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A new construction of random Colombeau distributions
Statistics & Probability Letters 54 (2001) 291 – 299
A new construction of random Colombeau distributions Ulu"g C$apar ∗ , H(useyin Aktu"glu Department of Mathematics, Eastern Mediterranean University, G. Magusa, Mersin-10, Turkey Received October 2000
Abstract In this paper a generalized random process is modeled through the randomization of a bilinear form between the space D of test functions and the Colombeau generalized functions. This results in a theory akin to Gelfand–Vilankin’s random Schwartz distributions. An extension theorem in Bochner–Badrikian style is proved under some continuity assumptions. c 2001 Elsevier Science An important application is a natural representation of nonlinear functionals of white noise. B.V. All rights reserved MSC: 28C20; 46F99; 60B05; 60G20 < Keywords: Random Colombeau distributions; C-duality; Extension of probability measures in generalized duals; Nonlinear functions of the white noise
1. Basic notions Colombeau’s new nonlinear theory of generalized functions has been introduced through the main texts (Colombeau, 1984; Rosinger, 1987; Egorov, 1990). The basic structure is the in@nite product algebra E() = (C∞ ())A0 , where is an open subset of Rn . DiCerent index sets Aq (q = 0; 1; 2; : : :) have been used in the literature. Here we adopt the index set system introduced by Biagioni (1990), which has the merit of being independent of the choice of bases in Rn and is given by ∞ 1 A0 (R) = 1 ∈ D(R) : 1 is even; constant in a O-neighborhood and (1) 1 (x) d x = 2 0 ∞ Aq (R) = 1 ∈ A0 : xj=m 1 (x) d x = 0; 1 6 j; m 6 q ; q = (1; 2; : : :); (2) 0
and
2n n n Aq (R ) = ∈ D(R ) : ∃1 ∈ Aq (R); (x) = 1 (|x| ) ; n n
(q = 0; 1; 2; : : :):
∗ Corresponding author. E-mail addresses: [email protected] (U. C$apar), [email protected] (H. Aktu"glu).
c 2001 Elsevier Science B.V. All rights reserved 0167-7152/01/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 1 ) 0 0 1 0 1 - 8
(3)
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(n : the surface area of the (n − 1)-dimensional ball centered at the origin). The Colombeau algebra of generalized functions is the factor algebra G() = EM ()=N();
(4)
where EM () is the subalgebra of E() consisting of families of moderate growth along all molli@ers (x) = −N (x= )( ∈ AN for some N ) and N() is the ideal of null members of EM (). Thus EM () has members which grow at most polynomially in 1= as → 0, along molli@ers and uniformly on compacts, whereas N() includes those elements of EM () vanishing faster than any power of (cf. Rosinger, 1987; Biagioni, 1990). If algebra operations and diCerentiation is de@ned representative-wise, G() becomes a diCerential algebra. For T ∈ [C∞ ()] (i.e. a Schwartz distribution of compact support) the mapping T → (T ∗ |) + N() is an injection of [C∞ ())] . Using the sheaf property of G (cf. Rosinger, 1987, p. 131) it can be extended to an injection of D () and this injection preserves C∞ () as a faithful subalgebra. Furthermore diCerentiation in G when restricted to D , coincides with the usual distributional derivative. As an example, using the above injection and diCerentiation rule we have the following representations of one dimensional delta and Heaviside functions: = T + N;
T (; x) = (−x);
H = TH + N;
TH (; x) =
∞
−x
∀ ∈ A0 (R);
(t) dt;
∀ ∈ A0 (R)
(cf. Rosinger, 1987). However the imbedding of the C0 () does not extend the point-wise product in accordance with Schwartz’s “impossibility result”. Unlike linear ones, generalized distributions have point values 0 belonging to the algebra C< of generalized constants (g.c.’s). C< = EM =J is roughly speaking the algebra of 0 Colombeau generalized functions (C.g.f.) on R = {0}. Because of the index set used there is a unique algebra of g.c.’s for any dimension n. For T ∈ G(), T = R + N, the point value T (x) is given by T (x) = Rx + J;
Rx () = R(In1 ; x);
∈ A0 (R)
(5)
where In1 : A0 (R) → A0 (Rn ) is the bijection described in Biagioni (1990, Proposition 1:2:11). (e.g. (0) = T; 0 + J; T; 0 () = (0)): < a There is a canonical injection of C into C< through z = h + J where h() = z for all ∈ A(R). In C, base of neighborhoods of the origin is introduced by the following sets: 0 Q := {z< ∈ C< : ∃h ∈ EM ; z< = h + J; such that |z()|¡ ∀ ∈ A0 }
(6)
or equivalently 0 ; z< = h + J; ∃N ¿ 0 such that ∀ ∈ AN (R)∃¿0 Q = {z< ∈ C< : ∃h ∈ EM
with |h( )|¡ if 0¡ ¡}:
(7)
< The related topology is discussed in the next section, but (6) induces a R-valued seminorm · − (some of the axioms may be satis@ed as +∞ = + ∞) inf {¿0 : z< ∈ Q } if ∃¿0; z< ∈ Q ; −
z
< := (8) +∞ if ∀¿0; z< ∈ Q ; As an example exp(i(0)) − = 1, but (0) − = + ∞. (8) also satis@es z
< − 6 z
< − w
< −.
