Noncommutative Integration of Generalized Diffusion PDE

: The article is devoted to the noncommutative integration of a diffusion partial differential equation (PDE). Its generalizations are also studied. This 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 the real or complex ﬁeld. This article is aimed at extending such approaches to PDEs of an order higher than 2. For this purpose, measures and random functions having values in modules over complexiﬁed Cayley–Dickson algebras are investigated. Noncommutative integrals of hypercomplex random functions are investigated. By using them, the noncommutative integration of the generalized diffusion PDE is scrutinized. Possibilities are indicated for a subsequent solution of higher-order PDEs using their decompositions and noncommutative integration.


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 φ(t) of a measure has the form φ(t) = exp(Q(t)) where Q(t) 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 H C for the solution of the Klein-Gordon hyperbolic PDE of second order with constant coefficients [8]. This is important in spin quantum mechanics, because U(2) ⊂ H. 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 A r over the real field R are natural generalizations of the complex field, where A 2 = H is the quaternion skew field, A 3 = O denotes the octonion algebra, A 0 = R, A 1 = C denotes the complex field. Then, each subsequent algebra A r+1 is obtained from Function P(t 1 , x 1 , t 2 , A) with values in the complexified Cayley-Dickson algebra A r,C for each t 1 < t 2 ∈ T, x 1 ∈ X t 1 , A ∈ U t 2 is called a transitional measure if it satisfies the following conditions: the set function ν x 1 ,t 1 ,t 2 (A) := P(t 1 , x 1 , t 2 , A) is a measure on (X t 2 , U t 2 ); (1) the function α t 1 ,t 2 ,A (x 1 ) := P(t 1 , x 1 , t 2 , A) of the variable x 1 is U t 1 -measurable, that is, α −1 t 1 ,t 2 ,A (B(A r,C )) ⊂ U t 1 ; (2) P(t 1 , x 1 , t 2 , A) = X z P(t, y, t 2 , A)P(t 1 , x 1 , t, dy) for each t 1 < t < t 2 ∈ T so that P(t, y, t 2 , A) as the function by y is in L 1 ((X t , U t ), ν x 1 ,t 1 ,t , A r,C ). A transition measure P(t 1 , x 1 , t 2 , A) is called unital if P(t 1 , x 1 , t 2 , X t 2 ) = 1 for each t 1 < t 2 ∈ T.
Then, for each finite set q = (t 0 , t 1 , . . . , t n+1 ) of points in T, such that t 0 < t 1 < . . . < t n+1 ; there is defined a measure in X g where g = q \ {t 0 }, variables x 1 , . . . , x n+1 are such that (x 1 , . . . , x n+1 ) ∈ D, x 0 ∈ X t 0 is fixed. Let the transitional measure P(t, x 1 , t 2 , dx 2 ) be unital. Then, for the product is fulfilled, where r = q \ {t j }. Equation (6) implies that for each v < q, where finite sets are ordered by inclusion: v < q if and only if v ⊂ q, where π q w : X g → X w is the natural projection, g = q \ {t 0 }, w = v \ {t 0 }. Υ T denotes the family of all finite linearly ordered subsets q in T, such that t 0 ∈ q ⊂ T, v ≤ q ∈ Υ T , π q :X T → X g is the natural projection, g = q \ {t 0 }. Hence, Conditions (4), (5), (7) imply that: {µ q x 0 ; π q v ; Υ T } is the consistent family of measures that induces a cylindrical distributionμ x 0 on the measurable space (X T ,Ũ) such that µ x 0 (π −1 q (D)) = µ q x 0 (D) (8) for each D ∈ U g . The cylindrical distribution given by Formulas (1)-(5), (7) and (8) is called the A r,C -valued Markov distribution with time t in T. Remark 1. Let X t = X for each t ∈ T,X t 0 ,x 0 := {x ∈X T : x(t 0 ) = x 0 }. Putπ q : x → x q for each x = x(t) inX T , where x q is defined on q = (t 0 , . . . , t n+1 ) ∈ Υ T such that x q (t) = x(t) for each t ∈ q. To an arbitrary function F :X T → A l r,C a function can be posed (S q F)(x) := F(x q ) = F q (y 0 , . . . , y n ), where y j = x(t j ), F q : X q → A l r,C , l ∈ N. Put can be defined whenever it converges.

Definition 2.
