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

Подождите немного. Документ загружается.

156 7 Stochastic Analysis on Manifolds

Remark 7.37. The hypothesis of Theorem 7.36 is satisﬁed for an auton-

omous smooth Itˆo vector ﬁeld on a compact manifold and any connection

Let (Ω, F, P) be a probability space. All σ-subalgebras of the σ-algebra F

that we use below in this section are assumed to be complete.

Theorem 7.38 Let the following objects be given: a non-decreasing family

of σ-subalgebras B

of F (t ∈ [0,l], l>0)andaWienerprocessw(t) in R

adapted to B

; a stochastic process a(t) in R

and a stochastic process A(t)

with values in the space L(R

) of linear automorphisms of R

; a connection H

on the manifold M, compatible with a Riemannian metric ·, ·;andaﬁeldof

linear operators E

: R

→ T

M smooth in m ∈ M and uniformly bounded

in the norm generated by ·, ·. Assume that the processes a(t) and A(t) are

non-anticipating with respect to B

, a.s. have continuous sample paths and

such that both a(t) and A(t) are a.s. bounded in norm, uniformly in t,bya

certain constant. Then for every initial condition ξ(0) = m

∈ M there exists

a strong and strongly unique solution ξ(t) of the equation

dξ(t) = exp

ξ(t)

a(t)dt + E

ξ(t)

A(t)dw(t)),

non-anticipating with respect to B

and a.s. having continuous sample paths,

that is well-deﬁned for t ∈ [0,l].

The equation (7.19) under consideration is reduced to an equation of type

(7.13) with random coeﬃcients. The remaining part of the proof is analogous

to that of [23, Theorem 2.2], modiﬁed as described in Remark 7.21.

Note that the equation in Theorem 7.38 is an analog of the equation in

Stratonovich form in Example 7.6. The proof of Theorem 7.38 can also be

obtained from the results of [66] for equations in Stratonovich form.

Remark 7.39. The conditions formulated in the hypotheses of Theorems

7.36 and 7.38 can be slightly weakened. Indeed, from formula (7.19)and

Theorem 7.20 it follows that in Theorem 7.36 instead of the compatibility of

connection and metric it is enough to require that the estimate

trΓ



(A(t, m



),A(t, m



)) <c

for m



∈ V

(r) holds on the balls V

(r) in the charts of a uniform Rieman-

nian atlas where c

is a certain constant independent of ball and chart. In

Theorem 7.38 the conditions can be weakened in an analogous way.

Now we are in a position to present an example of an equation in Belopols-

kaya-Daletskii form that is used below.

Example 7.40. Consider an Itˆo vector ﬁeld (a(t, m), A(t, m)) as in Exam-

ple 7.4 where a(t, m)andB(t, m) are smooth. For this ﬁeld the Itˆo equation

in Belopolskaya-Daletskii form

7.3 Itˆo Equations in Belopolskaya-Daletskii Form 157

dξ(t) = exp

ξ(t)

(a(t, ξ(t))dt + A(t, ξ(t)))dw(t) (7.23)

is well-deﬁned where w(t) is a Wiener process in R

(see Example 7.4) and exp

is the exponential mapping of a connection on M . Introduce a Riemannian

metric on M (for example, the ﬁrst fundamental form generated by the inner

product in R

). If M is a compact manifold, the connection and metric are

compatible in the sense of Deﬁnition 7.35. The existence of a strong solution

for all t ∈ [0, ∞) is proved as above.

We conclude this section with a description of a class of stochastic diﬀer-

ential equations on inﬁnite-dimensional Hilbert manifolds that will be used

below. This class is a particular case of the equations on inﬁnite-dimensional

manifolds considered in [23] (see also [35, 66], where equations in Stratonovich

form are considered).

Let M be a Hilbert manifold, H be a connection on M and G(·, ·)be

a strong Riemannian metric on M (the term “strong” means that G(·, ·)

determines the topology of the model space in tangent spaces to M;the

description of Riemannian metrics and connections on inﬁnite-dimensional

manifolds can be found, e.g., in [172], see also Chapter 10 for the particular

case of groups of diﬀeomorphisms). Notice that in this case Deﬁnition 7.9 of

a uniform Riemannian atlas and Deﬁnition 7.35 of compatible metrics and

connections remain valid.

