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

Stochastic antiderivational equations on Banach spaces over local non-Archimedean fields are investigated. Theorems about existence and uniqueness of the solutions are proved under definite conditions. In particular, Wiener processes are considered in relation to the non-Archimedean analog of the Gaussian measure.

1. Introduction.This paper continues the investigations of stochastic processes on non-Archimedean spaces [8].In the first part, stochastic processes were defined on Banach spaces over non-Archimedean local fields and the analogs of Itô formula were proved.This part is devoted to stochastic antiderivational equations.In the non-Archimedean case, antiderivational equations are used instead of stochastic integral or differential equations in the classical case.
In Section 2, suitable analogs of Gaussian measures are considered.Certainly they do not have any complete analogy with the classical one, some of their properties are similar and some are different.They are used for the definition of the standard (Wiener) stochastic process.Integration by parts formula for the non-Archimedean stochastic processes is studied.Some particular cases of the general Itô formula from [8] are discussed here more concretely.In Section 3, with the help of them, stochastic antiderivational equations are defined and investigated.Analogs of theorems about existence and uniqueness of solutions of stochastic antiderivational equations are proved.Generating operators of solutions of stochastic equations are investigated.All results of this paper are obtained for the first time.
In this paper the notations of [8] are also used.
Definition 2.1.A cylindrical measure µ on ᐁ P is called q-Gaussian if each of its one-dimensional projections is q-Gaussian, that is, µ g (dx) = C β,γ,q f β,γ,q v(dx), (2.2) where v is the Haar measure on Bf (K) with values in R, g is a continuous K-linear functional on H = c 0 (α, K) giving projection on one-dimensional subspace in H, C β,γ,q > 0 are constants such that µ g (K) = 1, β and γ may depend on g, q is independent of g where 1 ≤ q < ∞, and α ⊂ ω 0 where ω 0 is the first countable ordinal.
If µ is a measure on H, then μ denotes its characteristic functional, that is, μ(g) := H χ g (x)µ(dx), where g ∈ H * , χ g : H → C is the character of H as the additive group (see [8,Section 3.4]).
We have where C > 0 and C 1 > 0 are constants independent of ζ j for b 0 > p 3 and each r > b 0 , 1 ≤ q < ∞ is fixed (see also the proof of [6, Lemma 2.8], [7], and [2, Theorem II.2.1]).Evidently, g(γ) is correctly defined for each g ∈ c 0 (ω 0 , K) * if and only if γ ∈ c 0 (ω 0 , K).In this case the character χ g(γ) : Since K is the local field, there exists x 0 ∈ c 0 such that |g(x 0 )| = g and x 0 = 1.Put g j := g(e j ).Then g ≤ sup j |g j | since g(x) = j x j g j , where x = x j e j := j x j e j with x j ∈ K. Consequently, g = sup j |g j |.We denumerate the standard orthonormal basis {e j : j ∈ N} such that |g 1 | = g .There exists an operator E on c 0 with matrix elements E i,j = δ i,j for each i, j > 1, E 1,j = g j for each j ∈ N. Then | det P n EP n | = g for each n ∈ N, where P n are the standard projectors on sp K {e 1 ,...,e n } [9].When g ∈ {e * j : j ∈ ω 0 }, then evidently, µ g has the form given by (2.5) since µ i (K) = 1 for each i ∈ ω 0 , where e * j (e i ) = δ i,j for each i, j.Suppose now that J ∉ L q (c 0 ).For this, we consider µ g (K \ B(K, 0,r )) On the other hand, there exists a constant C 2 > 0 such that for b 0 > p 3 and for each r > b 0 , we have the following inequality: From the estimates of [2, Lemma II.1.1]and using the substitution z = y 1/2q for y > 0 and z = (−y) 1/2q for y < 0, we get that µ g is not σ -additive, consequently, µ is not σ -additive since P −1 g (A) are cylindrical Borel subsets for each A ∈ Bf (K), where P g z = g(z) is the induced projection on K for each z ∈ c 0 .
