1. Introduction
For the studies and analysis of dynamical systems and inverse problems, random functions are frequently used. They play a very important role in the integration of partial differential equations (PDEs), diffusion-type PDEs (for example, [
1,
2,
3,
4]). For these purposes, matrix or operator measures are studied and used [
1,
2,
3,
5,
6]. In [
1,
6], real and complex measures, stochastic PDEs, and their applications to solutions of second-order PDEs were described over real and complex fields. In [
2,
5], random functions, stochastic processes, Markov processes, and stochastic PDEs are described, and their applications to solutions of PDEs using evolutionary operators, generators of their semigroups are given. In [
3,
4], these themes are also provided, but the emphasis is on Feynman-type integrals, and their convergence in suitable domains of function spaces.
However, there are restrictions for these approaches because they work for partial differential operators (PDOs) of an order not higher than 2. Indeed, they are based on complex modifications of Gaussian measures. Nevertheless, if a characteristic function
of a measure has the form
where
is a polynomial, then its degree is not higher than 2 according to the Marcinkievich theorem (Chapter II, §12 in [
7]).
On the other side, hypercomplex numbers open new opportunities in these areas. For example, Dirac used the complexified quaternion algebra
for the solution of the Klein–Gordon hyperbolic PDE of second order with constant coefficients [
8]. This is important in spin quantum mechanics, because
. It was proved in [
9] that, in many variants, it is possible to reduce a PDE problem of a higher order to a subsequent solution of PDEs of an order not higher than 2 with hypercomplex coefficients. In general, the complex field is insufficient for this purpose.
On the other hand, algebras of hypercomplex numbers, in particular, the Cayley–Dickson algebras
over the real field
are natural generalizations of the complex field, where
is the quaternion skew field,
denotes the octonion algebra,
,
denotes the complex field. Then, each subsequent algebra
is obtained from the preceding algebra
with the doubling procedure using the doubling generator [
10,
11,
12] (see also
Appendix A).
They are widely applied in PDEs, noncommutative analysis, mathematical physics, quantum field theory, hydrodynamics, industrial and computational mathematics, and noncommutative geometry [
8,
13,
14,
15,
16,
17,
18,
19,
20,
21].
This article is motivated by the fact that many existing approaches for solutions of PDEs are based on evolutionary operators obtained as solutions of the corresponding stochastic PDEs. However, this is restricted to PDEs of an order not higher than 2 over a real or complex field. This article is aimed at extending such approaches to PDEs of an order higher than 2.
Previously, measures with values in the complexified Cayley–Dickson algebra
were studied in [
22]. They appear naturally with a solution of a second-order hyperbolic PDE with Cayley–Dickson coefficients. In this work, the results and notation of [
22] are used. They are recalled in
Appendix B. Relations between different forms of the diffusion PDE (such as backward Kolmogorov, Fokker-Planck-Kolmogorov, and stochastic) are discussed.
This article is devoted to dynamical systems such as hypercomplex generalized diffusion PDEs. For this purpose, measures and random functions having values in modules over the complexified Cayley–Dickson algebras are investigated. An integration of generalized diffusion processes is investigated. For their study, hypercomplex transition measures are used. Noncommutative integrals of hypercomplex random functions are studied. The existence of novel random functions and Markov processes over hypercomplex numbers is studied in Theorem 1, Corollary 1. Integrals of hypercomplex random functions and operators acting on them are investigated in Theorems 2–5. Properties of hypercomplex stochastic integrals are described in Propositions 1–3. In Theorem 6, their stochastic continuity is investigated. Necessary specific novel definitions are given. Notation is described in detail. Lemmas 1–5 are given in order to prove the theorems and propositions. These lemmas concern estimates of hypercomplex stochastic integrals, which was not performed before. In Theorems 7 and 8, and Corollary 2, solutions of generalized diffusion PDEs with hypercomplex random functions and operators are scrutinized. Ordered products of appearing operators are studied. Generators of semigroups of evolutionary operators are also studied for the generalized diffusion PDE in its stochastic form over the complexified Cayley–Dickson algebra. The stochastic Cauchy problem related with the generalized diffusion PDE is investigated for modules over complexified Cayley–Dickson algebras. Basics of hypercomplex numbers and measures are recalled in Appendices
Appendix A and
Appendix B (Formulas (A1)–(A40)). This opens new possibilities for a subsequent solution of higher-order PDEs using their decompositions and noncommutative integration, which is also discussed in the conclusion.
