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

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216 8 Mean Derivatives in Linear Spaces

From (8.50), (8.51), (8.52)and(8.53) we obtain that

0 = lim

k→∞





x(t + Δt) − x(t) −



t+Δt

(τ,x(·))dτ



(x(·))dμ

= lim

k→∞





x(t + Δt) − x(t) −



t+Δt

a(τ,x(·))dτ



(x(·))dμ



(x(t + Δt) − x(t)) −



t+Δt

a(τ,x(·))dτ

(x(·))dμ.

Thus



(x(t + Δt) − x(t)) −



t+Δt

a(s, x(·))ds

(x(·))dμ =0. (8.54)

Since (8.54) is valid for every g

, we have proved the following:

Lemma 8.47 The process ξ(t) −



a(s, ξ(s))ds is a martingale with respect

Deﬁne the measures ρ

on (

Ω,

F) by the relations dρ

=(1+ x(·)

)dμ

By Lemma 6.29 these measures weakly converge to the measure ρ deﬁned by

the relation dρ =(1+x(·)

)dμ.

Using an elementary modiﬁcation of the above arguments (in particular,

replacing the measures ν

by ρ

, a

by α

,1+x and 1 + x

, etc.) one

can easily show that for every bounded continuous function g

: Ω → R that

is measurable with respect to

, the relation

lim

i→∞



(x(t + Δt) − x(t))(x(t + Δt) − x(t))

∗

−



t+Δt

(s, x(·))ds

(x(·))dμ



(x(t + Δt) − x(t))(x(t + Δt) − x(t))

∗

−



t+Δt

α(s, x(·))ds

(x(·))dμ

holds. Besides, for every i

8.4 First Order Diﬀerential Equations and Inclusions with Mean Derivatives 217





(x(t + Δt) − x(t))(x(t + Δt) − x(t))

∗

−



t+Δt

(s, x(·))ds



(x(·))dμ

and so





(x(t + Δt) − x(t))(x(t + Δt) − x(t))

∗

−



t+Δt

α(s, x(·))ds



(x(·))dμ =0.

From this we obtain:

Lemma 8.48 For the coordinate process ξ(t) on the probability space

(

Ω,

F,μ) the process ξ(t)ξ

∗

(t) −



α(s, ξ(·))ds is a martingale with respect

From Lemmas 8.5 and 8.47 it immediately follows that



ξ(t) −



a(τ,ξ(τ))dτ



and so Dξ(t)=a(t, ξ(t)) ∈ a(t, ξ(t)).

Note that from the Deﬁnition 8.10 of the quadratic derivative it follows

that D

(ξ(t)ξ

∗

(t)) = D

ξ(t). Then from Lemma 8.48 we obtain that D

ξ(t)=

α(t, ξ(t)) ∈ α(t, ξ(t)).

Thus ξ(t) satisﬁes (8.45). From Lemma 8.47 it follows that ξ(t)isasemi-

martingale. 

Theorem 8.49 Suppose that α(t, x) takes values in the space

(n) of

positive semi-deﬁnite symmetric matrices, has closed convex values, is lower

semicontinuous and for each α ∈ α(t, x) the following estimate

tr α(t, x) <K(1 + x)

(8.55)

holds for some K>0.Letalsoa(t, x) be a Borel measurable set-valued

mapping and satisfy the estimate

a(t, x) <K(1 + x) (8.56)

for some K>0. Then for any initial condition ξ(0) = ξ

there exists a weak

solution of (8.45) that is well-deﬁned on the entire interval t ∈ [0,T].

Proof. From Michael’s theorem it follows that under the conditions of Theo-

rem 8.49 the set-valued mapping α(t, x) has a single-valued continuous selec-

218 8 Mean Derivatives in Linear Spaces

tor α(t, x). Obviously α(t, x) belongs to

(n) for all t, x. The Borel measur-

able set-valued mapping a(t, x) has a Borel measurable single-valued selector

a(t, x).