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293
A g.c. z< is said to be associated with complex number z, denoted by z< z if for some representation z< = h + J; lim → 0 h( ) = z; ∀ ∈ Aq (R) for all suQciently large q. It turns out that (0) has no associated complex number. On the other hand the g.c. de@ned by w< = g + J where g(’) = R |x|’(x) d x; ’ ∈ Ao (R) satis@es w< 0 also w
< − = 0, but w< = J (cf. Rosinger, 1987, p. 75). In fact we have the following implications between three types of zero-like behavior of g.c.’s: z< = J ⇒ z< 0 ⇒ z
< − = 0:
(9)
In G() for K (compact) ⊂ ; N ¿ 0 and ¿0 the set QK; N; := {T ∈ G() : ∃R ∈ EM (); T = R + N such that sup{|(@ R)(; x)| : ∈ N n ; | | 6 N; ∈ A0 (Rn ); x ∈ K} ¡}
(10)
forms a base of neighborhoods of the origin. < The following C-valued bilinear form is of central importance in the present paper: Suppose that T ∈ G() is represented by T = fT + N; fT ∈ EM (). Then for ∈ D(); ; T ∈ C< is the g.c. given by ; T := ( T )(x) d x = h + J; h() = fT (In1 ; x) (x) d x; (11)
where indicates the product in G. It is well-known that if T ∈ D (); then ; T = T ( ) ∈ C (e.g. Biagioni, 1990; 1.6.9 or Rosinger, 1987, Theorem 1:6). Unlike linear distributions ; T = J; ∀ ∈ D() does not imply that T = N (≡ 0). 2. Topology and duality Biagioni and Egorov state that so far the topologies in C< and G() have not played a role in applications (cf. Biagioni, 1990, p. 45; Egorov, 1990). At that times the applications comprised mostly nonlinear p.d.e. However the study of measure theoretical properties or probability distributions are intimately related to the topological structures. Therefore we @rst start by the analysis of properties of topologies in C< and G() proposed by Biagioni (cf. Biagioni, 1990, p. 40) but not further elaborated. Then utilizing (11), we try to establish a duality theory which is comparable to the one in topological vector spaces (t.v.s.) and which < the base of will be essential for the introduction of our version of generalized stochastic processes. In C, neighborhoods given by (6) (or (7)) de@nes a non-HausdorC topology which also results from the uniform < structure induced by the R-valued seminorm in (8). Since the base {Q : ¿0} consists of non-absorbing < sets, the topology in C is not a linear vector space topology. However in C< all Cauchy sequences converge (not necessarily to a unique limit as the space is not HausdorC). An essential feature of C< is that it is a disconnected space. The @nite and in@nite components are given by C
< − ¡∞} and
C<∞ = C< \ C
(12)
respectively. C
J1 = {z< : z
< − = 0} = · −1 {0}:
(13)
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In C<1 every Cauchy sequence converges to a unique limit and this is suQcient for the completeness of C<1 in the same manner as in a space with a pseudo-metric. The @nite component C<1; f of C<1 is now a Banach space. But since J1 is an ideal only in C<1; f , C<1 will not be an algebra. Nevertheless the equivalence relation induced by J1 , i.e. z<1 z<2 ⇔ z<1 − z<2 − = 0
(14)
will be particularly useful in the sequel (see Section 3). Substituting C< for the scalar @eld, a duality theory similar to that of a t.v.s. can be established. However ∗ this theory will have some pathological features. If E is a vector space, E< (resp. E< ) will denote C-vector < space of C-valued linear (resp. linear and continuous) functionals. It is also a C< (resp. C
(0) − = + ∞. < < A C-dual pair (E; F) is de@ned in relation to a C-bilinear form ·; · with the usual separation properties. ∗ < < Thus they are C-dual For (E; E< ) and (E; E< ) the separation axioms hold since a C-form is also a C-form. pairs. The weak topology in E determined by F, denoted %(E; < F) is the coarsest topology in E under which < < all C-forms fy (·) = ·; y; y ∈ F are continuous. Again unlike the classical theory, the continuous C-dual of < < E under %(E; < F) is not necessarily F. However if E is already a t.v.s. then the C-dual of E under %(E; < E) < take values in C
(16)
For T2 ∈ D ; T1 ≈ T2 is the association of a Colombeau distribution with a linear distribution. Utilizing (14) we can de@ne a new and less restrictive equivalence relation (iii) T1 T2 ⇔ [ (T1 − T2 )] J (i:e: ; T1 − T2 − = 0) ∀ ∈ D. ∗ Now G= := G0 is isometric to a subspace of all C<1 valued linear functionals, i.e. G0 () ,→ D< 0 : < We note that ∼ ⇒ ≈ ⇒ . (D; G=∼ ); (D; G=≈ ) and (D; G0 ) are all C-dual pairs. However except in the @rst pair the bilinear form should be considered as C<1 -valued. < Let (E; F) be a C-dual pair. The @nite part of E, denoted by Ef is de@ned as Ef = {x ∈ E : x; y ∈ C
< F). As D is a connected space under its own inductive limit topology we have Ef is a l.c.t.v.s. under %(E; ∗ ∗ ∗ ∗ ∗ also Gf ⊂ D< ⊂ D< f ⊂ D< in (D< ; D) and (D< ; D) dualities. Furthermore Gf ⊂ D< f and Go; f ⊂ D< 0; f . For a < @xed T ∈ G(), the bilinear C-form ’; T is not in general continuous on D(). However D< is suQciently rich and contains, besides Schwartz distributions, many of the important generalized functions:
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Proposition 2.1. ; (H m )(k) (k = 0; 1; 2; : : :) is continuous on D(R); (H : The Heaviside function and ( )(k) is di6erentiation in G(R)). For the proof we use the relation (m + 1)H m ≈ (cf. Rosinger; 1987; p. 94–97) and also the fact that the association relation is compatible with both continuity on D and di6erentiation. Proposition 2.2. If M is a :nite dimensional separated locally compact t.v.s. (i.e. isomorphic to a Euclidean < M< ; M ) coincide. Furthermore < M< 0 ; M ) and *( space); then on M< 0 the weak and the strong topologies; i.e. %( 0 M< 0 is metrizable. Theorem 2.3. Let E be a separated l.c.t.v.s. and M be a :nite dimensional subspace of E. Let us equip ⊥ ⊥ ⊥ E< = M< with the quotient topology %( < E< ; E) modulo the subspace M< : Then E< = M< ∼ = M< 0 algebraically and topologically . The proof follows classical lines and one half of HorvRath (1996), Proposition 3:13:2, p. 262. The annihilator ⊥ M< = {f ∈ F : x; f − = 0; ∀x ∈ M } is %(F; < E)-closed. 0 0 < If (E; F) is a C-dual pair and A ⊂ E, the polar A< of A in E is A< = {y ∈ F : x; y − 6 1; x ∈ A}. Note 0 ⊥ that M< = M< . 0
