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

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176 7 Stochastic Analysis on Manifolds

We also refer the reader to another paper of Grigoryan, [139], where ad-

ditional deep results concerning stochastic completeness are obtained.

Some criteria of stochastic completeness based on the use of atlases which

are in some sense uniform, and on the existence of a function with special

properties on M, can be found in [66].

7.6.3 Parallel translation along a stochastic process.

Itˆo processes on manifolds

Deﬁnition 7.82. A stochastic process on M is called an Itˆoprocessif it is

an Itˆo development of an Itˆo process in some tangent space.

Using the transformation of equations of Itˆo type into equations of

Stratonovich type and the expression of Itˆo processes in tangent spaces via

Stratonovich integrals it is possible to present Itˆo processes on manifolds as

the Eells-Elworthy development of a process with Stratonovich integral as in

Deﬁnition 7.60. For simplicity of presentation here we restrict ourselves to

the use of Itˆo developments.

It is possible to deﬁne the Riemannian parallel translation along Itˆopro-

cesses on M analogously to the standard (i.e., non-stochastic) construction.

Let η(t)beanItˆo process given for t ∈ [0,l]andlet

ξ(t)beahorizontal

lift of η(t)toOM with the initial condition

ξ(0) =

b.Letv ∈ T

η(0)

M be a

(random) vector.

Deﬁnition 7.83. The parallel translation of the vector v along η(·)atthe

point η(t) is the vector (

ξ(t) ◦ (

−1

v)) ∈ T

η(t)

It is obvious that parallel translation preserves the Riemannian norms

and does not depend on the choice of horizontal lift

ξ. From the construc-

tion of the process R

y(t)wherey(t)=



α(τ)dτ +



A(τ)dw(τ )inT

it is clear, in particular, that the vector TπE

ξ(t)

(

−1

α(t)) and the operator

TπE

ξ(t)

(

−1

A(t)) from Lemma 7.65 are parallel along η(t)=R

y(t), respec-

tively,tothevectorα(t) and to the operator A(t).

Remark 7.84. We mention the papers [58, 66, 151, 152] where the parallel

translation is constructed along stochastic processes of various sorts. It is not

hard to see that Itˆo processes in the sense of Deﬁnition 7.82 are local quasi-

martingales (i.e., special semi-martingales, see [176]) with continuous sample

paths, along which the parallel translation is constructed in [151], and the

parallel translations in the sense of [151] and in the sense of Deﬁnition 7.83 co-

incide. From this it follows, in particular, that the parallel translation along

an Itˆo process is the limit in probability of trajectory-wise parallel trans-

lations (in the ordinary “piecewise smooth” sense) along processes whose

sample paths are piecewise geodesic approximations of the trajectories of the

process η(t)(see[151]). It is clear that one can always select a sub-sequence

7.7 The Integral Approach to Stochastic Diﬀerential Equations on Manifolds 177

of these approximations such that the trajectory-wise parallel translations

converge almost surely. Hence the parallel translation along η(t) as in Deﬁ-

nition 7.83 is an extension of the classical parallel translation from the set of

piecewise smooth curves to a.s. all sample paths of η(t) that are continuous

but a.s. not smooth (cf. Remarks 7.66 and 7.67).

7.7 The Integral Approach to Stochastic Diﬀerential

Equations on Manifolds

In this Section we present an integral description of Itˆo stochastic diﬀerential

equations, analogous to the classical case in linear spaces as presented in

Section 6.2. The construction of the required integral operators is a stochastic

modiﬁcation of the integral operators with Riemannian parallel translation

from Section 3.2 based on constructions from Section 7.6. Then, making use

of parallel translation along a stochastic process, we modify the notion of

a stochastic diﬀerential equation of Itˆo type and describe a broader class

of equations for which the notion of a non-degenerate (in particular, unit)

diﬀusion coeﬃcient is well-deﬁned.

7.7.1 General constructions

Let y(t)=



b(τ)dτ +



B(τ )dw(τ)beanItˆo process in the tangent

space T

M to the Riemannian manifold M as in Section 7.6. Introduce

for the corresponding Itˆo process R

y(t)onM the new notation R

y(t)=

S(b(τ ),B(τ))(t). One can easily see that the operator S introduced in this

way is a stochastic analog of the operator S with parallel translation from

Section 3.2.

