AP International Journal of Pure and Applied Mathematics Volume 109 No. 3 2016, 539-556 ISSN:1311-8080(printedversion);ISSN:1314-3395(on-lineversion) url:http://www.ijpam.eu doi:10.12732/ijpam.v109i3.5 ijpam.eu NON-GAUSSIAN WICK CALCULUS BASED ON HYPERCOMPLEX SYSTEMS Abd-Allah Hyder1, M. Zakarya2§ 1Department of Mathematics Faculty of Science Al-Azhar University Naser City, Cairo, EGYPT 2Department of Mathematics Faculty of Science Al-Azhar University Assiut 71524, EGYPT Abstract: In this paper, we developed a non-Gaussian Wick calculus based on the theory of hypecomplex systems L1(Q,dm(x)). Using the Delsarte characters χn(x), we introduce a χ-Wickproduct,aχ-HermitetransformonthespaceofgeneralizedfunctionsHχ anddiscuss −q their properties. By means of the usual properties of complex analytic functions, we proved thecharacterizationtheoremforHχ . Moreover,wesetupaframeworktostudythestochastic −q partial differential equationsdriven by Hχ -processes, and apply this framework tosolve the −q χ-stochastic Poisson equation. AMS Subject Classification: 46FXX, 30G35, 91G80 Key Words: non-Gaussian, Wick product, Hermite transform, white noise, hypercomplex systems 1. Introduction White noise analysis is an important and popular theme which is intensively Received: April25, 2016 (cid:13)c 2016 Academic Publications, Ltd. Revised: July 27, 2016 url: www.acadpubl.eu Published: September30, 2016 § Correspondence author 540 A.-A. Hyder, M. Zakarya and extensively studied in many works (see e.g. the books [21], [22] and ref- erences therein). These investigations are essentially based on the concept of a Gaussian measure and the associated expansion into Hermite polynomials. Gaussian measures are remarkable objects, but they are not invariant (as a class of measures) with respect to nonlinear transformations. Moreover, in ap- plications to some problems of mathematical physics, we need to use measures which areobtained as very singular perturbationsof Gaussian ones (as inquan- tum field theory) or are constructed by the Gibbs approach (as in statistical physics) [23]. Theclassical whitenoiseanalysis (Gaussianwhitenoiseanalysis)itispossi- ble to understandas a theory of generalized functions of infinitemany variables with pairing between test and generalized functions provided by integration with respect to the Gaussian measure. Recently, some authors as Okb El Bab, Zabel,GhanyandHyder[10], Ghany[11],GhanyandHyder[12,13,14],Ghany and Zakarya [15, 16, 17] and Ghany and Qurashi [18], studied some important subjects related to Gaussian white noise analysis. Also, Okb El Bab, Zabel and Ghany [26], introduced some studies of harmonic analysis in hypercomplex systems. Furthermore, Okb El Bab, Ghany and Zakarya [27], studied some im- portant subjects related to a construction of non-Gaussian white noise analysis using the theory of hypercomplex systems. It is well known that there exist several approaches to the construction of such theory of generalized functions: the Berezansky-Samoilenko approach [3] and the Hida approach [20]. In the Berezansky-Samoilenko approach, spaces of test and generalized functions are constructed as infinite tensor products of one-dimensional spaces. The Hida approach consists in the construction of some rigging of a Fock space with sub- sequent application of the Wiener-Itˆo-Segal isomorphismto the spaces of this rigging. Afteranumberofyearsitbecameclear thattheHidaapproachismore