# A nonlinear stochastic equation of convolution type: solution and stochastic representation.

1. Introduction

The aim of this paper is to study the following nonlinear stochastic equation of convolution type

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.1)

where [??] is a [R.sub.n]-generalized vector field, [??] is a n-dimensional generalized function, v > 0 a real constant, t [member of] [0,[infinity]) the time parameter, x = ([x.sub.1], ..., [x.sub.n]) [member of] [R.sub.n] the spatial variable, [DELTA] the Laplacian operator in [R.sub.n], [nabla] the gradient and * the convolution product for generalized functions (see [1], [11] and Subsection 2.2 for more details), [??] * [??] is a driving term, [??] * [??] denotes the differential operator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and the initial condition [[??].sub.0] = ([u.sub.0,1], ..., [u.sub.0,n]) is a n-dimensional generalized stochastic process, see Section 3 for more details.

Problem (1.1) with * replaced by the usual product would coincide with the Burgers equation well known in the literature ([3], [5] and references therein). However the physical interpretation of (1.1) is quite different. This is easily seen by comparing the j-component of the Fourier transform of the nonlinear kinetic term ([??].[nabla]) [??] in the Burgers equation, namely

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.2)

with the j-component of the Fourier transform of the nonlinear term in (1.1), ([??] * [nabla]) [??]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.3)

[[??].sub.i] denotes the Fourier transform of [u.sub.i] .

In the Burgers case the expression (1.2) implies that the Fourier modes at length scale 2[pi]/q and 2[pi]/k-q control the eddies at scale 2[pi]/k, consistent with the phenomenological description of the inertial range in the turbulence cascade. However, in the convolution case the nonlocal nonlinearity corresponds to a self-interaction of the modes at each length scale. Therefore the convolution equation has a different physical interpretation. Nevertheless, nonlocal nonlinearities are also important in models of transport in magnetized plasmas, see [4] and also in the modeling of convection driven by density gradients as it arises in geophysical fluid flows, see [12], [13], [14] and [15].

By restricting oneself to solutions of gradient type, the Burgers equation may be linearized by the Cole-Hopf transformation. This provides the most general solution for (1 + 1)-dimensions but not for (1 + n)-dimensions. In contrast, for our equation (1.1), using the Laplace transform in a general setting, we obtain a general solution for (1 + n)-dimensions.

The paper is organized as follows: In Section 2 we provide the mathematical background needed to solve the Cauchy problem stated above, namely the spaces of test and generalized functions, the characterization theorem of generalized functions and the convolution product as well as some of its properties. In Section 3 we combine the convolution calculus and the characterization theorem in order to find an explicit solution of the problem (1.1).

Finally, in Section 4 we obtain a stochastic representation of the Laplace transform of the solution. Stochastic representations of differential equations and their solutions, not only provide a new interpretation of the solutions but are also useful for existence results and in the study of the fluctuations associated to the phenomena for which the equation represents a mean-field approximation. For a striking recent example of such use of stochastic representations we refer to thework done on the Navier-Stokes equation (see [16] and references therein).

2. Preliminaries

2.1. Test and Generalized Functions Spaces

In this section we introduce the framework needed later on. The starting point is the real Hilbert space H = [L.sup.2](R,[R.sup.d]) x [R.sup.r], d, r [member of] N with scalar product (*, *) and norm |*|. More precisely, if (f, x) = (([f.sub.1], ..., [f.sub.d]), ([x.sub.1], ..., [x.sub.r])) [member of] H, then

[ILLUSTRATION OMITTED]

Let us consider the real nuclear triplet

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

The pairing <*,*> between M' and M is given in terms of the scalar product in H, i.e., <([omega], x), ([xi], p) := ([omega], [xi])[L.sup.2] + (x, p) [R.sup.r], ([omega], x) [member of] M' and ([xi], p) [member of] M. Since M is a Frechet nuclear space, it can be represented as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [S.sub.n](R,[R.sup.d]) x [R.sup.r] is a Hilbert space with norm squared given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], see e.g. [8] or [2] and references therein. We will consider the complexification of the triple (2.1) and denote it by

N' [contains] Z [contains] N (2.2)

where N = M+ iM and Z = H + iH. On M' we have the standard Gaussian measure [gamma] given by Minlos's theorem via its characteristic functional, namely for every ([xi], p) [member of] M