A function F is called integrable relative to a Markov cylindrical distribution µ x 0 if the limit lim along the generalized net by finite subsets q = (t 0 , . . . , t n+1 ) ∈ Υ T of T exists (see (9)). This limit is called a functional integral relative to the Markov cylindrical distribution: Remark 2. Spatially homogeneous transition measure. Suppose that P(t, A) is an A r,Cvalued measure on (X, U) for each t ∈ T such that A − x ∈ U for each A ∈ U and x ∈ X, where A ∈ U, X is a locally R-convex space which is also a two-sided A r,C -module, U is an algebra of subsets of X. Suppose also that P is a spatially homogeneous transition measure: for each A ∈ U, t 1 < t 2 ∈ T and t 2 − t 1 ∈ T and every x 1 ∈ X, where P(t, A) also satisfies the following condition: for each t 1 < t 2 and t 1 + t 2 in T. Then, is the characteristic functional of transitional measure P(t 1 , x 1 , t 2 , dx) for each t 1 < t 2 ∈ T and each x 1 ∈ X, where X * R notates the topologically dual space of all continuous R-linear real-valued functionals y on X, y ∈ X * R . Particularly for P satisfying Conditions (12) and (13) with t 0 = 0 its characteristic functional φ satisfies the equalities: where ψ(t, y) := X P(t, dx) exp(iy(x)) and (16) for each t 1 < t 2 ∈ T and t 2 − t 1 ∈ T and t 1 + t 2 ∈ T respectively and y ∈ X * R , x 1 ∈ X, since Z(A r,C ) = C. Remark 3. If T is a T 1 ∩ T 3.5 topological space, we denote by C 0 b (T, H) the Banach space of all continuous bounded functions f : T → H supplied with the norm: where H is a Banach space over R that may also be a two-sided A r,C -module. If T is compact, then C 0 b (T, H) is isomorphic with the space C 0 (T, H) of all continuous functions f : T → H. For a set T and a complete locally R-convex space H that may also be a two-sided A r,C -module, consider product R-convex space H T := ∏ t∈T H t in the product topology, where H t := H for each t ∈ T.
Suppose that B is a separating algebra on the space either X := X(T, H) = L q (T, B(T), λ, H) or X := X(T, H) (X, B), where t ∈ T, ω ∈ Ω, (Ω, R, P) is a measure space with an A r,C -valued measure P, P : R → A r,C .
Events S 1 , . . . , S n are independent in total if P( l ∏ n k=1 S k ) = l ∏ n k=1 P(S k ). Subalgebras R k ⊂ R are independent if all collections of events S k ∈ R k are independent in total, where k = 1, . . . , n, n ∈ N. To each collection of random variables ξ γ on (Ω, R) with γ ∈ Υ is related the minimal algebra R Υ ⊂ R for which all ξ γ are measurable, where Υ is a set. Collections {ξ γ : For X = C 0 b (T, H) or X = H T define X(T, H; (t 1 , . . . , t n ); (z 1 , . . . , z n )) as a closed submanifold in X of all f : T → H, f ∈ X such that f (t 1 ) = z 1 , . . . , f (t n ) = z n , where t 1 , . . . , t n are pairwise distinct points in T and z 1 , . . . , z n are points in H. For n = 1 and t 0 ∈ T and z 1 = 0, we denote X 0 := X 0 (T, H) := X(T, H; t 0 ; 0).

Definition 3.
Suppose that H is a real Banach space that may also be a two-sided A r,C -module. Consider a random function w(t, ω) with values in the space H as a random variable such that: where µ is an A r,C -valued measure on (X(T, H), B), µ g (A) := µ(g −1 (A)) for g : X → H such that g −1 (R H ) ⊂ B and each A ∈ R H . Thereby, F t,u a R-linear operator F t,u : X → H is denoted, which is prescribed by the following formula: the vectors w(t m , ω) − w(t m−1 , ω), . . . , w(t 1 , ω) − w(0, ω) and w(0, ω) (20) are mutually independent for each chosen 0 < t 1 < . . . < t m in T and each m ≥ 2, where ω ∈ Ω. Then, {w(t) : t ∈ T} is the random function with independent increments, where w(t) is the shortened notation of w(t, ω).
In addition, Remark 4. Random function w(t, ω) satisfying Conditions (19)- (21) in Definition 3 possesses a Markovian property with transitional measure P(u, x, t, A) = µ F t,u (A − x) (see also (10)-(17)). As usual, it is put for the expectation of a random variable f : Ω → A h r,C whenever this integral exists, where P = P [r] is the A r,C -valued measure on a measure space (Ω [r] , [r] F ) shortly denoted by (Ω, F ), where f is (F , B(A h r,C ))measurable, h ∈ N, B(A h r,C ) denotes the Borel σ-algebra on A h r,C . If P is specified, it may be shortly written E instead of E P . If G is a sub-σ-algebra in the σ-algebra F and if there exists a random variable g : Ω → A h r,C such that g is (G, B(A h r,C ))-measurable and A f (ω)P(dω) = A g(ω)P(dω) for each A ∈ G, then g is called the conditional expectation relative to G and denoted by g = E( f |G). An operator J : A n r,C → A h r,C is called right A r,C -linear in the weak sense if J(xb + yc) = (Jx)b + (Jy)c (22) for each x and y in R n and b and c in A r,C , where real field R is canonically embedded into the complexified Cayley-Dickson algebra A r,C as Ri 0 , i 0 = 1. Over the algebra H C = A 2,C , this gives right linear operators J(xb + yc) = (Jx)b + (Jy)c for each x and y in A n 2,C and b and c in A 2,C , since H C is associative. For brevity, we omitted "in the weak sense". We notate such a set of operators with L r (A n r,C , A h r,C ). Then where z = (z 1 , . . . , z n ), z j ∈ A r,C for each j ∈ {1, . . . , n}, where a 2 = 2|b| 2 + 2|c| 2 for each a = b + ic in A r,C with b and c in A r (see also Remark 2.1 of [22]). In particular, it is useful to consider the following case: w = Jξ + p, where ξ is a R 2nvalued random variable on a measurable space (Ω [0] , [0] F ) and with a probability measure P [0] : [0] F → [0, 1], where p ∈ A n r,C , where R 2n is embedded into A n r,C as i 0 R n + i 0 iR n , where J ∈ L r (A n r,C , A n r,C ). This means that ξ is Assume that there is an injection θ : ( Then, it may be the case that P and P [r] are related by Formulas 2.4(2), 2.4(3) of [22] with the use of U = U [r] = J 2 and U [0] = I using the A r,C -analytic extension. If f = F(w), where F : A n r,C → A h r,C is a Borel measurable function; then, there exists a Borel measurable function G : R 2n → A h r,C such that G(ξ) = f . Therefore, if u : A h r,C → R is a Borel measurable function, using Formulas 2.4(2), 2.4(3) of [22] we put , then g is called the conditional expectation of u( f ) relative to G and denoted by E(u( f )|G) = g, since P( This convention is used if some other is not specified. Let L r,i (A n r,C , A h r,C ) denote a family of all right A r,C -linear operators J from A n r,C into A h r,C fulfilling the condition J(A n r ) ⊂ A h r .
