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

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226 9 Mean Derivatives on Manifolds

Remark 9.2. Let f : M → M

be a smooth mapping of manifolds. Since

the value of a mean derivative depends on the “now” σ-algebra of the pro-

cess, the tangent mapping Tf sends mean derivatives of a process η(t)

to mean derivatives of the process ξ(t)=f(η(t)) only in the following

form: Tf(Dη(t)) = D

(ξ(t)) or Tf(D

η(t)) = Dξ(t) but generally speaking

Tf(Dη(t)) = Dξ(t). An analogous fact in true for backward mean deriva-

tives: Tf(D

∗

η(t)) = D

∗

(ξ(t)) or Tf(D

∗

η(t)) = D

∗

ξ(t) but generally speaking

Tf(D

∗

η(t)) = D

∗

ξ(t).

Lemma 9.3 Let ξ(t) be a solution of equation (7.18).ThenY

(t, m)=

a(t, m) and so Dξ(t)=a(t, ξ(t)).

Proof. Let m ∈ M and consider a normal chart U

(m)ofH at m.Inthis

chart the local connector of H at m is equal to zero, i.e., ξ(t) is described by

equation (7.19) with Γ

(A, A) = 0. Applying Lemma 8.26(i) and Deﬁnition

9.1 we then obtain Dξ(t)

= a(t, m). Since here both sides of the equation

are vectors, the equality remains true in all charts. Using these arguments

for all m we obtain the formula Dξ(t)=a(t, ξ(t)). 

Recall that an invariant equation, independent of a choice of connection on

M,isanItˆo equation (ˆa, A), a cross-section of an Itˆo bundle (see Section 7.2).

If a connection is speciﬁed on M, we can pass to the canonically corresponding

Itˆo vector ﬁeld (a, A) and obtain the Itˆo equation in Belopolskaya-Daletskii

form (7.18) whose solutions are solutions of (ˆa, A) and vice versa. We also

use a connection for determining the forward mean derivatives on M.

Lemma 9.4 For a solution ξ(t) of the Itˆo equation (ˆa, A) its forward mean

derivative Dξ(t) with respect to a connection H satisﬁes the equality

ξ(t)=ˆa(t, ξ(t)) +

trΓ

ξ(t)

(A(t, ξ(t)),A(t, ξ(t)) = H(A)

where A is the generator of the ﬂow of equation (ˆa, A), H : τM → TM is

the mapping generated by the connection H by formula (2.45) and Γ is the

local connector of H.

Proof. By formula (7.17), ˆa(t, m)=a(t, m) −

trΓ

ξ(t)

(A(t, ξ(t)),A(t, ξ(t)).

Thus the equality D

ξ(t)=ˆa(t, ξ(t)) +

trΓ

ξ(t)

(A(t, ξ(t)),A(t, ξ(t)) follows

from Lemma 9.3.Thefactthat

a(t, m)=ˆa(t, ξ(t)) +

trΓ

ξ(t)

(A(t, ξ(t)),A(t, ξ(t)) = H(A)

follows from Lemma 7.26. 

From Lemmas 9.3 and 9.4 it follows that if we apply the same connection

both for the transition from (ˆa, A)to(a, A) (and hence to equation (7.18))

and for determining the mean derivative, we obtain for a solution ξ(t)that

Dξ(t)=a(t, ξ(t)). Moreover, if we change the connection, the Itˆo vector ﬁeld

9.1 Forward and Backward Mean Derivatives 227

(a, A) canonically corresponding to (ˆa, A) and the forward mean derivative

Dξ(t) will be changed but the equality Dξ(t)=a(t, ξ(t)) for those new values

will remain true.

For the sake of simplicity, if ξ(t) is a solution of (7.18)werenamethe

vector Y

∗

(t, m)asa

∗

(t, m), thus D

∗

ξ(t)=a

∗

(t, ξ(t)).

Let H be a connection on M . Let (a, A)beanItˆo vector ﬁeld on M .