Let ξ(t)beanItˆo process that a.s. exists for t ∈ [0,l]and(α(t, m),A(t, m))

be an Itˆo vector ﬁeld on M, t ∈ [0,l]. As in Section 2.2 (see Theorem 2.32),

denote by Γ

t,s

the operator of parallel translation along ξ(·) from the random

point ξ(s) to the random point ξ(t). For the sake of simplicity, if t =0,

i.e., if the translation is performed at the point ξ(0) rather than Γ

0,s

,we

shall often use the notation Γ analogous to that from Section 3.2.2.Thus

Γα(t, ξ(t)) is the random vector in T

ξ(0)

M obtained by parallel translation

of the random vector α(t, ξ(t)) along ξ(·)atthepointξ(0). Analogously,

ΓA(t, ξ(t)) is the random operator sending R

to T

ξ(0)

M that is obtained by

parallel translation of A(t, ξ(t)) along ξ(·)atξ(0).

Let an Itˆo vector ﬁeld (α(t, m),A(t, m)) be Borel measurable. Consider

the processes Γα(t, ξ(t)) and ΓA(t, ξ(t)). Using the properties of horizontal

lift, i.e., of a strong solution of equation (7.37), it is not hard to show that

these processes are non-anticipative with respect to the family B

that is used

in the deﬁnition of the Itˆo process ξ(t). Consider equation (7.37)onOM,in

which α is replaced by Γα and A by ΓA:

dξ(t)=e

ξ(t)



−1

Γα(t)



dt + E



−1

ΓA(t)dw(t)



. (7.42)

178 7 Stochastic Analysis on Manifolds

For further developments it is important to present conditions under which

(7.42) has global solutions. This is the case, if, say, on OM there exists a

proper function ϕ such that the values of the generator of the Itˆo equation

on OM, corresponding to (7.42), are uniformly bounded.

In [107] the existence of global solutions of (7.42)isprovedforα(t)and

A(t) (and hence for Γα(t)andΓA(t) since the parallel translation preserves

the norms) uniformly bounded under the additional assumption that M is a

uniformly complete Riemannian manifold.

Deﬁnition 7.85. A Riemannian manifold M is said to be uniformly com-

plete if the following two conditions are satisﬁed:

(1) there exists an induced metric on OM which possesses a uniform Rie-

mannian atlas;

(2) on the balls V

(r) of the atlas, the norm of the operator X → Γ



(X, X),

where X ∈ H



and b



∈ V

(r), as a norm of a quadratic operator, is

bounded by a constant C>0 independent of the chart and the ball.

Evidently compact Riemannian manifolds and Lie groups are examples of

uniformly complete Riemannian manifolds but the latter class of manifolds

is much broader than these two examples.

For uniformly complete manifolds and uniformly bounded Γα(t)and

ΓA(t) the solvability of (7.42) follows from Theorem 7.38.

Let, for any given initial data, equation (7.42) have a strong and strongly

unique solution that is well-deﬁned for all t ∈ [0,l]. It is clear that the pro-

jection of this solution to M is the Itˆo process S(Γα(τ,ξ(t)),ΓA(τ,ξ(τ)))(t).

Deﬁnition 7.86. The Itˆo process S(Γα(τ, ξ(τ)),ΓA(τ,ξ(τ)))(t)onM is

called the line Itˆo integral with Riemannian parallel translation of the ﬁeld

(α, A) along ξ(t).

S(Γα(τ, ξ),ΓA(τ,ξ))(t) is a direct analog of the ordinary line integral used

in the theory of Itˆo stochastic diﬀerential equations in Euclidean spaces. If

M is a Euclidean space, Γ is the identical mapping and

(Γα(τ,

ξ),ΓA(τ,ξ))(t)=



α(τ,ξ)dτ +



A(τ,ξ)dw(τ).

Like its classical analog, the integral S(Γα(τ,ξ),ΓA(τ, ξ))(t) is naturally

connected with the Itˆo equations.