convenient, and in most cases all investigations in white noise analysis and its generalizations are based on this approach. There exist many works dedicated to white noise analysis development. In recent years, two approaches to the generalization of the Gaussian infinite- dimensional analysis (white noise analysis) to non-Gaussian measures have ap- peared: one of them is based on spectral theory for families of commuting self-adjoint operators [4, 5] and the other proceeds from biorthogonal expan- sions [1]. Moreover, in the one-dimensional model case, the second approach can be extensively generalized if the characters of an hypercomplex system are used, in its construction, instead of exponential functions. Now,werecall,inbrief,thepropertiesandobjectsofhypercomplexsystems, which we need in our paper. For more details see [2]. NON-GAUSSIAN WICK CALCULUS BASED... 541 Consider a commutative hypercomplex system L (Q,dm(x)) of functions f 1 withlocally compactbasisQ,nonnegative structuremeasurec(A,B,x)(A,B ∈ B(Q),x ∈ Q) and multiplicative measure dm(x) (x ∈ Q); L (Q,dm(x)) is 1 assumed to be normal (Q ∋ x 7→ x∗ ∈ Q is the corresponding involution in the basis) and to have basis unity e∈ Q. We can introduce generalized translation operators for these hypercomplex systems, namely for each function f ∈ C(Q) we construct the function Q × Q ∋ hx,yi 7→ (T f)(y) ∈ C (which is equal x to f(x + y) in the case of ordinary translation on Q = R). We impose an additional condition on L (Q,dm(x)): let the function (T f)(y) be separately 1 x continuous. Using the structure measure, the convolution of functions on Q is defined for general hypecomplex system. But the above mentioned restrictions on L (Q,dm(x)) give the possibility to write this convolution in terms of the 1 operators T , as follows: x (f ∗g)(x) = f(y)(Ty∗g)(x)dm(y), f,g ∈ L1(Q,dm(x)). (1.1) ZQ The hypecomplex system L (Q,dm(x)) with convolution (1.1) is a commu- 1 tative normedalgebra with involution L (Q,dm(x)) ∋ f 7→ f∗ ∈ L (Q,dm(x)), 1 1 where f∗(x) =f(x∗) (x ∈ Q). A generalized character of thehypercomplex system is definedas afunction χ ∈ C(Q) such that (T χ)(y) = χ(x)χ(y), x,y ∈Q. (1.2) x The character is said to be Hermitian (ordinary) if χ(x∗) = χ(x) for x (if it is bounded). The function χ(x) = 1 (x ∈ Q) is always a character; χ(e) = 1 for any χ. Thispaperisdevotedtodevelopanon-GaussianWickcalculusbasedonthe theory of hypecomplex systems L (Q,dm(x)). Using the Delsarte characters 1 χ (x), weintroduceaχ-Wick productandχ-Hermitetransformonthespaceof n χ generalized functions H (with the zero space L (Q,dm(x))) and discuss their −q 2 properties. By means of the usual properties of complex analytic functions, we χ prove the characterization theorem for H . Moreover, we setup a framework −q χ to study the stochastic partial differential equations (SPDEs) driven by H - −q processes, and apply this framework to solve the χ-stochastic Poisson equation. This paper is organized as follows: In Section 2, we recall the rigging of the space L (Q,dρ(x)) using the Delsarte characters. In Section 3, we introduce 2 the χ-Wick product, the χ-Hermite transform and proved the characterization χ theorem for the space of generalized functions H . In Section 4, we give a −q χ framework to study the SPDEs driven by H -processes. In Section 5, we −q 542 A.-A. Hyder, M. Zakarya apply the framework presented in Section 4 to solve the χ-stochastic Poisson equation. 