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

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366 15 The Newton-Nelson Equation

μ on (

Ω,

F) by the relation dμ = θ(l)dν. On the probability space (

Ω,

F,μ)

the coordinate process W(t) turns into ξ(t)=x



a(s)ds+w(t)wherew(t)

is a “new” Wiener process on (

Ω,

F,μ), adapted to P

= P

(Girsanov’s

theorem). Note that by construction a(t) is not anticipative with respect to

, i.e., ξ(t) is a diﬀusion type process.

Theorem 15.14 The process ξ(t) satisﬁes equation (15.6) and the initial

conditions (15.23).

Lemma 15.15 The random element x

has the same probability distribution

on (

Ω,

F,ν) and on (

Ω,

F,μ).

Proof. Let f be an arbitrary bounded Borel measurable function. As in Sec-

tion 8.3 we denote by E

(ψ |B) the conditional expectation for ψ with

respect to B on the probability space (Ω,F,ν)andbyE



(ψ |B)thesame

conditional expectation on (Ω,F,μ). Then



[f(x

)] = E

[f(x

)θ(l)] = E

[f(x

θ(l)] = E

[f(ξ

)],

because E

θ(l)=θ(0) = 1 (since θ(t) is a martingale). The assertion of the

Lemma follows from the fact that f is arbitrary. 

Proof. [of Theorem 15.14] On the probability space (

Ω,

F,μ) equation (15.24)

turns into

a(t)=a



¯α

(s, ξ(s)) ds +



¯α

(s, ξ(s))a(s)ds



¯α

(s, ξ(s))dw(s)+



tr ∇¯α

(s, ξ(s)) ds



(s, ξ(s)) ds +



(a(s) ·∇)u

(s, ξ(s)) ds.

Now the proof is reduced to the veriﬁcation that the required formulae are

satisﬁed. This is done by application of formulae from Section 8.3 (for details,

see [128]and[129]). 

Let a

be an arbitrary bounded random element with values in R

Corollary 15.16 Let there exist a δ>0 such that for all x ∈ R

the in-

equality ρ

(x) >δholds and let in addition grad log ρ

be uniformly bounded.

Then for a

= a

+ u

(0,x

) the process ξ(t) from Theorem 15.14 satisﬁes

equation (15.6) with initial conditions ξ(0) = x

and D

ξ(0) = a

15.2 The Geometric Form of Stochastic Mechanics 367

15.2 The Geometric Form of Stochastic Mechanics

15.2.1 Some comments on stochastic mechanics on

Riemannian manifolds

Stochastic-mechanical systems are inﬂuenced by the geometry of conﬁgura-

tion space even more so than the systems of classical mechanics. As in the

classical case the Riemannian metric on the conﬁguration space determines

the kinetic energy of the system, but it also deﬁnes the ﬁeld of Wiener pro-

cesses in terms of which the motion is described, so that the quadratic deriva-

tive of a trajectory yields the (2, 0) metric tensor. In addition, the curvature

of the conﬁguration space is involved in the stochastic version of Newton’s

law.

Let M be a n-dimensional Riemannian manifold. We study the exponential

map, parallel translation and other geometric objects on M generated by the

Levi-Civit´a connection.

Let ξ(t) be a stochastic process in M and assume that the mean derivatives

in the sense of Deﬁnition 9.1 and formulae (9.14)and(9.15) exist for ξ(t).

Deﬁnition 15.17. The vector

(DD

∗

+ D

∗

D)ξ(t) is called the acceleration

of ξ(t) (cf. Deﬁnition 15.1).

As in Section 15.1.1 there exists a Borel vector ﬁeld on M (the regression)

such that the acceleration is the composition of that ﬁeld and ξ(t).