Consider an Itˆo vector ﬁeld (α(t, m),A(t, m)) on M and the corresponding

equation in Belopolskaya-Daletskii form

dη(t) = exp

η(t)

(α(t, η(t))dt + A(t, η(t))dw(t)),η(0) = m

, (7.43)

where exp is the exponential mapping of the Levi-Civit´a connection. It turns

out that its solution is an Itˆo process that satisﬁes the equation

7.7 The Integral Approach to Stochastic Diﬀerential Equations on Manifolds 179

η(t)=S(Γα(τ, η(τ)),ΓA(τ,η(τ)))(t). (7.44)

Indeed, construct a horizontal lift of η(t) in the following way. Introduce an Itˆo

vector ﬁeld (¯α(t, b),

A(t, b)) on OM by the formulae ¯α(t, b)=Tπ

−1

α(t, πb)

and

A(t, b)=Tπ

−1

A(t, πb)

. For every

b ∈ O

(M) let there exist a unique

strong global solution

ξ(t) of the equation

ξ(t)=e

ξ(t)



¯α



ξ(t)



dt +



ξ(t)



dw(t)



ξ(0) =

Consider the processes α(t)=

ξ(t)

−1

α(t, π

ξ(t))) in T

M and A(t)=

ξ(t)

−1

A(t, π

ξ(t))) in L(R

M). By construction we obtain that

ξ(t)is

a solution of equation (7.37)forα(t)andA(t). Thus

ξ(t) is a horizontal lift

of η(t). So, α(t)=Γα(t, η(t)),A(t)=ΓA(t, η(t)) and for η(t) relation (7.44)

is satisﬁed.

Equation (7.44) is an analog of the integral form of the Itˆo equation in

Euclidean space (see Section 6.2.3). For (7.44) the usual notion of strong and

weak solutions are introduced as in Section 6.2.3.

Theorem 7.87 Let equation (7.44) have a weak solution η(t).Thenη(t) is

a weak solution of equation (7.43).

Proof. Let

ξ(t) be the horizontal lift of η(t) with initial condition

(0) =

b ∈ O

(M). From (7.44) and from the construction of the operator S it

follows that for every t ∈ [0,l] a.s. α(t, η(t)) = TπE

ξ(t)

(

−1

Γα(t, η(t))) and

A(t, η(t)) = TπE

ξ(t)

(

−1

ΓA(t, η(t))). Hence by Lemma 7.65 equation (7.43)

is satisﬁed for η(t) a.s. for all t ∈ [0,l]. All other conditions of Deﬁnition 7.82

are fulﬁlled by the hypothesis since η(t) is a weak solution of (7.44). 

Corollary 7.88 If η(t) is a strong solution of (7.44), η(t) is a strong solution

of (7.43).

Proof. As in Theorem 7.87 it is proven that η(t) satisﬁes (7.43). Here all

requirements of Deﬁnition 6.23 are fulﬁlled since η(t) is a strong solution of

(7.44). 

As in the case of ordinary diﬀerential equations (see Section 3.2) the use of

integral operators with parallel translation allows one to reduce some ques-

tions to the investigation of stochastic diﬀerential equations in a single tan-

gent space.

Let (α(t, m),A(t, m)) be an Itˆo vector ﬁeld on M , t ∈ [0,l]. In T

consider the stochastic diﬀerential equation

z(t)=



Γα(τ,R

z(τ))dτ +



ΓA(τ,R

z(τ))dw(τ ) (7.45)

180 7 Stochastic Analysis on Manifolds

Theorem 7.89 An Itˆoprocessz(t) in T

M is a strong (weak) solution of

(7.45) if and only if η(t)=R

z(t) is a strong (weak, respectively) solution of

(7.43).

Proof. Assume that a Wiener process w(t) with values in R

and an Itˆopro-

cess z(t)=



α(τ)dτ +



A(τ)dw(τ ) with values in T

M are given on a

probability space (Ω, F, P) and that they satisfy (7.45). Then by Lemma 7.65,

by the construction of the mapping E and by the Deﬁnition 7.82 of par-

allel translation we obtain that for all t ∈ [0,l] equality (7.43) a.s. holds

for R

z(t). Conversely, let z(t)andw(t) be such that the development

η(t)=R

z(t) for all t a.s. satisﬁes (7.43). By the construction of the de-

velopment there is a horizontal lift

ξ(t)ofη(t),

ξ(0) =

b ∈ O

M and along

η(·) the parallel translation is well-deﬁned. From Lemma 7.65 and equa-

tion (7.43) it follows that for all t ∈ [0,l] a.s. TπE

ξ(t)

(

−1

α(t)) = α(t, η(t))

and TπE

ξ(t)

(

−1

A(t)) = A(t, η(t)). Applying parallel translation along η(·)

at the point m

to the latter equality, we obtain that for all t ∈ [0,l] a.s.