2. A rigging of L2(Q,dρ(x)) Using Delsarte Characters In this section, we consider a subclass of the above hypercomplex systems for which the set of generalized characters is in one-to-one correspondence with the complex plane: χ ←→ λ ∈C; denote this character by χ(x,λ). We assume that χ(x,0) = 1 (x ∈ Q), i.e., the unit character corresponds to λ = 0. The function Q×C ∋ hx,λi 7→ χ(x,λ) ∈ C is assumed to be continuous, and let the function C ∋ λ 7→ χ(x,λ) ∈ C be entire for each x ∈ Q. Thus, for each x ∈ Q the following expansion holds: ∞ λn χ(x,λ) = χ (x), λ ∈ C, (2.1) n n! n=0 X where χ ∈ C(Q) are some coefficients called the Delsarte characters [6, 7]. It n is possible, of course, to give a direct definition of the Delsarte characters if we will rewrite equality (2.1) in the terms of χ . n Let ρ be a fixed Bore1 probability measure on Q i.e., (ρ(Q) = 1), positive on open sets, which is not connected directly with the multiplicative measure m. We suppose that χ ’s belong to L (Q,dρ(x)), for any n and satisfy the n 2 estimate 1 2 ∃ C > 0 : kχ k = |χ (x)|2dρ(x) ≤ Cnn!, n ∈ N (2.2) n L2(Q,dρ(x)) n (cid:18)ZQ (cid:19) (explaining that always kχ k = 1), are linearly independent, and the 0 L2(Q,dρ(x)) system (χ )∞ is total in L (Q,dρ(x)). n n=0 2 The following theorem gives the expansions of functions on Q in the Del- sartecharacters andintroducesthecorrespondingspacesoftestandgeneralized functions. Theorem 2.1. There exists a quasinuclear rigging: Hχ ⊇ H ⊇ Hχ (2.3) −q 0 q where H = L (Q,dρ(x)), for all q ∈ N : 0 2 ∞ ∞ Hχ = ϕ = ϕ χ ∈H :kϕk2 = |ϕ |2(n!)2Kqn < ∞ , q ( n n 0 Hqχ n ) n=0 n=0 X X NON-GAUSSIAN WICK CALCULUS BASED... 543 and ∞ ∞ Hχ = ξ = ξ qχ : kξk2 = |ξ |2K−qn < ∞ for some q ∈N . −q ( n n H−χq n ) n=0 n=0 X X Here, the system (χ ,qχ)∞ , where qχ = I−1χ ∈ Hχ (I : Hχ → Hχ is the n n n=0 n n −q −q q canonical isometry), is a biorthogonal basis of the space H , and K is a fixed 0 sufficiently large number. Moreover, the dual action is given by ∞ (ξ,ϕ) = ξ ϕ (n!)2Kqn, ϕ ∈ Hχ, ξ ∈ Hχ . H0 n n q −q n=0 X Proof. The proof can be found in [7]. 3. χ-Wick Product, χ-Hermite Transform and Characterization χ Theorem for H −q The Wick product was first introduced by Wick [29] and used as a tool to renormalize certain infinite quantities in quantum field theory. Later on, the Wick product was considered, in a stochastic setting, by Hida and Ikeda [19]. In [9], Dobroshin and Minlos were comprehensively treated this subject both in mathematical physics and probability theory. Currently, the Wick product provides a useful concept for various applications, for example, it is important in the study of stochastic ordinary and partial differential equations (see, e.g., [24]). In this section, we define a new Wick product, called χ-Wick product, on χ the space H with respect to an arbitrary Borel probability measure ρ. Then, −q we give the definition of the χ-Hermite transform and apply it to establish a χ characterization theorem for the space H . −q Definition 3.1. Let ξ = ∞ ξ qχ, η = ∞ η qχ ∈ Hχ with ξ ,η m=0 m m n=0 n n −q m n ∈ C. The χ-Wick product of ξ, η, denoted by ξ⋄ η, is defined by the formula χ P P ∞ χ ξ⋄ η = ξ η q . (3.1) χ m n m+n m,n=0 X χ χ It is important to show that the spaces H ,H are closed under χ-Wick −q q product. χ χ Lemma 3.1. If ξ,η ∈ H and ϕ,ψ ∈ H , we have −q q 544 A.