The main results of this work were obtained for the first time. The noncommutative integration developed in this paper permits to subsequently analyze and integrate PDEs of orders higher than 2 of different types, including parabolic, elliptic, and hyperbolic. The obtained results open new opportunities for subsequent studies of PDEs and their solutions regarding inverse problems.
2. Generalized Diffusion PDEs
Definition 1. Suppose that
is an additive group contained in . Suppose also that T is a subset in
containing a point . Let be a locally -convex space that is also a two-sided -module for each , where . Then,for the product of measurable spaces, where is the Borel σ-algebra of , is an algebra of cylindrical subsets of generated by projections , where is a left-ordered direct product, is a finite subset of T, , for each in T. Function with values in the complexified Cayley–Dickson algebra for each , , is called a transitional measure if it satisfies the following conditions:so that as the function by y is in . A transition measure is called unital if Then, for each finite set of points in T, such that ; there is defined a measure in where , variables are such that , is fixed. Let the transitional measure be unital. Then, for the product , where , , the equalityis fulfilled, where Equation implies thatfor each , where finite sets are ordered by inclusion: if and only if , where is the natural projection, denotes the family of all finite linearly ordered subsets q in T, such that , , is the natural projection, . Hence, Conditions , , imply that: is the consistent family of measures that induces a cylindrical distribution on the measurable space such thatfor each . The cylindrical distribution given by Formulas –, and is called the -valued Markov distribution with time t in T.
Remark 1. Let for each , Put for each in , where is defined on such that for each . To an arbitrary function a function can be posed where , , . Put If , , , then the integralcan be defined whenever it converges. Definition 2. A function F is called integrable relative to a Markov cylindrical distribution if the limitalong the generalized net by finite subsets of T exists (see ). This limit is called a functional integral relative to the Markov cylindrical distribution: Remark 2. Spatially homogeneous transition measure.Suppose that is an -valued measure on for each such that for each and , where , X is a locally -convex space which is also a two-sided -module, is an algebra of subsets of X. Suppose also that P is a spatially homogeneous transition measure:for each , and and every , where also satisfies the following condition:for each and in T. Then,is the characteristic functional of transitional measure for each and each , where notates the topologically dual space of all continuous -linear real-valued functionals y on X, . Particularly for P satisfying Conditions and with its characteristic functional ϕ satisfies the equalities:wherefor each and and respectively and , , since . Remark 3. If T is a topological space, we denote by the Banach space of all continuous bounded functions supplied with the norm:where H is a Banach space over that may also be a two-sided -module. If T is compact, then is isomorphic with the space of all continuous functions . For a set T and a complete locally -convex space H that may also be a two-sided -module, consider product -convex space in the product topology, where for each .
Suppose that is a separating algebra on the space either or or on , where is a σ-additive measure on the Borel σ-algebra on T, . Consider a random variable with values in , where , , is a measure space with an -valued measure P, .
Events are independent in total if . Subalgebras are independent if all collections of events are independent in total, where , . To each collection of random variables on with is related the minimal algebra for which all are measurable, where Υ is a set. Collections are independent if , where for each .
For or define as a closed submanifold in X of all , such that , where are pairwise distinct points in T and are points in H. For and and , we denote .
Definition 3. Suppose that H is a real Banach space that may also be a two-sided -module. Consider a random function with values in the space H as a random variable such that:where μ is an -valued measure on , for such that and each . Thereby, a -linear operator is denoted, which is prescribed by the following formula:for each in T, where is a separating algebra of H such that for each in T, where with or , ;are mutually independent for each chosen in T and each , where Then, is the random function with independent increments, where is the shortened notation of .
Remark 4. Random function satisfying Conditions – in Definition 3 possesses a Markovian property with transitional measure
(see also -).
