© Hindawi Publishing Corp. STOCHASTIC PROCESSES ON NON-ARCHIMEDEAN BANACH SPACES

Non-Archimedean analogs of Markov quasimeasures and stochastic processes are investigated. They are used for the development of stochastic antiderivations. The non-Archimedean analog of the Ito formula is 
proved.


Introduction.
Stochastic differential equations on real Banach spaces and manifolds are widely used for the solutions of mathematical and physical problems and also for the construction and investigation of measures on them [4,17]. In particular, stochastic equations can be used for the construction of quasi-invariant measures on topological groups. On the other hand, non-Archimedean functional analysis has developed rapidly in recent years, as well as its applications in mathematical physics [1,19,21,22]. Wide classes of quasiinvariant measures including analogs of Gaussian type on non-Archimedean Banach spaces, loops, and diffeomorphisms groups were investigated in [7,8,9,11,12,15]. Quasi-invariant measures on topological groups and their configuration spaces can be used for the investigation of their unitary representations (see [10,12,13,15] and the references therein).
In view of this developments, non-Archimedean analogs of stochastic equations and diffusion processes need to be investigated. Some steps in this direction were made in [6]. There are different variants for such activity, for example, p-adic parameters analogous to time, but with stochastic processes on spaces of complex-valued functions.
At the same time measures may be real, complex, or with values in a non-Archimedean field.
This work treats the case that was not considered by other authors and that is suitable and helpful for the investigation of stochastic processes and quasiinvariant measures on non-Archimedean topological groups. Here are considered spaces of functions with values in Banach spaces over non-Archimedean local fields, in particular, with values in the field Q p of p-adic numbers. For this, non-Archimedean analogs of stochastic processes are considered on spaces of functions with values in the non-Archimedean infinite field with a nontrivial valuation such that a parameter analogous to the time is p-adic (see Section 4.1 and Definition 4.1). Their existence is proved in Theorem 4.2. Specific antiderivation operators generalizing Schikhof antiderivation operators on spaces of functions C n are investigated (see Section 2). Their continuity and differentiability properties are given in Lemmas 2.1, 2.2, and Theorem 2.12. Also operators analogous to nuclear operators are studied (see Propositions 2.7, 2.8, and 2.9). In Section 3, non-Archimedean analogs of Markov quasimeasures are defined and Propositions 3.2 and 3.3 about their boundedness and unboundedness are proved. The non-Archimedean stochastic integral is defined in Section 4.2. Its continuity as the operator on the corresponding spaces of functions is proved in Proposition 4.4. In Theorems 4.5, 4.7, and Corollary 4.6, analogs of the Itô formula are proved. Spaces of analytic functions lead to simpler expressions of the Itô formula analog, but the space of analytic functions is very narrow, and though it is helpful in non-Archimedean mathematical physics, it is insufficient for the solutions of all mathematical and physical problems. For example, in many cases of topological groups for non-Archimedean manifolds, spaces of analytic functions are insufficient. On the other hand, for spaces C n rather simple formulas are found. All results of this paper are obtained for the first time.

Specific antiderivation of operators
2.1. Let X := c 0 (α, K p ) be a Banach space over a local field (see [23]) K p such that K p ⊃ Q p , {e j : j ∈ α} denotes the standard orthonormal base in c 0 (α, K p ), where α is an ordinal [5] and e j = (0,...,0, 1, 0,...) with the unit on the jth place, j ∈ α [21]. The space c 0 (α, K p ) consists of vectors x = (x j : x j ∈ K p ,j ∈ α) such that for each > 0, a set {j : j ∈ α; |x j | > } is finite, x := sup j |x j |. It is convenient to supply the set α with the ordinal structure due to the Kuratwoski-Zorn lemma. Let F be a continuous function on B r ×C 0 (B r ,X) ⊗k with values in C 0 (B r ,X): where Z ⊗k = Z ⊗···⊗Z is the product of k copies of a normed space Z and Z ⊗k is supplied with the box (maximum norm) topology [5], B r := B(K p ,t 0 ,r ) is a ball in K p containing t 0 and of radius r , Banach spaces C t (M, X) of mappings f : M → X from a C ∞ -manifold M with clopen charts modelled on a Banach space Y over K p into X of class of smoothness C t with 0 ≤ t < ∞ are the same as in [10,12,15] with the supremum-norm ofΦ v f , 0 ≤ v ≤ t, when M is closed and bounded in the corresponding Banach space. Such mappings can be written in the form where F j ∈ C 0 (B r × C 0 (B r ,X) ⊗k , K p ) for each j ∈ α. In particular, let where L(X, Y ) denotes a Banach space of continuous linear operators A : X → Y supplied with the operator norm A := sup 0≠x∈X Ax Y / x X , and L(X) := L(X, X), A i (v) are continuous linear operators for each v ∈ B r such that There exists the following antiderivation of operators given by (2.3): where v n = σ n (t), {σ n : n = 0, 1, 2,...} is an approximation of the identity in B r , satisfying the following conditions: (i) σ 0 (t) = t 0 , (ii) σ m • σ n = σ n • σ m for each m ≥ n and there exists 0 < ρ < 1 such that from (iii) |x − y| < ρ n , it follows that σ n (x) = σ n (y) and (iv) |σ n (x) − x| < ρ n (see [19,Sections 62 and 79]).