Let α(t, m) be a vector ﬁeld and A(t, m) be a ﬁeld of linear operators

A(t, m):R

→ T

M where m ∈ M, t ∈ [0,l]andR

is the Euclidean space

in which a Wiener process w(t) takes values. As in the ﬁnite-dimensional

case the pair (α, A) is called an Itˆo vector ﬁeld and for this ﬁeld (and for the

exponential mapping exp of the connection H) equation (7.18) is well-deﬁned.

For convenience of reference we formulate an existence of solution theorem

for (7.18) in this case as a separate statement.

Theorem 7.41 Let the above mentioned objects G, H, α(t, m) and A(t, m)

be given on a Hilbert manifold M and let the conditions of Theorem 7.36 be

satisﬁed for them. Then for every m

∈ M there exists a strong and strongly

unique solution ξ(t) of equation (7.18) with initial condition ξ(0) = m

,well-

deﬁned for all t ∈ [0,l].

Note that Theorem 7.41 is valid for every ﬁnite-dimensional Euclidean

space R

. As for Theorem 7.36, the proof of Theorem 7.41 is reduced to

Theorem 7.20. Recall (see Remark 7.21) that the proof of Theorem 7.20 is

analogous to that of Theorem 2.2 in [23] (proved in the inﬁnite-dimensional

case). The modiﬁcation mentioned in Remark 7.21 is also valid here.

158 7 Stochastic Analysis on Manifolds

7.4 Completeness of Stochastic Flows

7.4.1 Setting up the problem and a necessary condition

for completeness

In this Section we follow [114]and[116].

Let M be a ﬁnite-dimensional non-compact manifold. Consider a stochas-

tic ﬂow ξ(s)onM generated by a stochastic diﬀerential equation in Itˆoorin

Stratonovich form with smooth coeﬃcients. Since the coeﬃcients are smooth,

we can pass from the Stratonovich to the Itˆo equation and vice versa. By A

we denote the generator of this ﬂow.

Consider the one-point compactiﬁcation M



{∞} of M where the system

of open neighborhoods of ∞ consists of complements to all compact sets

of M. Denote by ξ(s):M → M



{∞} the stochastic ﬂow. For any point

m ∈ M and time t the orbit ξ

t,m

(s) of this ﬂow is the unique solution of

the above-mentioned equation with initial conditions ξ

t,m

(t)=m.Asthe

coeﬃcients of the equation are smooth, this is a strong solution and so a

Markov diﬀusion process given on some random time interval. The point ∞

is the “cemetery” where the solution (deﬁned on a random time interval)

arrives after the explosion.

We refer the reader to [174] for more information on the behavior of a

diﬀusion process at inﬁnity.

Recall that the generator A is a second order vector (see Deﬁnition 2.74).

In local coordinates one can ﬁnd the matrix of its pure second order term,

which is symmetric and positive semi-deﬁnite.

For a stochastic ﬂow the generator plays the same role as the derivative in

the direction of a vector ﬁeld in the right-hand side of an ordinary diﬀerential

equation. The main result on completeness for stochastic ﬂows here is analo-

gous to Theorem 3.3 where the derivative in the direction of the vector ﬁeld

is replaced with the corresponding generator. However in the stochastic

case there is an additional diﬃculty that for a ﬂow with inverse time direction

the generator does not coincide with the generator for the ﬂow itself. This

is why we obtain a necessary and suﬃcient condition for completeness only

for ﬂows with the additional assumption that the ﬂow must be continuous at

inﬁnity (see the exact Deﬁnition 7.44 below).

We denote the probability space, where the ﬂow is deﬁned, by (Ω,F, P)

and assume that it is complete. We also deal with separable realizations of

all processes.

Let T be a positive real number.

Deﬁnition 7.42. The ﬂow ξ(s)iscomplete on [0,T]ifξ

t,m

(s) a.s. takes val-

ues in M for any pair (t, m) (with 0 ≤ t ≤ T ) and for all s ∈ [t, T ]. The

ﬂow ξ(s)iscomplete if it is complete on any interval [0,T] ⊂ R (cf. Deﬁni-

tion 6.30).