As usual, it is put for the expectationof a random variable whenever this integral exists, where is the -valued measure on a measure space shortly denoted by , where f is -measurable, , denotes the Borel σ-algebra on . If P is specified, it may be shortly written E instead of . If is a sub-σ-algebra in the σ-algebra and if there exists a random variable such that g is -measurable andfor each , then g is called the conditional expectation relative to and denoted by . An operator is called right -linear in the weak sense iffor each x and y in and b and c in , where real field is canonically embedded into the complexified Cayley–Dickson algebra as , . Over the algebra , this gives right linear operators for each x and y in and b and c in , since is associative. For brevity, we omitted “in the weak sense”. We notate such a set of operators with . Thenwhere , for each , where for each in with b and c in (see also Remark 2.1 of [22]). In particular, it is useful to consider the following case: , where ξ is a -valued random variable on a measurable space and with a probability measure , where , where is embedded into as , where . This means that ξ is -measurable, while w is -measurable, where is a measurable space, is a measure.
Assume that there is an injection and has an extension on such that , for each and . Then, it may be the case that and are related by Formulas 2.4, 2.4 of [22] with the use of and using the -analytic extension. If , where is a Borel measurable function; then, there exists a Borel measurable function such that . Therefore, if is a Borel measurable function, using Formulas 2.4, 2.4 of [22] we putIffor each , where is -measurable, , , then g is called the conditional expectation of relative to and denoted by , since and , where is a σ-subalgebra in . This convention is used if some other is not specified.
Let denote a family of all right -linear operators J from into fulfilling the condition Theorem 1. Suppose that either or , where with , , either with or . Then, there exists a family Ψ of pairwise inequivalent Markovian random functions with -valued transition measures of the type (see Definition 2.4 of [22]) on X of a cardinality , where , . Proof. Naturally, the algebra
, if considered to be a linear space over
, also possesses a structure of the
-linear space isomorphic with
. Therefore, the Borel
-algebra
of the algebra
is isomorphic with
. So, put
for each
and
, where an operator
U and a vector
p are marked, satisfying conditions of Definitions 2.4 and 2.3
of [
22].
Naturally, an embedding of into exists as , where . If is an -valued random function, J is a right -linear operator satisfying the condition , (see , in Remark 4), then generally, is an -valued random function, where , is a shortened notation of .
Operators
exist (see, for example, Chapter IX, Section 13 in [
23].), since
is positive definite for each
j. On the Cayley–Dickson algebra
, function
exists (see §3.7 and Lemma 5.16 in [
19]). It has an extension on
and its branch, such that
for each
can be specified by the following. Take an arbitrary
with
and
. Put
,
,
,
. If
and
,
a can be presented in the form
with
,
,
,
,
. Therefore, in the latter case
, since
. If
a is such that
and
; then, for
, there are
and
. On the other hand, for
a with
, equation
has a solution with
and
in
, since, by utilizing the standard basis of the complexified Cayley–Dickson algebra, this equation can be written as the quadratic system in
complex variables
. The latter system has a solution
in
, since each polynomial over
has zeros in
by the principal algebra theorem. Therefore, the initial equation has a solution in
. Thus, the operator
exists and it evidently belongs to
.
Particularly, J can be , while as , it is possible to take a Wiener process with the zero expectation and the unit covariance operator.
If , then defines a continuous -linear projection from X into H. Therefore, provides a continuous -linear projection from X into for each , where . These projections and Borel -algebras on for finite linearly ordered subsets q in T induce an algebra of X. Since is supplied with the product Tychonoff topology, a minimal -algebra generated by coincides with the Borel -algebra . Topological spaces T and H are separable and relative to the norm topology on ; is also obtained.
By virtue of Proposition 2.7 of [
22] and Formulas 2.4
and 2.4
of [
22], a characteristic functional of
fulfils Condition
. It is worth to associate with
a spatially homogeneous transition measure
according to Equation
in Remark 2. Representation 2.10
of [
22] implies that a bijective correspondence exists between
-additive norm-bounded
-valued measures and their characteristic functionals, since it is valid for each real-valued addendum
(see, for example, [
1,
7]) and
. Moreover, a characteristic functional of the ordered convolution
of two
-additive norm-bounded
-valued measures
and
is the ordered product
of their characteristic functionals
and
, respectively. Therefore, Conditions
–
in Definition 1 are satisfied.