2.2.
Suppose that X and Y are Banach spaces over a (complete relativeto-its-uniformity) local field K. Let X and Y be isomorphic with the Banach spaces c 0 (α, K) and c 0 (β, K) and there are given the standard orthonormal bases {e j : j ∈ α} in X and {q j : j ∈ β} in Y , respectively, then each E ∈ L(X, Y ) has its matrix realisation E j,k := q * k Ee j , where α and β are ordinals, q * k ∈ Y * is a continuous K-linear functional q * k : Y → K corresponding to q k under the natural embedding Y Y * associated with the chosen basis, and Y * is a topologically conjugated or dual space of K-linear functionals on Y .

Let
A be a commutative Banach algebra and A + denote the Gelfand space of A, that is, A + = Sp(A), where Sp(A) in another words spectrum of A was defined in [21,Chapter 6]. Let C ∞ (A + , K) be the same space as in [16,21].
for a locally compact Hausdorff totally disconnected topological space X, where f +g and f g are defined pointwise for each f ,g ∈ C ∞ (X, K). ) (see also [21]).
It was proved in [20], in the case of F with the discrete valuation group, that each continuous F-linear operator A : where P n := P n,A , {P n : n ≥ 0} is a family of projections and P n P m = 0 for each n ≠ m, P n ≤ 1 and P 2 n = P n for each n, U is a partially isometric operator, that is, U| cl( n Pn(E)) is isometric, U | E cl( n Pn(E)) = 0, ker(U) ⊃ ker(A), Im(U) = cl(Im(A)), π ∈ F, |π | < 1 and π is the generator of the valuation group of F.
We restrict our attention to the case of the local field F; consequently, F has the discrete valuation group. If A > 1, we get where λ A ∈ F and |λ A | = A . In view of [16], this is the particular case of the spectral integration on the discrete topological space X. Evidently, for each where J has the spectral decomposition given by formula (2.16), s n := |λ J ||π | n P n . Using this result, it is possible to give the following definition.
Definition 2.6. Let E and H be two normed F-linear spaces, where F is an infinite spherically complete field with a nontrivial non-Archimedean valuation. The F-linear operator A ∈ L(E, H) is called of class L q (E, H) if there exist a n ∈ E * and y n ∈ H for each n ∈ N such that ∞ n=1 a n q E * y n q H < ∞, (2.18) and A has the form Ax = ∞ n=1 a n (x)y n (2.19) for each x ∈ E, where 0 < q < ∞. For each such A, we put where the infimum is taken by all such representations (2.19) of A, Proof. Let A ∈ L q (E, H) and 1 ≤ q < ∞ since the case q = ∞ follows from its definition. Then, A has the representation (2.16). Then due to the ultrametric inequality, due to the Hölder inequality. The case 0 < q < 1 is analogous to the classical one given in [18].
Proposition 2.8. If J ∈ L q (H) and S ∈ L r (H) are commuting operators, the field F is with the discrete valuation group, and 1/q Proof. Since F is with the discrete valuation, then J and S have the decompositions (2.16). Certainly, each projector P n,J and P m,S belongs to L 1 (H) and have the decomposition (2.19). The F-linear span of n,m range(P n,J P m,S ) is dense in H. In particular, for each x ∈ range(P n,J P m,S ), we have J k S l x = λ k J λ l S π nk+ml P n,J P m,S x. Applying Remark 2.5 to the commuting operators J k and S l for each k, l ∈ N and using the base of H, we get projectors P n,J and P m,S which commute for each n and m; consequently, Proposition 2.9. If E is the normed space and H is the Banach space over the field F (complete relative to its uniformity), then Proof. In view of Proposition 2.7, it remains to prove that a n,k (x)y n,k with ∞ n=1 a n,k r y n,k r < 2 −k−2 . Therefore, for each p ∈ N, consequently, using convergence while p tends to ∞, we get Proof. For each δ > 0, there are b n ∈ H * and z n ∈ G such that Sy = ∞ n=1 b n (y)z n for each y ∈ H and ∞ n=1 b n r z n r ≤ ν r r (S) + δ. Therefore, The operator T * exists due to the Hahn-Banach theorem for normed spaces over the spherically complete field F [21].
where E and H are over the spherically complete field F.