Denote by ∇A the covariant derivative of the ﬁeld A(t, m) with respect to the

connection H. ∇A is a ﬁeld of bilinear operators ∇A(t, m)(·, ·):T

M ×R

→

M. Consider the ﬁeld ∇A(t, m)(A·, ·):R

× R

→ T

M and the related

vector ﬁeld

tr ∇A(A)(t, m)=tr∇A(t, m)(A(t, m)(·), ·) (9.1)

(see the description of the trace in formula (7.9)). Determine on M the fol-

lowing equation

dξ(t) = exp

ξ(t)

(a(t, ξ(t))dt +tr∇A(A)(t, ξ(t))dt

−A(t, ξ(t))D

∗

w(t)dt + A(t, ξ(t))dw(t)), (9.2)

where (a(t, m)dt +tr∇A(A)(t, m)dt − A(t, m)D

∗

w(t)dt + A(t, m)dw(t)) de-

notes the class of stochastic processes in T

M consisting of the solutions of

all the equations of the form

X(t + r)=



t+r

˜a(s, X(s))ds +



t+r



(

A(s, X(s))ds

−



t+r

A(s, X(s))D

∗

(s)ds +



t+r

A(s, X(s))dw(s), (9.3)

here r>0, a(s, X)andA(s, X)) are analogous to those in Deﬁnition 7.27 with

the additional assumption that A(s, X) is smooth, and A



: T

M × R

→

M is the ordinary derivative of A in the vector space T

Let us represent (9.2) in local coordinates in the same manner as (7.18)was

represented in the form (7.19). Note the formula (exp



=exp



(·,

A(·))+

exp



(·, ·), (where the primes denote derivatives) and the equalities

exp



(0) = I and exp



(0)(·, ·)=−Γ

(·, ·) that follow from formula (7.20).

Thus exp

sends the vector tr



(

A(t, 0)), tangent to T

M,tothevector

exp



A(t, 0)(·), ·)=trA



(t, m)(A(·), ·)+trΓ

(A, A)=tr∇A(A)(t, m),

tangent to M (this clariﬁes the notation in (9.2)) and

A(t, 0)D

∗

w(t) turns

into A(t, m)D

∗

w(t). Summarizing the above formulae we obtain the presen-

tation for (9.2) in the local coordinates of a chart U

as follows:

dξ(t)=a(t, ξ(t))dt +tr∇A(A)(t, ξ(t))dt (9.4)

−A(t, ξ(t))D

∗

w(t)dt −

tr Γ

ξ(t)

(A, A)dt + A(t, ξ(t))dw(t).

Direct veriﬁcation shows that (9.4) transforms correctly (covariantly) under

changes of coordinates. This means that equation (9.2) is well-deﬁned.

228 9 Mean Derivatives on Manifolds

Theorem 9.5 Let ξ(t), ξ(0) = m

, be a strong solution of (9.4).Then

∗

ξ(t)=a(t, ξ(t)) for t ∈ (0,l].

Proof. For the sake of simplicity consider ξ(t) in the normal chart of the

initial point m

.Letξ(t) = exp

X(t)where

X(t)=



˜a(s, X(s))ds +



A(s, X(s)))ds

−



A(s, X(s)))D

∗

w(s)ds +



A(s, X(s))dw(s)

is a process in T

M which exists by construction. Since D

∗

ξ(t) is a vector,

it suﬃces to show that D

∗

X(t)=˜a(t, X(t)). The latter is a consequence

of (8.29). 

Deﬁnition 9.6. The backward stochastic diﬀerential

a(t, m)

∗

t + A(t, m)d

∗

(t)

of a process ξ(t) determined by the Itˆo vector ﬁeld (a

∗

,A) is a class of stochas-

tic processes in the tangent space T

M consisting of solutions of the (back-

ward) stochastic equations

X(t − r)=



t−r

˜a(s, X(s))ds +



t−r

A(s, X(s))d

∗

(s), (9.5)

where r>0, ˜a(t, m)and

A(t, m) are analogous to the corresponding terms

in Deﬁnition 7.27, with the additional assumption that

A(t, m)issmooth.