Theorem 2.4. If E is a separated l.c.t.v.s. and U is a neighborhood of the origin; then U< is %( < E< ; E)-compact. < Theorem 2.5. Let (E; F) be a C
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Example. Random Dirac distribution: Let (-; .; P) be given by - = (0; 1); . = B ∩ (0; 1), P = m. For ∈ D(R) let T (!) = ; ! , where ! denotes the Dirac measure at ! ∈ (0; 1) (see also the De@nition 4:3). If ! = g(!) + N, then ! is found as g(!; ’; x) = ’(! − x). Therefore ; ! = k(!) + J where k(!; ’) = (x)’(! − x) d x. Here since ! is a Schwartz distribution Y (-) ⊂ C ⊂ C
De$nition 3.3. Y : - → C< is called representative integrable if it has an ordinary Lebesgue integrable repre sentative z(!; ’) such that - z(!; ’) dP(!) is moderate. Then we de@ne E(Y ) - z(!; ’) dP(!) + J . E(Y ) is independent of the choice of integrable representative. In the following special case which will be important in the next section, the two types of integrals coincide. Theorem 3.4. For ∈ D(R)n let Y; T (!) be given by Y; T (!) = ; T (!) = [T (!)]() where T : - → D (Rn ) is a weakly measurable mapping. Let K ⊂⊂ Rn be such that Supp ⊂ K 0 = int K and assume that there exist p ∈ Nn and f : - → C 0 (Rn ) satisfying T (!)|K 0 = (Dp f(!)|K0 for all ! ∈ - (Dp denotes the pth order distributional derivative). Then the two concepts of integrability coincide for Y; T . Note: The critical assumption that p does not depend on ! is valid in many important examples including the white noise set-up. < Theorem 3.5. Let Y (!) be a strongly integrable C-valued r.v. with the approximating sequence {Ym } of simple functions.Suppose there exist faithful representatives zm (!; ’) of Ym and a representative z(!; ’) of Y (!) satisfying |zm (!; ’) − z(!; ’)| dP → 0 for all ’ ∈ A0 as m → ∞. Then Y dP(!) = + J E(Y ); where (’) = - z(!; ’) dP(!). Corollary 3.6. Let z(!; ’) be a representative of Y (!). Suppose that there exist ordinary simple functions zm (!; ’) = .j zm; j (’)IAm; j (!) which satisfy for all !; |zm (!; ’)−z(!; ’)| → 0 and - |zm (!; ’)−z(!; ’)| dP(!) → 0 uniformly in ’ as m → ∞. Then Y (!) is strongly integrable and the two integrals coincide. The strong integrability can not be characterized in terms of the integrability of · − like in the classical − Bochner’s integration, since · may take on +∞ values but limm → ∞ Ym dP(!) may still exist. We have the following modi@cation of the “if ” part of the above characterization: Theorem 3.7. Suppose that Y is a C<1 -valued r.v. and there exists z<0 ∈ C<1 such that
Y (w) − dP(w) +
Y (w) − z<0 − dP(w)¡∞: Y −1 (C
Then Y (w) is strong integrable.
Y −1 (C<∞ )
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4. Random generalized functions and the extension problem The earliest works on the utilization of (C.g.f.) in stochastics were C $ apar (1993a) and C$apar (1993b). Later on the other authors’ main concern was the application of the new theory to the solution of stochastic p.d.e.’s. Oberguggenberger (1995) and Oberguggenberger and Russo (1996) de@ned Colombeau algebras of generalized stochastic processes as ordinary processes (with index set [0; ∞)) with paths being C.g.f.’s. Martias (1996) and Martias (1998) attempted to de@ne C.g.f. valued processes and in@nite dimensional C.g.f. replacing Rn with a Wiener space. < Our approach is to randomize the bilinear C-form ’; T . This has many merits. In the @rst place if T (!) ∈ D , then the theory reduces to the Gelfand’s linear theory with index set D (in place of [0; ∞)). < Besides C-linearity allows functional analytic techniques comparable to the linear theory. Let (-; .; P) be a complete probability space. De$nition 4.1. A Colombeau random generalized function (C.r.g.f ) is a measurable map T : - → G(). The measurability is in the weak sense; i.e. for every ∈ D(); ; T (!) is measurable (usually representativewise). The set of all C.r.g.f.’s is denoted by G- (). If T ∈ D- (); ; T (!) ∈ C yields the imbedding < < < .; P), the vector space of all C-valued .; P) and L(-; (representative D- () → G- (). We denote by L(-; measurable) r.v.’s and their quotient space under the equivalence relation of P-a.s. equality respectively. < De$nition 4.2. A generalized process (g.p.) is a mapping X : D() → L(-; .; P) which is linear when con< .; P). sidered as a mapping into L(-; De$nition 4.3. The generalized process generated by a C.r.g.f.: Let T (!) be a C.r.g.f. Then it generates a < g.p. in the sense of De:nition 4:2, i.e. ; T (!); ∈ D. By De:nition 4:1 ; T (!) ∈ L(-; .; P). DiCerent modes of continuity of the process are de@ned the same way as in the classical theory, replacing < norms by · − . Let F be a @nite dimensional subspace of D() and let X : F → L(-; .; P) be a P-a.s. continuous g.p. with index set F. Then proceeding as in Badrikian (1974), we @nd that the process induces a < if it is furnished by the Borel %-algebra generated by %( < ; F). If X denotes < F probability measure 9F in F < of the projective system the class of all @nite dimensional subspaces of D, then the projective limit lim F ←
< }F∈: under the canonical restrictions, is isomorphic to the @nite part D< ∗f of the algebraic C<1 -dual of D. {F This projective system possesses the sequential maximality (s.m.) property. Besides the measures 9F on these members have the Radon property, a fact which follows from Proposition 2:2, and Theorems 2:4 and 2:5. Thus we can reproduce a proof parallel to exposRe 3 in Badrikian (1974) to establish. < }F∈X resulting from Theorem 4.4. Let {9F }F∈: be the projective system of probability measures on {F ∗ < ∼ an a.s. continuous g.p. Then there exists a unique extension 9 to lim F = D< f equipped with the smallest ←
∗ < measurable; such that 9F = ;F (9); %-algebra H making all of the canonical projections ;F of D< f onto F ∀F ∈ :. ∗ ∗ < . If pF denotes the < D< f ; D) of D< f () and Gf |F ∼ Now Gf () is dense in the weak topology %( = F restriction of ;F to Gf , the same family {9F }F∈X can be used to create a compatible family of measures −1 (F)}, i.e. a cylindrical measure in G1; f . As a result, 9-outer measure of Gf () is 1, thus 9 induces on {pF ∗ a probability measure on the trace of %-algebra H of D< f over Gf .