Proof.
For each δ > 0, there are a n ∈ E * and y n ∈ H such that T x = ∞ n=1 a n (x)y n for each x ∈ E and ∞ n=1 a n r y n r ≤ ν r , that is correct due to the Hahn-Banach theorem for E and H over the spherically complete field F [21]. Therefore, ν r r (T * ) ≤ ∞ n=1 y n r a n r ≤ ν r r (T ) + δ since y * H * = y H for each y ∈ H.

For a space
Suppose that (Ω, Ꮾ,λ) is a probability space (with a nonnegative measure λ), where Ꮾ is a σ -algebra of subsets of Ω. We define a K-linear Banach space L q (Ω, Ꮾ,λ; L k,r (H 1 ,...,H k ; H)) and L q (Ω, Ꮾ,λ; L k (H 1 ,...,H k ; H)) as a completion of a family of mappings n j=1 A j Ch W j with A j ∈ L k,r (H 1 ,...,H k ; H) or A j ∈ L k (H 1 ,...,H k ; H), respectively, and W j ∈ Ꮾ and n ∈ N, where Ch W is the characteristic function of a subset W . That is, as consisting of those mappings Ω ν A(ν) ∈ L k,r (H 1 ,...,H k ; H) for which A(ν) r is λ-measurable and (2.29)

2.5.
We consider a C ∞ -manifold X with an atlas At( H) is a completion of a set of all functions f : where H is a Banach space over K. Then, C n (X, H) is the set of all functions f : X → H such that for each x ∈ X there exists a neighborhood x ∈ U ⊂ X for which f | U ∈ C n b (U , H). By L s (Ω, Ꮾ,λ; C n (X, H)), we denote a completion of a space of simple functions n j=1 ξ j (x)Ch W j (ν) with ξ j (x) ∈ C n (X, H), W j ∈ Ꮾ and n ∈ N, relative to the norm where X is the C ∞ Banach manifold on c 0 (α, K) and ξ(x, ν) C n (X,H) is attached to ξ as a function by x ∈ X with parameter ν ∈ Ω such that ξ(x, ν) C n (X,H) is a measurable function by ν. H)).

Markov quasimeasures for a non-Archimedean Banach space
Remark 3.1. Let H = c 0 (α, K) be a Banach space over a local field K. Let ᐁ ᏼ be a cylindrical algebra generated by projections on finite-dimensional over K subspaces F in H and Borel σ -algebras Bf (F ). Denote by ᐁ the minimal σalgebra σ (ᐁ ᏼ ) generated by ᐁ ᏼ . Each vector x ∈ H is considered as continuous linear functional on H by the formula x(y) = j x j y j for each y ∈ H, so there is the natural embedding H H * = l ∞ (α, K), where x = j x j e j , x j ∈ K.

Notes and definitions.
Let T = B(K,t 0 ,r ) and X t = X be a locally Kconvex space for each t ∈ T . Let (X T ,ᐁ) := t∈T (X t , ᐁ t ) be a product of measurable spaces, where ᐁ t is a σ -algebra of subsets of X t ,ᐁ is the σ -algebra of cylindrical subsets ofX T generated by projectionsπ q :X t → X q , X q := t∈q X t , and q ⊂ T is a finite subset of T (see [

4, Section I.1.3]). A function P (t 1 ,x 1 ,t 2 ,A)
with values in C for each t 1 ≠ t 2 ∈ T , x 1 ∈ X t 1 , A ∈ ᐁ t 2 is called a transition measure if it satisfies the following conditions: (i) the set function is a σ -additive measure on (X t 2 , ᐁ t 2 ); (ii) the function For each set q = (t 0 ,t 1 ,...,t n+1 ) of pairwise distinct points in T , there is a measure in X s := t∈s X t defined by the formula where s = q \ {t 0 }, variables x 1 ,...,x n+1 are such that (x 1 ,...,x n+1 ) ∈ E, and x 0 ∈ X t 0 is fixed.

