# Evolution equation associated with the power of the Gross Laplacian.

Abstract

We study an evolution equation associated with the power of the Gross Laplacian [[DELTA].sup.p.sub.G] and a potential function V on an infinite dimensional space. The initial condition is a generalized function. The main technique we use is the representation of the Gross Laplacian as a convolution operator. This representation enables us to apply the convolution calculus on a suitable distribution space to obtain the explicit solution of the perturbed evolution equation. Our results generalize those previously obtained by Hochberg [7] in the one dimensional case with V = 0, as well as by Barhoumi-Kuo-Ouerdiane for the case p = 1

AMS Subject Classification: Primary 60H40; secondary 60H15, 46F25, 46G20.

Keywords: Evolution equation, Gross Laplacian, potential function, white noise analysis, generalized functions, convolution operator, Laplace transform, duality theorem.

1. Introduction

In the paper [7] Hochberg studied the parabolic partial differential equation

[partial derivative]u/[partial derivative]t = [(-1).sup.n+1] [[partial derivative].sup.2n]u/ [partial derivative][x.sup.2n], n [greater than or equal to] 2.

He showed that the fundamental solution of this equation is the density of a finitely additive signed measure of unbounded variation. In this paper we will consider the infinite dimensional generalization of the above equation with an additional potential function

[partial derivative][U.sub.t]/[partial derivative]t = [(-1).sup.p+1] 1/2 [[DELTA].sup.p.sub.G][U.sub.t] + [V.sub.t], p [member of] N, (1.1)

where [[DELTA].sub.G] is the Gross Laplacian [5] [8]. We will show that the solutions are generalized functions (rather than ordinary function), which reflect the above fact that the corresponding "measure" are finitely additive signed measure (rather than countably additive measure) of unbounded variation. On the other hand, Equation (1.1) with p = 1 has been studied in the paper [2].

In Section 2 we will briefly described the infinite dimensional framework in order to setup Equation (1.1). In Section 3 we will study this equation with an initial condition and show that the unique solution is a generalized function. The main tool is the interpretation of the Gross Laplacian as a convolution operator. In Section 4 we will give concluding remarks.

2. Background

First we review from the paper [4] basic concepts, notation, and some results which will be needed in the present paper. Independent development for similar results can be found in the paper [1].

Let X be a real nuclear Frechet space with topology given by an increasing family [{|x|}.sub.p]; p [member of] [N.sub.0]} of Hilbertian norms, [N.sub.0] being the set of nonnegative integers. Then X is represented as X = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [X.sub.p], where [X.sub.p] is the completion of X with respect to the norm [|x|.sub.p]. We use [X.sub.-p] to denote the dual space of [X.sub.p]. Then the dual space X' of X can be represented as X' = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [X.sub.-p] and is equipped with the inductive limit topology.

Let N = X + iX and [N.sub.p] = [X.sub.p] + [iX.sub.p], p [member of] Z, be the complexifications of X and [X.sub.p], respectively. For n [member of] [N.sub.0] we denote by [N.sup.[??]n] the n-fold symmetric tensor product of N equipped with the [pi]-topology and by [N.sup.[??]n.sub.p] the n-fold symmetric Hilbertian tensor product of [N.sub.p]. We will preserve the notation [|x|.sub.p] and [|x|.sub.-p] for the norms on [N.sup.[??]n.sub.p] and [N.sup.[??]n.sub.-p], respectively.

Let [theta] be a Young function, i.e., it is a continuous, convex, and increasing function defined on [R.sub.+] such that [theta](0) = 0 and [lim.sub.x[right arrow][infinity]] [theta](x)=x = 1. We define the conjugate function [theta]* of [theta] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For a Young function [theta], we denote by [F.sub.[theta]](N') the space of holomorphic functions on N0 with exponential growth of order [theta] and of minimal type. Similarly, let G[theta](N) denote the space of holomorphic functions on N with exponential growth of order [theta] and of arbitrary type. Moreover, for each p [member of] Z and m > 0, define Exp([N.sub.p], [theta], m) to be the space of entire functions f on [N.sub.p] satisfying the condition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the spaces [F.sub.[theta]](N') and G[theta](N) can be represented as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and are equipped with the projective limit topology and the inductive limit topology, respectively. The space [F.sub.[theta]](N') is called the space of test functions on N'. Its dual space [F'.sub.[theta]](N'), equipped with the strong topology, is called the space of distributions on N'.