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5. Applications 5.1. A Stochastic model for pure impulse-jump systems The following model can be used for describing physical or economical systems which undergo ran∞ dom change of states and occasionally such changes occur with impulse jumps. Consider T = i=1 Bi H=i (1 + i Ci =i ) where {Bi }; {Ci } are two independent sequences of i.i.d. r.v.’s. {=i } is a sequence of positive, a.s. non-decreasing and non-explosive r.v.’s, i independent 0; 1 Rademacher r.v.’s. and H=i and =i are the ∞ and Dirac functions with singularities at =i . As a C.g.f. T (!) = fT (!) + N; fT (!; ’; x) = ∞Heaviside B (!) ’(y) dy[1 + i (!)Ci (!)’(=i (!) − x)]: Then it is easily determined that T (!) ∈ G(R) and i i =i (!)−x measurable in the sense of De@nition 4:1. Thus in view of Proposition 2:1. ; T (!) is an a.s. trajectory continuous g.p. generated by T (!). 5.2. Processes de:ned as nonlinear functions of the white noise For the white noise we use a canonical probability space (-; .; 9) where - = S (Rn ) (tempered distributions), .: the Borel %-algebra in S generated by the weak star topology and 9 is provided by the Bochner–Minlos theorem which satis@es S ei ’; ! d9(!) = e−1=2 ’ L2 ; ’ ∈ S(Rn ): Then the white noise is the linear generalized process ?, given by @n ? : - → D (Rn ); ; ?(!) = ; ! = (−1)n ;W ; @x1 · · · @x n where W is an n-parameter Wiener process and ∈ D(Rn ). Viewed in the sense of De@nition 4:3, it is of the form ; T? where C.r.g.f. T? has a representative f? (!; ’; x) = ’(· − x); !; ’ ∈ A0 (Rn ). The distribution ∗ of the process is supported by S (Rn ) and we note that S (Rn ) ⊂ D (Rn ) ⊂ Gf (Rn ) ⊂ D< f : The nonlinear functions of the white noise (e.g. ?2 ) are no more random linear distributions. However they have simple representations within the scope of this paper without resorting to more sophisticated machinery of Hida, Watanabe distributions etc. To illustrate we consider one dimensional case. Note that all representatives used in the following are scalar integrable functions, so that Theorem 3.5 is in force. Let W (t); t ∈ R+ be a one parameter Wiener process and let ?( ) = ; T? , where T? = f? + N ∈ G. Suppressing ! we have by the above f? (’; x) = ?t (’(t − x)) = − Wt (’ (t − x)) = − R W (t)’ (t − x) dt. Thus if ?( ) = ; T? = g + J we have g (’) = f? (’; x) (x) d x = − W (t)’ (t − x) (x) dt d x; ’ ∈ A0 ; ∈ D(R+ ): (18) R2
R
The C<1 -valued mean functional M< ? ( ) := E[ ; T? ] = m?; + J and the covariance functional K< ? ( 1 ; 2 ) := E[ 1 ; T? 2 ; T? ] = k?; 1 2 + J can easily be calculated by Theorem 3:5, and (18). However the equalities should be interpreted as : m?; (’) = E − W (t)’ (t − x) (x) dt d x = 0; R2
k?;
1; 2
(’) = E
R4
W (t1 )W (t2 )’ (t1 − x)’ (t2 − y) 1 (x) 2 (y) dt1 dt2 d x dy
+ min(t1 ; t2 )’ (t1 − x)’ (t2 − y) 1 (x) 2 (y) dt1 dt2 d x dy; 1 ; 2 ∈ D(R ): We note that k?; 1 ; 2 − R 1 (t) 2 (t) dt ∈ J1 , so that it is in accordance with the linear theory. Similarly for ?2 = (dW=dt)2 we have
=
R4
?2 ( ) = ; T?2 = h + J;
T?2 = f?2 + N = (f? )2 + N
U. C & apar, H. Aktuglu / Statistics & Probability Letters 54 (2001) 291 – 299
where
f?2 (’; x) =
and
h (’) =
R3
2
R
W (t)’ (t − x) dt
=
R2
299
W (t1 )W (t2 )’ (t1 − x)’ (t2 − x) dt1 dt2
W (t1 )W (t2 )’ (t1 − x)’ (t2 − x) (x) dt1 dt2 d x:
Hence M< ?2 ;( ) = m?2 ; + J1 ; min(t1 ; t2 )’ (t1 − x)’ (t2 − x) (x) dt1 dt2 d x; m?2 ; (’) = K< ?2 ( 1 ;
R3
2 ) = k ?2 ;
where
1; 2
k?2 ;
1;
(’) = 2
R6
E
∈ D(R+ );
+ J1 ;
4
i=1
W (ti ) ’ (t1 − x)’ (t2 − x)’ (t3 − y)’ (t4 − y) 1 (x) 2 (y)
4
dti d x dy;
i=1
∈ D(R+ ): 4 4 Using the joint density of i=1 W (ti ); E[ i=1 W (ti )] is obtained, following a tedious calculation, as t(1) (2t(2) + t(3) ) (t(i) denotes the ordered ti ). Likewise for any slowly increasing function F ∈ C∞ (R) (cf. Biagioni, 1990, p. 27), the functional F(?) of the white noise is well de@ned as a C.r.g.f. In many applications t we prefer white t noise being de@ned as a genuine process. This can be done by considering ?t ( ) = 0 (s) dW (s) = − 0 W (s) d (s); which is a usual process with values in D . Now the t Colombeau version of this is given by T?t = f?t +N where f?t (!; ’; x) =−?t; s (!)(’(s−x)) =− 0 W (!; s)’ (s− x) ds; ’ ∈ A0 : Thus T?t represents the white noise as a G(R+ )-valued stochastic process. Then the nonlinear functionals are treated in the same manner as above. 1;
2
References Badrikian, A., 1974. SReminaire sur les functions alReatoire linReaires et les mesures cylindrique. Lecture Notes in Math., Vol. 379. Springer, Berlin. Biagioni, H.A., 1990. A Nonlinear Theory of Generalized Functions. Lecture Notes in Math., Vol. 1421. Springer, Berlin. C$apar, U., 1993a. Introduction to random nonlinear distributions, Proceedings of the 22nd Conference on Stochastic Processes and their Applications. Vrije University, Amsterdam, pp. 25 –27. C$apar, U., 1993b. Random generalized functions nonlinear in the sense of Colombeau, Proceedings of the VI. national symposium of mathematics. Eastern Mediterranean University, G. Magusa, Mersin-Turkey, pp. 85 –89. Colombeau, J.F., 1984. New Generalized Functions and Multiplication of Distributions. North-Holland Math. Studies, Vol. 84. North-Holland, Amsterdam. Egorov, Yu.V., 1990. A contribution to the theory of generalized functions. Russian Math. Surveys 45 (5), 1– 49. HorvRath, J., 1996. Topological Vector Spaces and Distribution. Vol. 1. Addison-Wesley, Reading, MA. Martias, C., 1996. Stochastic Integration on Generalized Function Spaces and its Applications. Stochastic Stochastic Rep. 57, 289–301. Martias, C., 1998. Colombeau’s generalized function on abstract Wiener spaces, Preprint. Oberguggenberger, M., 1995. Generalized functions and stochastic processes. Progress in Probability, Vol. 36. Birk(auser, Basel, Switzerland, pp. 215 –229. Oberguggenberger, M., Russo, F., 1996. Nonlinear SPDE’s: Colombeau solutions and path-wise limits. Proceedings of the Sixth Oslo–Silivri Workshop. Birkh(aser, Basel, Switzerland. Robertson, A.P., Robertson, W.J., 1966. Topological Vector Spaces. Cambridge University Press, Cambridge. Rosinger, E.E., 1987. Generalized Solutions of Nonlinear Partial DiCerential Equations. North-Holland Math. Studies, Vol. 146. North-Holland, Amsterdam.