Let ε

→ 0 be a positive sequence. Deﬁne α

(t, x)=α(t, x)+ε

I where

I is the unit n × n matrix. Clearly the α

are strictly positive deﬁnite and

continuous. Then by Lemma 8.40 there exists a continuous A

(t, x) such that

(t, x)A

∗

(t, x)=α

(t, x). Recall that tr α

(t, x) is equal to the sum of the

squares of the elements of A

(t, x), i.e., it is the square of the Euclidean norm

in L(R

, R

). Since in the ﬁnite-dimensional linear space S(n) all norms are

equivalent, from (8.55) it immediately follows that A(t, x) <K(1+x)for

some K>0. As α

(t, x) is positive deﬁnite, the matrix A

(t, x) is invertible

for all t, x.Sincea(t, x) is measurable and satisﬁes (8.55), under the above-

mentioned properties of A

(t, x)by[83, Theorem III.3.3] there exists a weak

solution of the stochastic diﬀerential equation

(t)=ξ



a(s, ξ

(s))ds +



(s, ξ

(s))dw(s), (8.57)

well-deﬁned on the entire interval t ∈ [0,T]. Denote this solution by ξ

(t).

(t) determines a measure μ

on (

Ω,

F)where(

Ω,

F) was introduced in the

proof of Theorem 8.46.

The rest of the proof is analogous to that of Theorem 7.51. All equa-

tions (8.57) satisfy the hypothesis of Lemma 6.27. The set of measures

{μ

} is weakly compact so that there exists a subsequence that weakly con-

verges to some measure μ. Denote by ξ(t) the coordinate process on the

probability space (

Ω,

F,μ). Construct A(t, x(·)) by analogy with Theorem

7.51, i.e., as a weak limit in the corresponding L

space of the bounded

(and so weakly compact) set A

. The process ξ(t) satisﬁes the equality

ξ(t)=ξ



a(s, ξ(s))ds +



A(s, ξ(·))dw where w(t) is some Wiener pro-

cess. Since by construction α

converges to α uniformly, one can easily show

that E

(AA

∗

)=α.ByTheorem8.7 and Theorem 8.12, this means that ξ(t)

is the weak solution of (8.45) that we are looking for. 

Equations and inclusions with backward mean derivatives arise in the de-

scription of some special stochastic processes of mathematical physics. For

example (see, e.g., [113, 106, 115]) a second order equation in backward mean

derivatives of the group of Sobolev diﬀeomorphisms may be derived that de-

scribes a process whose expectation is a ﬂow of a viscous incompressible ﬂuid.

It should be pointed out that the study of such equations and inclusions is

generally much more complicated than that of equations and inclusions with

forward mean derivatives. Nevertheless there exists a simple method which

uses the inverse time direction to solve equations and inclusions with forward

mean derivatives, allowing one to obtain results for the case of backward

mean derivatives. We refer the reader to [7] for some statements of this sort.

As mentioned in Section 8.1, the notion of current velocity is analogous to

that of ordinary velocity for a non-random process. Thus, from the physical

8.4 First Order Diﬀerential Equations and Inclusions with Mean Derivatives 219

point of view, it is most natural to study equations and inclusions with current

velocities.

The system



ξ(t)=v(t, ξ(t))

ξ(t)=α(t, ξ(t))

(8.58)

is called the ﬁrst order diﬀerential equation with current velocities.

Theorem 8.50 Let v :[0,T] × R

→ R

be smooth and α : R

→ S

(n) be

smooth and autonomous (so it determines the Riemannian metric α(·, ·) on

, introduced in Section 8.1). Suppose in addition that v and α satisfy the

estimates

v(t, x) <K(1 + x), (8.59)

tr α(x) <K(1 + x

) (8.60)

for some K>0.Letξ

be a random element with values in R

whose prob-

ability density ρ

with respect to the volume form Λ

of α(·, ·) on R

(see

Section 8.1) is smooth and nowhere equal to zero. Then for the initial condi-

tion ξ(0) = ξ

equation (8.58) has a weak solution that is well-deﬁned on the

entire interval t ∈ [0,T].

Proof. Since v(t, x) is smooth and the estimate (8.59) is fulﬁlled, its ﬂow g

is well-deﬁned on the entire interval t ∈ [0,T]. By g

(x) we denote the orbit

of the ﬂow (i.e., the solution of the equation x



(t)=v(t, x)) with the initial

condition g

(x)=x.Sincev(t, x) is smooth, its ﬂow is also smooth.