Theorem 1. Suppose that either X = C 0 b (T, H) or X = H T , where H = A n r,C with n ∈ N, 2 ≤ r < ∞, either T = [0, s] with 0 < s < ∞ or T = [0, ∞). Then, there exists a family Ψ of pairwise inequivalent Markovian random functions with A r,C -valued transition measures of the type µ Ut,pt (see Definition 2.4 of [22]) on X of a cardinality card(Ψ) = c, where c = 2 ℵ 0 , Proof. Naturally, the algebra A n r,C = ⊗ n j=1 A r,C , if considered to be a linear space over R, also possesses a structure of the R-linear space isomorphic with R 2 r+1 n . Therefore, the Borel σ-algebra B(A n r,C ) of the algebra A n r,C is isomorphic with B(R 2 r+1 n ). So, put P(t, A) = µ Ut,pt (A) for each 0 < t ∈ T and A ∈ B(H), where an operator U and a vector p are marked, satisfying conditions of Definitions 2.4 and 2.3(α) of [22].
Naturally, an embedding of R n into A n r,C exists as i 0 R n , where i 0 = 1. If ξ(t) is an R n -valued random function, J is a right A r,C -linear operator J : A n r,C → A n r,C satisfying the condition J(A n r ) ⊂ A n r , v ∈ A n r,C (see (22), (23) in Remark 4), then generally, w(t) = Jξ(t) + vt is an A n r,C -valued random function, where 0 ≤ t ∈ T, w(t) is a shortened notation of w(t, ω).
Operators B ±1/2 j exist (see, for example, Chapter IX, Section 13 in [23].), since B j is positive definite for each j. On the Cayley-Dickson algebra A r , function √ a exists (see §3.7 and Lemma 5.16 in [19]). It has an extension on A r,C and its branch, such that √ a > 0 for each a > 0 can be specified by the following. Take an arbitrary a = a 0 + ia 1 On the other hand, for a with a 1,0 = 0, equation (γ + iδ) 2 = a 0 + ia 1 has a solution with γ and δ in A r , since, by utilizing the standard basis of the complexified Cayley-Dickson algebra, this equation can be written as the quadratic system in 2 r complex variables γ 0 + iδ 0 , . . . , γ 2 r −1 + iδ 2 r −1 . The latter system has a solution (γ, δ) in A 2 r , since each polynomial over C has zeros in C by the principal algebra theorem. Therefore, the initial equation has a solution in A r,C . Thus, the operator U 1/2 = m j=1 a 1/2 j B 1/2 j exists and it evidently belongs to L r (A n r,C , A n r,C ). Particularly, J can be J = U 1/2 , while as ξ(t), it is possible to take a Wiener process with the zero expectation and the unit covariance operator.
If f ∈ X, then T t → f (t) defines a continuous R-linear projection π t from X into H. Therefore, π t n × (π t n−1 × . . . × π t 1 ) provides a continuous R-linear projection π q from X into H q for each 0 < t 1 < . . . < t n ∈ T, where q = {t 1 , . . . , t n }. These projections and Borel σ-algebras B(H q ) on H q for finite linearly ordered subsets q in T induce an algebra R(X) of X. Since H T is supplied with the product Tychonoff topology, a minimal σ-algebra R σ (H T ) generated by R(H T ) coincides with the Borel σ-algebra B(H T ). Topological spaces T and H are separable and relative to the norm topology on By virtue of Proposition 2.7 of [22] and Formulas 2.4(2) and 2.4(3) of [22], a characteristic functional of P U,p (t, A) := µ Ut,pt fulfils Condition (17). It is worth to associate with P U,p (t, A) a spatially homogeneous transition measure P U,p (t 1 , x 1 , t 2 , A) according to Equation (12) in Remark 2. Representation 2.10(2) of [22] implies that a bijective correspondence exists between σ-additive norm-bounded A r,C -valued measures and their characteristic functionals, since it is valid for each real-valued addendum µ j,k (see, for example, [1,7]) and Z(A r,C ) = C. Moreover, a characteristic functional of the ordered convolution (µ * ν) of two σ-additive norm-bounded A r,C -valued measures µ and ν is the ordered productμ ·ν of their characteristic functionalsμ andν, respectively. Therefore, Conditions (1)-(4) in Definition 1 are satisfied.