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In order to solve the (1+n)-dimensional equation of convolution type (1.1) we need to introduce an appropriate space of vectorial generalized functions. We borrow this construction from [9]. Let [theta] = ([[theta].sub.1], [[theta].sub.2]) : [R.sup.2.sub.+] [right arrow] R, ([t.sub.1], [t.sub.2]) [right arrow] [[theta].sub.1] ([t.sub.1]) + [[theta].sub.1] [t.sub.2] where [[theta].sub.1], [[theta].sub.2] are twoYoung functions, i.e., [[theta].sub.i] : [R.sub.+] [right arrow] [R.sub.+] continuous convex strictly increasing function and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For every pair m = ([m.sub.1],[m.sub.2]) with [m.sub.1],[m.sub.2] .]0,[infinity][, we define the Banach space [F.sub.[theta]],m([N.sub.-n]), n [member of] N by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Here, for each z = ([omega], x) we have [theta]([m|z|.sub.-n]) := [[theta].sub.1]([m.sub.1]|[omega]|-n) + [[theta].sub.2]([m.sub.2]|x|). Now we consider as test function space the space of entire functions on N' of ([[theta].sub.1], [[theta].sub.2])-exponential growth and minimal type

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

endowed with the projective limit topology. We would like to use [F.sub.[theta]] (N') to construct a triple centered in the complex Hilbert space [L.sup.2](M', [gamma]). To this end we need another condition on the pair ofYoung functions ([[theta].sub.1], [[theta].sub.2]). Namely,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.3)

This is enough to obtain the following Gelfand triple

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (2.4)

where [F.sub.[theta]] (N') is the topological dual of [F.sub.[theta]] (N') with respect to [L.sup.2](M', [gamma]) endowed with the inductive limit topology.

In applications it is very important to have the characterization of generalized functions from [F'.sub.[theta]] (N'). First we define the Laplace transform of an element in [F'.sub.[theta]] (N').

For every fixed element ([xi], p) [member of] N the exponential function exp(([xi], p)) is a well defined element in [F.sub.[theta]] (N'), see [7]. The Laplace transform L of a generalized function [PHI] [member of] [F'.sub.[theta]] (N') is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (2.5)

We are ready to state to characterization theorem (see e.g., [7] and [1] for the proof) which is the main tool in our further consideration.

Theorem 2.1.

1. The Laplace transform is a topological isomorphism between [F.sub.[theta]] (N') and the space [G.sub.[theta]] * (N), where [G.sub.[theta]] * (N) is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and [G.sub.[theta] *,m]([N.sub.n]) is the space of entire functions on [N.sub.n] with the following [theta]-exponential growth condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Here [[theta].sup. *] = ([[theta].sup.*.sub.1], [[theta].sup.*.sub.2]), where [[theta].sup.*.sub.i] = sup (tx - [[theta].sub.i] (t)) is the Legendre transform associated to the function [[theta].sub.i], i = 1, 2 sup t [greater than or equal to] 0.

2. In the particular case [theta](x) = ([[theta].sub.1](x), [[theta].sub.2](x)) = (x, x), we denote the space [F'.sub.[theta]] (N') by [F.sub.'x](N'). Then the Laplace transform realizes a topological isomorphism between the distributions space [F'.sub.x](N') and the space Hol0(N) of holomorphic function on a neighborhood of zero of N.

2.2. The Convolution Product *

It is well known that in infinite dimensional complex analysis the convolution operator on a general function space F is defined as a continuous operator which commutes with the translation operator. Let us define the convolution between a generalized and a test function. Let [PHI] [member of] [F'.sub.[theta]] (N]) and [phi] [member of] [F.sub.[theta]] (N') be given, then the convolution *. is defined by

([PHI] * [phi])([omega], x) := <<[PHI], [[tau].sub.-([omega],[phi]>>.

where [[tau].sub.-([omega],x) is the translation operator, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

It is not hard the see that [PHI] * [phi] [member of] [F.sub.[theta]] (N'), cf. [7]. The convolution product is given in terms of the dual pairing as ([PHI] * [phi])(0,0) = <<[PHI], [phi]>> for any [PHI] [member of] [F'.sub. [theta]] (N') and [phi] [member of] [F.sub.[theta]] (N').

We can generalize the above convolution product for generalized functions as follows. Let [PHI], [PSI] [member of] [F'.sub.[theta]] (N') be given, then [PHI] * [PSI] is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (2.6)

This definition of convolution product for generalized functions will be used later for the solution of the equation (1.1). We have the following equality, (see [11], Proposition 3):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

As a consequence of the above equality and definition (2.6) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.7)

which says that the Laplace transform maps the convolution product on [F'.sub.[theta]] (N') into the usual pointwise product in the algebra of functions [G.sub.[theta] *](N). Therefore we may use Theorem 2.1 to define the convolution product between two generalized functions as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

This allows us to introduce the convolution exponential of a generalized function. In fact, for every [PHI] [member of] [F'.sub.[theta] (N') we may easily check that exp [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Using the inverse Laplace transform and the fact that any Young function [theta] verifies the property ([theta] *) * = [theta] we obtain that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Now we give the definition of the convolution exponential of [PHI] [member of] [F'.sub.[theta] (N'), denoted by exp * [PHI]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Notice that exp * [PHI] is a well defined element in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and therefore the distribution exp * [PHI] is given in terms of a convergent series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (2.8)

where [[PHI].sup.* n] is the convolution of [PHI] with itself n times, [[PHI].sup. * 0] := [[delta].sub.0] by convention with [[delta].sub.0] denoting the Dirac distribution at 0. We refer to [1] for more details concerning convolution product on [F'.sub.[theta]] (N').