The continuity equation (8.21) can clearly be transformed into the form

∂ρ

∂t

= −α(v, Grad ρ) −ρ Div v. (8.61)

Suppose that ρ(t, x) is nowhere zero in [0,T]×R

. Then we can divide (8.61)

by ρ so that it is transformed into the equation

∂p

∂t

= −α(v, Grad p) −Div v (8.62)

where p = log ρ.Letp

= log ρ

We show that the solution of (8.62) with initial condition p(0) = p

described by the formula p(t, x)=p

−t

(x)) −



(Div v)(s, g

−t

(x))ds.

Consider the function p

as given on the level surface (0, R

) of the product

[0,T] ×R

. Consider the vector ﬁeld (1,v(t, x)) on [0,T] ×R

. The orbits of

its ﬂow ˆg

, starting at the points of (0, R

), have the form ˆg

(0,x)=(t, g

(x))

and, like g

, the ﬂow ˆg

is smooth. Introduce on [0,T] ×R

the Riemannian

metric ˆα(·, ·) by the formula ˆα((X

), (X

)) = X

+α(Y

). Notice

that for any (t, x) the point ˆg

−t

(t, x) belongs to (0, R

) where the function

is given. Thus on the one hand (1,v)p(t, x), the derivative of p(t, x)inthe

direction of (1,v), by construction equals −Div v(t, x), and on the other hand

220 8 Mean Derivatives in Linear Spaces

one can easily calculate that (1,v)p(t, x)=

∂

∂t

p(t, x)+α(v(t, x), Grad p(t, x)).

Thus (8.62) is satisﬁed.

Note that ρ =e

is indeed nowhere zero, i.e., our arguments are well-

deﬁned.

Since ρ(t, x) is well-deﬁned for all t ∈ [0,T], it determines a process ξ(t)

with this probability density and so with initial density ρ

. By construction

ξ(t)=v(t, ξ(t)).

Let u =

Grad p = Grad log

√

ρ and a(t, x)=v(t, x)+u(t, x).

From Lemma 8.40 and from the hypothesis of the Theorem it follows that

there exists a smooth A(x) such that A(x)A

∗

(x)=α(x) and the relation

A(x) <K(1 + x) holds. Then ξ(t) satisﬁes the stochastic diﬀerential

equation

ξ(t)=ξ



a(s, ξ(s))ds +



A(ξ(s))dw(s) (8.63)

and so by Theorem 8.12 D

ξ(t)=α(ξ(t)). 

Lemma 8.51 Let α(x), ρ(t, x) and Λ

be as in Theorem 8.50 and let the

vector ﬁeld v from Theorem 8.50 be autonomous. Then the ﬂow ˆg

of the

vector ﬁeld (1,v(x)) on [0,T] × R

preserves the volume form ρ(t, x)dt ∧ Λ

(i.e., ˆg

∗

(ρ(t, x)dt ∧Λ

)=ρ

(x)dt ∧ Λ

where ˆg

∗

is the pull-back) and so for

any measurable set Q ⊂ R

and for any t ∈ [0,T] the relation



(x)Λ



(Q)

ρ(t, x)Λ

holds.

Proof. It is enough to show that L

(1,v)

(ρ(t, x)dt ∧Λ

)=0whereL

(1,v)

is the

Lie derivative along (1,v). Clearly L

(1,v)

(ρ(t, x)dt ∧Λ

)=(L

(1,v)

ρ(t, x))dt ∧

+ ρ(t, x)(L

(1,v)

dt ∧ Λ

). For a function the Lie derivative coincides with

the derivative in the direction of the vector ﬁeld, hence L

(1,v)

ρ(t, x)=

∂ρ

∂t

α(v, Grad ρ) (see the proof of Theorem 8.50)andso(L

(1,v)

ρ(t, x))dt ∧ Λ

(

∂ρ

∂t

+ α(v, Grad ρ))dt ∧ Λ

. Since neither the form Λ

nor the vector ﬁeld

v(x) depend on t, L

(1,v)

dt ∧ Λ

=dt ∧ (L

)=Divv (dt ∧ Λ

)astheLie

derivative along v of the volume form Λ

equals (Div v)Λ

(see Section 1.7).

Taking into account (8.61), we obtain L

(1,v)

(ρ(t, x)dt ∧ Λ

)=0. 

We refer the reader to [7] where diﬀerential inclusions with current veloc-

ities are considered and a certain existence of solution result is obtained.

8.5 The Case of P-mean Derivatives

In what follows, for the sake of simplicity of presentation, we deal with pro-

cesses given on a ﬁnite time interval t ∈ [0,T] ⊂ R.