Then, Formulas (5), (7) and (8) in Definition 1 together with the data above describe an A r,C -valued Markov cylindrical distribution P U,p on X (see Corollary 2.6 of [22] and 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 P U,p 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 P U,p on a completion R P (X) of R(X), where R σ (X) = B(X).
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 c of pairwise nonequivalent and orthogonal measures of such type P U,p on X since each P has the representation 2.10(2) of [22].
Let Ω = Ω [r] be the set of all elementary events can consider a Markovian random function corresponding to P U,p (see Definition 3).
Proof. By virtue of Theorem 1, random function w(t, ω) has the transitional measure . Therefore, Formulas (24) and (25) follow from Proposition 2.8 and Theorem 2.9 of [22]. Definition 4. Let (Ω, F , P) be a measure space with an A r,C -valued σ-additive norm-bounded measure P on a σ-algebra F of a set Ω with P(Ω) = 1. There is a filtration {F t : t ∈ T}, Then, if for each t ∈ T a random variable u(t) : Ω → X with values in a topological space X is (F t , B(X))-measurable, random function {u(t) : t ∈ T} and filtration {F t : t ∈ T} are adapted, where B(X) denotes the minimal σ-algebra on X containing all open subsets of X (i.e., the Borel σ-algebra). Let G be a minimal σ-algebra on T × Ω generated by sets (v, t] × A with A ∈ F v , also {0} × A with A ∈ F 0 . Let also µ be a σ-additive measure on (T × Ω, G) induced by the measure product λ × P, where λ is the Lebesgue measure on T. If u : T × Ω → X is (G µ , B(X))-measurable, then u is called a predictable random function, where G µ denotes the completion of G by |µ|-null sets, where |µ| is the variation of µ (see Definition 2.10 in [22]).
The random function given by Corollary 1 is called an A n r,C -valued (U, p)-random function or, in short, U-random function for p = 0.
Remark 5. Random functions described in the proof of Theorem 1 are A r,C generalizations of the classical Brownian motion processes and of the Wiener processes.
Let w(t) be the A n r,C -valued (U, p)-random function provided by Theorem 1 and Corollary 1. Let a normal filtration {F t : t ∈ T} on (Ω, F , P) be induced by w(t). Therefore, w(t) is for each t 1 and t 1 + t 2 in T with t 2 > 0. In view of Theorem 1 and Corollary 1, conditions Remark 4). Suppose that {S(t) : t ∈ T} is an L r (A n r,C , A h r,C ) valued random function (that is, random operator), S(t) = S(t, ω), ω ∈ Ω (see also the notation in Remark 4). It is called elementary if a finite partition 0 = t 0 < t 1 < . . . < t k = s exists, so that where S l : A stochastic integral relative to w(t) and the elementary random function S(t) is defined by the formula: where t ∧ t = min(t, t ) for each t and t in T. Similarly, elementary L r,i (A n r,C , A h r,C ) random functions and their stochastic integrals are defined. Put for each x and y in A h r,C , where y = (y 1 , . . . , y h ) with y l ∈ A r,C for each l,z = z 0 − z for each z = z 0 + z in A r,C with z 0 ∈ R and z ∈ A r,C , Re(z ) = 0.
Q * denotes an adjoint operator of an R-linear operator Q : A n r,C → A h r,C , such that < Qx, y >=< x, Q * y > (29) for each x ∈ A n r,C and y ∈ A h r,C . Then, we put for Q = A + iB with A and B in L r,i (A n r,C , A h r,C )

and let
(ii) w = w 0 + iw 1 be an A n r,C -valued random function with U 0 -and U 1 -random functions w 0 and w 1 , respectively, having values in A n r , so that U 0 and U 1 belong to L r,i (A n r,C , A n r,C ), and operator U = U 0 + iU 1 fulfils Conditions 2.3(α) and of Definition 2.4 of [22], where w 0 and w 1 are independent; 0 ≤ a < b < ∞, [a, b] ⊂ T (see Definitions 2.10 of [22] , Remarks 4 and 5 above). Then Proof. This follows from Corollary 1 (24), and Formulas (26) and (27) Proof. Since A and B belong to L r,i (A n r,C , A h r,C ), then by Formula (30), where Tr(AA * ) denotes the trace of operator AA * , as usual. On the other side, Since A ∈ L r,i (A n r,C , A h r,C ), then < Ae k , e l >∈ A r for each k = 1, . . . , n, l = 1, . . . , h, where {e k : k = 1, . . . , m} denotes the standard orthonormal base in the Euclidean space R m , where m = max(n, h); R n is embedded into A n r,C as i 0 R n . Therefore, we deduce using Formulas (28), (29), and (33) that since Tr(AA * ) = ∑ l < AA * e l , e l >= ∑ l,k < A * e l , e k >< e k , A * e l >. This implies Formula (31). From the Cauchy-Bunyakovskii-Schwarz inequality, Remark 4, Formulas (31) and (34), one obtains Inequality (32). Theorem 2. If S(t) is an elementary random function with values in L r,i (A n r,C , A h r,C ) and w(t) is an U-random function in A n r as in Definition 4 with U ∈ L r,i (A n r,C , A n r,C ), then P-almost everywhere for each 0 ≤ a < t ∈ T.