3.2.
Evidently condition (3.8) of Proposition 3.2 is satisfied for the nonnegative normalized transition measure.

3.3.
Let X t = X for each t ∈ T ,X t 0 ,x 0 := {x ∈X T : x(t 0 ) = x 0 }. We define a projection operatorπ q : x x q , where x q is defined on q = (t 0 ,...,t n+1 ) such that x q (t) = x(t) for each t ∈ q, that is, x q = x| q . For every F :X T → C, there corresponds (S q F )(x) := F(x q ) = F q (y 0 ,...,y n ), where y j = x(t j ). F q : X q → C. We put Ᏺ := {F | F :X T → C, S q F are ᐁ q -measurable}. If F ∈ Ᏺ, τ = t 0 ∈ q, then there exists an integral (3.11) Definition 3.4. A function F is called integrable with respect to the Markov quasimeasure µ x 0 if the limit lim q J q (F ) =: J(F) (3.12) along the generalized net by finite subsets q of T exists. This limit is called a "functional integral" with respect to the Markov quasimeasure Remark 3.5. Consider a complex-valued measure P (t,A) on (X, ᐁ) for each t ∈ T := B(K, 0,R) such that A−x ∈ ᐁ for each A ∈ ᐁ and x ∈ X, where A ∈ ᐁ, X is a locally K-convex space, and ᐁ is a σ -algebra of X. Suppose that P is a spatially homogeneous transition measure (see also Section 3.1), that is, for each A ∈ ᐁ, t 1 ≠ t 2 ∈ T , and every x 1 ∈ X, where P (t,A) satisfies the condition The transition measure P (t 1 ,x 1 ,t 2 ,A) is called homogeneous. In particular, for T = Z p we have If P (t,A) is a continuous function by t ∈ T for each fixed A ∈ ᐁ, then (3.16) defines P (t,A) for each t ∈ T , when P (1,A) is given since Z is dense in Z p .

Remark and definition.
Let (Ω, Ᏺ,λ) be a probability space. Points ω ∈ Ω are called "elementary events" and values λ(S) probabilities of events S ∈ Ᏺ. A measurable map ξ : (Ω, Ᏺ) → (X, Ꮾ) is called a random variable with values in X, where Ꮾ is the σ -algebra of a locally K-convex space X. The random variable ξ induces a normalized measure ν ξ (A) := λ(ξ −1 (A)) in X and a new probability space (X, Ꮾ,ν ξ ). We take X = C 0 (T , H) (see Section 2.1) and the σ -algebra Ꮾ which is the subalgebra of the Borel σ -algebra Bf (X) of X, where H is a Banach space over K, T = B(K,t 0 ,R) =: B R , 0 < R < ∞, K is the local field. A random variable ξ : ω ξ(t, ω) with values in (X, Ꮾ) is called a (non-Archimedean) stochastic process on T with values in H.
Events S 1 ,...,S n are called independent in total if P ( n k=1 S k ) = n k=1 P (S k ). Then σ -subalgebras Ᏺ k ⊂ Ᏺ are said to be independent if all collections of events S k ∈ Ᏺ k are independent in total, where k = 1,...,n, n ∈ N. To each collection of random variables ξ γ on (Ω, Ᏺ) with γ ∈ Υ is related the minimal σ -algebra Ᏺ Υ ⊂ Ᏺ with respect to which all ξ γ are measurable, where Υ is a set. The collections {ξ γ : γ ∈ Υ j } are called independent if so are Ᏺ Υ j , where Υ j ⊂ Υ for each j = 1,...,n, n ∈ N.
This definition is justified by the following theorem. Proof. Since H is over the local field, then H has a projection π 0 on its Banach subspace H 0 of separable type over K (see its definition in [21]), that is, H 0 is isomorphic with c 0 (α, K) with countable α. Therefore, a σ -additive measure µ 0 on (H 0 ,Bf (H 0 )) induces a σ -additive measure µ on (H, π −1 Therefore, it is sufficient to consider the case of H of separable type over K. If w is the real-valued nonnegative Haar measure on K with w(B(K, 0, 1)) = 1, then it does not have any atoms since it is defined on Bf (K), each singleton {x} is the Borel subset and w(y + A) = w(A) for each A ∈ Bf (K). Indeed, if w had some atom E, then it would be a singleton since K is the complete separable metric space, and for each disjoint w-measurable subsets A and S in e j is the standard orthonormal base in c 0 (ω 0 , K), and ω 0 is the first countable ordinal, since K is the local field (see [8,9,21]).
Let Z be a compact subset without isolated points in a local field K, for example, Z = B(K,t 0 , 1). Then the Banach space C 0 (Z, K) has the Amice polynomial orthonormal base Q m (x), where x ∈ Z, m ∈ N o := {0, 1, 2,...} [1]. Each f ∈ C 0 has a decomposition f (x) = m a m (f )Q m (x) such that lim m→∞ a m = 0, where a m ∈ K. These decompositions establish the isometric isomorphism If H = c 0 (ω 0 , K), then the Banach space C 0 (T , H) is isomorphic with the tensor product C 0 (T , K)⊗H (see [21,Section 4 [21,Theorem 4.33]). If u i are roots of basic polynomials Q m as in [1], then Q m (u i ) = 0 for each m > i. The set {u i : i} is dense in T . Put Y 1 = C 0 (T , K) and Y 2 = H and J := J 1 ⊗J 2 ∈ L 1 (Y 1 ⊗Y 2 ), where J 1 Q m := α m Q m such that α m ≠ 0 for each m and i |α i | < ∞. Take J 2 to be also nondegenerate.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable:

Manuscript Due
December 1, 2008 First Round of Reviews March 1, 2009