α(t)=Γα(t, η(t)) and A(t)=ΓA(t, η(t)). From this it follows that z(t)and

w(t) satisfy (7.45). It is easy to see that z(t)andw(t) are non-anticipative

with respect to the common family of σ-subalgebras: P

in the case of a

strong solution and B

(to which z(t) is adapted) in the case of a weak one.



As an example of the application of equation (7.45) to the investigation of

equation (7.43) we present a statement on the existence of a weak solution

of (7.43).

Theorem 7.90 Let an Itˆo vector ﬁeld (α(t, m),A(t, m)), t ∈ [0,l],onauni-

formly complete Riemannian manifold M (see Deﬁnition 7.85) be jointly con-

tinuous in t and m and uniformly bounded in the norm generated by the Rie-

mannian metric. Then for every initial condition m

∈ M equation (7.43)

has a weak solution.

Proof. We use the martingale approach to the construction of solutions

[83, 84, 162]. In the case under consideration we need some preliminary con-

structions that take into account the speciﬁc features of the equations. Since

(α, A) is uniformly bounded on [0,l]×M , one can easily construct a sequence

of smooth approximations (α

) that converge uniformly on [0,l] × M to

(α, A). Note that all (α

) are uniformly bounded by a common constant

since (α, A) is uniformly bounded. Let η

be a strong solution of the equation

dη

(t) = exp

(t)

(α

(t, η

(t))dt + A

(t, η

(t))dw(t)),η

(0) = m

that exists by Theorem 7.36. From the above statements it follows that the

processes z

(t)=



Γα

(τ,η

(τ))dτ +



ΓA

(τ,η

(τ))dw(τ ) are strong solu-

tions of the equations

7.7 The Integral Approach to Stochastic Diﬀerential Equations on Manifolds 181

(t)=



Γα

(τ,R

(τ))dτ +



ΓA

(τ,R

(τ))dw(τ ).

Let

Ω = C

([0,l],T

M) be the Banach space of continuous mappings

from the interval [0,l]toT

M (i.e., continuous curves in T

M),

F be the σ-

algebra in

Ω generated by cylinder sets and B

be the σ-subalgebra generated

by cylinder sets with bases on [0,t], t ∈ [0,l] (cf. Sections 6.1.1 and 6.2.1).

Recall that all σ-algebras are assumed to be complete (contain all sets of

measure zero). Denote by μ

the probability measure on (

Ω,

F) generated by

the process z

. Consider the probability space (

Ω,

F,μ

) where the elementary

events are continuous curves x(·) ∈ C

([0,l],T

M) and the realization of

as the coordinate process on (

Ω,

F,μ

):z

(t, x(·)) = x(t). Note that

the coordinate process is not anticipative with respect to B

. Taking into

account Remarks 7.67 and 7.84 we obtain that for a continuous curve x(·) ∈

Ω the processes R

x(t)andY (R

x(t)) for Y ∈ T

x(t)

M are μ

-a.s. well-

deﬁned since R

and Γ are extensions of the inverse of Cartan’s development

and parallel translation, respectively, from the set of smooth curves to μ

a.s. all continuous curves (sample paths of z

and R

). This is true for

every i, i.e., for every j and for all measures μ

the processes Γα

(t, R

x(t))

and ΓA

(t, R

x(t)) are μ

-a.s. well-deﬁned. From the uniform convergence

of (α

)to(α, A) and from the properties of parallel translation it follows

that Γα

(t, R

x(t)) and ΓA

(t, R

x(t)) converge as j →∞, μ

-a.s. uniformly

in t for all i,toΓα(t, R

x(t)) and ΓA(t, R

x(t)), respectively. From the fact

that the ﬁelds (α

) are uniformly bounded by a common constant one can

easily deduce that the set of measures μ

is weakly compact.

Let μ be a limit measure. Consider the coordinate process z(t, x(·)) = x(t)

on the probability space (

Ω,

F,μ). By construction z(t) is not anticipative

with respect to B

. Using Prokhorov’s Theorem 6.6 one can easily show that

the processes Γα

(t, R

x(t)) and ΓA

(t, R

x(t)) are μ-a.s. well-deﬁned and μ-

a.s. converge uniformly in t to Γα(t, R

x(t)) and ΓA(t, R

x(t)), respectively.