A one parameter generalized stochastic process with values in [F'.sub.[theta]] (N') is a family of generalized functions {[PHI](t), t [greater than or equal to] 0} [subset] [F'.sub.[theta]] (N'). The process [PHI](t) is said to be continuous if the map t [right arrow] [PHI] (t) is continuous. For a given continuous generalized stochastic process [(X(t)).sub.t[greater than or equal to] 0 we define the generalized stochastic process

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.9)

The process Y(t,[omega], x) is differentiable and we have [partial derivative/[partial derivative]t] Y (t,[omega],x) = X (t,[omega],x). The details of the proof can be seen in [10], Proposition 11.

2.3. Convolution Inverse of Distributions

Let [PHI] a fixed element on the distribution space [F'/sub.[theta] (N') and consider the following convolution equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.10)

Applying the Laplace transform to the convolution equation (2.10) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

If [??] ([xi], q) [not equal to] 0 for every ([xi], q) [member of] N, then using the division result in the space [G.sub.[theta]*](N) (see [6]) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Moreover, by the Laplace transform isomorphism (see Theorem 2.1), we prove the existence and uniqueness of the solution [PSI] [member of] [F'.sub.[theta] (N') in the equation (2.10). If we denote this solution [PSI] by [[PHI].sup.*-1], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] .

This division result is also true in the limit case [theta] (x) = (x, x); i. e., [??] [member of] [G.sub.[theta]*](N) = [Hol.sub.0](N).

3. Solution of the n-dimensional Convolution Equation We are now ready to solve the Cauchy problem stated in (1.1) which we recall for the reader convenience, namely

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.1)

The different terms in (3.1) are as follows: [[??].sub.0](x) = ([u.sub.0],1(x), ..., [u.sub.0,n](x)) is a generalized function; [u.sub.0,j (x) [member of] [F'.sub.[theta]] (N'), v > 0 a real constant, x = ([x.sub.1], ..., [x.sub.n]) [member of] [R.sup.n],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

We are now ready to prove the main result of this paper, namely we obtain the explicit solution of the equation (3.1). using the tools from Section 2.

Theorem 3.1. Let [[??].sub.0](x) = ([u.sub.0,1](x), ..., [u.sub.0,n](x)) and [??] = ([f.sub.1], ..., [f.sub.n]) be such that [u.sub.0,k], [f.sub.k] [member of] [F'.sub.x](N'), k = 0, ..., n. Then the solution [??](t, [omega], x) of the nonlinear equation of convolution type (3.1) is given explicitly by the following system:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.2)

with k = 1, ..., n and [[delta].sub.0] is the Dirac measure at point zero and [[gamma].sub.2vt] is the Gaussian measure with variance 2vt on [R.sub.n].

Proof. We denote [[partial derivative].sub.t] = [partial derivative]/[[partial derivative].sub.t] such that the n-dimensional equation (3.1) may be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.3)

where k = 1, ..., n.

We denote by [v.sub.k] = [v.sub.k](t, [xi], q), [g.sub.k] = [g.sub.k](t, [xi], q) and [v.sub.0,k] = [v.sub.0,k]([xi], q), [xi] [member of] S (R,[R.sup.d), q [member of] [R.sub.n], the Laplace transforms of the generalized functions [u.sub.k] = [u.sub.k](t, [omega], x), [f.sub.k] = [f.sub.k](t, [omega], x) and the initial condition [u.sub.0,k] = [u.sub.0,k]([omega], x), respectively, for k = 1, ..., n. Applying the Laplace transform to the system (3.3) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.4)

Changing the variables, Sk = 1/[v.sub.k] = 1, ..., n, the system (3.4) is equivalent to the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.5)

We denote [S.sub.1] .... [[??].sub.k] ... [S.sub.n] = [S.sub.1] ...[S.sub.k-1[S.sub.k]+1] ... [S.sub.n], for k = 1, ..., n. Ifwe multiply the first equation of the system (3.5) by [S.sub.1] .... [S.sub.k] ...[S.sub.n], we deduce

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.6)

Let us denote such equation by ([E.sub.k]), k = 1, ..., n. If we fixe k, then the difference ([E.sub.1]) - ([E.sub.k]) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.7)