7.4 Completeness of Stochastic Flows 159

We start with a suﬃcient condition for completeness of a stochastic ﬂow

analogous to conditions for completeness of ODE ﬂows with one-sided esti-

mates. This is a simple version of a rather general suﬃcient condition [66,

Theorem IX. 6A]. We use the notion of a proper function introduced in Def-

inition 3.2.

Theorem 7.43 Let there exist a smooth proper function ϕ on M such that

A(t, m)ϕ<C for some C>0 at all t ∈ [0, +∞) and m ∈ M. Then the ﬂow

ξ(t, s) is complete.

Proof. Consider the collection of subsets W

= ϕ

−1

([0,k)) of M where k

ranges over the positive integers. Since ϕ is proper, these sets are relatively

compact and



= M. Besides, by construction W

⊂ W

i+1

, i =1, 2,....

Let t ∈ [0, +∞)andm ∈ M and consider the orbit ξ

t,m

(s). Denote by

the ﬁrst time ξ

t,m

(s) hits the boundary of W

. Transform ϕ(ξ

t,m

(s ∧τ

))

by the Itˆo formula. Since W

is relatively compact, the Itˆointegralonthe

interval [t, s ∧ τ

) is a martingale and so its expectation is equal to 0. Then

Eϕ(ξ

t,m

(s ∧τ

)) = ϕ(m)+

s∧τ



(Aϕ)(θ, ξ

t,m

(θ))dθ<ϕ(m)+Cs,

since A(t, m)ϕ<C and s ≥ s ∧ τ

Consider the set Ω

= {ω ∈ Ω | s<τ

}. Obviously k(1 − P(Ω

)) <

Eϕ(ξ

t,m

(s ∧ τ

)), since for ω/∈ Ω

we get ξ

t,m

(s ∧ τ

,ω)=ξ

t,m

(τ

,ω), i.e.,

ϕ(ξ

t,m

(s ∧τ

,ω)) = k.Thus,

1 −P(Ω

) <

ϕ(m)+Cs

. (7.24)

Hence lim

k→∞

(1−P(Ω

)) = 0. However by construction, lim

k→∞

∞



i=1

= Ω,

i.e., for any given s ≥ t the value ξ

t,m

(s)existsinM with probability 1. 

7.4.2 A necessary and suﬃcient condition for

completeness of ﬂows continuous at inﬁnity

In this Section the maximal assumption on the stochastic ﬂow is that its gen-

erator A(t, x) is smooth and strictly elliptic (i.e., in a local coordinate system

its pure second order term is described by a positive deﬁnite, i.e., invertible,

matrix). This assumption allows us to apply the machinery from [12]. Notice

that using this machinery we can reduce the condition that the stochastic

equation is C

∞

-smooth to the assumption that it is H¨older continuous. How-

ever, in some statements in this Section we use weaker assumptions, indicated

in the hypotheses.

Let ξ(s) be a (not necessarily complete) stochastic ﬂow.

160 7 Stochastic Analysis on Manifolds

Deﬁnition 7.44. We say that the ﬂow ξ(s)iscontinuous at inﬁnity if for

any 0 ≤ t ≤ T and any compact K ⊂ M the equality

lim

m→+∞

P(ξ

t,m

(T )) ∈ K) = 0 (7.25)

holds.

One can easily see that continuity at inﬁnity according to Deﬁnition 7.44

means that for any t ∈ [0, +∞) and for all s ∈ [t, +∞) the correspon-

dence (m, s) → ξ

t,m

(s) is continuous (in probability) at the point (s, {∞}) ∈

[t, ∞) × (M



{∞}). See [203, 206] for details.

As an example we mention that a ﬂow whose diﬀusion semigroup has the

so-called C

property is continuous at inﬁnity. We refer the reader to [206]

for relations between continuity of a ﬂow on M



{∞} and the C

property of

the corresponding diﬀusion semigroup (see also, e.g., [12]and[67] for details

on the C

-property).

Our next task is to construct a special proper function associated to a

complete stochastic ﬂow ξ(s).

Consider an expanding sequence of compact sets M

such that M

⊂ M

i+1

for all i and



= M. Let (T

) be an increasing sequence of real numbers

tending to +∞.