The concluding arguments are exactly the same as in the classical existence of

weak solution theorem for equations with continuous coeﬃcients [83]. Using

the above-mentioned convergencies a Wiener process ˜w(t), adapted to B

,is

constructed on (

Ω,F,μ)sothatz(t)and ˜w(t) satisfy (7.45) for all t almost

surely. By Theorem 7.89, R

z(t) is a weak solution of (7.43). 

Remark 7.91.

(i) In the constructions and applications of operators with parallel trans-

lation in this section we have not used the fact that the torsion of the Levi-

Civit´a connection equals zero. Thus all constructions and applications remain

true if we use an arbitrary Riemannian connection on M under the condition

analogous for that connection to, say, the condition of unform completeness.

Note that for some special choice of connection on a Lie group the above-

mentioned constructions yield the well-known multiplicative integral.

(ii) Let (ˆα, A)beanItˆo equation (cross-section of an Itˆo bundle). Recall

that its solution is described by equation (7.13). Using diﬀerent Riemannian

182 7 Stochastic Analysis on Manifolds

metrics and connections on M , one can describe this equation in the form

(7.43) with corresponding (strictly speaking, diﬀerent) Itˆo vector ﬁelds that

canonically correspond to (ˆα, A) with respect to the chosen connections. Inte-

gral operators with parallel translation and the presentation of the equation

(ˆα, A)inintegralform(7.44)or(7.45) also depend on the choice of metric

and connection. We emphasize that a solution of (7.43)orof(7.44)doesnot

depend on the choice of metric and connection since this solution is a solution

of (7.13).

7.7.2 Stochastic diﬀerential equations in terms of

Wiener processes in tangent spaces

Let a be a vector ﬁeld and A be a (1,1)-tensor ﬁeld on M, i.e., A

is a

linear operator T

M → T

M for every m ∈ M. Note that the ﬁelds may

be time-dependent and in this case we shall denote them by a

t,m

and A

t,m

respectively. Using these ﬁelds we construct a modiﬁcation of equation (7.18)

in the following way. Assume that in every tangent space T

M a realization

(t) of a Wiener process is given. Then the equation

dξ(t) = exp

ξ(t)



t,ξ(t)

dt + A

t,ξ(t)

dw(t)



(7.46)

is well-deﬁned where exp is the exponential mapping of the Levi-Civit´acon-

nection. Indeed, let b

be a ﬁeld of orthonormal frames that determines the

realizations w

(t)=b

w(t) of a Wiener process in tangent spaces. Then

t,m

)isanItˆo vector ﬁeld (unlike the pair (a, A)).

We have changed the notation for equations of type (7.46) to avoid con-

fusion with equations of type (7.18).

For equations of type (7.46) it is necessary to modify the notion of solution.

Deﬁnition 7.92. We say that equation (7.46)hasastrong solution ξ(t)if,

for any Wiener process w(t)inR

,thereisaprocessξ(t)inM deﬁned on

the same probability space as w(t) and non-anticipative with respect to P

and there is a realization b

ξ(t)

w(t) of the Wiener process at ξ(t) such that

the processes w

ξ(t)

(t)=b

ξ(t)

w(t)andξ(t) satisfy (7.46) for every t.

Deﬁnition 7.93. Equation (7.46)hasaweak solution if there are:

(1) a probability space (Ω,F, P) with a non-decreasing family B

of com-

plete σ-subalgebras of F;

(2) a stochastic process ξ(t)onM, non-anticipative with respect to B

;

(3) a Wiener process w(t)inR

relative to B

;

(4) realizations w

(t)=b

ξ(t)

w(t)ofw(t)inT

ξ(t)

such that w

ξ(t)

(t)andξ(t) a.s. satisfy (7.46) for every t, as in Deﬁnition 7.28.

7.7 The Integral Approach to Stochastic Diﬀerential Equations on Manifolds 183

Using (7.19) and Deﬁnitions 7.92 and 7.93 it is easy to show that in a

chart equation (7.46) takes the form

dξ(t)=a



t, ξ(t)



dt −

tr Γ

ξ(t)

(A, A)dt + A

ξ(t)



ξ(t)

dw(t)



, (7.47)

where Γ

(··) is the local connector and tr Γ

(A, A)=trΓ

)

where b

∈ O

(M) is a cross-section of OM, i.e., a ﬁeld of orthonormal

frames b

: R

→ T

M, m ∈ U. Observe that tr Γ

(A, A) is independent

of b (consistently with the notation), because the trace is invariant under the

action of the orthogonal group. Using this fact and the results of Section 7.3,

it is not hard to show that (7.47) is covariant with respect to changes of

coordinates. Below b

ξ(t)

will usually arise as the horizontal lift of ξ(t).