After simplification, we divide by [S.sub.1]Sk and obtain the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.8)

which can be integrated. The solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.9)

where for simplification of notation gj (s) = gj (s, [xi], q), j = 1, ..., n. Then we have the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.10)

Introducing the expression of [S.sub.k]/[S.sub.j] in (3.5) we deduce the following linear system of equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.11)

For any fixed k, the solution of the homogeneous equation is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (3.12)

where [lambda] is a constant. Then the solution of (3.11) is given by the method of variation of constants as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.13)

where the constant [lambda] is determined by the initial conditions; [S.sub.k](0, [xi], q) = [S.sub.0,k] = [lambda]. Then (3.13) may be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.14)

Since [S.sub.k](t, [xi], q) = 1/[v.sub.k] (t,[xi],q), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.15)

In fact, it is easy to show that for every t [greater than or equal to] 0, the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

belongs to the space [Hol.sub.0](N) and satisfy Y(t, 0, 0) = 1 [not equal to] 0. Then there exists U a neighborhood of (0, 0) of N, such that Y(t, q, [xi]) [not equal to] 0 for every ([xi], q) [member of] U. Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Finally, to obtain the solution of the equation (3.1) we use the following equalities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

and then [u.sub.k](t, x), k = 1, ..., n is given by the Laplace inverse transform according to the theorem 2.1, as in (3.2).

Corollary 3.2. If the potential [??] in the equation (3.1) does not depend of the time variable t, i.e., [??] = [??] (x), then the solution is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

with k = 1, ..., n. In particular if f = 0, the solution has the from

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

with k = 1, ..., n and [nabla] * represents the divergence operator.

4. Stochastic Representation of Solution of the n-dimensional Convolution Equation

In the previous section we obtained an explicit solution of the equation (1.1). Here, we are interested in a stochastic representation of that solution. Let us start with a simple and useful Lemma:

Lemma 4.1. Let b, [alpha] [member of] R with b [not equal to] = 0. We can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (4.1)

where [N.sub.b] is a random variable with Poisson distribution with intensity b, defined on a probability space ([OMEGA], P), E denoting the mathematical expectation.

Proof. Since Nb is a random variable with Poisson distribution with intensity b, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (4.2)

which implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

We use the same notations as in section 3. First we take f = 0 in equation (1.1) and obtain a stochastic representation for the Laplace transfom v of its solution u.

Theorem 4.2. Let v be the Laplace transform of the solution of equation (1.1), with f = 0, obtained in Theorem 3.1. There are stochastic processes [X.sup.k.sub.t], k = 1, ..., n such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Proof. According to (3.14), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Using Lemma 4.1, we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Defining the stochastic processes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

for k = 1, ..., n, the result of the theorem follows.

Let us denote by [[eta].sup.[lambda]]. an exponential random variable with parameter [lambda] defined on a probability space.

Theorem 4.3. Let v be the Laplace transform of solution of equation (1.1) obtained in Theorem 3.1. There are stochastic processes [X.sup.k.sub.t], [Y.sup.k.sub.t], k = 1, ..., n such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (4.3)

Proof. Since [v.sub.k] = 1/[S.sub.k], with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (4.4)

and [S.sub.0,k] = 1/[v.sub.0,k]. Denoting Gk(s) = [[integral].sup.s.sub.0] [g.sub.k] ([tau])d[tau], we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Using Lemma 4.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

We define the stochastic processes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Now, we consider a random variable [[eta].sup.[vq.sup.2]] with exponential distribution of parameter [vq.sup.2] which is independent of [N.sub.vq.sup.2]t], therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

We obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Assuming that [[eta].sup.[vq.sup.2] is independent of the Poisson random variable [N.sub.[vq.sup.2]t], we deduce

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (4.5)

and the result of the theorem follows.

Acknowledgment

We thank Martin Grothaus for useful discussions. Financial support by GRICES, Portugal/ Tunisia, 2004 and FCT, POCTI - Programa Operacional Ciencia, Tecnologia e Inovacao, FEDER are gratefully acknowledged.

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F. Cipriano

GFM e Dep. de Matematica FCT-UNL, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal E-mail: cipriano@gfm.cii.fc.ul.pt

H. Ouerdiane

Departement de Mathematiques, Faculte des Sciences de Tunis, 1060 Tunis, Tunisia, E-mail: habib.ouerdiane@fst.rnu.tn

J. L. Silva

University of Madeira, CCM 9000-390 Funchal, Portugal E-mail: luis@uma.pt

R. Vilela Mendes

CMAF, Complexo Interdisciplinar, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal E-mail: vilela@cii.fc.ul.pt

The aim of this paper is to study the following nonlinear stochastic equation of convolution type

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.1)

where [??] is a [R.sub.n]-generalized vector field, [??] is a n-dimensional generalized function, v > 0 a real constant, t [member of] [0,[infinity]) the time parameter, x = ([x.sub.1], ..., [x.sub.n]) [member of] [R.sub.n] the spatial variable, [DELTA] the Laplacian operator in [R.sub.n], [nabla] the gradient and * the convolution product for generalized functions (see [1], [11] and Subsection 2.2 for more details), [??] * [??] is a driving term, [??] * [??] denotes the differential operator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and the initial condition [[??].sub.0] = ([u.sub.0,1], ..., [u.sub.0,n]) is a n-dimensional generalized stochastic process, see Section 3 for more details.