As in the previous section, we use the measure space (

Ω,

F) and the family

introduced in Section 6.1.1.

Let a :[0,T] ×

Ω → R

and α :[0,T] ×

Ω →

(n) be measurable.

8.5 The Case of P-mean Derivatives 221

Deﬁnition 8.52. An equation with P-mean derivatives is a system of the

form



ξ(t)=a(t, ξ(·)),

ξ(t)=α(t, ξ(·)).

(8.64)

Deﬁnition 8.53. We say that the equation (8.64)hasaweak solution ξ(t)if

there exists a probability space (Ω,F, P) and a stochastic process ξ(t), given

on (Ω,

F, P) and taking values in R

, such that equation (8.64) is fulﬁlled

P-a.s.

For simplicity we deal with deterministic initial conditions only.

Lemma 8.54 For a continuous (measurable, C

-smooth, k ≥ 1) mapping

α :[0,T] ×

Ω → S

(n) satisfying Condition 4.12, there exists a continuous

(measurable, C

-smooth, respectively) mapping A :[0,T] ×

Ω → L(R

, R

)

that satisﬁes Condition 4.12 and such that α(t, x(·)) = A(t, x(·))A

∗

(t, x(·))

for each (t, x(·)) ∈ R ×

Ω.

The proof of Lemma 8.54 is a simple modiﬁcation of that for Lemma 8.40.

The fact that ζ(t, x(·)),

√

δ(t, x(·)), and hence A(t, x(·)), satisﬁes Condition

4.12 follows from the construction.

Theorem 8.55 Let a :[0,T] ×

Ω → R

and α :[0,T] ×

Ω → S

(n) be

jointly continuous in t, x(·) and satisfy Condition 4.12. Let also the following

estimates hold:

tr α(t, x(·)) <K

(1 + x(·))

, (8.65)

a(t, x(·)) <K

(1 + x(·)). (8.66)

Then for every initial condition ξ

∈ R

equation (8.64) has a weak solution

that is well-deﬁned on the entire interval [0,T].

Proof. Note that α(t, x(·)) satisﬁes the hypothesis of Lemma 8.54 and so there

exists a continuous A(t, x(·)) such that A(t, x(·))A

∗

(t, x(·)) = α(t, x(·)) and

A(t, x(·)) satisﬁes Condition 4.12. Immediately from the deﬁnition of trace in

this case it follows that tr α(t, x(·)) equals the sum of the squares of all the

elements of the matrix A(t, x(·)), i.e., it is the square of the Euclidean norm

in L(R

, R

). Since in the ﬁnite dimensional vector space all norms are equiv-

alent, from estimate (8.65) it follows that A(t, x(·)) <K

(1 + x(·))for

some K

> 0. Recall that a(t, x(·)) is continuous and satisﬁes Condition 4.12

and the estimate (8.66). Under all these conditions, by [83, Theorem III.2.4]

there exists a weak solution ξ(t) of the diﬀusion type stochastic diﬀerential

equation

ξ(t)=ξ



a(s, ξ(·))ds +



A(s, ξ(·))dw(s),

that is a diﬀusion type process, well-deﬁned on the entire interval [0,T]. From

Theorems 8.8 and 8.13 it follows that ξ(t) a.s. satisﬁes (8.64). 

222 8 Mean Derivatives in Linear Spaces

A more general existence result where α(t, x(·)) may take values in

(n)

is obtained in Theorem 8.59 below.

Consider set-valued mappings a(t, x)andα(t, x) sending [0,T] ×

Ω to

and S

(n), respectively, and satisfying Condition 4.12.Thediﬀerential

inclusion with forward P-mean derivatives is a system of the form



ξ(t) ∈ a(t, ξ(·)),

ξ(t) ∈ α(t, ξ(·)).

(8.67)

Deﬁnition 8.56. We say that the inclusion (8.67)hasaweak solution with

initial condition ξ

∈ R

if there exists a probability space and a stochastic

process ξ(t) given on it, taking values in R

, such that ξ(0) = ξ

and a.s. ξ(t)

satisﬁes inclusion (8.67).

As in the case of equations with P-mean derivatives, we deal with deter-

ministic initial conditions only.