By the deﬁnition of a Wiener process on a Riemannian manifold (see

Section 7.6.2), we have:

Theorem 7.94 AWienerprocess ˜w(t) on M is a strong solution of the

equation d˜w(t) = exp

˜w(t)

(dw). In local coordinates, this equation reads

d˜w(t)=−

tr Γ

˜w(t)

(I,I)dt + b

w(t)

dw(t).

Assume that M is uniformly complete. To construct the integral operators

needed to study (7.46), we have to alter the construction of Section 7.7.1.

Let the probability space (Ω,F,P), the family B

, the manifold M,and

the functions a(t)inT

M and A(t)inL(T

M) be as in Section 7.7.1.

Specify a realization bw of the Wiener process w in T

M. It is clear that

the operator S from Section 7.7.1 is applicable to the pair (a, Ab).

Let a(t, m)andA(t, m) be a vector ﬁeld and a (1, 1)-tensor ﬁeld on M ,

respectively, and let η(t), t ∈ [0,l], be an Itˆo process on M. Consider the

vector and tensor ﬁelds Γa



t, η(t)



and ΓA



t, η(t)



obtained by the parallel

translation of a



t, η(t)



and, respectively, A



t, η(t)



along η(·)toη(0). The

operator S can be applied to the pair



Γa



t, η(t)



,ΓA



t, η(t)



, provided the

ﬁelds a(t, m)andA(t, m)

are bounded and Borel measurable jointly in t and

Therefore, we can deﬁne the Itˆo integral and the line integral with par-

allel translation in terms of the ﬁeld of Wiener processes. To distinguish

these integrals from those of Section 7.7.1, we denote them by S



a(τ)dτ +

A(τ)dw(τ)



(t)andS



Γa



τ,η(τ)



dτ + ΓA



τ,η(τ)



dw(τ)



(t), respectively.

Then (7.44) is to be replaced by the following equation

ξ(t)=S



Γa



τ,ξ(τ)



dτ + ΓA



τ,ξ(τ)



dw(τ)



(t). (7.48)

Let b

w be the initial realization of the Wiener process in T

M. Observe

that the parallel translation of b

along a solution ξ(·)of(7.48) gives rise to

a realization of the Wiener process at ξ(t). (See Deﬁnitions 7.92 and 7.93.)

The equation

184 7 Stochastic Analysis on Manifolds

z(t)=



Γa



τ,R

z(τ)



dτ +



ΓA



τ,R

z(τ)



(τ) (7.49)

is an analog of (7.45). Similar results to Theorems 7.36, 7.87 and 7.90 hold

for equations (7.46), (7.48)and(7.49).

7.7.3 Equations with unit diﬀusion coeﬃcients

Generalizing the classical notion, it is natural to call (7.46) an equation with

a non-degenerate diﬀusion coeﬃcient if the operator A

(t,m)

: T

M → T

is non-degenerate for all m ∈ M and t ∈ [0,l] (cf. Deﬁnition 6.18).

Among equations with non-degenerate diﬀusion coeﬃcients we are espe-

cially interested in those with A = I, i.e., with the diﬀusion coeﬃcient equal

to the identity operator. Then the equation can be written down in the form:

dξ(t) = exp

ξ(t)



t,ξ(t)

dt +dw(t)



. (7.50)

Note that in this form the equations with a smooth ﬁeld of linear operators

A(m):R

→ T

M such that A(m)A

∗

(m)=I can also be represented.

Solutions of (7.50) will play a crucial role in Chapter 15 and in Section 14.4.

Note that for (7.50) the local expression (7.47) turns into the equality

dξ(t)=a



t, ξ(t)



dt −

tr Γ

ξ(t)

(I,I)dt + b

ξ(t)

dw(t).

Theorem 7.95 Assume that the Riemannian manifold M is stochastically

complete and the vector ﬁeld a(t, m) is Borel measurable jointly in (t, m) ∈

[0,l] × M and uniformly bounded. Then there exists a weakly unique weak

solution ξ(t) of (7.50) for any initial condition ξ(0) = m

that is well-deﬁned

on [0,l] .