Deﬁnition 9.7. An Itˆo equation in backward diﬀerentials on M is an expres-

sion of the form

∗

ξ(t) = exp

ξ(t)

∗

(t, ξ(t))d

∗

t + A(t, ξ(t))d

∗

(t)). (9.6)

This means that for each t in the domain of ξ(t) the process ξ(t − r),

r>0, a.s. coincides with a process from the class exp

ξ(t)

∗

(t, ξ(t))d

∗

t +

A(t, ξ(t))d

∗

(t)) until ξ(t − r) leaves some neighborhood of ξ(t).

From formula (8.29) it follows that (9.5) is equivalent to the following

equation in T

M of type (9.3)

X(t − r)=



t−r

˜a(s, X(s))ds +



t−r



(

A(s, X(s))ds (9.7)

−



t−r

A(s, X(s)) ◦ D

∗

w(s)ds +



t−r

A(s, X(s))dw(s).

Let us describe (9.6) in the local coordinates of a chart U

.Todo

this consider (9.7) and then make a transition to the corresponding ex-

9.1 Forward and Backward Mean Derivatives 229

pression of type (9.4) in local coordinates. Then replace tr ∇A(A)(t, m)by

tr A



(t, m)(A(·), ·)+trΓ

(A, A) (see the derivation of equation (9.4)) and



t−r

tr A



(s, m)(A(·), ·)ds −



t−r

A(s, m) ◦ D

∗

w(s)ds +



t−r

A(s, m)dw(s)



t−r

A(s, m)d

∗

(s) (cf. formula (8.29)). So we obtain the formula

∗

ξ(t)=a

∗

(t, ξ(t))d

∗

t +

tr Γ

ξ(t)

(A, A)d

∗

t + A(t, ξ(t))d

∗

(t) (9.8)

which is the description of (9.6) in local coordinates that we have been search-

ing for.

It should be pointed out that equation (9.6) plays an auxiliary role (as

equation (8.30) does in linear spaces). However, it can be considered as an

invariant form of formula (9.8) that is convenient for applications.

Let a process ξ(t) be a solution of equation (7.18), t ∈ [0,T]. Fix t ∈ [0,T].

From the above arguments we obtain the following:

Theorem 9.8 The process η(t) satisfying the Itˆo equation in backward dif-

ferentials (9.6) with a

∗

(t, m)=a(t, m) − tr ∇A(A·, ·)+A(t, m)D

∗

w(t),such

that η(t)=ξ(t), has the same backward mean derivative at t as ξ(·).

Thus for small enough s<tsuch η(s) approximates ξ(s).

Remark 9.9. Note the description of mean derivatives for diﬀusion processes

on Riemannian manifolds given in [190]. There, the expressions in local coor-

dinates include Christoﬀel symbols (i.e., the local connectors) and lead to for-

mulas similar to (7.19)and(9.8). However that presentation is not connected

with stochastic diﬀerential equations. We have shown that Ito equations in

Belopolskaya-Daletskii form are naturally compatible with the machinery of

mean derivatives.

We introduce the notation ˆa

∗

(t, m)=a

∗

(t, ξ(t)) +

tr Γ

ξ(t)

(A, A). Taking

into account formula (2.19), as in the proof of Lemma 7.25 we obtain that

(X, Y )=−ϕ



αn

(X, Y )whereϕ

αn

is the change of coordinates from the

normal chart to another chart U

. Thus analogously to the proof of Lemma

7.25 one can easily see that under the change of coordinates ϕ

βα

between the

charts U

and U

thetriple(m, (ˆa

∗

,A)) transforms according formula (7.16)

in the form

, (ˆa

∗

)) →



βα





βα

ˆa

∗

−

tr ϕ



βα

),ϕ



βα

)



(9.9)

Note that (9.9) can be obtained from Lemma 8.33 by replacing f by ϕ

βα

Thus (ˆa

∗

,A)isabackwardItˆo equation according to Deﬁnition 7.23.

230 9 Mean Derivatives on Manifolds

Recall that the process ξ(t) is described by the Itˆo equation (ˆa, A) corre-

sponding to (7.18).