A new construction of random Colombeau distributions Ulu"g C$apar ∗ , H(useyin Aktu"glu Department of Mathematics, Eastern Mediterranean University, G. Magusa, Mersin-10, Turkey Received October 2000
Abstract In this paper a generalized random process is modeled through the randomization of a bilinear form between the space D of test functions and the Colombeau generalized functions. This results in a theory akin to Gelfand–Vilankin’s random Schwartz distributions. An extension theorem in Bochner–Badrikian style is proved under some continuity assumptions. c 2001 Elsevier Science An important application is a natural representation of nonlinear functionals of white noise. B.V. All rights reserved MSC: 28C20; 46F99; 60B05; 60G20 < Keywords: Random Colombeau distributions; C-duality; Extension of probability measures in generalized duals; Nonlinear functions of the white noise
1. Basic notions Colombeau’s new nonlinear theory of generalized functions has been introduced through the main texts (Colombeau, 1984; Rosinger, 1987; Egorov, 1990). The basic structure is the in@nite product algebra E() = (C∞ ())A0 , where is an open subset of Rn . DiCerent index sets Aq (q = 0; 1; 2; : : :) have been used in the literature. Here we adopt the index set system introduced by Biagioni (1990), which has the merit of being independent of the choice of bases in Rn and is given by ∞ 1 A0 (R) = 1 ∈ D(R) : 1 is even; constant in a O-neighborhood and (1) 1 (x) d x = 2 0 ∞ Aq (R) = 1 ∈ A0 : xj=m 1 (x) d x = 0; 1 6 j; m 6 q ; q = (1; 2; : : :); (2) 0
and
2n n n Aq (R ) = ∈ D(R ) : ∃1 ∈ Aq (R); (x) = 1 (|x| ) ; n n
(q = 0; 1; 2; : : :):
∗ Corresponding author. E-mail addresses: [email protected] (U. C$apar), [email protected] (H. Aktu"glu).
c 2001 Elsevier Science B.V. All rights reserved 0167-7152/01/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 1 ) 0 0 1 0 1 - 8
(3)
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(n : the surface area of the (n − 1)-dimensional ball centered at the origin). The Colombeau algebra of generalized functions is the factor algebra G() = EM ()=N();
(4)
where EM () is the subalgebra of E() consisting of families of moderate growth along all molli@ers (x) = −N (x= )( ∈ AN for some N ) and N() is the ideal of null members of EM (). Thus EM () has members which grow at most polynomially in 1= as → 0, along molli@ers and uniformly on compacts, whereas N() includes those elements of EM () vanishing faster than any power of (cf. Rosinger, 1987; Biagioni, 1990). If algebra operations and diCerentiation is de@ned representative-wise, G() becomes a diCerential algebra. For T ∈ [C∞ ()] (i.e. a Schwartz distribution of compact support) the mapping T → (T ∗ |) + N() is an injection of [C∞ ())] . Using the sheaf property of G (cf. Rosinger, 1987, p. 131) it can be extended to an injection of D () and this injection preserves C∞ () as a faithful subalgebra. Furthermore diCerentiation in G when restricted to D , coincides with the usual distributional derivative. As an example, using the above injection and diCerentiation rule we have the following representations of one dimensional delta and Heaviside functions: = T + N;
T (; x) = (−x);
H = TH + N;
TH (; x) =
∞
−x
∀ ∈ A0 (R);
(t) dt;
∀ ∈ A0 (R)
(cf. Rosinger, 1987). However the imbedding of the C0 () does not extend the point-wise product in accordance with Schwartz’s “impossibility result”. Unlike linear ones, generalized distributions have point values 0 belonging to the algebra C< of generalized constants (g.c.’s). C< = EM =J is roughly speaking the algebra of 0 Colombeau generalized functions (C.g.f.) on R = {0}. Because of the index set used there is a unique algebra of g.c.’s for any dimension n. For T ∈ G(), T = R + N, the point value T (x) is given by T (x) = Rx + J;
Rx () = R(In1 ; x);
∈ A0 (R)
(5)
where In1 : A0 (R) → A0 (Rn ) is the bijection described in Biagioni (1990, Proposition 1:2:11). (e.g. (0) = T; 0 + J; T; 0 () = (0)): < a There is a canonical injection of C into C< through z = h + J where h() = z for all ∈ A(R). In C, base of neighborhoods of the origin is introduced by the following sets: 0 Q := {z< ∈ C< : ∃h ∈ EM ; z< = h + J; such that |z()|¡ ∀ ∈ A0 }
(6)
or equivalently 0 ; z< = h + J; ∃N ¿ 0 such that ∀ ∈ AN (R)∃¿0 Q = {z< ∈ C< : ∃h ∈ EM
with |h( )|¡ if 0¡ ¡}:
(7)
< The related topology is discussed in the next section, but (6) induces a R-valued seminorm · − (some of the axioms may be satis@ed as +∞ = + ∞) inf {¿0 : z< ∈ Q } if ∃¿0; z< ∈ Q ; −
z
< := (8) +∞ if ∀¿0; z< ∈ Q ; As an example exp(i(0)) − = 1, but (0) − = + ∞. (8) also satis@es z
< − 6 z
< − w
< −.