On determining the covariant mean derivatives D

(D + D

∗

)and

(D −D

∗

), we then obtain the following analog of formula (15.1)

(DD

∗

+ D

∗

D)ξ(t)=(D

−D

)ξ(t)=D

(t) −D

(t). (15.25)

Now suppose that D

ξ(t)=σ

¯g where D

is the quadratic mean derivative,

σ>0isaconstantand¯g isthemetric(2, 0)-tensor. According to the material

of Section 9.4 this means that we determine the forward and backward mean

derivatives with respect to the Levi-Civit´a connection of Riemannian metric

given by D

ξ(t). In this case we derive from formulas (9.17)and(9.18)the

following analogs of formulas (15.2)and(15.3):

(t)=

∂

∂t

(t)+∇

(t)

(t), (15.26)

(t)=∇

(t)

(t)+

∇

(t), (15.27)

where ∇ is the covariant derivative of the Levi-Civit´a connection and ∇

is the Laplace-Beltrami operator (see Deﬁnition 2.58). Thus from formulae

(15.25)–(15.27) it follows that for such ξ(t) the following formula

368 15 The Newton-Nelson Equation

(DD

∗

D)ξ(t)=



∂

∂t

(t)+∇

(t)



−



∇

(t)+∇

(t)



(15.28)

holds (this is the analog of formula (15.4)).

Let M be the conﬁguration space of a mechanical system as in Section 11.1

with a force ﬁeld ¯α(t, m, X), i.e., the trajectory of the system is governed

by equation (11.2). It seems to be natural to determine Newton’s law for

stochastic mechanics in this case by complete analogy with equation (15.6),

i.e., by setting the acceleration equal to the force. But in doing this we will

not obtain the correspondence with the solutions of the Schr¨odinger equations

analogous to those described in Section 15.1.1. More precisely, the situation

is as follows. One can easily make a minimal modiﬁcation of the construction

and prove an analog of Theorem 15.4 (we leave this to the reader as a simple

exercise) but one obtains the Laplace-Beltrami operator ∇

in the analog of

equation (15.7), while physicists would normally use the Laplace-de Rham

operator Δ =dδ +δd from Deﬁnition 1.73 (this diﬀerence seems to have been

ﬁrst highlighted in [45]).

So, in order to obtain the correspondence mentioned above we might re-

place the Laplace-Beltrami operator by the Laplace-de Rham operator in the

right hand side of (15.28), i.e., change the above deﬁnition of acceleration (a

variant of such a change is discussed below in Remark 15.19). Instead we take

into account Weitzenbock’s formula (2.37) and deﬁne a stochastic mechanical

system on M as follows (cf. Deﬁnition 15.2):

Deﬁnition 15.18. A process ξ(t)inM is called a stochastic-mechanical tra-

jectory in M of a particle with mass m, under the action of the force ﬁeld

¯α(t, m, X), if it satisﬁes the system

⎧

⎨

⎩

(DD

∗

+ D

∗

D)ξ(t)=

¯α(t, ξ(t),v

(t)) +





Ric(ξ(t)) ◦ u

(t),

ξ(t)=



¯g,

(15.29)

where ¯g is an autonomous positive deﬁnite symmetric (2, 0)-tensor and all

mean derivatives and the Ricci tensor in the ﬁrst equality of (15.29)arede-

termined with respect to the Levi-Civit´a connection of the Riemannian (0, 2)-

metric tensor inverse to ¯g. In this case we say that a stochastic-mechanical

system with force ¯α(t, m, X) is given on M . Relation (15.29) is called the

Newton-Nelson equation on M.

Note that (15.29) determines the Riemannian metric which deﬁnes the

kinetic energy for a classical mechanical system whose quantization is de-

scribed by (15.29). In particular, the Newton law (11.2) for the latter system

is given in terms of the covariant derivative of the Levi-Civit´a connection of

that metric.