Deﬁnition 9.10. The Itˆo equation (ˆa, A) and the backward Itˆo equation

(ˆa

∗

,A) introduced above are said to be coupled to each other.

Denote by

∗

the backward generator of (ˆa

∗

,A) coupled with (ˆa, A)that

describes the process ξ(t). In local coordinates it is clearly expressed in the

form

∗

= −ˆa

∗

∂

∂q

(AA

∗

)

∂

∂q

. (9.10)

Lemma 9.11 D

∗

ξ(t)=−H(

∗

) where H is the mapping generated as in

formula (2.45) by the connection H, with respect to which the mean derivatives

are calculated.

Proof. Note that D

∗

ξ(t)=a

∗

(t, ξ(t)) and ˆa

∗

(t, ξ(t)) = a

∗

tr Γ

ξ(t)

(A, A)=

ˆa

∗

∂

∂q

+ Γ

(AA

∗

)

∂

∂q

. From formulae (2.45)and(9.10) we obtain that

∗

)=−ˆa

∗

∂

∂q

+ Γ

(AA

∗

)

∂

∂q

= −a

∗

∂

∂q

− Γ

(AA

∗

)

∂

∂q

+ Γ

(AA

∗

)

∂

∂q

= −a

∗

∂

∂q

. 2

Lemma 9.11 is “symmetric” to Lemma 9.4.

9.2 Current and Osmotic Velocities

Now consider the current velocity D

ξ(t)=

(a(t, ξ(t)) + a

∗

(t, ξ(t)) =

(t, ξ(t)) where v

(t, m)=

(a(t, m)+a

∗

(t, m)) is the regression.

Theorem 9.12 v

(t, m) is a vector tangent to M, independent of the choice

of connection, with respect to which the forward and backward mean deriva-

tives are calculated.

Proof. Indeed,

(t, m)=

(a(t, m)+a

∗

(t, m))



a(t, m) −

tr Γ

ξ(t)

(A, A)+

tr Γ

ξ(t)

(A, A)+a

∗

(t, m)



(ˆa(t, m)+ˆa

∗

(t, m)).

Under the change of coordinates ϕ

βα

between charts U

and (U)

the trans-

formation rule for ˆa(t, m) is described in formula (7.12) while ˆa

∗

(t, m)trans-

forms according to formula (7.16). Thus

9.2 Current and Osmotic Velocities 231

(t, m)

(ˆa(t, m)

+ˆa

∗

(t, m)

)





βα

ˆa(t, m)

tr ϕ



βα

)+ϕ



βα

ˆa

∗

(t, m)

−

trϕ



βα

)





βα

(ˆa(t, m)

+ˆa

∗

(t, m)

)=ϕ



βα

(t, m)

Hence under coordinate changes v

(t, m) transforms by formula (1.1)andso

it is a tangent vector, independent of the choice of connection. 

For current velocity (and osmotic velocity below) there is a simple gener-

alization of the presentation that we used in Section 8.1 for R

in the case of

a smooth non-degenerate diﬀusion term of a diﬀusion process.

Let A(m):R

→ T

M be smooth, autonomous and have rank equal

to dim M at every m ∈ M.Thenα(m)=A(m)A

∗

(m) is smooth, sym-

metric and non-degenerate. Denote the matrix of α by (α

). Its inverse

(α

) is smooth, symmetric and non-degenerate, hence it determines on M

a Riemannian metric, which we denote by α(·, ·). Denote by ρ

(t, x)the

probability density of ξ(t) with respect to the volume form dt ∧ Λ



det(α

)dt ∧ dx

∧ dx

∧···∧dx

(see Section 8.1).

Theorem 9.13 For v

(t, m) and ρ

(t, m) the following generalization of for-

mula (8.19) (equation of continuity) holds

∂ρ

(t, x)

∂t

= −Div(v

(t, x)ρ

(t, x)), (9.11)

where Div denotes divergence with respect to the Riemannian metric α(·, ·).

Theorem 9.13 generalizes formulae (8.19)and(8.21). The proof of Theo-

rem 9.13 is a modiﬁcation of that for Lemma 8.18 (cf. the proof in [190]).