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293
A g.c. z< is said to be associated with complex number z, denoted by z< z if for some representation z< = h + J; lim → 0 h( ) = z; ∀ ∈ Aq (R) for all suQciently large q. It turns out that (0) has no associated complex number. On the other hand the g.c. de@ned by w< = g + J where g(’) = R |x|’(x) d x; ’ ∈ Ao (R) satis@es w< 0 also w
< − = 0, but w< = J (cf. Rosinger, 1987, p. 75). In fact we have the following implications between three types of zero-like behavior of g.c.’s: z< = J ⇒ z< 0 ⇒ z
< − = 0:
(9)
In G() for K (compact) ⊂ ; N ¿ 0 and ¿0 the set QK; N; := {T ∈ G() : ∃R ∈ EM (); T = R + N such that sup{|(@ R)(; x)| : ∈ N n ; | | 6 N; ∈ A0 (Rn ); x ∈ K} ¡}
(10)
forms a base of neighborhoods of the origin. < The following C-valued bilinear form is of central importance in the present paper: Suppose that T ∈ G() is represented by T = fT + N; fT ∈ EM (). Then for ∈ D(); ; T ∈ C< is the g.c. given by ; T := ( T )(x) d x = h + J; h() = fT (In1 ; x) (x) d x; (11)
where indicates the product in G. It is well-known that if T ∈ D (); then ; T = T ( ) ∈ C (e.g. Biagioni, 1990; 1.6.9 or Rosinger, 1987, Theorem 1:6). Unlike linear distributions ; T = J; ∀ ∈ D() does not imply that T = N (≡ 0). 2. Topology and duality Biagioni and Egorov state that so far the topologies in C< and G() have not played a role in applications (cf. Biagioni, 1990, p. 45; Egorov, 1990). At that times the applications comprised mostly nonlinear p.d.e. However the study of measure theoretical properties or probability distributions are intimately related to the topological structures. Therefore we @rst start by the analysis of properties of topologies in C< and G() proposed by Biagioni (cf. Biagioni, 1990, p. 40) but not further elaborated. Then utilizing (11), we try to establish a duality theory which is comparable to the one in topological vector spaces (t.v.s.) and which < the base of will be essential for the introduction of our version of generalized stochastic processes. In C, neighborhoods given by (6) (or (7)) de@nes a non-HausdorC topology which also results from the uniform < structure induced by the R-valued seminorm in (8). Since the base {Q : ¿0} consists of non-absorbing < sets, the topology in C is not a linear vector space topology. However in C< all Cauchy sequences converge (not necessarily to a unique limit as the space is not HausdorC). An essential feature of C< is that it is a disconnected space. The @nite and in@nite components are given by C
< − ¡∞} and
C<∞ = C< \ C
(12)
respectively. C
J1 = {z< : z
< − = 0} = · −1 {0}:
(13)
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In C<1 every Cauchy sequence converges to a unique limit and this is suQcient for the completeness of C<1 in the same manner as in a space with a pseudo-metric. The @nite component C<1; f of C<1 is now a Banach space. But since J1 is an ideal only in C<1; f , C<1 will not be an algebra. Nevertheless the equivalence relation induced by J1 , i.e. z<1 z<2 ⇔ z<1 − z<2 − = 0
(14)
will be particularly useful in the sequel (see Section 3). Substituting C< for the scalar @eld, a duality theory similar to that of a t.v.s. can be established. However ∗ this theory will have some pathological features. If E is a vector space, E< (resp. E< ) will denote C-vector < space of C-valued linear (resp. linear and continuous) functionals. It is also a C< (resp. C
(0) − = + ∞. < < A C-dual pair (E; F) is de@ned in relation to a C-bilinear form ·; · with the usual separation properties. ∗ < < Thus they are C-dual For (E; E< ) and (E; E< ) the separation axioms hold since a C-form is also a C-form. pairs. The weak topology in E determined by F, denoted %(E; < F) is the coarsest topology in E under which < < all C-forms fy (·) = ·; y; y ∈ F are continuous. Again unlike the classical theory, the continuous C-dual of < < E under %(E; < F) is not necessarily F. However if E is already a t.v.s. then the C-dual of E under %(E; < E) < take values in C
(16)
For T2 ∈ D ; T1 ≈ T2 is the association of a Colombeau distribution with a linear distribution. Utilizing (14) we can de@ne a new and less restrictive equivalence relation (iii) T1 T2 ⇔ [ (T1 − T2 )] J (i:e: ; T1 − T2 − = 0) ∀ ∈ D. ∗ Now G= := G0 is isometric to a subspace of all C<1 valued linear functionals, i.e. G0 () ,→ D< 0 : < We note that ∼ ⇒ ≈ ⇒ . (D; G=∼ ); (D; G=≈ ) and (D; G0 ) are all C-dual pairs. However except in the @rst pair the bilinear form should be considered as C<1 -valued. < Let (E; F) be a C-dual pair. The @nite part of E, denoted by Ef is de@ned as Ef = {x ∈ E : x; y ∈ C
< F). As D is a connected space under its own inductive limit topology we have Ef is a l.c.t.v.s. under %(E; ∗ ∗ ∗ ∗ ∗ also Gf ⊂ D< ⊂ D< f ⊂ D< in (D< ; D) and (D< ; D) dualities. Furthermore Gf ⊂ D< f and Go; f ⊂ D< 0; f . For a < @xed T ∈ G(), the bilinear C-form ’; T is not in general continuous on D(). However D< is suQciently rich and contains, besides Schwartz distributions, many of the important generalized functions:
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295
Proposition 2.1. ; (H m )(k) (k = 0; 1; 2; : : :) is continuous on D(R); (H : The Heaviside function and ( )(k) is di6erentiation in G(R)). For the proof we use the relation (m + 1)H m ≈ (cf. Rosinger; 1987; p. 94–97) and also the fact that the association relation is compatible with both continuity on D and di6erentiation. Proposition 2.2. If M is a :nite dimensional separated locally compact t.v.s. (i.e. isomorphic to a Euclidean < M< ; M ) coincide. Furthermore < M< 0 ; M ) and *( space); then on M< 0 the weak and the strong topologies; i.e. %( 0 M< 0 is metrizable. Theorem 2.3. Let E be a separated l.c.t.v.s. and M be a :nite dimensional subspace of E. Let us equip ⊥ ⊥ ⊥ E< = M< with the quotient topology %( < E< ; E) modulo the subspace M< : Then E< = M< ∼ = M< 0 algebraically and topologically . The proof follows classical lines and one half of HorvRath (1996), Proposition 3:13:2, p. 262. The annihilator ⊥ M< = {f ∈ F : x; f − = 0; ∀x ∈ M } is %(F; < E)-closed. 0 0 < If (E; F) is a C-dual pair and A ⊂ E, the polar A< of A in E is A< = {y ∈ F : x; y − 6 1; x ∈ A}. Note 0 ⊥ that M< = M< . 0