In what follows we shall look for solutions of (15.29) in the class of Itˆo

processes that are Itˆo developments of processes in tangent spaces of the form

15.2 The Geometric Form of Stochastic Mechanics 369

ζ(t)=



a(s)ds + σw(t)whereσ =





(see Section 9.3). In this case it is

evident that if



≈ 0 (i.e., if the mass is very big in comparison with Plank’s

constant), similarly to the trajectories in R

, ξ(t) turns into a deterministic

curve and the ﬁrst equality of (15.29) becomes the classical Newton law

(11.2) if the metric determined from the second equality of (15.29) remains

unchanged (recall that for a deterministic curve its quadratic derivative is

zero).

We shall not deal with limits as



→ 0 and so without loss of generality

in what follows we shall suppose the mass



=1.

Now the correspondence between stochastic mechanics and ordinary quan-

tum mechanics on manifolds is analogous to that of Section 15.1.1 and is

proved via the same argument (cf. [190]).

Remark 15.19. In order to make natural the deﬁnition of acceleration in

the form (

∂

∂t

(t)+∇

(t)

(t)) − (−

Δu + ∇

(t)

(t)) with the Laplace-

de Rham operator Δ (see above), in [53, 54] the construction of the parallel

translation along stochastic processes was modiﬁed so that the parallel trans-

lation obtained takes into account the deviation of geodesics. Having deﬁned

D and

∗

by formula (9.15), where the new parallel translation is involved,

one obtains the Newton-Nelson equation in the form

(

∗

D)ξ(t)=¯α(t, ξ(t),v

(t)), (15.30)

completely analogous to the usual form. Of course, (15.30) is equivalent to

(15.29). Note that the Newton-Nelson equation in the form (15.30) is given

in [53, 54, 190]. We do not use the form (15.30) since our constructions below

are based on the usual parallel translation.

15.2.2 Existence theorems

The main purpose of this section is to generalize the existence theorem of Sec-

tion 15.1.2 to a rather broad class of Riemannian manifolds, not necessarily

Euclidean spaces (see [106, 107, 115]).

Our generalization follows the same scheme as the basic existence theo-

rems of Section 15.1.2, the necessary modiﬁcation is based on the methods

developed in Chapters 8 and 9. Using parallel translation along stochastic

processes we construct a special stochastic equation in the tangent space at

the initial conﬁguration of the motion (a certain stochastic version of the

velocity hodograph equation (11.20), diﬀerent from (14.10)) and prove the

existence of its solutions. Then we show that the developments of the solu-

tions satisfy the Newton-Nelson equation (at least after a certain non-zero

instant, ﬁxed in advance, if the initial conﬁguration is deterministic). Note

370 15 The Newton-Nelson Equation

that the trajectory is an Itˆo process of diﬀusion type on a manifold, not

necessarily a diﬀusion process.

We consider a vector force ﬁeld ¯α of the same kind as in Section 15.1.2,

namely of the form ¯α(t, m, X)=¯α

(t, m)+¯α

(t, m)X where ¯α

(t, m)isa

vector ﬁeld on M depending on t ∈ [0,l]and¯α

(t, m) is a linear operator in

M depending on the parameter t ∈ [0,l], i.e., ¯α

(t, m)isa(1, 1)-tensor

ﬁeld on M.Examplesofsuch ¯α include potential and gyroscopic force ﬁelds

(e.g., a magnetic ﬁeld or an electromagnetic ﬁeld on a general relativistic

space-time).

As in Section 15.1.2, for the sake of simplicity (and without loss of gener-

ality) we set



= 1 (i.e., we work in the system of units with this property).

Let us introduce some notation. For a (1, 1)-tensor ﬁeld υ(t, m) we will deal

with its covariant derivative, the (1, 2)-tensor ﬁeld ∇υ(t, m)(·, ·):T

M ×

M → T

M, as well as with the tensor ∇υ(t, m)(υ(t, m)·, ·):T

M ×

M → T

M and its trace, the vector ﬁeld

tr∇υ(t, m)(υ)=



i=1

∇υ(t, m)(υ(t, m)e

where e

,...,e

is an orthonormal frame in T

M (see formula 6.11).