If, say, a(t, x)andα(t, x) are lower semi-continuous and have closed convex

values, then by Michael’s theorem they have continuous selectors a(t, x(·))

and α(t, x(·)), respectively. If those selectors satisfy the conditions of Theorem

8.55, the weak solution of (8.64) with coeﬃcients a(t, x(·)) and α(t, x(·)), that

exists by Theorem 8.55, is obviously a weak solution of (8.67).

Theorem 8.57 Let α(t, x) be an upper semi-continuous set-valued mapping

from [0,T] ×

Ω to

(n) with closed convex values that satisﬁes Condition

4.12 and let for every α(t, x(·)) ∈ α(t, x(·)) the estimate

tr α(t, x(·)) <K

(1 + x(·))

(8.68)

hold for some K

> 0.

Let a(t, x(·)) be an upper semi-continuous set-valued mapping from [0,T]×

Ω to R

with closed convex values that satisﬁes Condition 4.12 and let the

estimate

a(t, x(·)) <K

(1 + x(·)) (8.69)

hold for some K

> 0.

Then for any initial condition ξ(0) = ξ

∈ R

inclusion (8.67) has a weak

solution.

Proof. Choose a sequence of positive numbers ε

→ 0. The set-valued map-

ping a(t, x(·)) satisﬁes the conditions of Lemma 4.14 and so there exists a

sequence of continuous single-valued mappings a

:[0,T] ×

Ω → R

that

point-wise converges to a measurable selector a(t, x(·)) of a(t, x(·)) and every

(t, x(·)) satisﬁes both Condition 4.12 and the estimate

a

(t, x(·)) <K

(1 + x(·)). (8.70)

The mapping α(t, x(·)) that takes values in the closed convex set

(n)

in the space of all symmetric n × n matrices also satisﬁes the conditions

8.5 The Case of P-mean Derivatives 223

of Lemma 4.14 and so there exists a sequence of continuous single-valued

mappings ˜α

:[0,T] ×

Ω →

(n) that point-wise converges to a measurable

selector α(t, x(·)) of α(t, x(·)) and every ˜α

(t, x(·)) satisﬁes both Condition

4.12 and the estimate

tr ˜α

(t, x(·)) <K

(1 + x(·))

. (8.71)

Deﬁne another sequence α

(t, x(·)) = ˜α

(t, x(·)) + ε

I where I is the unit

matrix, which evidently also point-wise converges to α(t, x(·)). All mappings

(t, x(·)) are continuous, satisfy Condition 4.12 and estimate (8.71)–at

least for k large enough – and in addition they all take values in the convex

open set S

(n) of positive deﬁnite symmetric matrices. Thus by Lemma 8.54

for every α

(t, x(·)) there exist continuous A

:[0,T] ×

Ω :→ L(R

, R

)such

that α

(t, x(·)) = A

(t, x(·))A

∗

(t, x(·)) and all A

(t, x(·)) satisfy Condition

4.12.

As in Theorem 8.42, immediately from the deﬁnition of trace in this case it

follows that tr α

(t, x(·)) equals the sum of the squares of all elements of the

matrix A

(t, x(·)), i.e., it is the square of the Euclidean norm of A

(t, x(·))

in L(R

, R

). Hence, (8.71)meansthatA

(t, x(·)) <K

(1 + x(·))and

so from the latter estimate and (8.70) it follows that for each k the pair

(t, x(·)),A

(t, x(·))) satisﬁes the Itˆo condition (6.22) with K>0the

same for all k.

Consider the sequence of diﬀusion type Itˆo stochastic diﬀerential equations

(t)=ξ



(s, ξ

(·))ds +



(s, ξ

(·))dw(s). (8.72)

Since their coeﬃcients are continuous and satisfy Condition 4.12 and esti-

mate (6.22) with the same K,byTheorem6.26 they all have weak solutions

(t), well-deﬁned on the entire interval [0,T], and by Lemma 6.28 the set

of measures μ

generated by ξ

(t)on(

Ω,

F) is weakly compact. Hence we

can choose a subsequence (we retain the notation μ

for this subsequence)

that weakly converges to a probability measure μ. Denote by ξ(t) the coor-

dinate process on the probability space (

Ω,

F,μ). Note that P

is the “past”

σ-algebra of ξ(t).