Proof. Since Ew(t) = 0 and U : A n r,C → A n r,C , U ∈ L r,i (A n r,C , A n r,C ) by the conditions of this theorem, a j ∈ A r \ {0} for each j and hence U 1/2 : A n r,C → A n r,C and U 1/2 ∈ L r,i (A n r,C , A n r,C ), since U satisfies the conditions of Definition 2.4 and 2.3(α) [22] (see also Theorem 1). Therefore, w(t, ω) ∈ A n r ; hence, S(t, ω)w(t, ω) ∈ A h r for each t ∈ T and P-almost all ω ∈ Ω, where w(t) is a shortening of w(t, ω), while S(t) is that of S(t, ω). On the other hand, Let e l ∈ A n r,C and f l ∈ A h r,C , where e l = (δ l,k : k = 1, . . . , n) and f l = (δ l,k : k = 1, . . . , h), where δ l,k is the Kronecker delta. Then, for an operator J in L r,i (A n r,C , A h r,C ) and each x ∈ A n r,C , the representation is valid: where x = x 1 e 1 + . . . + x n e n , x k ∈ A r,C and J l,k ∈ A r for each k and l.
belong to L r,i (A n r,C , A n r,C ), since the positive definite matrix [B j ] 1/2 with real matrix elements corresponds to the positive definite operator B j for each j, and z 1/2 ∈ A r for each z ∈ A r . By virtue of Proposition 2.5, and Formulas 2.8(2) and 2.8(3) in [22], µ Ut,0 is the A rvalued measure for each t > 0, since the Cayley-Dickson algebra A r is power-associative and exp l (z) = exp(z) for each z ∈ A r .
Random function S(t)w(t) is obtained from the standard Wiener process ξ in R n with the zero expectation and the unit covariance operator with the use of operator U 1/2 : 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 (36)-(39) above and Remarks 4 and 5.

Lemma 3.
If conditions (i) in Theorem 3, (ii) in Lemma 1 are satisfied, then Proof. According to Formula (26) S(t) = S(t l ) for each t l < t ≤ t l+1 , where a = t 0 < t 1 < . . . < t k = b. Since S(t) is (F t l , B(L r (A n r,C , A h r,C ))-measurable for each t ∈ (t l , t l+1 ], then t l+1 a S(t) 2 2 dt is (F t l , B([0, ∞]))-measurable. We consider a modified elementary random 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) for each 0 ≤ a < t in T, where operator U is specified in Definition 2.4 [22], such that U ∈ L r,i (A n r,C , A n r,C ); then, a sequence {S κ (t) : κ ∈ N} of elementary random functions exists with t ∈ T such that for each 0 ≤ a < t in T.

Theorem 5.
If w fulfills Condition (ii) in Lemma 1 and S(t) is a L r (A n r,C , A h r,C )-valued predictable random function satisfying the following inequality: Then, a sequence {S κ (t) : κ ∈ N} of elementary random functions exists with t ∈ T, such that The proof is analogous to that of Theorem 4 with the use of Formula (41), using (49), (50) and (51), since E(E(ζ|F a )) = Eζ with ζ = b a F(S; U 0 , U 1 )(τ)dτ, ζ ≥ 0 Palmost everywhere.

Definition 5.
A sequence {S κ (t) : κ ∈ N} of elementary L r (A n r,C , A h r,C )-valued random functions with t ∈ T is mean absolute square convergent to a predictable L r (A n r,C , A h r,C )-valued random function {S(t) : t ∈ T}, where w satisfies Condition (ii) in Lemma 1, if Condition (51) in Theorem 5 is satisfied. The corresponding mean absolute square limit is induced by Formulas (41) and (51), and is denoted by l.i.m.. The family of all predictable L r (A n r,C , A h r,C )-valued random functions where w = w 0 + iw 1 is an A n r,C -valued random function with U 0 and U 1 random functions w 0 and w 1 , respectively, having values in A n r , where 0 ≤ a ≤ t ≤ b in T, where w satisfies Condition (ii) in Lemma 1.
P-almost everywhere for each 0 ≤ a < t ∈ T.