For (t, m) ∈ [0,T

]×M

the distribution function μ

t,m,s

of random elements

t,m

(s), s ∈ [t, T

], on M form a weakly compact set of measures. Indeed,

take an arbitrary sequence of random elements ξ

) with corresponding

measures μ

. Since [0,T

]×M

×[0,T

] is compact, it is possible to select

a subsequence (t

) of the sequence (t

) which converges (to

), say). It is a well-known fact that the function Ef(ξ

t,m

(s)) is

continuous jointly in t, m, s for any bounded continuous function f : M →

R. Then we obtain that E(f (ξ

))) → E(f(ξ

))), i.e., from

any sequence of measures described above it is possible to select a weakly

converging subsequence.

Take a monotonically decreasing sequence of positive numbers ε

→ 0such

that the series

∞



i=1

√

converges. From Prokhorov’s theorem it follows that

for the measures corresponding to ξ

t,m

(s), s ∈ [t, T

], (t, m) ∈ [0,T

] × M

mentioned above, there exists a compact Ξ

⊂ M such that for all μ

t,m,s

the inequality μ

t,m,s

(M\Ξ

) <ε

holds. Construct an expanding system of

compact sets Θ

⊃



k=0

for any i, being closures of open domains in M

with smooth boundary and such that Θ

⊂ Θ

i+1

for any i and



= M.By

construction, for s ∈ [0,T

], (t, m) ∈ [0,T

] ×M

the relation μ

t,m,s

(M\Θ

) <

holds. In particular, μ

t,m,s

(Θ

i+1

\Θ

) <ε

Choose neighborhoods U

⊂

of the set Θ

that are proper subset of

i+1

and consider a smooth function ψ

that equals 0 on U

,equals1on

i+1

and takes values between 0 and 1 on

. Construct the function

7.4 Completeness of Stochastic Flows 161

θ on M setting its value on Θ

i+1

\Θ

equal to ψ

√

+(1− ψ

)

√

i−1

. Notice

that on Θ

i+1

\Θ

the values of θ are not greater than

√

Immediately from the above construction we obtain the following:

Lemma 7.45 For a complete ﬂow ξ(s) the function θ, constructed above, is

smooth, positive and proper.

Theorem 7.46 If the ﬂow ξ(t) is complete, for every (t, m) and every T>t

the inequality Eθ(ξ

t,m

(s)) < ∞ holds for each s ∈ [t, T ].

Proof. Take i such that [0,T] ⊂ [0,T

], t ∈ [0,T

]andm ∈ M

.Then

t,m,s

(M\Θ

) <ε

and so μ

t,m,s

(Θ

) > (1 − ε

). By construction, the values

of the continuous function θ on compact Θ

are bounded by the constant

√

i−1

. Then also by construction

Eθ(ξ

t,m

(s)) ≤

√

i−1

∞



k=i

√

i−1

∞



k=i

√

<C<+∞ (7.26)

for some positive constant C since by deﬁnition the series

∞



k=i+1

√

con-

verges. 

Corollary 7.47 The function Eθ(ξ

t,m

(s)) is integrable in s ∈ [t, T ].

Proof. From the construction in Theorem 7.46 it follows that for given t, m,

estimate (7.26) is valid with the same C for all s ∈ [t, T ]. 

Theorem 7.48 For every stochastic ﬂow ξ(s) on a manifold M that is com-

plete on an interval [0,T], there exists a proper positive function θ on M such

that for all t ∈ [0,T], m ∈ M and s ∈ [t, T ] the inequality Eθ(ξ

t,m

(s)) < ∞

holds.

Theorem 7.48 follows from Lemma 7.45 and Theorem 7.46.

Let T>0 and consider the direct product M

=[0,T] × M . Denote by

: M

→ M the natural projection: π

(t, m)=m.

Theorem 7.49 Let the generator A of the complete ﬂow be smooth and

strictly elliptic. Then the function u(t, m)=Eθ(ξ

t,m

(T )) on M

is C

smooth in t ∈ [0,T], C

-smooth in m ∈ M and satisﬁes the equation



∂

∂t

+ A



u = 0 (7.27)

Proof. Since M is locally compact and satisﬁes the second countability axiom

(and hence is paracompact, see Section 1.1) we can choose a countable locally

ﬁnite open covering {V

}

∞

i=1

of M such that all V

have compact closures.