Problem (1.1) with * replaced by the usual product would coincide with the Burgers equation well known in the literature ([3], [5] and references therein). However the physical interpretation of (1.1) is quite different. This is easily seen by comparing the j-component of the Fourier transform of the nonlinear kinetic term ([??].[nabla]) [??] in the Burgers equation, namely

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.2)

with the j-component of the Fourier transform of the nonlinear term in (1.1), ([??] * [nabla]) [??]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.3)

[[??].sub.i] denotes the Fourier transform of [u.sub.i] .

In the Burgers case the expression (1.2) implies that the Fourier modes at length scale 2[pi]/q and 2[pi]/k-q control the eddies at scale 2[pi]/k, consistent with the phenomenological description of the inertial range in the turbulence cascade. However, in the convolution case the nonlocal nonlinearity corresponds to a self-interaction of the modes at each length scale. Therefore the convolution equation has a different physical interpretation. Nevertheless, nonlocal nonlinearities are also important in models of transport in magnetized plasmas, see [4] and also in the modeling of convection driven by density gradients as it arises in geophysical fluid flows, see [12], [13], [14] and [15].

By restricting oneself to solutions of gradient type, the Burgers equation may be linearized by the Cole-Hopf transformation. This provides the most general solution for (1 + 1)-dimensions but not for (1 + n)-dimensions. In contrast, for our equation (1.1), using the Laplace transform in a general setting, we obtain a general solution for (1 + n)-dimensions.

The paper is organized as follows: In Section 2 we provide the mathematical background needed to solve the Cauchy problem stated above, namely the spaces of test and generalized functions, the characterization theorem of generalized functions and the convolution product as well as some of its properties. In Section 3 we combine the convolution calculus and the characterization theorem in order to find an explicit solution of the problem (1.1).

Finally, in Section 4 we obtain a stochastic representation of the Laplace transform of the solution. Stochastic representations of differential equations and their solutions, not only provide a new interpretation of the solutions but are also useful for existence results and in the study of the fluctuations associated to the phenomena for which the equation represents a mean-field approximation. For a striking recent example of such use of stochastic representations we refer to thework done on the Navier-Stokes equation (see [16] and references therein).

2. Preliminaries

2.1. Test and Generalized Functions Spaces

In this section we introduce the framework needed later on. The starting point is the real Hilbert space H = [L.sup.2](R,[R.sup.d]) x [R.sup.r], d, r [member of] N with scalar product (*, *) and norm |*|. More precisely, if (f, x) = (([f.sub.1], ..., [f.sub.d]), ([x.sub.1], ..., [x.sub.r])) [member of] H, then

[ILLUSTRATION OMITTED]

Let us consider the real nuclear triplet

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

The pairing <*,*> between M' and M is given in terms of the scalar product in H, i.e., <([omega], x), ([xi], p) := ([omega], [xi])[L.sup.2] + (x, p) [R.sup.r], ([omega], x) [member of] M' and ([xi], p) [member of] M. Since M is a Frechet nuclear space, it can be represented as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [S.sub.n](R,[R.sup.d]) x [R.sup.r] is a Hilbert space with norm squared given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], see e.g. [8] or [2] and references therein. We will consider the complexification of the triple (2.1) and denote it by

N' [contains] Z [contains] N (2.2)

where N = M+ iM and Z = H + iH. On M' we have the standard Gaussian measure [gamma] given by Minlos's theorem via its characteristic functional, namely for every ([xi], p) [member of] M

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In order to solve the (1+n)-dimensional equation of convolution type (1.1) we need to introduce an appropriate space of vectorial generalized functions. We borrow this construction from [9]. Let [theta] = ([[theta].sub.1], [[theta].sub.2]) : [R.sup.2.sub.+] [right arrow] R, ([t.sub.1], [t.sub.2]) [right arrow] [[theta].sub.1] ([t.sub.1]) + [[theta].sub.1] [t.sub.2] where [[theta].sub.1], [[theta].sub.2] are twoYoung functions, i.e., [[theta].sub.i] : [R.sub.+] [right arrow] [R.sub.+] continuous convex strictly increasing function and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For every pair m = ([m.sub.1],[m.sub.2]) with [m.sub.1],[m.sub.2] .]0,[infinity][, we define the Banach space [F.sub.[theta]],m([N.sub.-n]), n [member of] N by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Here, for each z = ([omega], x) we have [theta]([m|z|.sub.-n]) := [[theta].sub.1]([m.sub.1]|[omega]|-n) + [[theta].sub.2]([m.sub.2]|x|). Now we consider as test function space the space of entire functions on N' of ([[theta].sub.1], [[theta].sub.2])-exponential growth and minimal type