Consider also the osmotic velocity D

ξ(t)=u

(t)=

(a(t, ξ(t)) −

∗

(t, ξ(t)) whose regression u

(t, m) takes the form u

(t, m)=

(a(t, m) −

∗

(t, m)). In the case under consideration the following generalization of for-

mulae (8.18)and(8.20) holds:

Theorem 9.14 For u

(t, m) the equality

(t, x)=

Grad log ρ

(t, x) = Grad log



(t, x) (9.12)

is valid where Grad denotes the gradient with respect to the Riemannian

metric α(·, ·).

The proof of Theorem 9.14 is a modiﬁcation of that for Lemma 8.17 (cf.

the proof in [190]).

232 9 Mean Derivatives on Manifolds

9.3 Mean Derivatives of Vector Fields Along Stochastic

Processes

The construction of mean derivatives for vector ﬁelds along a stochastic pro-

cess needs a modiﬁcation that is typical in the transition from vector spaces

to manifolds.

Let Y (t, m) be a vector ﬁeld on M. Consider the invariant mean derivatives

DY (t, ξ(t)) = lim

Δt→+0



Y (t + Δt, ξ(t + Δt)) − Y (t, ξ(t))

Δt



∗

Y (t, ξ(t)) = lim

Δt→+0



Y (t, ξ(t)) − Y (t − Δt, ξ(t −Δt))

Δt



, (9.13)

taking values in TTM. Specify a connection H and denote by K : TTM →

TM its connector (see Deﬁnition 2.13). Introduce the covariant mean deriva-

tives of Y (t, m) along ξ(t) by analogy with the ordinary covariant derivatives

via the formulae

DY (t, ξ(t)) = K ◦ lim

Δt→+0



Y (t + Δt, ξ(t + Δt)) − Y (t, ξ(t))

Δt



= K ◦DY (t, ξ(t)),

∗

Y (t, ξ(t)) = K ◦ lim

Δt→+0



Y (t, ξ(t)) − Y (t − Δt, ξ(t −Δt))

Δt



= K ◦D

∗

Y (t, ξ(t)). (9.14)

For an Itˆo process ξ(t)onM (see Deﬁnition 7.82) denote by Γ

s,t

the

operator of parallel translation along ξ(·)fromξ(t)toξ(s) (see the deﬁnition

and notation in Section 7.7.1). Let Y (t, m)beaC

-smooth vector ﬁeld on

M. It is easy to see that the covariant mean derivatives DY (t, ξ(t)) and

∗

Y (t, ξ(t)) deﬁned by formulae (9.14) can be equivalently described in this

case by the formulae

DY (t, ξ(t)) = lim

Δt→+0



t,t+Δt

Y (t + Δt, ξ(t + Δt)) − Y (t, ξ(t))

Δt



∗

Y (t, ξ(t)) = lim

Δt→+0



Y (t, ξ(t) − Γ

t,t−Δt

Y (t − Δt, ξ(t − Δt))

Δt



.(9.15)

Recall that the vector ﬁeld Y can be considered as a mapping Y : M →

TM with the additional condition πY =idwhereπ : TM → M is the natural

projection and id is the identity mapping (see Deﬁnition 1.4). In particular

the tangent mapping TY =(Y,dY ) sends TM to TTM.Letm ∈ M.The

restriction of the diﬀerential dY at T

M is the derivative (a linear operator)



: T

M → T

(m,Y (m))

TM. Denote by Y



the second derivative (a bilinear

mapping) that sends T

M × T

M to T

(m,Y (m))

TM.

9.3 Mean Derivatives of Vector Fields Along Stochastic Processes 233

Let ξ(t) be a process on M that is given by an Itˆo equation (ˆa, A)where

A(m) is autonomous, smooth and has maximal rank at every m as a mapping

A(m):R

→ R

for some k ≥ n. Then, as above, α(m)=A(m)A

∗

(m) has in

local coordinates a symmetric, smooth and non-degenerate matrix (α

)and

its inverse (α

) deﬁnes on M the Riemannian metric α(·, ·). Below in this

construction we deal with the Levi-Civit´a connection H of this Riemannian

metric. In particular, in formulae (9.14) K is the connector of this connection,

and all covariant derivatives, the Laplace-Beltrami operator and all forward

and backward mean derivatives of ξ(t) are calculated with respect to this

connection.