Consider a smooth partition of unity {ϕ

}

∞

i=1

adapted to this covering. Then

at any point m ∈ M the equality θ(m)=



∞

i=1

(m)θ(m) holds.

162 7 Stochastic Analysis on Manifolds

Introduce the functions v

(m)=ϕ

(m)θ(m) and the functions u

(t, m)=

(ξ

t,m

(T )) and θ

(t, m)=



i=0

(t, m). Notice that all v

(m)aresmooth

and bounded. Then all v

(m) satisfy the conditions of [81, Theorem VIII.4.1]

and so all u

(t, m)areC

-smooth in t, C

-smooth in m and satisfy the relation

∂

∂t

+ Au

=0. Hence all functions θ

(t, m), being ﬁnite sums of functions

(t, m), are also C

-smooth in t, C

-smooth in m and satisfy

∂

∂t

+Aθ

=0.

In addition it is evident that θ(t, m) is the limit of θ

(t, m)ask →∞and

the functions θ

(t, m) form an increasing locally bounded sequence. Then,

since A is strictly elliptic, the assertion of the Theorem follows from standard

Schauder estimates. (For autonomous A, see [12, Lemma 1.8].) 

Theorem 7.50 If a complete ﬂow ξ(s) is continuous at inﬁnity, the function

u(t, m)=Eθ(ξ

t,m

(T )) on M

is proper.

Proof. Let ξ(s) be continuous at inﬁnity. To prove the properness of u(t, m)

it is suﬃcient to show that u(t, m) →∞as θ(m) →∞, i.e., that for any

C>0thereexistsΞ>0 such that θ(m) >Ξyields u(t, m) >Cfor any

t ∈ [0,T]. Since θ is proper, K = θ

−1

([0, 2C]) is compact. From formula

(7.25) in the Deﬁnition 7.44 of continuity at inﬁnity it follows that for any

t ∈ [0,T] there exists a Ξ such that P(ξ

t,m

(T ) /∈ K) >

for θ(m) >Ξ.Then

u(t, m)=Eθ(ξ

t,m

(T )) > 2C ·

= C.Sincet belongs to the compact set [0,T]

and u(t, m) is continuous in t, this completes the proof. 

On the manifold M

consider the ﬂow η(s)=(s, ξ(s)). Obviously for

(t, m) ∈ M

the trajectory of η

(t,m)

(s) satisﬁes the relation π

(η

(t,m)

(s)) =

t,m

(s). It is clear that η(s) is the ﬂow with inﬁnitesimal generator A

de-

termined by the formula:

(t,m)

= A(t, m)+

∂

∂t

. (7.28)

is a direct analog of the diﬀerentiation in the direction of X

described

in Theorem 3.3.

Theorem 7.51 Aﬂowξ(s) on M, continuous at inﬁnity and having a

smooth strictly elliptic generator, is complete on [0,T] if and only if there ex-

ists a positive proper function u

: M

→ R that is C

-smooth in t ∈ [0,T],

-smooth in m ∈ M and such that A

<C for some constant C>0 at

all points (t, m) ∈ M

Proof. Let there exist a smooth proper positive function u

(t, m)onM

such that A

<Cat all points of M

. Then from Theorem 7.43 it follows

that η(s) is complete. Thus ξ(s) is also complete.

Let ξ(s) be complete. Consider the function θ(m)onM introduced above

and the function u

(t, m)=Eθ(ξ

t,m

(T )) on M

.Sinceξ(s) is continuous at

inﬁnity, u

(t, m)isproperbyTheorem7.50.ByTheorem7.49 it is also C

in t, C

in m and satisﬁes the relation (

∂

∂t

+ A)u

= A

=0. 

