t the σ-subalgebra

is generated by cylinder sets with bases on [0,t].

Lemma 14.4 The process Sv(t) is non-anticipative with respect to

Proof. Indeed, if the curves v

(·)andv

(·)from

Ω = C

([0,l],T

M) coin-

cide at all t ∈ [0,l

]where0<l

<l,thenSv

(t) coincides with Sv

(t)for

t ∈ [0,l

] by construction of the operator S.By[83, Chapter III, Section 2]

this is the assertion of Lemma 14.4. 

Consider the vector ﬁeld

ξ(t) along ξ(t)=Sv(t).

Theorem 14.5

ξ(t)=E

(Γ

t,0

b(t)).

Proof. From the properties of parallel translation and from the construction

of ξ(t) it follows that

(Γ

t,t+t

ξ(t + t) −

ξ(t))

= E



t,0





t+t

b(s)ds +



t+t

B(s)dw(s)



Note that N

is a σ-subalgebra in P

. Since the Itˆo integral



t+t

B(s)dw(s)

is a martingale with respect to P

, by the properties of conditional expecta-

tion we obtain that





t+t

B(s)dw(s)



=0.

The Theorem follows. 

Along a process ξ(t)=Sv(t) as above we can deﬁne the covariant

quadratic mean derivative of

ξ(t) as follows. Introduce the notation 

ξ(t)=

t,t+t

ξ(t + t) −

ξ(t) where (as above in this section) Γ

t,s

is the ordinary

parallel translation along C

-smooth curves.

Deﬁnition 14.6. The quadratic mean derivative of

ξ(t) along ξ(t)=Sv(t)

on M at time t is an L

random element of the form

ξ(t) = lim

t↓0





ξ(t) ⊗

ξ(t)

t



where ⊗ is the tensor product and Γ

t,s

is the ordinary parallel translation

along C

-smooth curves.