Let ξ(t) have mean derivatives Dξ(t)=a(t, ξ(t)) and D

∗

ξ(t)=a

∗

(t, ξ(t)),

i.e., the vector ﬁelds a(t, m)anda

∗

(t, m) are the regressions of Dξ(t)and

∗

ξ(t), respectively. Then taking into account the forward and backward Itˆo

formulae (6.10)and(6.13) as well as the construction of mean derivatives one

can easily see that DY (t, ξ(t)) is a vector in T

(ξ(t),Y (t,ξ(t))

TM of the form

∂Y

∂t

+ Y



(a(t, ξ(t)) +

trY



(I,I)

and D

∗

Y (t, ξ(t)) is a vector in the same space of the form

∂Y

∂t

+ Y



∗

(t, ξ(t)) −

trY



(I,I).

By the Deﬁnition 2.22 of the covariant derivative K(Y



(a(t, m)) = ∇

a(t,m)

and K(Y



∗

(t, m))) = ∇

∗

(t,m)

Y . One can also easily derive that

trY



(I,I)=

∇

Y where ∇

is the Laplace-Beltrami operator. Thus

from the above formulae we obtain the following description of the regressions

DY and D

∗

Y of the covariant derivatives (9.14):

DY =

∂Y

∂t

+ ∇

Y +

∇

Y, (9.16)

∗

Y =

∂Y

∂t

+ ∇

∗

Y −

∇

Formulae (9.16)and(9.17) are natural analogs of (8.24)and(8.25), respec-

tively.

We now turn to the case where we have to use a Riemannian metric spec-

iﬁed apriorion M (i.e., not the metric generated by the diﬀusion coeﬃcient

of an equation). In this case we ﬁnd formulae for covariant mean derivatives

along an Itˆo process, using a modiﬁcation of the construction of covariant

derivatives based on parallel translation. We use Itˆo processes and parallel

translation with respect to the Levi-Civit´a connection of the above metric.

Let ξ(t)beanItˆo development of the Itˆo process ζ(t) given in a certain

tangent space by the formula ζ(t)=



a(s)ds + σw(t)wherea(t) satisﬁes

(7.54). Let Dξ(t)=a(t, ξ(t)) and D

∗

ξ(t)=a

∗

(t, ξ(t)). Applying the Itˆo

234 9 Mean Derivatives on Manifolds

formulae together with formulae (9.15), we obtain the following modiﬁcations

of formulae (9.16)and(9.17):

DY =

∂Y

∂t

+ ∇

Y +

∇

Y, (9.17)

∗

Y =

∂Y

∂t

+ ∇

∗

Y −

∇

9.4 The Quadratic Mean Derivative

For a stochastic process ξ(t) on a probability space (Ω, F,P) with values

in a manifold M we introduce its quadratic derivative as follows (cf. Deﬁni-

tion 8.10 for the case of linear spaces). Take any chart U and consider in it

the L

random variable determined by the rule

ξ(t) = lim

t→+0



(ξ(t + t) − ξ(t)) ⊗ (ξ(t + t) −ξ(t))

t



, (9.18)

where the limit is assumed to exist in L

(Ω,F,P).

Deﬁnition 9.15. D

ξ(t) is called the quadratic mean derivative of the pro-

cess ξ(t)onM at time t.

Notice that for D

ξ(t) there exists a regression in any chart, i.e., a mea-

surable ﬁeld α

(t, m) such that D

ξ(t)=α

(t, ξ(t)).

An important geometric feature of the quadratic mean derivative is that

(like current velocity) it is independent of the choice of connection; its re-

gression is a (2, 0)-tensor ﬁeld.

Let ξ(t) be given by an Itˆo equation (ˆa, A) (see Section 7.3), i.e., in par-

ticular, it is Markovian. Recall that here A(t, m) is a ﬁeld of linear operators

A(t, m):R

→ T

M with k suﬃciently large.