For the verification of formula (2.6), it is sufficient at first to consider the measure µ on the algebra ᐁ P of cylindrical subsets in c 0 .Then for each projection µ g , where g ∈ sp K (e 1 ,...,e m ) * , we have where e = (1,...,1) for each x ∈ sp K (e 1 ,...,e m ).Since J ∈ L q , then µ is the Radon measure, consequently, the continuation of µ from ᐁ P produces µ on the Borel σ -algebra of c 0 , hence lim m→∞ μQmg (h) = μg (h), where Q m is the natural projection on sp K (e 1 ,...,e m ) * for each m ∈ N such that Q m (g) = (g 1 ,...,g m ).Using expressions of μi , we get formula (2.6).From this, it follows that if J ∈ L q , then μ(g) exists for each g ∈ c * 0 if and only if γ ∈ c 0 , since μg (h) = μ(gh) for each h ∈ K and g ∈ c * 0 .
Remark 2.4.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 0 := {0, 1, 2,...} [1].Suppose that P n−1 : C n−1 (Z, K) → C n (Z, K) are antiderivations from [11,Section 80], where n ∈ N.Each f ∈ C 0 has a decomposition f (x) = m a m (f )Q m (x), where a m ∈ K.These decompositions establish the isometric isomorphism θ : is a linear injective compact operator such that P 1 P 0 ∈ L 1 , where P j here corresponds to Pj+1 : C j → C j+1 antiderivation operator by Schikhof (see also [11,Sections 54 and 80] and [6, Section I.2.1]).The Banach space C 2 (Z, K) is dense in C 0 (Z, K).Using Theorem 2.2 above and [8, Note 2.3] for q ≥ 1, we get a q-Gaussian measure on C 0 (Z, K), where P 1 P 0 f = j λ j P j f and Jf = j ζ j P j f for each f ∈ C 0 , we put [12,Section 4.R]).Therefore, the antiderivation P n on C n (Z, K) induces the antiderivation P n on C n (Z, H).
where µ i are q-Gaussian measures on Y i induced by J i as above.In particular, for q = 1 we can also take J 1 = P 1 P 0 .The 1-Gaussian measure on C 0 (Z, H) induced by J = J 1 ⊗ J 2 ∈ L 1 with we get the 1-Gaussian measures µ on it also, where t 0 ∈ Z is a marked point.Certainly, we can take other operators J 1 ∈ L q (Y 1 ) not related with the antiderivation as above.

Non-Archimedean stochastic antiderivational equations
3.1.We define a (non-Archimedean) Wiener process w(t, ω) with values in H as a stochastic process (see [8,Definition 4.1]) such that (ii) the random variable w(t, ω)−w(u, ω) has a distribution µ F t,u , where µ is a probability Gaussian measure on C 0 (T , H) described in Definition 2.1.
In the non-Archimedean case, the formula (see [10,Lemma 3.5]) is not valid, but it has another analog.Let X be a locally compact Hausdorff space and let BC c (X, H) denote a subspace of C 0 (X, H) consisting of bounded continuous functions f such that for each > 0 there exists a compact subset V ⊂ X for which f (u) H < for each u ∈ X \ V .In particular, for X ⊂ K, e * ∈ H * , and a fixed t ∈ X in accordance with [12, Theorem 7.22], there exists a K-valued tight measure µ t,ω,e * ,b,k on the σ -algebra Bco(X) of clopen subsets in X such that where H * is a topologically conjugate space, 1 ≤ r ,q ≤ ∞, and 1/r and H = K (so that e * = 1), (3.4) takes a simpler form, which can be considered as another analog of the classical formula.For the evaluation of appearing integrals, tables from [13, Section 1.5.5]can be used.Another important result is the following theorem.
For each y ∈ H and each γ ∈ K, the function Mχ γ (gψ(t, ω)y) is continuous by t ∈ T , consequently, there exists a continuous function φ : T → H such that Mχ γ (gψ(t, ω)y) = χ γ (gφ(t)y) for each y ∈ H and t ∈ T since characters χ γ are continuous from K to C and χ γ (h) = χ 1 (γh) for each 0 ≠ γ ∈ K and h ∈ K and the C-linear span of the family {χ γ : γ ∈ K} of characters is dense in C 0 (K, C) by the Stone-Weierstrass theorem [3,4].On the other hand, when lim j a j = 0 for a sequence a j in K. Therefore, From the equality χ a+b (c) = χ a (c)χ b (c) for each a, b, c ∈ K, the statement of this theorem follows for each γ ∈ K.