For p [member of] [N.sub.0] and m > 0, we define the Hilbert spaces

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[theta].sub.n] = [inf.sub.r>0] [e.sup.[theta](r)]/[r.sup.n], n [member of] [N.sub.0]. Put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The space [F.sub.[theta]](N) equipped with the projective limit topology is a nuclear Frechet space [4]. The space [G.sub.[theta]](N') carries the dual topology of [F.sub.[theta]](N) with respect to the C-bilinear pairing given by

<<[??], [??]>> = [summation over (n[greater than or equal to]0)] n! <[[PHI].sub.n], [[psi].sub.n]>,(2.1)

where [??] = [([[PHI].sub.n]).sup.[infinity].sub.n=0] [member of] [G.sub.[theta]](N') and [??] = [([[psi].sub.n]).sup.[infinity].sub.n=0] [member of] [F.sub.[theta]](N).

It was proved in [4] that the Taylor map defined by

T : [psi] [??] (1/n! [[psi].sup.(n)](0)).sup.[infinity].sub.n=0]

is a topological isomorphism from [F.sub.[theta]](N') onto [F.sub.[theta]](N). The Taylor map T is also a topological isomorphism from [G.sub.[theta]*](N)) onto [G.sub.[theta]](N')). The action of a distribution [PHI] [member of] [F'.sub. [theta]](N') on a test function [psi] [member of] [F.sub.[theta]](N') can be expressed in terms of the Taylor map as follows:

<<[PHI], [psi]>> = <<[??], [??]>>

where [??] = [(T*).sup.-1][PHI] and [??] = T[psi].

It is easy to see that for each [xi] 2 N, the exponential function

[e.sub.[xi]](z) = [e.sup.<z,[xi]>, z [member of] N',

is a test function in the space [F.sub.[theta]](N') for any Young function [theta]. Thus we can define the Laplace transform of a distribution [PHI] [member of] [F'.sub.[theta]](N') by

[??]([xi]) = <<[PHI], [e.sub.[xi]]>>, [xi] [member of] N. (2.2)

[??]From the paper [4], we have the duality theorem which says that the Laplace transform is a topological isomorphism from [F'.sub.[theta]](N') onto [G.sub.[theta]*](N).

For [psi] [member of] [F.sub.[theta]](N'), the translation [t.sub.x] [psi] of [psi] by x [member of] N' is defined by

[t.sub.x][psi] (y) = [psi] (y - x), y [member of] N'.

The translation operator [t.sub.x] is a continuous linear operator from [F.sub.[theta]](N') into itself for any x [member of] N'. By a convolution operator on the space [F.sub.[theta]](N') of test functions we mean a continuous linear operator from [F.sub.[theta]](N') into itself which commutes with translation operators [t.sub.x] for all x [member of] N'.

We define the convolution [PHI] * ' of a distribution [PHI] [member of] [F'.sub.[theta]](N') and a test function [psi] [member of] [F.sub.[theta]](N') to be the function

([PHI] * [psi])(x) = <<[PHI], t-x[psi]>>; x [member of] N'.

Direct calculations show that [PHI]*[psi] [member of] [F.sub.[theta]](N') for any [psi] [member of] [F.sub.[theta]](N') and that the mapping [T.sub.[PHI]] defined by

[T.sub.[PHI]] : [psi] [??] [PHI] * [psi], [psi] [member of] [F.sub.[theta]](N'),

is a convolution linear operator on [F.sub.[theta]](N'). Conversely, it was proved in [3] that all convolution operators on [F.sub.[theta]](N') occur this way, i.e., if T is a convolution operator on [F.sub.[theta]](N'), then there exists a unique [PHI] [member of] [F'.sub.[theta]](N') such that T = [T.sub.[PHI]], or equivalently,

T([psi]) = [T.sub.[PHI]](') = [PHI] * [psi], [psi] [member of] [F.sub.[theta]](N'). (2.3)