Theorem 2.4. If E is a separated l.c.t.v.s. and U is a neighborhood of the origin; then U< is %( < E< ; E)-compact. < Theorem 2.5. Let (E; F) be a C
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Example. Random Dirac distribution: Let (-; .; P) be given by - = (0; 1); . = B ∩ (0; 1), P = m. For ∈ D(R) let T (!) = ; ! , where ! denotes the Dirac measure at ! ∈ (0; 1) (see also the De@nition 4:3). If ! = g(!) + N, then ! is found as g(!; ’; x) = ’(! − x). Therefore ; ! = k(!) + J where k(!; ’) = (x)’(! − x) d x. Here since ! is a Schwartz distribution Y (-) ⊂ C ⊂ C
De$nition 3.3. Y : - → C< is called representative integrable if it has an ordinary Lebesgue integrable repre sentative z(!; ’) such that - z(!; ’) dP(!) is moderate. Then we de@ne E(Y ) - z(!; ’) dP(!) + J . E(Y ) is independent of the choice of integrable representative. In the following special case which will be important in the next section, the two types of integrals coincide. Theorem 3.4. For ∈ D(R)n let Y; T (!) be given by Y; T (!) = ; T (!) = [T (!)]() where T : - → D (Rn ) is a weakly measurable mapping. Let K ⊂⊂ Rn be such that Supp ⊂ K 0 = int K and assume that there exist p ∈ Nn and f : - → C 0 (Rn ) satisfying T (!)|K 0 = (Dp f(!)|K0 for all ! ∈ - (Dp denotes the pth order distributional derivative). Then the two concepts of integrability coincide for Y; T . Note: The critical assumption that p does not depend on ! is valid in many important examples including the white noise set-up. < Theorem 3.5. Let Y (!) be a strongly integrable C-valued r.v. with the approximating sequence {Ym } of simple functions.Suppose there exist faithful representatives zm (!; ’) of Ym and a representative z(!; ’) of Y (!) satisfying |zm (!; ’) − z(!; ’)| dP → 0 for all ’ ∈ A0 as m → ∞. Then Y dP(!) = + J E(Y ); where (’) = - z(!; ’) dP(!). Corollary 3.6. Let z(!; ’) be a representative of Y (!). Suppose that there exist ordinary simple functions zm (!; ’) = .j zm; j (’)IAm; j (!) which satisfy for all !; |zm (!; ’)−z(!; ’)| → 0 and - |zm (!; ’)−z(!; ’)| dP(!) → 0 uniformly in ’ as m → ∞. Then Y (!) is strongly integrable and the two integrals coincide. The strong integrability can not be characterized in terms of the integrability of · − like in the classical − Bochner’s integration, since · may take on +∞ values but limm → ∞ Ym dP(!) may still exist. We have the following modi@cation of the “if ” part of the above characterization: Theorem 3.7. Suppose that Y is a C<1 -valued r.v. and there exists z<0 ∈ C<1 such that
Y (w) − dP(w) +
Y (w) − z<0 − dP(w)¡∞: Y −1 (C
Then Y (w) is strong integrable.
Y −1 (C<∞ )
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297
4. Random generalized functions and the extension problem The earliest works on the utilization of (C.g.f.) in stochastics were C $ apar (1993a) and C$apar (1993b). Later on the other authors’ main concern was the application of the new theory to the solution of stochastic p.d.e.’s. Oberguggenberger (1995) and Oberguggenberger and Russo (1996) de@ned Colombeau algebras of generalized stochastic processes as ordinary processes (with index set [0; ∞)) with paths being C.g.f.’s. Martias (1996) and Martias (1998) attempted to de@ne C.g.f. valued processes and in@nite dimensional C.g.f. replacing Rn with a Wiener space. < Our approach is to randomize the bilinear C-form ’; T . This has many merits. In the @rst place if T (!) ∈ D , then the theory reduces to the Gelfand’s linear theory with index set D (in place of [0; ∞)). < Besides C-linearity allows functional analytic techniques comparable to the linear theory. Let (-; .; P) be a complete probability space. De$nition 4.1. A Colombeau random generalized function (C.r.g.f ) is a measurable map T : - → G(). The measurability is in the weak sense; i.e. for every ∈ D(); ; T (!) is measurable (usually representativewise). The set of all C.r.g.f.’s is denoted by G- (). If T ∈ D- (); ; T (!) ∈ C yields the imbedding < < < .; P), the vector space of all C-valued .; P) and L(-; (representative D- () → G- (). We denote by L(-; measurable) r.v.’s and their quotient space under the equivalence relation of P-a.s. equality respectively. < De$nition 4.2. A generalized process (g.p.) is a mapping X : D() → L(-; .; P) which is linear when con< .; P). sidered as a mapping into L(-; De$nition 4.3. The generalized process generated by a C.r.g.f.: Let T (!) be a C.r.g.f. Then it generates a < g.p. in the sense of De:nition 4:2, i.e. ; T (!); ∈ D. By De:nition 4:1 ; T (!) ∈ L(-; .; P). DiCerent modes of continuity of the process are de@ned the same way as in the classical theory, replacing < norms by · − . Let F be a @nite dimensional subspace of D() and let X : F → L(-; .; P) be a P-a.s. continuous g.p. with index set F. Then proceeding as in Badrikian (1974), we @nd that the process induces a < if it is furnished by the Borel %-algebra generated by %( < ; F). If X denotes < F probability measure 9F in F < of the projective system the class of all @nite dimensional subspaces of D, then the projective limit lim F ←
< }F∈: under the canonical restrictions, is isomorphic to the @nite part D< ∗f of the algebraic C<1 -dual of D. {F This projective system possesses the sequential maximality (s.m.) property. Besides the measures 9F on these members have the Radon property, a fact which follows from Proposition 2:2, and Theorems 2:4 and 2:5. Thus we can reproduce a proof parallel to exposRe 3 in Badrikian (1974) to establish. < }F∈X resulting from Theorem 4.4. Let {9F }F∈: be the projective system of probability measures on {F ∗ < ∼ an a.s. continuous g.p. Then there exists a unique extension 9 to lim F = D< f equipped with the smallest ←
∗ < measurable; such that 9F = ;F (9); %-algebra H making all of the canonical projections ;F of D< f onto F ∀F ∈ :. ∗ ∗ < . If pF denotes the < D< f ; D) of D< f () and Gf |F ∼ Now Gf () is dense in the weak topology %( = F restriction of ;F to Gf , the same family {9F }F∈X can be used to create a compatible family of measures −1 (F)}, i.e. a cylindrical measure in G1; f . As a result, 9-outer measure of Gf () is 1, thus 9 induces on {pF ∗ a probability measure on the trace of %-algebra H of D< f over Gf .