Then, Formulas
,
and
in Definition 1 together with the data above describe an
-valued Markov cylindrical distribution
on
X (see Corollary 2.6 of [
22] and Definition 1), since
for each
. The space
H is Radon by the Theorem I.1.2 of [
1], since
H is separable and complete as the metric space. From Theorem 2.3 and Proposition 2.7 of [
22], it follows that
is uniformly norm-bounded. In view of Theorem 2.15 and Corollary 2.17 [
22], this cylindrical distribution has an extension to a norm-bounded measure
on a completion
of
, where
.
Considering different operators
U and vectors
p, and utilizing the Kakutani theorem (see, for example, in [
1]), we infer that there is a family of the cardinality
of pairwise nonequivalent and orthogonal measures of such type
on
X since each
P has the representation 2.10
of [
22].
Let
be the set of all elementary events
where
is a finite subset of
,
,
is a subset of
(see Remarks 1 and 3), where
, where
for each
in
. Hence, an algebra
exists of cylindrical subsets of
induced by the projections
where
is a subset in
. This procedure induces algebra
of
. So, one can consider a Markovian random function corresponding to
(see Definition 3). □
Corollary 1. Let be a random function given by Theorem 1 with the transitional measure for each , thenandfor each k and h in , where , and , , , where E means the expectation relative to . Proof. By virtue of Theorem 1, random function has the transitional measure
, where
. Therefore, Formulas
and
follow from Proposition 2.8 and Theorem 2.9 of [
22]. □
Definition 4. Let be a measure space with an -valued σ-additive norm-bounded measure P on a σ-algebra of a set Ω with . There is a filtration , if for each in T, where is a σ-algebra for each , where either with or . A filtration is called normal if and for each .
Then, if for each a random variable with values in a topological space X is -measurable, random function and filtration are adapted, where denotes the minimal σ-algebra on X containing all open subsets of X (i.e., the Borel σ-algebra). Let be a minimal σ-algebra on generated by sets with , also with . Let also μ be a σ-additive measure on induced by the measure product , where λ is the Lebesgue measure on T. If is -measurable, then u is called a predictable random function, where denotes the completion of by -null sets, where is the variation of μ (see Definition 2.10 in [22]). The random function given by Corollary 1 is called an -valued -random function or, in short, U-random function for .
Remark 5. Random functions described in the proof of Theorem 1 are generalizations of the classical Brownian motion processes and of the Wiener processes.
Let be the -valued -random function provided by Theorem 1 and Corollary 1. Let a normal filtration on be induced by . Therefore, is -measurable for all ; is independent of any for each and in T with . In view of Theorem 1 and Corollary 1, conditions and are satisfied, where , (see Remark 4).
Suppose that is an valued random function (that is, random operator), , (see also the notation in Remark 4). It is called elementary if a finite partition exists, so thatwhere is -measurable for each , where n and h are natural numbers, where denotes the characteristic function of the segment , . A stochastic integral relative to and the elementary random function is defined by the formula:where for each t and in T. Similarly, elementary random functions and their stochastic integrals are defined. Putfor each x and y in ,where with for each l, for each in with and , . denotes an adjoint operator of an -linear operator , such thatfor each and . Then, we put for with A and B in Lemma 1. Let
be an elementary -valued random variable with -almost everywhere on for each , and let
be an -valued random function with - and - random functions and , respectively, having values in , so that and belong to , and operator fulfils Conditions 2.3 and of Definition 2.4 of [22], where and are independent; , (see Definitions 2.10 of [22] , Remarks 4 and 5 above). Then, -almost everywhere on .