Proof. From Lemmas 1 and 2, and Proposition 1, Identity (56) follows. Then, Theorem 3 and Proposition 1 imply Inequality (57), since E(E(ζ|F a )) = Eζ with ζ = b a F(S; U 0 , U 1 )(t)dt and since for each t ∈ [a, b]. From Proposition 3, it follows that η(t) is defined P-almost everywhere. By virtue of Theorem IV.2.1 in [5], η(t) is the separable random function up to the stochastic equivalence since (A h r,C , | · |) is the metric space. Therefore, η(t) is considered to be the separable random function.

Definition 6.
Let ζ(t), t ∈ T, be a L h r,C -valued random function adapted to the filtration {F t : t ∈ T} of σ-algebras F t and let E|ζ(t)| < ∞ for each t ∈ T. If E(ζ(t)|F s ) = ζ(s) for each s < t in T, then the family {ζ(t), F t : t ∈ T} is called a martingale. If ζ(t) ∈ R for each t ∈ T and E(ζ(t)|F s ) ≥ ζ(s) for each s < t in T, then {ζ(t), F t : t ∈ T} is called a sub-martingale.
For each S ∈ V 2,1 (U 0 , U 1 , a, b, n, h) according to Definition 5 and the Fubini theorem b a E(F(S; U 0 , U 1 ))(t)dt < ∞. By virtue of Theorem 5, there exists a sequence {S κ (t) : κ ∈ N} of elementary L r (A n r,C , A h r,C )-valued random functions, such that Limit (51) is satisfied. From Lemma 5 and the Fubini theorem, we infer that Therefore, there exists a sequence { κ : κ ∈ N} with lim κ→∞ κ = 0 and a sequence {n k ∈ N : k ∈ N}, such that In view of the Borel-Cantelli lemma (see, for example, Chapter II, Section 10 [7]) a natural number k 0 ∈ N exists, such that satisfying Condition (ii) in Lemma 1, where n and h are natural numbers. A stochastic Cauchy problem over A r,C is: where Y(t) is an A h r,C -valued random function, ζ is an A h r,C -valued random variable which is F a -measurable, t ∈ [a, b] ⊂ T, 0 ≤ a < b, where H, G, w are as in (66)-(68). Problem (69) is understood as the following integral equation: Then, the random function Y(t) is called a solution if it satisfies Conditions (71)-(73): where Y(t) is a shortened notation of Y(t, ω).

Theorem 7.
Let G(t, y) and H(t, y) be Borel functions, w satisfy Condition (ii) in Lemma 1, and K = const > 0 be such that Then, a solution Y of Equation (70) exists (see Definition 7); if Y and Y 1 are two stochastically continuous solutions, then Proof. We consider a Banach space In view of Proposition 2, there exists operator Q on B 2,∞ such that for each t ∈ [a, b], since G and H satisfy Condition (ii) of this theorem. Then, QX(t) is (F t , B(A h r,C ))-measurable for each t ∈ [a, b], since G and H are Borel functions and X ∈ B 2,∞ . By virtue of Proposition 3, using the inequality (α + β + γ) 2 ≤ 3(α 2 + β 2 + γ 2 ) for each α, β and γ in R, the Cauchy-Bunyakovskii-Schwarz inequality, (75), (76), and Condition (ii) of this theorem, we infer that Then, using the Cauchy-Bunyakovskii-Schwarz inequality, 2.3(12) of [22], Proposition 3, Condition (i) of this theorem, and inequality (α + β) 2 ≤ 2(α 2 + β 2 ) for each α and β in R, we deduce that /m! for each X and X 1 in B 2,∞ , m = 1, 2, 3, . . .. Therefore, for each m = 1, 2, 3, . . .. Hence, the series ∑ ∞ m=1 Q m+1 X − Q m X B 2,∞ converges. Thus, the following limit exists lim m→∞ Q m X(t) =: Y(t) in B 2,∞ . From the continuity of Q, it follows that lim m→∞ Q(Q m X) = QY, hence QY = Y. Thus, . This means that Y(t) is the solution of Equation (70). In view of Theorem 6 and Condition (ii) of this theorem, solution Y(t) is stochastically continuous up to the stochastic equivalence. Now, let Y and Y 1 be two stochastically continuous solutions of Equation (70). We consider a random function q N (t), such that q On the other hand, Then, using the Fubini theorem, 2.3(12) of [22], Proposition 3, Lemma 5, we deduce that The Gronwall inequality (see Lemma 3.15 in [2], Lemma 1 in Chapter 8, Section 2 in [5] Random functions Y(t) and Y 1 (t) are stochastically continuous and hence stochastically bounded. Consequently, Therefore, random functions Y(t) and Y 1 (t) are stochastically equivalent. This implies Equality (74). Corollary 2. Let operators G and H be G ∈ L r (A h r,C , A h r,C ) and H ∈ L r (A n r,C , A h r,C ) such that G be a generator of a semigroup {S(t) : t ∈ [0, ∞)}. Let w(t) also be a random function fulfilling Condition (ii) in Lemma 1. Then, the Cauchy problem where t ∈ T, E[ ζ 2 ] < ∞, has a solution where q k : A h r,C → A r,C , u k : Ω → A r,C , m ∈ N. This implies that u k (ω) is independent of F t for each k. Therefore, using (82), we deduce that for q of the form (83). This implies that where [22], since g and f are bounded, where f C := sup z∈A h r,C f (z) < ∞. Therefore, for each > 0, there exists ω)) has the decomposition of type (83) and such that E[ q ( ) (Y(t), ω) − q(Y(t), ω) 2 ] < /C 2 g . Taking ↓ 0, one obtains that Formulas (84) and (85) are accomplished for each f ∈ C 0 b (A h r,C , A r,C ). Therefore, Thus, Equality (80) is proven.