, where a and E satisfy the local Lipschitz condition (LLC).Suppose there is a stochastic process of the type Proof of Theorem 3.4.We have max( a(x)−a(y) g , E(x)−E(y) g ) ≤ K x − y g , hence max( a(x) g , E(x) g ) ≤ K 1 ( x g + 1) for each x, y ∈ H and for each 1 ≤ g < ∞, t ∈ B R , and each ω ∈ Ω, where K and K 1 are positive constants, a(x) and E(x) are short notations of a(t, ω, x) and E(t, ω, x) for x = ξ(t, ω), respectively.Let X 0 (t) = x,..., where in general Pa(u,ξ) 1| u=t = a t, ξ(t, ω) − a t 0 ,ξ t 0 ,ω ≠ Pu a(u, ξ) = j a t j ,ξ t j ,ω t j+1 − t j , (3.11) t j = σ j (t) for each j = 0, 1, 2,.... Let M(η) be a mean value of a real-valued distribution η(ω) by ω ∈ Ω.Then where On the other hand, Pu b+m−l ,w(u,ω) l a m−l+b,l u, x(u) • I ⊗b ⊗ a ⊗(m−l) ⊗ E ⊗l g u=t . (3.14) Due to condition (ii) of Theorem 3.4 for each > 0 and 0 < R 2 < ∞, there exists Therefore, there exists a unique solution on each B since sup u X 1 (u)−X 0 (u) < ∞ and lim l→∞ c l C = 0 for each C > 0, hence there exists lim n→∞ X n , where n ∈ N, C j ∈ K, T = B R , since B R has a disjoint covering by balls B(K,x j , r j ), on each such ball there exists a unique solution with a given initial condition on it (i.e., in a chosen point x j such that C j and B(K,x j ,r j ) are independent of ω).If S is a polyhomogeneous function, then there exists n = deg(S) < ∞ such that differentials D m S = 0 for each m > n, but its antiderivative P has D n+1 P S ≠ 0. If S 1 > S 2 , then P S 1 > P S 2 , which we can apply to a convergent series considering terms D m P S (mod p k ) for each k ∈ N. Therefore, where the function ψ is locally constant by t and independent of ω.The term u, X), E(u, X), and a k−l,l (u, X) depending on X locally polynomially or polyhomogeneously for each u, but such locally polynomial or polyhomogeneous functions by X are dense in The proof of Theorem 3.3 is a particular case of the latter proof.
Proposition 3.5.Let ξ be the Wiener process given by (3.8) with the 1-Gaussian measure associated with the operator P 1 P 0 as in Remark 2.4 and let also max a(t, ω, x) − a(v, ω, x) , E(t, ω, x) − E(v, ω, x) for each t and v ∈ B(K,t 0 ,R) λ-a.e. by ω ∈ Ω, where b, C 1 , and C 2 are nonnegative constants.Then ξ has a C 2 -modification with probability 1 and for each t ∈ B(K,t 0 ,R), where q(t) Proof.For the function f (t,x) = x s in accordance with [8, Theorem 4.5], we have hence since |t j − v j | ≤ |t − v|+ρ j for each j ∈ N, where 0 < ρ < 1, hence d( P s * ) ≤ 1, since f ∈ C s as a function by x and ( Φs g)(x; h 1 ,...,h s ;0,..., 0) = D s x g(x)•(h 1 ,...,h s )/s! for each g ∈ C s and due to the definition of g C s .Considering in particular polyhomogeneous g on which d( P s * ) takes its maximum value, we get d( P s * ) = 1.Since P (C 2 ) = 1 for the Markov measure P induced by the transition measures P (v,x,t,S) := µ F t,v (S|ξ(v) = x) for t ≠ v of the non-Archimedean Wiener process (see Theorem 2.2), then ξ has a C 2modification with probability 1.
Note.If we consider a general stochastic process as in [8, Theorem 4.2], then from the proof of Proposition 3.5 it follows that ξ has a modification in the space J(C 0 0 (T , H)) with the probability 1, where J is a nondegenerate correlation operator of the product measure µ on C 0 0 (T , H).