Proof. This follows from Corollary 1, and Formulas and , since -almost everywhere and for each in for the U-random function w. □
Lemma 2. Let , with A and B belonging to , where , , . Then, Proof. Since
A and
B belong to
, then
by Formula
, where
denotes the trace of operator
, as usual. On the other side,
Since
, then
for each
,
, where
denotes the standard orthonormal base in the Euclidean space
, where
;
is embedded into
as
. Therefore, we deduce using Formulas
,
, and
that
since
.
This implies Formula . From the Cauchy–Bunyakovskii–Schwarz inequality, Remark 4, Formulas and , one obtains Inequality . □
Theorem 2. If is an elementary random function with values in and is an U-random function in as in Definition 4 with , then-almost everywhere for each . Proof. Since
and
,
by the conditions of this theorem,
for each
j and hence
and
, since
U satisfies the conditions of Definition 2.4 and 2.3
[
22] (see also Theorem 1). Therefore,
; hence,
for each
and
-almost all
, where
is a shortening of
, while
is that of
. On the other hand,
for each
, where
for each
z in the Cayley–Dickson algebra
, where
with
for each
l,
is the standard basis of
.
Let
and
, where
and
, where
is the Kronecker delta. Then, for an operator
J in
and each
, the representation is valid:
where
,
and
for each
k and
l.
From the conditions imposed on
U (see Definition 2.4 of [
22]), it follows that
U and
belong to
, since the positive definite matrix
with real matrix elements corresponds to the positive definite operator
for each
j, and
for each
.
By virtue of Proposition 2.5, and Formulas 2.8
and 2.8
in [
22],
is the
-valued measure for each
, since the Cayley–Dickson algebra
is power-associative and
for each
.
Random function
is obtained from the standard Wiener process
in
with the zero expectation and the unit covariance operator with the use of operator
:
according to Theorem 1. Therefore, the statement of this theorem follows from the Ito isometry theorem (see, for example, Proposition 1.2 in [
1], Theorem 3.6 in [
2], XII in Chapter VIII, Section 1 in [
5] ), Formulas
–
above and Remarks 4 and 5. □
Theorem 3. Suppose that
is an elementary valued random function and
is an -valued random function satisfying Condition in Lemma 1. Then,-almost everywhere for each . Proof. We consider the following representation:
of
S with
for every
and
with
x and
y in
. For each
, we have
(see Remark 2.1 of [
22] and Formula
in Theorem 2 above). On the other hand,
for each
. For two operators
G and
H in
, the inequality is valid
due to Representation
. Applying Theorem 2 and Lemma 2 (see also Remarks 4 and 5) to
and
, where
and
, we infer that
-almost everywhere for each , since and for each a and b in . □
Lemma 3. If conditions in Theorem 3, in Lemma 1 are satisfied, thenfor each , , , . Proof. According to Formula
for each
, where
. Since
is
-measurable for each
, then
is
-measurable. We consider a modified elementary random function
such that
for each
if
; otherwise
for each
if
for some
l. Therefore,
for each
; hence,
Then, we deduce that
by Chebyshëv inequality (see, for example, in Section II.6 [
7]), Equality (43) above, Formulas 2.10(1) and (2) in [
22]. By virtue of Theorem 3 (see also Formulas (40) and (41))
since
for a random variable
which is
-measurable (Section II.7 [
7]). This implies Inequality (42). □
Theorem 4. If w is a U-random function and is an -valued predictable random function satisfying the conditionfor each in T, where operator U is specified in Definition 2.4 [22], such that ; then, a sequence of elementary random functions exists with such thatfor each in T. Proof. for each
, since
implying
; hence,
for each
j. In view of Formulas (35), (37) random function
having values in
has the decomposition into a finite
-linear combination
of real random functions
using vectors
,
and the standard basis
of the Cayley–Dickson algebra
over
. For each real-valued random function, the condition
is fulfilled for each
in
T by (44); hence, a sequence of real-valued random functions
exists, such that
for each
. Thus, Formulas (46), (47), and (48) imply (45). □
Theorem 5. If w fulfills Condition in Lemma 1 and is a -valued predictable random function satisfying the following inequality:for each in T, where Then, a sequence of elementary random functions exists with , such thatfor every in T. The proof is analogous to that of Theorem 4 with the use of Formula (41), using (49), (50) and (51), since with , -almost everywhere.