-A. Hyder, M. Zakarya χ (i) ξ⋄ η ∈ H , χ −q χ (ii) ϕ⋄ ψ ∈ H . χ q ∞ ∞ Proof. If ξ = ξ qχ, η = η qχ ∈ Hχ , then for some q ∈ N we have m m n n −q 1 m=0 n=0 P P ∞ ∞ |ξ |2K−q1m < ∞ and |η |2K−q1n < ∞. (3.2) m n m=0 n=0 X X We note that ∞ ∞ ∞ ∞ χ χ χ ξ⋄ η = ξ η q = ξ η q = ζ q , (3.3) χ m n m+n m n l l l ! m,n=0 l=0 m+n=l l=0 X X X X ∞ where ζ = ξ η . With q = q +p we have l m n 1 m+n=l P ∞ ∞ ∞ 2 |ζ |2K−ql = ξ η K−q1lK−pl l m n (cid:12) (cid:12) Xl=0 Xl=0(cid:12)mX+n=l (cid:12) (cid:12) (cid:12) ∞ ∞ ∞ (cid:12) (cid:12) ≤ (cid:12) |ξ |(cid:12)2K−q1m |η |2K−q1n K−pl m n ! ! l=0 m+n=l m+n=l X X X ∞ ∞ ∞ ≤ K−pl |ξ |2K−q1m |η |2K−q1n m n ! ! ! l=0 m=0 n=0 X X X < ∞, (3.4) which proves (i). The proof of (ii) is similar. The following important algebraic properties of the χ-Wick product follow directly from Definition 3.1. χ Lemma 3.2. For each ξ,η,ζ ∈ H , we get −q (i) ξ⋄ η = η⋄ ξ (Commutative law), χ χ (ii) ξ⋄ (η⋄ ζ)= (ξ ⋄ η)⋄ ζ (Associative law), χ χ χ χ (iii) ξ⋄ (η+ζ)= ξ⋄ η+ξ⋄ ζ (Distributive law). χ χ χ NON-GAUSSIAN WICK CALCULUS BASED... 545 Remark. According to Lemmas 3.1 and 3.2, we can conclude that the χ χ spaces H and H form topological algebras with respect to the χ-Wick prod- −q q uct. As shown in Lemmas 3.1 and 3.2, the χ-Wick product satisfies all the or- dinary algebraic rules for multiplication. Therefore, one can carry out calcu- lations in much the same way as with usual products. But, there are some problems when limit operations are involved. To treat these situations it is convenient to apply a transformation, called the χ-Hermite transform, which converts χ-Wick productsinto ordinary(complex) productsand convergence in Hχ into bounded, pointwise convergence in a certain neighborhood of 0 in C. −q The original Hermite transform, which first appeared in Lindstrøm et al. [25], has been applied by the authors in many different connections. Now, we give the definition of the χ-Hermite transform and discuss its basic properties. Definition 3.2. Let ξ = ∞ ξ qχ ∈ Hχ with ξ ∈ C. Then, the n=0 n n −q n χ-Hermite transform of ξ, denoted by H ξ, is defined by χ P ∞ H ξ(z) = ξ zn ∈ C (when convergent). (3.5) χ n n=0 X In the following, we define for 0 < M,q < ∞ the neighborhoods O (M) of q zero in C by ∞ O (M) = z ∈ C : |zn|2Kqn < M2 . (3.6) q ( ) n=0 X It is easy to see that q ≤p, N ≤ M ⇒ O (N) ⊆ O (M). (3.7) p q Note that if ξ = ∞ ξ qχ ∈ Hχ , z ∈ O (M) for some 0 < M,q < ∞, we n=0 n n −q q have the estimate P ∞ ∞ |ξn||zn| = |ξn||zn|K−q2nKq2n n=0 n=0 X X 1 1 ∞ 2 ∞ 2 ≤ |ξ |2K−qn |zn|2Kqn n ! ! n=0 n=0 X X 1 ∞ 2 < M |ξ |2K−qn n ! n=0 X 546 A.-A. Hyder, M. Zakarya < ∞. (3.8) The conclusion above can be stated as follows: Proposition 3.1. If ξ ∈ Hχ , then H ξ converges for all z ∈ O (M) for −q χ q all q,M < ∞. A usefulproperty of theχ-Hermite transform is that itconverts theχ-Wick product into ordinary (complex) product. χ Proposition 3.2. If ξ,η ∈ H , then −q H (ξ⋄ η)(z) = H ξ(z).H η(z). (3.9) χ χ χ χ for all z such that H ξ and H η exist. χ χ Proof. TheproofisanimmediateconsequenceofDefinitions3.1and3.2. Let ξ = ∞ ξ qχ ∈ Hχ , with ξ ∈ R. Then, the number ξ = H ξ(0) ∈ n=0 n n −q n 0 χ R is called the generalized expectation of ξ and is denoted by E(ξ). Suppose P that V ∋ z 7→ f(z) ∈ C is an analytic function, where V is a neighborhood of E(ξ). Assume that the Taylor series of f around E(ξ) has coefficients in R. Then, the χ-Wick version f⋄χ of f is defined by Hχ ∋ ξ 7→ f⋄χ(ξ) = H−1 f ◦H (ξ) ∈ Hχ . (3.10) −q χ −q (cid:0) (cid:1) Example 3.1. If the function f : C → C is entire, then f⋄χ is defined for χ all ξ ∈ H . For example, the χ-Wick exponential is defined by −q ∞ 1 exp⋄χ(ξ) = ξ⋄χn. (3.11) j! j=0 X Using χ-Hermite transform, we see that χ-Wick exponential has the same algebraic properties as the usual exponential. For instance, exp⋄χ(ξ +η) = exp⋄χ(ξ)⋄ exp⋄χ(η), ξ,η ∈ Hχ . (3.12) χ −q From Proposition 3.1, we can deduce that χ-Hermite transform of any ξ ∈ Hχ is a complex-valued analytic function on O (M) for all q,M < ∞. −q q Moreover, the converse of this deduction is true, i.e., every complex-valued an- alytic function on O (M) (for some q,M < ∞) is the χ-Hermite transform of q χ someelementinH . Toprovethis,weneedthefollowingtwoauxiliaryresults. −q NON-GAUSSIAN WICK CALCULUS BASED... 547 Lemma 3.3. Let f(z) = ∞ η zn be an analytic function in z ∈ C n=0 n such that there exists M < ∞,C > 0 and δ > 0 such that |f(z)| ≤ M when P z ∈O := {z ∈C : C|z|≤ δ2}. Then |η zn| ≤ M for all z ∈ O and n ∈ N. n Proof. See [24], Lemma 2.6.10. Proposition 3.3. Let f(z) = ∞ η zn, η ∈ C be a formal power series n=0 n n in z ∈C. Suppose there exist q,M <∞ and δ > 0 such that f(z) is convergent P for z ∈ O (δ) and |f(z)| ≤ M for all z ∈O (δ). Then q q ∞ |η zn| ≤ MA(q) for all z ∈ O (δ) (3.13) n 3q n=0 X where ∞ A(q) := K−qn < ∞ (Note that K >1). (3.14) n=0 X Proof. It is evident that z ∈ O (δ) implies Kqz ∈ O (δ). According to 3q q Lemma 3.3, we get 1 1 ∞ ∞ 2 ∞ 2 |η zn| ≤ |η |2|zn|2Kqn K−qn n n ! ! n=0 n=0 n=0 X X X 1 1 ∞ 2 ∞ 2 = |η |2|(Kqz)n|2K−qn K−qn n ! ! n=0 n=0 X X ∞ ≤ M K−qn. (3.15) n=0 X Theorem 3.1. (Characterization Theorem for Hχ ) If ξ = ∞ ξ qχ ∈ −q n=0 n n Hχ , where ξ ∈C, then there exist q < ∞ and R < ∞ such that −q n q P 1 ∞ 2 |H ξ(z)| ≤ R |zn|2Kqn ∀z ∈ C. (3.16) χ q ! n=0 X In particular, H ξ is a bounded analytic function on O (M) for all M < ∞. χ q Conversely, suppose f(z)= ∞ η zn is a givenanalytic power seriesof z ∈C n=0 n with η ∈ C such that there exist q < ∞ and δ > 0, such that f(z) is absolutely n P convergent when z ∈ O (δ) and q sup |f(z)| < ∞. (3.17) z∈Oq(δ) 548 A.-A. Hyder, M. Zakarya χ Then, there exists a unique η ∈ H such that H η = f, namely −q χ ∞ η = η qχ. (3.18) n n n=0 X Proof. For each z ∈ C, we have 1 1 ∞ ∞ 2 ∞ 2 |H ξ(z)| ≤ |ξ ||zn|≤ |ξ |2K−qn |zn|2Kqn .(3.19) χ n n ! ! n=0 n=0 n=0 X X X Since ξ ∈Hχ , we see that R2 := ∞ |ξ |2K−qn < ∞ for all q < ∞. −q q n=0 n Conversely, Since K > 1, then K−r ∈ O (δ) for all r < ∞ and for some r P δ < ∞. By virtue of Proposition 3.3, we have ∞ |η ||zn| ≤ MA(q) for all z ∈ O (δ). (3.20) n 3q n=0 X Hence, for r ≥ 3q and z ∈ O (δ), we get 3q ∞ ∞ ∞ |η |2K−rn ≤ C |η |K−rn ≤ C |η ||zn| ≤ CMA(q)< ∞. (3.21) n n n n=0 n=0 n=0 X X X where C := sup{|η | : n ∈ N}, and hence η := ∞ η qχ ∈ Hχ , as claimed. n n=0 n n −q P 4. Framework for χ-SPDEs In the Gaussian case, if the objects of a differential equation are regarded as (S) -valued ((S) is the Kondratiev space of stochastic distributions con- −1 −1 structed upon Gaussian measure), we often obtain a more realistic mathemat- ical model of the situation. This model called a Wick-type stochastic differ- ential equation (see [24] for more details). Analogously, we can introduce a χ non-Gaussian Wick-type stochastic model by the replacement of (S) on H −1 −q and the Wick product associated with the Gaussian measure on the χ-Wick product. χ One of the many useful properties of H it that it contains the χ-white −q noise, which is defined by the formal expansion ∞ W (t,x) = χ (t,x)qχ, t ∈ R, x ∈ Q. (4.1) χ n n n=0 X