Proof. Here, we are using a method based on a change of probability measure

[83, 84, 162]. Consider the standard Wiener process ˜w on T

M, i.e., the

coordinate process ˜w



t, x(·)



= x(t) on the probability space (

Ω,

F,ν), where

Ω = C



[0,l],T



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

is the Wiener measure. Recall that the elementary event in

Ω is a continuous

curve x(·) ∈ C



[0,l],T



. Observe that ˜w(t) is non-anticipative with

respect to the family of σ-subalgebras B

generated by the cylinder sets with

bases over [0,t], t ∈ [0,l]. (See Sections 6.2.1 and 7.7.)

Since M is stochastically complete, the development R

˜w(t) is well-

deﬁned. Taking into account Remarks 7.67 and 7.84,weseethatR

x(t)

and Γa



t, R

x(t)



exist for ν-almost all x ∈

Ω. Furthermore, it follows from

the properties of parallel translation, of R

and of a(t, m) that the stochastic

process Γa



t, R

˜w(t)



is uniformly bounded and non-anticipative with re-

spect to B

. Consider the measure μ on (

Ω,

F) with density ρ with respect

to ν given by

7.7 The Integral Approach to Stochastic Diﬀerential Equations on Manifolds 185

ρ (x(·)) = exp





Γa(t, R

x(t)) , d˜w(t)−



Γa(t, R

x(t))



(7.51)

It is known (see [83, 162]) that under the hypotheses of the theorem



ρ dν =1, (7.52)

i.e., μ is a probability measure, and, furthermore, w



t, x(·)



= x(t) −



Γa



τ,R

x(τ)



dτ is a Wiener process on (

Ω,

F,μ) relative to B

.Itis

not hard to show that ρ>0 everywhere, i.e., ν is absolutely continuous with

respect to μ and has density ρ

−1

. In other words, the probability measures μ

and ν are equivalent. The coordinate process z



t, x(·)



= x(t)on(

Ω,

F,μ)is

non-anticipative with respect to B

and, moreover, B

= P

. Since the mea-

sures μ and ν are equivalent, R

(z(t, x(·)) = R

x(t)existsμ-a.s. Thus, z(t)

and w(t) are related via the equation

dz(t)=Γa(t, R

z(t)) dt +dw (7.53)

on T

M.Inotherwords,z(t) is a weak solution of (7.53). It is shown in

[83, 162] that every solution of (7.53) gives rise to a probability measure on

(

Ω,

F) with density ρ. This means that a solution of (7.53) is weakly unique.

By deﬁnition, the process R

(t)existson(

Ω,

F,μ). This process is, in fact,

a weak solution of (7.50). Hence the solution is weakly unique. 

We point out that w(t), deﬁned as in the proof of Theorem 7.95, is a Wiener

process relative to the family P

generated by the weak solution z(t).

Theorem 7.96 Assume that M is stochastically complete, a(t, m) is Borel

measurable jointly in (t, m) ∈ [0,l] × M, the inequality



a (t, m(t))

dt<

∞ holds for any continuous curve m(·): [0,l] → M, m(0) = m

∈ M,and

the density ρ deﬁned by (7.51) satisﬁes (7.52). Then there exist a weakly

unique weak solution ξ of (7.50) with the initial condition ξ(0) = m

that is

well-deﬁned on [0,l].

This result is a simple generalization of Theorem 7.95. The only reﬁne-

ment needed in the proof is as follows. Even though the hypothesis of The-

orem 7.96 does not guarantee that Γa



t, R

x(t)



is uniformly bounded,

ν-almost all x(·) ∈

Ω with x(0) = 0 ∈ T

M satisfy the inequality



Γa(t, R

x(t))

dt<∞. Arguing in the same way as in the proof of

Theorem 7.95, we see that this inequality together with (7.52) yield the ex-

istence and weak uniqueness of a solution of (7.53)asin[83, 162]. The rest

of the proof of Theorem 7.95 remains unchanged.

Corollary 7.97 Assume that



a (t, m(t))

dt<∞ for any continuous

curve m(·): [0,l] → M. Then, under the hypotheses of Theorem 7.96, the

assertion of the theorem holds for any initial condition ξ(0) = m ∈ M.