Lemma 9.16 Suppose that ξ(t) is given by an Itˆo equation (ˆa, A).Then

ξ(t)=A(t, ξ(t))A

∗

(t, ξ(t)) = 2(QA)(t, ξ(t)) (9.19)

where A

∗

is the conjugate operator, A is the corresponding generator and Q

is deﬁned by formula (2.44). A(t, m)A

∗

(t, m) is a (2, 0)-tensor ﬁeld on M.In

particular, D

ξ(t)=0if and only if ξ(t) has C

-smooth sample paths.

Proof. Let H be a connection and represent (ˆa, A)asanItˆo equation in

Belopolskaya-Daletskii form in terms of the Itˆo vector ﬁeld (a, A) canonically

corresponding to (ˆa, A) with respect to H (see Section 7.3). Then in a local

chart ξ(t) is expressed in Baxendale form

9.4 The Quadratic Mean Derivative 235

ξ(t + t) − ξ(t)=



t+t

a(s, ξ(s))ds

−



t+t

trΓ

ξ(t)

(A(t, ξ(t)),A(t, ξ(t)))ds



t+t

A(s, ξ(s))dw(s)

where Γ

(·, ·) is the local connector of H in that chart. As in the proof

of Theorem 8.12, by direct calculation it follows that the components of

(ξ(t + t) − ξ(t)) ⊗ (ξ(t + t) − ξ(t)) are elements of the matrix (ξ(t +

t) − ξ(t))(ξ(t + t) −ξ(t))

∗

where we use the matrix multiplication of the

column-vector (ξ(t + t) −ξ(t)) and the row-vector (ξ(t + t) −ξ(t))

∗

(i.e.,

the transpose of (ξ(t + t) − ξ(t))). In particular, this matrix product is a

symmetric positive semi-deﬁnite matrix.

Taking into account the properties of the Lebesgue and Itˆo integrals one

can see that (ξ(t + t) −ξ(t))(ξ(t + t) −ξ(t))

∗

is approximated by the sum

a(t, ξ(t))(a(t, ξ(t)))

∗

(Δt)

−

a(t, ξ(t)) (trΓ (A(t, ξ(t)),A(t, ξ(t))))

∗

(Δt)

+ a(t, ξ(t))Δt)(A(t, ξ(t))Δw(t))

∗

−

trΓ (A(t, ξ(t)),A(t, ξ(t)))(a(t, ξ(t)))

∗

(Δt)

trΓ (A(t, ξ(t)),A(t, ξ(t)))(trΓ (A(t, ξ(t)),A(t, ξ(t))))

∗

(Δt)

+(A(t, ξ(t))Δw(t))(a(t, ξ(t))Δt)

∗

+ A(t, ξ(t))(A(t, ξ(t)))

∗

Δt.

In this expression only the last summand is an inﬁnitesimal of the same

order as Δt while the others are inﬁnitesimals of higher order. Then applying

formula (9.18) we obtain that D

ξ(t)=E

(A(t, ξ(t))A

∗

(t, ξ(t))). Thus (9.19)

follows and it is obvious that this expression is independent of H. Recall (see

Lemma 7.2)thatAA

∗

is a (2, 0)-tensor ﬁeld.

The fact that D

ξ(t)=2QA now follows from the deﬁnitions.

From (9.19) it evidently follows that the equality D

ξ(t)=0meansthat

A(t, m) = 0 and so the sample paths of ξ(t)areC

-smooth. If they are C

smooth, from the deﬁnition of quadratic mean derivative (see Theorem 8.12)

we have D

ξ(t) = 0 (as always for C

-curves). 

Recall that A(t, m)A

∗

(t, m) is the diﬀusion coeﬃcient of ξ(t). Note also

that if AA

∗

is non-degenerate, smooth and autonomous, it can be considered

as a metric (2, 0)-tensor so that its inverse is a Riemannian metric on M.In

particular, this metric can be used to determine forward and backward mean

derivatives of ξ(t). These derivatives turn out to have many uses.