Examples of the above (1, 1)-tensor ﬁelds are



Ric(m)and¯α

(t, m), for

which we consider the vector ﬁelds (traces) tr∇



Ric(m)(



Ric) and

tr∇¯α

(t, m)(¯α

). Note that the ﬁeld tr∇



Ric(m)(



Ric) is C

∞

-smooth since

the tensor ﬁeld



Ric is also C

∞

-smooth.

In what follows in this Section we assume the next two conditions to be

fulﬁlled.

Condition 15.20 The Riemannian manifold M is complete in the usual

sense (see Deﬁnition 1.49 and the Hopf-Rinow Theorem (Theorem 3.68)).

The Ricci tensor, the linear operator



Ric(m):T

M → T

M,isbounded

uniformly in m with respect to the operator norm generated in the tangent

spaces by the Riemannian metric ·, ·. The vector ﬁeld tr∇



Ric(



Ric) on M is

also uniformly bounded with respect to the norm generated by the Riemannian

metric.

Condition 15.21 The vector ﬁeld ¯α

(t, m) and the tensor ﬁeld ¯α

(t, m) are

Borel measurable jointly in t and m, the vector ﬁeld tr∇¯α

(t, x)(¯α

) exists

and is also Borel measurable jointly in t and m, and there exists a constant

C>0 such that





¯α

(t, m(t))

+ ¯α

(t, m(t))

+ tr∇¯α

(t, x)(¯α

)



dt<C

for any continuous curve m(t) on M , t ∈ [0,l],andwhere¯α

(t, m) is the

operator norm.

15.2 The Geometric Form of Stochastic Mechanics 371

Remark 15.22. From Theorem 7.80 it follows that under Condition 15.20

the Riemannian manifold M is stochastically complete.

Let m

∈ M and introduce the probability space (

Ω,

F,ν)where

Ω =

([0,l],T

M),

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

the Wiener measure (recall that T

M is an n-dimensional Euclidean space

with respect to the metric tensor ·, · at m

). Denote by

the σ-algebra

generated by the cylinder sets with bases over [0,t], all the

being completed

by the sets of ν-measure zero.

Recall that the coordinate process W (t, ω(·)) = ω(t), ω(·) ∈

Ω, is a Wiener

process in T

M.SincebyRemark15.22 M is stochastically complete, the

Itˆo development R

W (t) is well-deﬁned for t ∈ [0,l] and Riemannian parallel

translation is also well-deﬁned along R

W (t) (see Section 7.6.1). Further on

we will use the operator Γ

t,s

of parallel translation introduced in Section

7.7.1.

Thus for any vector or tensor ﬁeld υ(t, m) we can consider the vector

(tensor) ﬁeld Γ

0,t

υ(t, R

W (t)) at m

to be the result of parallel translation

of υ(t, R

W (t)) along R

W (·) from the (random) point R

W (t) to the point

W (0) = m

.SinceR

W (t) is an extension of Cartan’s development R

from the class of piecewise smooth curves onto ν-almost all continuous curves

in T

M and the analogous fact is valid for parallel translation along R

W (·)

(see Remarks 7.67 and 7.84), Γ

0,t

υ(t, R

W (t) is determined along ν-almost

all continuous curves in T

M. This ﬁeld along ω(·) ∈

Ω will be denoted by

(Γ

0,t

υ)(t, ω(·)). Note that it may depend on t (which is compatible with the

notation) even if υ(m) is autonomous.

Pick t

∈ (0,l)andfort ∈ [0,l] consider the function t

(t) deﬁned by

formula (15.13).