Definition 5. A sequence of elementary -valued random functions with is mean absolute square convergent to a predictable -valued random function , where w satisfies Condition in Lemma 1, if Condition in Theorem 5 is satisfied. The corresponding mean absolute square limit is induced by Formulas and , and is denoted by . The family of all predictable -valued random functions satisfying Condition (49) is denoted by
A stochastic integral of is:where is an -valued random function with and random functions and , respectively, having values in , where in T, where w satisfies Condition in Lemma 1. Proposition 1. Let the conditions of Theorem 5 be satisfied, and let , . Then, there exists for each and Proof. In view of Theorem 5, Definitions 4 and 5, and Remark 5, there exists for each . Formula (53) for elementary random functions for each follows from Formula (27). Hence, taking , we infer Equality (53) for by Theorem 5. □
Proposition 2. If , for each , w satisfies Condition in Lemma 1, andwhere , then there exists Proof. In view of Proposition 1, stochastic integrals and exist for each . Then, Equality (55) follows from Theorem 5, Equality (54), and Formula (52) in Definition 5. □
Proposition 3. If , and if w satisfies Condition in Lemma 1, where , then-almost everywhere for each . Proof. From Lemmas 1 and 2, and Proposition 1, Identity (56) follows. Then, Theorem 3 and Proposition 1 imply Inequality (57), since
with
and since
-almost everywhere. □
Remark 6. Let for each , and for each be a characteristic function of , . Then, for each , if , where . It is putfor each . From Proposition 3, it follows that is defined -almost everywhere. By virtue of Theorem IV.2.1 in [5], is the separable random function up to the stochastic equivalence since is the metric space. Therefore, is considered to be the separable random function. Definition 6. Let , , be a -valued random function adapted to the filtration of σ-algebras and let for each . If for each in T, then the family is called a martingale. If for each and for each in T, then is called a sub-martingale.
Lemma 4. Assume that and w satisfies Condition in Lemma 1, , andand is provided by Formula ; then, is a martingale and is the submartingale. Proof. By virtue of Proposition 3 is -measurable and for each . Hence is the martingale.
Random function
has the decomposition:
with
, for each
k,
j,
l, where
is the standard orthonormal basis of the Euclidean space
, where
is embedded into
as
. Therefore, each random function
is the martingale. Then,
By virtue of Theorem 1 and Corollary 2 in Chapter III, Section 1 [
5], Inequality
and Formula
above
is the submartingale for each
k,
j,
l. Consequently,
is the submartingale by Formulas (60) and (61). □
Lemma 5. Let and w satisfy Condition in Lemma 1 such that Proof. Inequality follows from Inequality . Therefore, remains to be proven. We take an arbitrary partition of . Then, we consider . In view of Lemma 4 is the martingale and is the submartingale.
Therefore, from Theorem 5 in Chapter III, Section 1 [
5], Formulas 2.10
,
of [
22] and Inequality
, we deduce that
(see also Remark 5). Together with Proposition 3 above and the Fubini theorem (II.6.8 [
7]), this implies that
Random function
is separable (see Remark 6); hence Inequality
follows from Inequality
. □
Theorem 6. Let be a predictable -valued random function, let w satisfy Condition in Lemma 1, . Then, random function is stochastically continuous, where .
Proof. If is an elementary -valued random function, then is stochastically continuous by Formula , since is stochastically continuous.
For each
according to Definition 5 and the Fubini theorem
. By virtue of Theorem 5, there exists a sequence
of elementary
-valued random functions, such that Limit
is satisfied. From Lemma 5 and the Fubini theorem, we infer that
Therefore, there exists a sequence
with
and a sequence
, such that
In view of the Borel–Cantelli lemma (see, for example, Chapter II, Section 10 [
7]) a natural number
exists, such that
for each
. Hence,
is stochastically continuous since
is stochastically continuous for each
. □
Definition 7. The generalized Cauchy problem over the complexified Cayley–Dickson algebra .Letsatisfying Condition in Lemma 1, where n and h are natural numbers. A stochastic Cauchy problem over is:where is an -valued random function, ζ is an -valued random variable which is -measurable, , , where H, G, w are as in –. Problem is understood as the following integral equation: Then, the random function is called a solution if it satisfies Conditions –:where is a shortened notation of . Theorem 7. Let and be Borel functions, w satisfy Condition in Lemma 1, and be such that
- (i)
and
- (ii)
for each x and y in , , where ,
- (iii)
.