Conclusions
The results obtained in this paper, namely, random functions and measures in modules over the complexified Cayley-Dickson algebras, and the integration of the generalized diffusion PDE, open new opportunities for the integration of PDEs of an order higher than 2. Indeed, a solution of a stochastic PDE with real or complex coefficients of an order higher than 2 can be decomposed into a solution of a sequence of PDEs of order 1 or 2 with A r,C coefficients [9,24]. They can be used for further studies of random functions and integration of stochastic differential equations over octonions and the complexified Cayley-Dickson algebra A r,C . Equations of the type (70) are related with generalized diffusion PDEs of the second order. For example, this approach can be applied to PDEs describing nonequilibrium heat transfer, fourth order Schrödinger-or Klein-Gordon-type PDEs.
Another application of obtained results is for the implementation of the plan described in [22]. It is related with investigations of analogs of Feynman integrals over the complexified Cayley-Dickson algebra A r,C for solutions of PDEs of orders higher than 2.

Appendix A. Basics on Hypercomplex Numbers
Remark A1. Quaternions and octonions (over the real field R) are the particular cases of hypercomplex numbers. The algebra O of octonions (octaves, the Cayley algebra) is defined as an eight-dimensional nonassociative algebra over R with a basis, for example, b 3 := b := {1, i, j, k, l, il, jl, kl} such that (A1) is the multiplication law in the octonion algebra O for each α, β, γ, δ ∈ H, where H denotes the quaternion skew field, ξ : The octonion algebra is neither commutative nor associative, since (ij)l = kl, i(jl) = −kl, but it is distributive and R1 is its center. If ξ := α + βl ∈ O, theñ is called the adjoint element of ξ, where α, β ∈ H. Then (ξη)˜. =ηξ,ξ +η = (ξ + η)˜. and ξξ = |α| 2 + |β| 2 , where |α| 2 = αα such that ξξ =: |ξ| 2 and |ξ| (A6) is the norm in O. Therefore, |ξη| = |ξ||η| (A7) Consequently, O does not contain divisors of zero (see also [12,[25][26][27]). The multiplication of octonions satisfies Equations (A8) and (A9) below: which forms the alternative system. In particular, (ξξ)ξ = ξ(ξξ). We consider also the Cayley-Dickson algebras A n over R, where 2 n is its dimension over R. They are constructed by induction starting from R such that A n+1 is obtained from A n with the help of the doubling procedure, in particular, A 0 := R, A 1 = C, A 2 = H, A 3 = O and A 4 is known as the sedenion algebra [10,28]. The Cayley-Dickson algebras are * -algebras, that is, there is a real-linear mapping A n a → a * ∈ A n such that for each a, b ∈ A n . Then, they are nicely normed, that is, a + a * =: 2Re(a) ∈ R and (A12) The norm in it is defined by the equation: We also denote a * byã. Each nonzero Cayley-Dickson number 0 = a ∈ A n has a multiplicative inverse given by a −1 = a * /|a| 2 .
The doubling procedure is as follows. Each z ∈ A n+1 is written in the form z = a + bl, where l 2 = −1, l / ∈ A n , a, b ∈ A n . The addition is component-wise. The conjugate of a Cayley-Dickson number z is prescribed by the formula: Multiplication is given by: (a + bl)(c + dl) = (ac −db) + (da + bc)l (A16) for each a, b, c, d in A n .
where x ∈ U, t ∈ V. Evidently, H k,l (U × V, λ n+1 , A r,C ) has a structure of a Hilbert space over R, also of a two-sided A r,C -module. Particularly, H 0 (U, λ n , A r,C ) = L 2 (U, λ n , A r,C ). Using the change in variables, we consider operators with constant coefficientŝ for each f ∈ H 2 (R n , λ n , A r,C ), where b u,k;j ∈ R for every u, k, j, β j = m 0 + . . . + m j , m 0 = 0, β 0 = 0. [B j ] denotes a matrix with matrix elements b u,k;j ∈ R for every u and k in {1, . . . , m j }, where j = 1, . . . , m. B j notates a linear operator B j : R m j → R m j prescribed by its matrix [B j ].