Let a

∈ T

M and consider in T

M the equation

a(t)=a



(Γ

0,s

¯α

)(s, W (·))ds −





0,s

tr∇



Ric





Ric



(s, W (·))ds



(Γ

0,s

tr∇¯α

(¯α

))(s, W (s))ds





0,s



Ric



(s, W (·)) − t

(s)



a(s)ds





(Γ

0,s

¯α

)(s, W (·)) −



0,s



Ric



(s, W (·)) +

(s)



dW (s)

−

(t)W (t) (15.31)

(an analog of equation (15.14)).

Equation (15.31) has a unique strong solution for t ∈ [0,l]. Indeed, the

coeﬃcients of (15.31) are either Lipschitz continuous and have linear growth

with respect to a,ordonotdependona. Since the solution is strong, it

exists for any Wiener process and is non-anticipative with respect to P

.In

372 15 The Newton-Nelson Equation

what follows we will consider a(t) for the realization of W (t) as the coordinate

process on (

Ω,

F,ν). From Condition 15.21 it follows that a(t) satisﬁes (7.54).

Consider the probability density θ(l)on(

Ω,

F) deﬁned for the above a

by formula (8.33) (cf. Section 15.1.2). Introduce the measure μ

= θdν on

(

Ω,

F) and denote by ζ(t) the coordinate process on (

Ω,

F,μ

). Then (see

Sections 8.3 and 15.1.2) ζ(t) is expressed in the form

ζ(t)=



a(s)ds + w(t) (15.32)

where w(t) is a Wiener process on (

Ω,

F,μ

) adapted to P

= P

= B

Since ζ(t)andW (t) coincide as coordinate processes on (

Ω,

F), (15.31) turns

into

a(t)=a



(Γ

0,s

¯α

)(s, ζ(·))ds −





0,s

tr∇



Ric





Ric



(s, ζ(·))ds



(Γ

0,s

tr∇¯α

(¯α

))(s, ζ(·))ds +



(Γ

0,s

¯α

)(s, ζ(·)) ◦a(s)ds



((Γ

0,s

¯α

)(s, ζ(·)) −



0,s



Ric



(s, ζ(·)) +

(s))dw(s)

−

(t)ζ(t) (15.33)

(an analog of equation (15.16)).

Since by Remark 15.22 M is stochastically complete and since a(t) satisﬁes

(7.54), from Theorem 7.98 it follows that for ζ(t)theItˆo development ξ(t)=

ζ(t) is well-deﬁned on the entire interval t ∈ [0,l].

Theorem 15.23 For t ∈ (t

,l) the above-mentioned process ξ(t) satisﬁes

(15.29) with the force ¯α(t, m, X)=¯α

(t, m)+¯α

(t, m)X, i.e., it is a trajectory

of the stochastic mechanical system with that force.

Proof. By construction ξ(t) satisﬁes equation (9.22). This allows us to derive

some technical statements which follow from the results of Section 7.7 and

from those on the calculation of mean derivatives in Chapters 8 and 9.

Lemma 15.24

(i) Dξ(t)=E

(Γ

t,0

a(t)).

(ii) D

∗

ξ(t)=E

(Γ

t,0

a(t)) + E



0,t



ζ(t)

− κ(t)



,whereκ(t) is as de-

ﬁned in Lemma 8.35.

Proof. Assertion (i) is in fact formula (9.25), and (ii) is (9.29). 

Corollary 15.25

(i) D

ξ(t)=E

(Γ

t,0

a(t)) +



t,0



ζ(t)

− κ(t)



15.2 The Geometric Form of Stochastic Mechanics 373

(ii) D

ξ(t)=−



t,0



ζ(t)

− κ(t)



Corollary 15.25 can be obtained from formulae (9.30)and(9.31).

Lemma 15.26

(i) DD

∗

ξ(t)=D

(Γ

t,0

a(t)) + E



t,0

a(t)



− E



t,0

ζ(t)



(ii) D

∗

Dξ(t)=D

∗

(Γ

t,0

a(t)).