Proposition 3.6.Let ξ be a stochastic process given by (3.8) and let for each t and v ∈ B(K,t 0 ,R) λ-a.e. by ω ∈ Ω, where b, C 1 , and C 2 are nonnegative constants.Then two solutions ξ 1 and ξ 2 with initial conditions ξ 1,0 and ξ 2,0 satisfy the following inequality: for each t ∈ B(K,t 0 ,R), where y(t) Proof.From Proposition 3.5, it follows that Remark 3.7.Let X t = X 0 + Pt a + Pw v and Y t = Y 0 + Pt q + Pw s be two stochastic processes corresponding to E = I and a Banach algebra H over K in [8,Section 4.3].Then  1, where the last term corresponds to (dw t )(dw t ) ≠ 0. This means that d(w 2 ) = 2wdw + (dw)(dw).For X t = w t and Y t = t, the integration by parts formula gives Pw t t = w t t − Pt w t − P(t,w t ) 1 such that P(t,w t ) 1 = j t j [w t j+1 − w t j ] − w t t + j w t j [t j+1 − t j ] ≠ 0, for example, for t = 1, w ∈ C 0 0 (T , H), T = Z p and t 0 = 0 this gives P(t,w t ) 1 = w 1 − w 0 = w 1 .Therefore, (dt)(dw t ) ≠ 0, that is the important difference of the non-Archimedean and classical cases (cf.[10,Exercise 4.3 and Theorem 4.5]).
If H is a Banach space over the local field K and f (x,y) = x * y is a K-bilinear functional on it, where x * is an image of x ∈ H under an embedding H H * associated with the standard orthonormal base {e j } in H, then has a solution under milder conditions, for example, A(t) is weakly continuous, that is, e * A(t)η is continuous for each e * ∈ Y * and η ∈ Y .Then e * U(t,s)η is differentiable by t and U(t,s) satisfies (3.31) in the weak sense and there exists a weak solution of (3.32) coinciding with U(t,s).If to substitute A(t) on another operator Ã(t), then for the corresponding evolution operator Ũ(t,s), there is the following inequality: where M := 1 + sup s,t∈B(K,0,R) U(t,s) and M is for Ũ.

.35)
If MCR<1, then there exists a sequence Ũn (t, s) which is also uniformly bounded.
If there exists U n (t, s) strongly and uniformly converging to U(t,s) in B(K, 0,R), then Ũn (t, s) also can be chosen to be strongly and uniformly convergent.
Proof.From the use of (3.30) and (3.33) iteratively for U n (σ j+1 (t), σ j (t)), for U n (σ j (t), s), and also for Ũn and taking Ũn − U n , it follows that (t, s)x is uniformly convergent to U(t,s)x, then for each > 0 there exist δ > 0 and m ∈ N such that sup t,s∈B(K,0,R) U n (t + h, s + v)x n − U n (t, s)x n < , for each n > m, and max(|h|, |v|) < δ due to equality (3.36).Proposition 3.11.Let a, a m−l+b,l , and E be the same as in Theorem 3.4.Then Theorem 3.4(i) has the unique solution ξ in B R for each initial value ξ(t 0 ,ω) ∈ L q (Ω, Ᏺ,λ; H) and it can be represented in the following form: where T (t,v; ω) is the multiplicative operator functional.
Proof.In view of Theorem 3.4, Definition 3.8, Remark 3.9, and Proposition 3.10 with the use of a parameter ω ∈ Ω, the statement of Proposition 3.11 follows.

3.
3. Now we consider the case J(C 0 0 (T , H)) ⊂ C 1 (T , H) (see Proposition 3.5), for example, the standard Wiener process.Corollary 3.12.Let a function f (t,x) satisfy conditions of [8,Theorem 4.7], then a generating operator of an evolution family T (t,v) of a stochastic process η = f (t, ξ(t, ω)) is given by the following equation:

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
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128) where u, t ∈ T .Hence d(X t Y t ) = X t dY t + (dX t )Y t + (dX t )(dY t ).Therefore, PXt Y t = X t Y t − X 0 Y 0 − PY t X t − P(X t ,Y t )1, which is the non-Archimedean analog of the integration by parts formula, where in all terms X t is displayed on the left of Y t .For two C 1 functions f and g, we have (f g) = f g + f g or d(f g) = gdf + f dg, that is, terms with (dt)(dt) are absent, consequently, (dt)(dt) = 0.In a particular case X t = Y t = w t , this leads to w 2 Proposition 3.10.Let B(t) and two sequences A n (t) and B n (t) be given and strongly continuous on B(K, 0,R) bounded linear operators, and let Ũ(t,s) be evolution operators corresponding to Ãn (t) = A n (t) + B n (t), where sup