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

endowed with the projective limit topology. We would like to use [F.sub.[theta]] (N') to construct a triple centered in the complex Hilbert space [L.sup.2](M', [gamma]). To this end we need another condition on the pair ofYoung functions ([[theta].sub.1], [[theta].sub.2]). Namely,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.3)

This is enough to obtain the following Gelfand triple

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (2.4)

where [F.sub.[theta]] (N') is the topological dual of [F.sub.[theta]] (N') with respect to [L.sup.2](M', [gamma]) endowed with the inductive limit topology.

In applications it is very important to have the characterization of generalized functions from [F'.sub.[theta]] (N'). First we define the Laplace transform of an element in [F'.sub.[theta]] (N').

For every fixed element ([xi], p) [member of] N the exponential function exp(([xi], p)) is a well defined element in [F.sub.[theta]] (N'), see [7]. The Laplace transform L of a generalized function [PHI] [member of] [F'.sub.[theta]] (N') is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (2.5)

We are ready to state to characterization theorem (see e.g., [7] and [1] for the proof) which is the main tool in our further consideration.

Theorem 2.1.

1. The Laplace transform is a topological isomorphism between [F.sub.[theta]] (N') and the space [G.sub.[theta]] * (N), where [G.sub.[theta]] * (N) is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and [G.sub.[theta] *,m]([N.sub.n]) is the space of entire functions on [N.sub.n] with the following [theta]-exponential growth condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Here [[theta].sup. *] = ([[theta].sup.*.sub.1], [[theta].sup.*.sub.2]), where [[theta].sup.*.sub.i] = sup (tx - [[theta].sub.i] (t)) is the Legendre transform associated to the function [[theta].sub.i], i = 1, 2 sup t [greater than or equal to] 0.

2. In the particular case [theta](x) = ([[theta].sub.1](x), [[theta].sub.2](x)) = (x, x), we denote the space [F'.sub.[theta]] (N') by [F.sub.'x](N'). Then the Laplace transform realizes a topological isomorphism between the distributions space [F'.sub.x](N') and the space Hol0(N) of holomorphic function on a neighborhood of zero of N.

2.2. The Convolution Product *

It is well known that in infinite dimensional complex analysis the convolution operator on a general function space F is defined as a continuous operator which commutes with the translation operator. Let us define the convolution between a generalized and a test function. Let [PHI] [member of] [F'.sub.[theta]] (N]) and [phi] [member of] [F.sub.[theta]] (N') be given, then the convolution *. is defined by

([PHI] * [phi])([omega], x) := <<[PHI], [[tau].sub.-([omega],[phi]>>.

where [[tau].sub.-([omega],x) is the translation operator, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

It is not hard the see that [PHI] * [phi] [member of] [F.sub.[theta]] (N'), cf. [7]. The convolution product is given in terms of the dual pairing as ([PHI] * [phi])(0,0) = <<[PHI], [phi]>> for any [PHI] [member of] [F'.sub. [theta]] (N') and [phi] [member of] [F.sub.[theta]] (N').

We can generalize the above convolution product for generalized functions as follows. Let [PHI], [PSI] [member of] [F'.sub.[theta]] (N') be given, then [PHI] * [PSI] is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (2.6)

This definition of convolution product for generalized functions will be used later for the solution of the equation (1.1). We have the following equality, (see [11], Proposition 3):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

As a consequence of the above equality and definition (2.6) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.7)

which says that the Laplace transform maps the convolution product on [F'.sub.[theta]] (N') into the usual pointwise product in the algebra of functions [G.sub.[theta] *](N). Therefore we may use Theorem 2.1 to define the convolution product between two generalized functions as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

This allows us to introduce the convolution exponential of a generalized function. In fact, for every [PHI] [member of] [F'.sub.[theta] (N') we may easily check that exp [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Using the inverse Laplace transform and the fact that any Young function [theta] verifies the property ([theta] *) * = [theta] we obtain that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Now we give the definition of the convolution exponential of [PHI] [member of] [F'.sub.[theta] (N'), denoted by exp * [PHI]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Notice that exp * [PHI] is a well defined element in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and therefore the distribution exp * [PHI] is given in terms of a convergent series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (2.8)

where [[PHI].sup.* n] is the convolution of [PHI] with itself n times, [[PHI].sup. * 0] := [[delta].sub.0] by convention with [[delta].sub.0] denoting the Dirac distribution at 0. We refer to [1] for more details concerning convolution product on [F'.sub.[theta]] (N').