The assertion of Lemma 15.26 follows from Lemma 15.24, Lemma 8.36

(which can evidently be generalized to the processes in M) and Lemma 8.38.

So the proof of Theorem 15.23 is reduced to the calculation of mean deriva-

tives for Γ

t,0

a(t) which can be done by calculating the derivatives for the

summands in (15.33).

Lemma 15.27 For t ≥ t

(i) D

∗

t,0





(s)dw(s)



= E



t,0



ζ(t)

−

κ(t)



;

(ii) D

t,0



ζ(t)



= E



t,0



a(t)

−

ζ(t)



;

(iii) D

∗

ξΓ

t,0



ζ(t)



= E



t,0



a(t)

−

κ(t)



The proof of Lemma 15.27(i) is analogous to that of Lemma 15.8(ii) with

a modiﬁcation based on formulae (9.15). The proofs of (ii) and (iii) are anal-

ogous to that of Lemma 8.37.

Lemma 15.28

(i) D

∗

t,0





(Γ

0,t

¯α

)(s, ζ(·)) dw(s)



− Γ

t,0

(Γ

0,t

tr∇¯α

(¯α

)) (t, ζ(t))

+ Γ

t,0

(Γ

0,t

¯α

)(t, ζ(·)) ◦ E



t,0



ζ(t)

− κ(t)



(ii) D

∗

t,0







0,t



Ric



(s, ζ(·)) dw(s)



− Γ

t,0



0,t

tr∇



Ric





Ric



(s, ζ(s))

+ Γ

t,0



0,t



Ric



(t, ζ(·)) ◦ E



t,0



ζ(t)

− κ(t)



To prove Lemma 15.28 one should apply formulae (9.15), the deﬁnitions of

the integrands and the expression for Itˆo integrals via the backward integrals

(6.7)and(6.26) (cf. Lemma 15.8(i)).

All other summands in (15.32) are diﬀerentiated directly according to

(9.15).

By the deﬁnition of the processes ζ(t)andξ(t) and of the operator of

parallel translation Γ

t,s

the following relations hold:

t,0

(Γ

0,t

¯α)(t, ζ(·)) = ¯α(t, ξ(t));

t,0

(Γ

0,t



Ric)(t, ζ(·)) =



Ric(ξ(t)).

374 15 The Newton-Nelson Equation

Thus by Lemmas 15.26–15.28 and formula (15.33)

(DD

∗

+ D

∗

D)ξ(t)

=¯α

(t, ξ(t)) + ¯α

(t, ξ(t)) ◦ a(t)+



t,0



ζ(t)

− κ(t)





Ric(ξ(t)) ◦



−E



t,0



ζ(t)

− κ(t)



. (15.34)

Taking into account Corollary 15.25 one sees that (15.34) coincides with

(15.29). Theorem 15.23 follows. 

Remark 15.29. As in Remark 15.11 we should point out that ξ(t)doesnot

satisfy (15.29)fort ∈ (0,t

) because here t

(t)=

and E

(Γ

t,0

ζ(t)

t·t

) =

(Γ

t,0

ζ(t)

). Thus ξ(t) can be interpreted only as a trajectory of the stochas-

tic mechanical system beginning at the instant t

from the random conﬁgu-

ration ξ(t

) with the initial mean forward derivative E

(Γ

t,0

a(t

)) (see the

details in Remark 15.10). A certain analog of this situation with ‘big bang’

will be described below in Section 15.3.2, which is devoted to general relativ-

ity.

Remark 15.30. One can easily see that E

(a(t)) is the hodograph of the

forward mean derivative for the process ξ(t) and that equation (15.33)isa

direct analog of the velocity hodograph equation (11.20).

We now turn to the proof of an existence theorem analogous to Theo-

rem 15.14.