Then, a solution Y of Equation exists (see Definition 7); if Y and are two stochastically continuous solutions, then Proof. We consider a Banach space
consisting of all predictable random functions
such that
is
-measurable for each
and
with the norm
In view of Proposition 2, there exists operator
Q on
such that
for each
, since
G and
H satisfy Condition
of this theorem. Then,
is
-measurable for each
, since
G and
H are Borel functions and
. By virtue of Proposition 3, using the inequality
for each
,
and
in
, the Cauchy–Bunyakovskii–Schwarz inequality,
,
, and Condition
of this theorem, we infer that
Thus,
. Then, using the Cauchy–Bunyakovskii–Schwarz inequality, 2.3
of [
22], Proposition 3, Condition
of this theorem, and inequality
for each
and
in
, we deduce that
for each
X and
in
,
, where
. Therefore, the operator
is continuous. Then, we infer that
for each
X and
in
,
. Therefore,
for each
. Hence, the series
converges. Thus, the following limit exists
in
. From the continuity of
Q, it follows that
, hence
. Thus,
. Consequently, for each . This means that is the solution of Equation . In view of Theorem 6 and Condition of this theorem, solution is stochastically continuous up to the stochastic equivalence.
Now, let
Y and
be two stochastically continuous solutions of Equation
. We consider a random function
, such that
if
and
for each
,
in the opposite case where
,
. Therefore,
for each
in
; consequently,
On the other hand,
by Condition
. This implies that
. Then, using the Fubini theorem, 2.3
of [
22], Proposition 3, Lemma 5, we deduce that
Thus, a constant
exists, such that
The Gronwall inequality (see Lemma 3.15 in [
2], Lemma 1 in Chapter 8, Section 2 in [
5]) implies that
. Consequently,
Random functions
and
are stochastically continuous and hence stochastically bounded. Consequently,
Therefore, random functions and are stochastically equivalent. This implies Equality . □
Corollary 2. Let operators G and H be and such that G be a generator of a semigroup . Let also be a random function fulfilling Condition in Lemma 1. Then, the Cauchy problemwhere , , has a solutionfor each . Proof. Condition
implies that
where
,
,
for each
k. As a realization of the semigroup
, it is possible to take
since
G is a bounded operator and
for each
by Formulas 2.1
and 2.3
in [
22]. Therefore, from Theorem 7 applied to Equation
, Assertion
of this corollary follows. □
Theorem 8. Let G, H, and w satisfy conditions of Theorem 7, and be an -valued random function satisfying the following equation:where , in , . Then, random function Y satisfying Equation is Markovian with the following transitional measure:for each . Proof. Random function
is
-measurable for each
. On the other hand,
is induced by the random function
for each
, where
is independent of
. Therefore,
is independent of
and each
(see
). By virtue of Theorem 7,
is the unique (up to stochastic equivalence) solution of the following equation:
and
is also its solution. Consequently,
.
Let
, where
denotes the family of all bounded continuous functions from
into
. Let
, where
denotes the family of all random variables
such that there exists
for which
, where
may depend on
g. We put
Hence,
, where
is a shortening of
as above,
(see
). Assume first that
q has the following decomposition:
where
,
,
. This implies that
is independent of
for each
k. Therefore, using
, we deduce that
for
q of the form
. This implies that
where
.
Then,
for each
by 2.3
[
22], since
g and
f are bounded, where
. Therefore, for each
, there exists
for which
has the decomposition of type
and such that
. Taking
, one obtains that Formulas
and
are accomplished for each
. Therefore,
for each
,
in
, since the families
and
of all such
g and
f are separate points in
. This implies that
for each
, where
. Thus, Equality
is proven. □