7.4 Completeness of Stochastic Flows 163

Corollary 7.52 Aﬂowξ(s) on M as in Theorem 7.51 is complete if and

only if for all T>0 there exists a positive proper function u

: M

→ R

on M

that is C

-smooth in t ∈ [0,T], C

-smooth in m ∈ M and such that

u(t, m) <C for some constant C>0 at all points (t, m) ∈ M

7.4.3 Remarks on L

-complete stochastic ﬂows

For a stochastic ﬂow ξ(s) in the Euclidean space R

there is a property

stronger than ordinary completeness, where ξ(s) is complete and in addi-

tion each of its orbits ξ

t,x

(s)ateverys>tbelongs to the functional space

((Ω,F, P), R

), i.e., Eξ

t,x

(s) < ∞.

In the general case of ﬂows on manifolds it is natural to replace the norm

by a proper function, i.e., to suppose that there exists a positive proper

function ψ on M such that Eψ(ξ

t,x

(s)) < ∞ for all s>t(note that both the

norm in R

and the distance with respect to a complete Riemannian metric

are proper functions). Moreover, by Theorem 7.48 such a function exists for

every complete ﬂow.

In order not to lose some useful properties of Eξ

t,x

(s) (which are not

possessed by the function from Theorem 7.48 without additional assump-

tions), in [120, 121] we introduced the notion of an L

-complete stochastic

ﬂow as follows:

Deﬁnition 7.53. Aﬂowξ(s) on a ﬁnite-dimensional manifold M is called

-complete on [0,T] if the following conditions are fulﬁlled:

(i) ξ(s) is complete on [0,T];

(ii) there exists a smooth proper positive function v : M → R such that

Ev(ξ

t,m

(T )) < ∞ for all m ∈ M, t ∈ [0,T];

(iii) for each K>0thereexistsacompactC

K,T

⊂ M , depending on K

and T , such that the inequality Ev(ξ

t,m

(T )) <K yields m ∈ C

K,T

;

(iv) the function f(t, m)=Ev(ξ

t,m

(T )) is C

-smooth in t and C

-smooth

in m.

AﬂowisL

-complete if it is L

-complete on every interval [0,T] ⊂ [0, ∞).

As above, introduce M

=[0,T] × M and the process η

(t,m)

(s)=

(s, ξ

t,m

(s)) on M

.ByA

we denote the generator of the ﬂow η(s). Let

u : M

→ R be a proper function. Consider the sequence of compact subsets

= u

−1

([0,k]) of M

. Specify a point (t, m) ∈ M

and for k such that

(t, m) ∈ W

, denote by τ

the time η

(t,m)

(s) ﬁrst hits the boundary of W

In [120, 121] the following necessary and suﬃcient conditions for L

completeness are obtained.

Theorem 7.54 ([120, Theorem 3.1]) The ﬂow ξ(s) on M is L

-complete on

[0,T], T>0, if and only if there exists a smooth proper positive function

u(t, m) on M

such that for all (t, m) ∈ M

the equality A

(t,m)

u = C holds

164 7 Stochastic Analysis on Manifolds

where C is some constant, and for all (t, m) ∈ M

the random variables

u(η

(t,m)

(T ∧ τ

)) are uniformly integrable.

Theorem 7.55 The ﬂow ξ(s) on M is L

-complete on [0,T], T>0,ifand

only if there exists a smooth proper positive function u on M

such that at

every (t, m) ∈ M

the following conditions are satisﬁed:

1) A

u ≤ C where C is a positive constant;

2) Eu(η

(t,m)

(T )) < ∞ and |Eu(η

(t,m)

(T )) − u(t, m)| <C

where C

is a

positive constant;

3) the function Eu(η

(t,m)

(T )) is C

-smooth in t and C

-smooth in m.

The assertion of Theorem 7.55 follows from [121, Theorem 3.6].

The constructions from Section 7.4.2 allow us to obtain the following suf-

ﬁcient condition of L

-completeness.

Theorem 7.56 A complete ﬂow, continuous at inﬁnity and having a smooth

strictly elliptic generator, is L

-complete.