Suppose [[PHI].sub.1], [[PHI].sub.2] [member of] [F'.sub.[theta]](N'). Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the convolution operators given by [[PHI].sub.1] and [[PHI].sub.2], respectively, as in Equation (2.3). It is clear that the composition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [??] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is also a convolution operator on [F.sub.[theta]](N'). Hence there exists a unique distribution, denoted by [[PHI].sub.1] * [[PHI].sub.2], in [F'.sub.[theta]](N') such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

The distribution [[PHI].sub.1] * [[PHI].sub.2] in Equation (2.4) is called the convolution of [[PHI].sub.1] and [[PHI].sub.2]. From Proposition 1 of the paper [3] we have the following equality for any [[PHI].sub.1],[[PHI].sub.2] [member of] [F'.sub.[theta]](N'),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

Let [gamma] be the standard Gaussian measure on the dual space X0 of the real nuclear space X, namely, its characteristic function is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [|x|.sub.0]is the norm [|x|.sub.p] on X for p = 0. Suppose that the Young function [theta] satisfies [lim.sub.r[right arrow]+[infinity]] [theta](r)/[r.sup.2] < +1. Then we obtain the Gel'fand triple [4]

[F.sub.[theta]](N') [??] [L.sup.2](X', [gamma]) [??] [F'.sub.[theta]](N').

It is worthwhile to mention that the S-transform, defined on [F'.sub.[theta]](N'), is related to the Laplace transform by

S([PHI])([xi]) = [??]([xi])[e.sup. <h[xi],[xi]>/2], [xi] [member of] N, [PHI] [member of] [F'.sub.[theta]](N'). (2.6)

Let [beta] be a continuous, convex, and increasing function on [R.sub.+]. Suppose f is function in Exp(C, [beta], m) for some m > 0. For each distribution [PHI] in [F'.sub.[theta]](N'), we define the convolution composition [f.sup.*]([PHI]) of f and [PHI] by

([f.sup.*]([PHI]))[??]b= f([??]). (2.7)

It was proved in [3] that [f.sup.*]([PHI]) belongs to [F'.sub.[lambda]](N') with [greater than or equal to] = [([beta] [??] [e.sup.[theta]*]).sup.*].

In particular, when [beta](x) = x, x [member of] [R.sub.+], and f(z) = [e.sup.z], z [member of] C, we get a distribution [e.sup.*[PHI]] in the space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](N') for each [PHI] [member of] [F'.sub.[theta]](N'). Moreover, by Equation (2.7), we have

([e.sup.*[PHI]])[??]= [e.sup.[??]]. (2.8)

The distribution [e.sup.*[PHI]] has the following series expansion

[e.sup.*[PHI]] = [[infinity].summation over (n=0)] 1/n! [[PHI].sup.*n],

where [[PHI].sup.*n] = [PHI]*[PHI]* ... *[PHI] (n times) and the convergence is in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (N') with respect to the strong topology.

3. Initial-valued Evolution equation

Let I [subset] R be an interval containing the origin. Consider a family {[[PHI].sub.t], [member of] I} of distributions in [F'.sub.[theta]](N'). We assume that the function t [??] [[PHI].sub.t] is continuous from I into [F'.sub.[theta]](N'). Then the function t [??] [[??].sub.t] is continuous from I into [G.sub.[theta]*](N). Thus for each t [member of] I, the set {[[??].sub.s], s [member of] [0, t]} is a compact subset of [G.sub.[theta]*](N). In particular, it is bounded in [G.sub.[theta]*](N). Hence there exist constants p [member of] N', m > 0, and [C.sub.t] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This inequality shows that the function [xi] [??] [[integral].sup.t.sub.0] c[PHI]s([xi]) ds belongs to the space [G.sub.[theta]*](N). Hence there exists a unique distribution, denoted by [[integral].sup.t.sub.0] [PHI]s ds, in [F'.sub.[theta]](N') satisfying

([[integral].sup.t.sub.0] [[PHI].sub.s] ds)[??] ([xi]) = [[integral].sup.t.sub.0] [[??].sub.s]([xi]) ds, [xi] [member of] N.

Moreover, the process [E.sub.t] = [[integral].sup.t.sub.0] [PHI]s ds; t 2 I; is differentiable in [F'.sub.[theta]](N') and satisfies the equation

[partial derivative] [[partial derivative].sub.t] [E.sub.t] = [[PHI].sub.t].