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5. Applications 5.1. A Stochastic model for pure impulse-jump systems The following model can be used for describing physical or economical systems which undergo ran∞ dom change of states and occasionally such changes occur with impulse jumps. Consider T = i=1 Bi H=i (1 + i Ci =i ) where {Bi }; {Ci } are two independent sequences of i.i.d. r.v.’s. {=i } is a sequence of positive, a.s. non-decreasing and non-explosive r.v.’s, i independent 0; 1 Rademacher r.v.’s. and H=i and =i are the ∞ and Dirac functions with singularities at =i . As a C.g.f. T (!) = fT (!) + N; fT (!; ’; x) = ∞Heaviside B (!) ’(y) dy[1 + i (!)Ci (!)’(=i (!) − x)]: Then it is easily determined that T (!) ∈ G(R) and i i =i (!)−x measurable in the sense of De@nition 4:1. Thus in view of Proposition 2:1. ; T (!) is an a.s. trajectory continuous g.p. generated by T (!). 5.2. Processes de:ned as nonlinear functions of the white noise For the white noise we use a canonical probability space (-; .; 9) where - = S (Rn ) (tempered distributions), .: the Borel %-algebra in S generated by the weak star topology and 9 is provided by the Bochner–Minlos theorem which satis@es S ei ’; ! d9(!) = e−1=2 ’ L2 ; ’ ∈ S(Rn ): Then the white noise is the linear generalized process ?, given by @n ? : - → D (Rn ); ; ?(!) = ; ! = (−1)n ;W ; @x1 · · · @x n where W is an n-parameter Wiener process and ∈ D(Rn ). Viewed in the sense of De@nition 4:3, it is of the form ; T? where C.r.g.f. T? has a representative f? (!; ’; x) = ’(· − x); !; ’ ∈ A0 (Rn ). The distribution ∗ of the process is supported by S (Rn ) and we note that S (Rn ) ⊂ D (Rn ) ⊂ Gf (Rn ) ⊂ D< f : The nonlinear functions of the white noise (e.g. ?2 ) are no more random linear distributions. However they have simple representations within the scope of this paper without resorting to more sophisticated machinery of Hida, Watanabe distributions etc. To illustrate we consider one dimensional case. Note that all representatives used in the following are scalar integrable functions, so that Theorem 3.5 is in force. Let W (t); t ∈ R+ be a one parameter Wiener process and let ?( ) = ; T? , where T? = f? + N ∈ G. Suppressing ! we have by the above f? (’; x) = ?t (’(t − x)) = − Wt (’ (t − x)) = − R W (t)’ (t − x) dt. Thus if ?( ) = ; T? = g + J we have g (’) = f? (’; x) (x) d x = − W (t)’ (t − x) (x) dt d x; ’ ∈ A0 ; ∈ D(R+ ): (18) R2
R
The C<1 -valued mean functional M< ? ( ) := E[ ; T? ] = m?; + J and the covariance functional K< ? ( 1 ; 2 ) := E[ 1 ; T? 2 ; T? ] = k?; 1 2 + J can easily be calculated by Theorem 3:5, and (18). However the equalities should be interpreted as : m?; (’) = E − W (t)’ (t − x) (x) dt d x = 0; R2
k?;
1; 2
(’) = E
R4
W (t1 )W (t2 )’ (t1 − x)’ (t2 − y) 1 (x) 2 (y) dt1 dt2 d x dy
+ min(t1 ; t2 )’ (t1 − x)’ (t2 − y) 1 (x) 2 (y) dt1 dt2 d x dy; 1 ; 2 ∈ D(R ): We note that k?; 1 ; 2 − R 1 (t) 2 (t) dt ∈ J1 , so that it is in accordance with the linear theory. Similarly for ?2 = (dW=dt)2 we have
=
R4
?2 ( ) = ; T?2 = h + J;
T?2 = f?2 + N = (f? )2 + N
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where
f?2 (’; x) =
and
h (’) =
R3
2
R
W (t)’ (t − x) dt
=
R2
299
W (t1 )W (t2 )’ (t1 − x)’ (t2 − x) dt1 dt2
W (t1 )W (t2 )’ (t1 − x)’ (t2 − x) (x) dt1 dt2 d x:
Hence M< ?2 ;( ) = m?2 ; + J1 ; min(t1 ; t2 )’ (t1 − x)’ (t2 − x) (x) dt1 dt2 d x; m?2 ; (’) = K< ?2 ( 1 ;
R3
2 ) = k ?2 ;
where
1; 2
k?2 ;
1;
(’) = 2
R6
E
∈ D(R+ );
+ J1 ;
4
i=1
W (ti ) ’ (t1 − x)’ (t2 − x)’ (t3 − y)’ (t4 − y) 1 (x) 2 (y)
4
dti d x dy;
i=1
∈ D(R+ ): 4 4 Using the joint density of i=1 W (ti ); E[ i=1 W (ti )] is obtained, following a tedious calculation, as t(1) (2t(2) + t(3) ) (t(i) denotes the ordered ti ). Likewise for any slowly increasing function F ∈ C∞ (R) (cf. Biagioni, 1990, p. 27), the functional F(?) of the white noise is well de@ned as a C.r.g.f. In many applications t we prefer white t noise being de@ned as a genuine process. This can be done by considering ?t ( ) = 0 (s) dW (s) = − 0 W (s) d (s); which is a usual process with values in D . Now the t Colombeau version of this is given by T?t = f?t +N where f?t (!; ’; x) =−?t; s (!)(’(s−x)) =− 0 W (!; s)’ (s− x) ds; ’ ∈ A0 : Thus T?t represents the white noise as a G(R+ )-valued stochastic process. Then the nonlinear functionals are treated in the same manner as above. 1;
2
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