A one parameter generalized stochastic process with values in [F'.sub.[theta]] (N') is a family of generalized functions {[PHI](t), t [greater than or equal to] 0} [subset] [F'.sub.[theta]] (N'). The process [PHI](t) is said to be continuous if the map t [right arrow] [PHI] (t) is continuous. For a given continuous generalized stochastic process [(X(t)).sub.t[greater than or equal to] 0 we define the generalized stochastic process

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.9)

The process Y(t,[omega], x) is differentiable and we have [partial derivative/[partial derivative]t] Y (t,[omega],x) = X (t,[omega],x). The details of the proof can be seen in [10], Proposition 11.

2.3. Convolution Inverse of Distributions

Let [PHI] a fixed element on the distribution space [F'/sub.[theta] (N') and consider the following convolution equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.10)

Applying the Laplace transform to the convolution equation (2.10) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

If [??] ([xi], q) [not equal to] 0 for every ([xi], q) [member of] N, then using the division result in the space [G.sub.[theta]*](N) (see [6]) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Moreover, by the Laplace transform isomorphism (see Theorem 2.1), we prove the existence and uniqueness of the solution [PSI] [member of] [F'.sub.[theta] (N') in the equation (2.10). If we denote this solution [PSI] by [[PHI].sup.*-1], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] .

This division result is also true in the limit case [theta] (x) = (x, x); i. e., [??] [member of] [G.sub.[theta]*](N) = [Hol.sub.0](N).

3. Solution of the n-dimensional Convolution Equation We are now ready to solve the Cauchy problem stated in (1.1) which we recall for the reader convenience, namely

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.1)

The different terms in (3.1) are as follows: [[??].sub.0](x) = ([u.sub.0],1(x), ..., [u.sub.0,n](x)) is a generalized function; [u.sub.0,j (x) [member of] [F'.sub.[theta]] (N'), v > 0 a real constant, x = ([x.sub.1], ..., [x.sub.n]) [member of] [R.sup.n],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

We are now ready to prove the main result of this paper, namely we obtain the explicit solution of the equation (3.1). using the tools from Section 2.

Theorem 3.1. Let [[??].sub.0](x) = ([u.sub.0,1](x), ..., [u.sub.0,n](x)) and [??] = ([f.sub.1], ..., [f.sub.n]) be such that [u.sub.0,k], [f.sub.k] [member of] [F'.sub.x](N'), k = 0, ..., n. Then the solution [??](t, [omega], x) of the nonlinear equation of convolution type (3.1) is given explicitly by the following system:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.2)

with k = 1, ..., n and [[delta].sub.0] is the Dirac measure at point zero and [[gamma].sub.2vt] is the Gaussian measure with variance 2vt on [R.sub.n].

Proof. We denote [[partial derivative].sub.t] = [partial derivative]/[[partial derivative].sub.t] such that the n-dimensional equation (3.1) may be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.3)

where k = 1, ..., n.

We denote by [v.sub.k] = [v.sub.k](t, [xi], q), [g.sub.k] = [g.sub.k](t, [xi], q) and [v.sub.0,k] = [v.sub.0,k]([xi], q), [xi] [member of] S (R,[R.sup.d), q [member of] [R.sub.n], the Laplace transforms of the generalized functions [u.sub.k] = [u.sub.k](t, [omega], x), [f.sub.k] = [f.sub.k](t, [omega], x) and the initial condition [u.sub.0,k] = [u.sub.0,k]([omega], x), respectively, for k = 1, ..., n. Applying the Laplace transform to the system (3.3) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.4)

Changing the variables, Sk = 1/[v.sub.k] = 1, ..., n, the system (3.4) is equivalent to the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.5)

We denote [S.sub.1] .... [[??].sub.k] ... [S.sub.n] = [S.sub.1] ...[S.sub.k-1[S.sub.k]+1] ... [S.sub.n], for k = 1, ..., n. Ifwe multiply the first equation of the system (3.5) by [S.sub.1] .... [S.sub.k] ...[S.sub.n], we deduce

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.6)

Let us denote such equation by ([E.sub.k]), k = 1, ..., n. If we fixe k, then the difference ([E.sub.1]) - ([E.sub.k]) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.7)

After simplification, we divide by [S.sub.1]Sk and obtain the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.8)

which can be integrated. The solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.9)

where for simplification of notation gj (s) = gj (s, [xi], q), j = 1, ..., n. Then we have the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.10)