Theorem 15.31 Let Conditions 15.20 and 15.21 be fulﬁlled. Let η be a ran-

dom element taking values in M with density ρ(m) =0for all m ∈ M.Leta

be a bounded Borel measurable vector ﬁeld on M. Then there exists a solution

ξ(t) of (15.29) with initial conditions ξ(0) = η and Dξ(0) = a(η). If there

exists a δ>0 such that ρ

(m) >δfor all m ∈ M and Grad log ρ

(see (8.20))

is bounded, then there exists a solution ξ(t) of (15.29) with initial conditions

ξ(0) = η and D

ξ(0) = a(η).

Proof. Denote by W (t) the coordinate process on the probability space

([0,l], R

), F,ν) taking values in R

where F is the σ-algebra gener-

ated by cylinder sets and ν is a Wiener measure. As above, it is clear that

W (t) is a standard Wiener process in R

. Take some Borel measurable cross-

section b

of the bundle OM and consider the Itˆo development R

W (t) with

initial data b

(η) (see Deﬁnition 7.68). R

W (t)existsfort ∈ [0,l]byThe-

orem 7.99 since, by Remark 15.22, from Condition 15.20 it follows that M

is stochastically complete. Consider the following objects along R

W (t): the

vector ﬁeld u

(t, m) constructed for R

W (t) by formula (8.20), the vector

ﬁeld D

(t, m) and the tensor ﬁeld ∇u

(t, m)=K ◦ T (u

(t, m)) (the

15.2 The Geometric Form of Stochastic Mechanics 375

operator of covariant derivative applied to u

(t, m)). Using the construc-

tion of parallel translation along R

W (t) (see Section 7.6), in this section

the operator Γ

t,0

is applied to a tensor υ(t, R

W (t)) yielding the tensor ﬁeld

t,0

υ(t, W (t)) along W (t)inR

, which is well-deﬁned along a.s. all continu-

ouscurvesinR

with respect to Wiener measure (see Section 7.6).

Denote by a

the vector b(0)

−1

[a(η)+u

(0,W(0))] and consider the fol-

lowing stochastic diﬀerential equation in R

a(t)=a



0,s

(s, W (s))ds +



0,s

(s, W (s)) ◦ dW (s)



0,s

(s, W (s))ds +





0,s

∇u

(s, W (s)) ◦ a(s)



−



0,s



Ric(s, W (s)) ◦ dW (s), (15.35)

where ◦dW (s) denotes Stratonovich integration in composition with parallel

translation, i.e.,



0,s

A(s, W (s)) ◦ dW



0,s

A(s, W (s))dW (s)+



0,s

tr∇A(t, m)(A)ds

for a (1, 1)-tensor ﬁeld A on M.

Since (15.35) is linear in a, it has a unique solution a(t). From Condi-

tion 15.21 it follows that a(t) satisﬁes (7.54). Set

θ(t) = exp



−



a(s)

ds +



a ·dW.



Under these assumptions θ(t) is a martingale with Eθ(t) = 1. Thus we can

introduce a new probability measure μ on (C

([0,l], R

), F) by the relation

dμ = θ(l)dν. Denote by ζ(t) the coordinate process on (C

([0,l], R

), F,μ).

By Girsanov’s theorem ζ(t)=



a(s)ds + w(t)wherew(t) is a Wiener pro-

cess on C

([0,l], R

), F,μ) adapted to ζ(t)anda(t) is not anticipative with

respect to P

, i.e., ζ is a process satisfying the hypothesis of Theorem 7.99.

Since M is stochastically complete and a(t) satisﬁes (7.54), there exists an

Itˆo development ξ(t)=R

ζ(t) with initial condition η.

Lemma 15.32 The process ξ(t) is the solution of equation (15.29) with ini-

tial conditions ξ(0) = η and D

ξ(0) = a(η).

Proof. From the deﬁnition of Itˆo development it evidently follows that

t,0

0,t

(t, ξ(t)) = u

(t, ξ(t)); Γ

t,0

0,t

∇u

◦ a = ∇

t,0