Indeed, from Lemma 7.45 and Theorems 7.46, 7.49 and 7.50 it follows that

the function θ constructed in Section 7.4.2, under the hypotheses of the The-

orem, meets all requirements set up for the function v by Deﬁnition 7.53.In

particular, Condition (iii) of Deﬁnition 7.53 is fulﬁlled since by Theorem 7.50

the function u(t, m)=Eθ(ξ

t,m

(T )) is proper on M

7.5 A Condition for Weak Compactness of Measures

Corresponding to Solutions of Stochastic

Diﬀerential Equations

Lemma 7.57 Consider on M a sequence of smooth Itˆo equations (ˆa

(t, m),

(t, m)), t ∈ [0,T] ⊂ R, with generators A

(t, m), respectively, such that:

(a) over each compact set K ⊂ M the images (ˆa

([0,T], K),A

([0,T], K))

for all q belongtoacompactsetinI(M );

(b) there exists a C

-smooth proper function ψ : M → R such that for all

q the inequality

ψ| <C (7.29)

holds for some constant C>0 independent of q, t and m.

Then

(i) for every m

∈ M there exist strong solutions ξ

(t) with initial con-

dition ξ

(0) = m

of all Itˆo equations (ˆa

(t, m),A

(t, m)) that are

strongly unique and well-deﬁned on the entire interval [0,T];

(ii) the set of measures {μ

} corresponding to ξ

(·) on the Banach manifold

Ω = C

([0,T],M) of continuous curves in M, equipped with the σ-

algebra

F generated by cylinder sets, is weakly compact.

7.5 A condition for weak compactness of measures 165

Proof. Recall that a solution that exists up to the ﬁrst time the boundary of

some chart including m

is hit is said to be a local solution. The existence

and strong uniqueness of strong local solutions ξ

(·) for all q follows from the

fact that all (ˆa

(t, m),A

(t, m)) are smooth and hence are bounded on every

relatively compact chart. The global existence follows from Condition (b)by

Theorem 7.43.

For every integer p consider the set W

= ψ

−1

([0,p]). Since ψ is proper,

these sets are compact and



= M. Besides, by construction W

⊂ W

p+1

for all p =1, 2,....Thusm

belongs to W

for suﬃciently large p.

Foracurvex(·) ∈ C

([0,T],M) denote by θ

x(·)

the time the boundary of

is ﬁrst hit. Introduce the subset in

Ω of the form Ω

= {x(·) | T<θ

x(·)

}

(i.e., every x(t)fromΩ

lies in W

for all t from t =0tot = T ). Taking into

account condition (b) it follows from the proof of Theorem 7.43 (see (7.24))

that μ

(Ω

) > 1 −

ψ(m

)+CT

for all q. Thus for every ε>0 there exists a p

large enough such that

(Ω

) > 1 −

(7.30)

for all q.

Employing the routine machinery of unit decomposition, one can easily

construct a sequence of smooth Itˆo equations (˜a

(t, m),

(t, m)) such that

(˜a

(t, m),

(t, m)) coincides with (ˆa

(t, m),A

(t, m)) for m ∈ W

, for all q,

and (˜a

(t, m),

(t, m)) equals zero outside some relatively compact neigh-

borhood V

of W

for all q.

Now choose an arbitrary complete Riemannian metric g(·, ·)onM and

by Nash’s Theorem 1.46 embed M isometrically into a Euclidean space R

with K large enough. Introduce an Itˆo vector ﬁeld (˘a

(t, m),

(t, m)) on

by setting

(t, m)=

(t, m) and in each chart U in V

by setting

˘a

(t, m)=˜a

(t, m)+Γ

˜α

where Γ

are Christoﬀel symbols of the second

kind of the Levi-Civit´a connection of the metric g(·, ·)inU,(˜α

)=(

)(

)

∗

Let N(M) be the normal bundle of M in R

with ﬁbers N

, m ∈ M.

Denote by Θ a relatively compact tubular neighborhood of M over V

(it exists since V

is relatively compact) and by r : Θ → V

the smooth

retraction of Θ onto V

along the ﬁbers of N(M).

Recall that Θ has the structure of a direct product

Θ = V

× O, (7.31)

where O is an open ball in R

K−n

that at any point m ∈ V

can be identiﬁed

with the normal space N

At any point (m, x) ∈ Θ the presentation (7.31) yields the presentation

of a tangent space to R

of the form T

(m,x)

= T

M × T

O. Introduce a

new Riemannian metric g

(·, ·)onΘ by transferring the Riemannian inner

product from T

M into the factor T

M in the above product by determining

the inner product in the factor T

O as the restriction of the Euclidean inner