Introducing the expression of [S.sub.k]/[S.sub.j] in (3.5) we deduce the following linear system of equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.11)

For any fixed k, the solution of the homogeneous equation is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (3.12)

where [lambda] is a constant. Then the solution of (3.11) is given by the method of variation of constants as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.13)

where the constant [lambda] is determined by the initial conditions; [S.sub.k](0, [xi], q) = [S.sub.0,k] = [lambda]. Then (3.13) may be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.14)

Since [S.sub.k](t, [xi], q) = 1/[v.sub.k] (t,[xi],q), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.15)

In fact, it is easy to show that for every t [greater than or equal to] 0, the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

belongs to the space [Hol.sub.0](N) and satisfy Y(t, 0, 0) = 1 [not equal to] 0. Then there exists U a neighborhood of (0, 0) of N, such that Y(t, q, [xi]) [not equal to] 0 for every ([xi], q) [member of] U. Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Finally, to obtain the solution of the equation (3.1) we use the following equalities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

and then [u.sub.k](t, x), k = 1, ..., n is given by the Laplace inverse transform according to the theorem 2.1, as in (3.2).

Corollary 3.2. If the potential [??] in the equation (3.1) does not depend of the time variable t, i.e., [??] = [??] (x), then the solution is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

with k = 1, ..., n. In particular if f = 0, the solution has the from

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

with k = 1, ..., n and [nabla] * represents the divergence operator.

4. Stochastic Representation of Solution of the n-dimensional Convolution Equation

In the previous section we obtained an explicit solution of the equation (1.1). Here, we are interested in a stochastic representation of that solution. Let us start with a simple and useful Lemma:

Lemma 4.1. Let b, [alpha] [member of] R with b [not equal to] = 0. We can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (4.1)

where [N.sub.b] is a random variable with Poisson distribution with intensity b, defined on a probability space ([OMEGA], P), E denoting the mathematical expectation.

Proof. Since Nb is a random variable with Poisson distribution with intensity b, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (4.2)

which implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

We use the same notations as in section 3. First we take f = 0 in equation (1.1) and obtain a stochastic representation for the Laplace transfom v of its solution u.

Theorem 4.2. Let v be the Laplace transform of the solution of equation (1.1), with f = 0, obtained in Theorem 3.1. There are stochastic processes [X.sup.k.sub.t], k = 1, ..., n such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Proof. According to (3.14), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Using Lemma 4.1, we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Defining the stochastic processes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

for k = 1, ..., n, the result of the theorem follows.

Let us denote by [[eta].sup.[lambda]]. an exponential random variable with parameter [lambda] defined on a probability space.

Theorem 4.3. Let v be the Laplace transform of solution of equation (1.1) obtained in Theorem 3.1. There are stochastic processes [X.sup.k.sub.t], [Y.sup.k.sub.t], k = 1, ..., n such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (4.3)

Proof. Since [v.sub.k] = 1/[S.sub.k], with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (4.4)

and [S.sub.0,k] = 1/[v.sub.0,k]. Denoting Gk(s) = [[integral].sup.s.sub.0] [g.sub.k] ([tau])d[tau], we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Using Lemma 4.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

We define the stochastic processes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Now, we consider a random variable [[eta].sup.[vq.sup.2]] with exponential distribution of parameter [vq.sup.2] which is independent of [N.sub.vq.sup.2]t], therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

We obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Assuming that [[eta].sup.[vq.sup.2] is independent of the Poisson random variable [N.sub.[vq.sup.2]t], we deduce

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (4.5)

and the result of the theorem follows.

Acknowledgment

We thank Martin Grothaus for useful discussions. Financial support by GRICES, Portugal/ Tunisia, 2004 and FCT, POCTI - Programa Operacional Ciencia, Tecnologia e Inovacao, FEDER are gratefully acknowledged.

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F. Cipriano

GFM e Dep. de Matematica FCT-UNL, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal E-mail: cipriano@gfm.cii.fc.ul.pt

H. Ouerdiane

Departement de Mathematiques, Faculte des Sciences de Tunis, 1060 Tunis, Tunisia, E-mail: habib.ouerdiane@fst.rnu.tn

J. L. Silva

University of Madeira, CCM 9000-390 Funchal, Portugal E-mail: luis@uma.pt

R. Vilela Mendes

CMAF, Complexo Interdisciplinar, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal E-mail: vilela@cii.fc.ul.pt

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Author: | Cipriano, F.; Ouerdiane, H.; Silva, J.L.; Mendes, R. Vilela |
---|---|

Publication: | Global Journal of Pure and Applied Mathematics |

Article Type: | Technical report |

Geographic Code: | 1USA |

Date: | Apr 1, 2008 |

Words: | 4009 |

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