The fact that ξ(t)−



a(s, ξ(·))ds is a martingale on (

Ω,

F,μ) with respect

and hence



[ξ(t + Δt) − ξ(t)] −



t+Δt

a(s, ξ(·))ds





= 0 (8.73)

is proved by analogy with Lemma 8.47.From(8.73) it follows that

ξ(t)=a(t, ξ(·)) ⊂ a(t, ξ(·)). (8.74)

Now turn to A

(t, x(·)). Recall that α

(t, x(·)) = A

(t, x(·))A

∗

(t, x(·))

point-wise converges to α(t, x(·)), a measurable selector of α(t, x(·)).

224 8 Mean Derivatives in Linear Spaces

The fact that the process [ ξ(t) ⊗ ξ(t)]−



α(s, ξ(·))ds is a martingale on

(

Ω,

F,μ) with respect to

and hence



[(ξ(t + Δt) − ξ(t)) ⊗ (ξ(t + Δt) −ξ(t))] −



t+Δt

α(s, ξ(·)))ds





=0,

from which it follows that

ξ(t)=α(t, ξ(·)) ⊂ α(t, ξ(·)), (8.75)

is proved by analogy with Lemma 8.48.

Relations (8.74)and(8.75) imply that ξ(t) is the solution of (8.67)that

we are looking for. 

Remark 8.58. From (8.73) it follows that the solution ξ(t)of(8.67)ob-

tained in Theorem 8.57 is a semi-martingale with respect to

since ξ(t) −



a(s, ξ(·))ds is a martingale with respect to

Theorem 8.59 Let a(t, x(·)) and α(t, x(·)) be as in Theorem 8.55 with the

exception that α sends [0,T] ×

Ω to

(n) instead of S

(n). Then for every

initial condition ξ

∈ R

equation (8.64) has a weak solution that is well-

deﬁned on the entire interval [0,T].

Indeed, we can construct a sequence of continuous single-valued mappings

= α + ε

I :[0,T] ×

Ω → S

(n) satisfying Condition 4.12 that converge

to α. The proof of Theorem 8.59 then follows the same argument as that of

Theorem 8.57.

Chapter 9

Mean Derivatives on Manifolds

9.1 Forward and Backward Mean Derivatives

Let a connection H be given on a manifold M .Letξ(t) be a stochastic process

on M . According to formulae (8.1)and(8.2) we can introduce mean forward

and mean backward derivatives of ξ(t), if they exist, in any chart. However,

from formula (7.19) it follows that for solutions of (7.18) we would obtain

the mean derivatives depending on the local connector of the connection H in

the chart and even on A, while for physical reasons the derivatives should be

vectors. This is why we modify the deﬁnition of mean derivatives as follows.

Consider the Borel ﬁelds Y

(t, ·)

and Y

∗

(t, ·)

on a chart U

such that

the forward (backward, respectively) mean derivative of ξ(t)att, calculated

in U

, is presented in the form Y

(t, ξ(t))

∗

(t, ξ(t))

, respectively); see

Section 8.1.OfcourseY

(t, ·)

and Y

∗

(t, ·)

do not transform as vectors

under changes of coordinates. Now construct the vector ﬁeld Y

(t, ·)(and

∗

(t, ·)) on M whose vector at any m ∈ M coincides with Y

(t, m)

(with

∗

(t, m)

, respectively), where Y

(t, m)

∗

(t, m)

, respectively) is calcu-

lated in the normal chart U

(m)ofH at m. Clearly the ﬁelds Y

and Y

∗

are

Borel measurable cross-sections of the tangent bundle TM.

Deﬁnition 9.1. D

ξ(t)=Y

(t, ξ(t)) and D

∗

ξ(t)=Y

∗

(t, ξ(t)) are called

the forward and backward, respectively, mean derivatives of ξ(·)att on M

with respect to H; D

ξ(t)=v

(t, ξ(t)) and D

ξ(t)=u

(t, ξ(t)) are called

the current and osmotic velocities, respectively, of ξ(·)wherev

(t, m)=

(t, m)+Y

∗

(t, m)) and u

(t, m)=

(t, m) − Y

∗

(t, m)).

The current and osmotic velocities do not depend on the connection (see

Theorem 9.12 below) and so we do not indicate the connection in the notation.

If H is speciﬁed, we shall not indicate it in the notation of mean forward and

backward derivatives either.

In the same manner as in Deﬁnition 9.1 we modify the deﬁnitions of D

x(t)

and D

∗

x(t)(see(8.7)and(8.8)).

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

to Mathematical Physics, Theoretical and Mathematical Physics,

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

225