Since the operatorB j is elliptic, then without loss of generality, matrix [B j ] is symmetric and positive definite. Then, using a variable change, it is also frequently possible to impose the condition Re(ψ k;j ψ * i;l ) = 0 if either k = i or j = l. Let A be a unital normed algebra over R, where A may be nonassociative, and let its center Z(A) contain the real field R. Then, by l ∏ m k=1 u k , we denote an ordered product from right to left, such that for each m ≥ 2, where l ∏ 1 k=1 u k = u 1 ; u 1 , . . . , u m are elements of A. Then, we put where l (z n ) = l ∏ n k=1 z, which corresponds to the ordered product from right to left (see above (A28)), z ∈ A, that is, for the particular case u 1 = z,. . . .,u n = z.
Definition A1. Let X be a right module over A r,C such that X = X 0 ⊕ X 1 i 1 ⊕ . . . ⊕ X 2 r −1 i 2 r −1 , where X 0 ,. . . ,X 2 r −1 are pairwise isomorphic vector spaces over C. If an addition x + y in X is jointly continuous in x and y and a right multiplication xb is jointly continuous in x ∈ X, and b ∈ A r,C and X j is a topological vector space for each j ∈ {0, 1, . . . , 2 r − 1}, then X is a topological right module over A r,C .
For the right module X over A r,C an operator h from X into A r,C is called right A r,C -linear in a weak sense if and only if it h( f b) = (h( f ))b for each f ∈ X 0 and b ∈ A r,C . Then, X * r denotes a family of all continuous right A r,C -linear operators h : X → A r,C in the weak sense on the topological right module X over A r,C .
An operator h : X → A r,C is right A r,C -linear if and only if h( f b) = (h( f ))b for each f ∈ X and b ∈ A r,C .
Symmetrically, on a left module Y over A r,C such that where Y 0 ,. . . ,Y 2 r −1 are pairwise isomorphic vector spaces over C are defined left A r,C -linear operators and left A r,C -linear in a weak sense operators. A family of all continuous left A r,C -linear operators g : Y → A r,C on the topological left module Y over A r,C in the weak sense is denoted by Y * l . X is a two-sided module over the complexified Cayley-Dickson algebra A r,C if and only if it is a left and right module over A r,C and i j x j = x j i j for each x j ∈ X j and j ∈ {0, 1, . . . , 2 r − 1}.
Theorem A1. Let a PDOŜ be of the form (A25), fulfilling the condition for each t ∈ R and t 1 ∈ R. Definition A3. Let Ω be a set with an algebra R of its subsets and an A r,C -valued measure µ : R → A r,C , where 2 ≤ r, Ω ∈ R. Then is called a variation, and |µ|(Ω) is a norm of the measure µ, where is the decomposition of the measure µ. µ j,k : R → R, |µ j,k | denotes the variation of a real-valued measure µ j,k for each j = 0, 1, . . . , 2 r − 1 and k = 0, 1, |µ| : R → [0, ∞).
A class G of subsets of a set Ω is compact if, for any sequence G k of its elements fulfilling ∞ k=1 G k = ∅, a natural number l exists so that l k=1 G k = ∅. An A r,C -valued measure µ (not necessarily σ-additive, i.e., a premeasure in another terminology) on an algebra R of subsets of the set Ω is approximated from below by a class H, where H ⊂ R, if for each A ∈ R and > 0 a subset B belonging to the class H exists, such that B ⊂ A and |µ|(A \ B) < (see Formula (A39)).
The A r,C -valued measure µ on the algebra R is called Radon if it is approximated from below by the compact class H. In this case, the measure space (Ω, R, µ) is called Radon.

Remark A4. Different forms of the diffusion PDE.
In the classical case over the real field R, different forms of the diffusion PDE such as backward Kolmogorov, Fokker-Planck-Kolmogorov, and stochastic are provided by Theorems 6 and 7 in Chapter I, Section 4, Theorem 4 in Chapter VIII, Section 2 in [5], or by Theorems 3.7, 3.11 in Chapter 3, Section 3.8 in [2]. The stochastic PDE ξ t,x (s) = x + s t a(u, ξ t,x (u))du + m ∑ k=1 s t b k (u, ξ t,x (u))dw k (u) is considered to be the diffusion PDE with m variables in Equation (14) in Chapter VIII, Section 2 in [5], where (b 1 , . . . , b m ) denotes the diffusion operator reduced to the diagonal form, a is the transition (generally may be nonlinear shift) operator, (w 1 , . . . , w k ) denotes the Gaussian-Wiener process with values in the Euclidean space R m . Solutions of the diffusion PDE in its stochastic form provide evolutionary operators and their generators serving for solutions of backward Kolmogorov or Fokker-Planck-Kolmogorov PDEs (see [1,2,5]).
Following this terminology, a generalized analog of the Fokker-Planck-Kolmogorov PDE or backward Kolmogorov is obtained by substituting their partial differential operator by the partial differential operatorŜ given by Formula (A25) in Remark A3. The generalized diffusion PDE itself (in the stochastic form) is Equation (70) in Definition 7 above.

Conflicts of Interest:
The authors declare no conflict of interest.