Let {[[PHI].sub.t]} and {[M.sub.t]} be two continuous [F'.sub.[theta]](N')-processes. Consider the initial value problem

d[X.sub.t]/dt = [[PHI].sub.t] * [X.sub.t] + [M.sub.t], [X.sub.0] = F [member of] [F'.sub.[theta]](N'). (3.1)

The next theorem is from Theorem 4 of the paper [3].

Theorem 3.1 The stochastic differential equation (3.1) has a unique solution in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (N') given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

We can apply Theorem 3.1 to study an evolution equation for a power of the Gross Laplacian and a generalized potential function with the initial condition being a Generalized function. Let [psi] [member of] [F.sub.[theta]](N') be represented by

[psi] (x) = [summation over (n[greater than or equal to]0)] <[x.sup.[cross product]n], [psi](n)>.

The Gross Laplacian ([[DELTA].sub.G] [psi])(x) of [psi] at x [member of] N' is defined to be

([[DELTA].sub.G] [psi])(x) = [summation over (n[greater than or equal to]0)] (n + 2)(n + 1) <[x.sup.[cross product]n],<[tau], [[psi].sup.(n+2)]>>,

where [tau] is the trace operator, namely,

<[tau], [xi] [cross product] [eta]> = <[xi], [eta]>, [xi], [eta] [member of] N.

For more information on the Gross Laplacian, see [5] [6] [8] [9] [10].

It turns out that the Gross Laplacian [[DELTA].sub.G] can be extended to be a continuous linear operator from [F'.sub.[theta]](N') into itself and the extension is a convolution operator given by

[[DELTA].sub.G][PSI] = T * [PSI], [PSI] [member of] [F'.sub.[theta]](N'), (3.3)

where T is the generalized function in [F'.sub.[theta]](N') with the Laplace transform given by [??] = (0, 0, [tau], 0; ...) [member of] [G.sub.[theta]](N') as in Equation (2.1).

Proposition 3.2 For every positive integer p we have

[[DELTA].sup.p.sub.G][PSI] = (T *p) * [PSI], [PSI] [member of] [F'.sub.[theta]](N'). (3.4)

Moreover, the generalized function associated with [[DELTA].sup.p.sub.G] is given by

[??] = (0, 0, ..., [[tau].sup.[cross product]p], 0, ...). (3.5)

Proof. Using Equations (2.4) and (3.3), we obtain

[[DELTA].sup.p.sub.G][PSI] = T * p * [PSI]

But the Laplace transform of T is given by

[??] ([xi]) = <[tau], [[xi].sup.[cross product]2> = <[xi], [xi]> = [|[xi]|.sup.2.sub.0].

Hence we have

[[??]) = <[tau], [[xi].sup.[cross product]2]>.sup.p] = [<[xi], [xi]>.sup.p] = [|[xi]|.sup.2p.sub.0].

For any positive integer p, let S = T * p and let the formal power series associated with S be given by [??] = ([S.sub.0], [S.sub.1], ..., [S.sub.n], ...). Then we can use the definition of the Laplace transform and the bilinear pairing between test functions and distributions in Equation (2.1) to deduce the following relationship

[[??]) = <[T.sup.*p], [e.sup.[xi]]> = [summation over (n[greater than or equal to]0)] n! <[S.sub.n], [[xi].sup.[cross product]n]/n!> = <[xi], [xi]>.sup.p],

which implies that [S.sub.n] = 0 for all n [not equal to] 2p and [S.sub.2p] = [[tau].sup.[cross product]p]. Therefore,

[??] = [??] = (0, 0, ..., [[tau].sup.[cross product]p], 0, ...).

This proves Equation (3.5).

Theorem 3.3 Let [theta] be a Young function such that [lim.sub.r[right arrow][infinity]] [theta](r)=r2 < 1 and let F 2 [F'.sub.[theta]](N') and fVtg be a continuous [F'.sub.[theta]](N')-processes. Then the following evolution equation associated with the p-th power of the Gross Laplacian and a potential function [V.sub.t]

[partial derivative][U.sub.t]/[partial derivative]t = [(-1).sup.p+1] 1/2 [[DELTA].sup.p.sub.G][U.sub.t] + [V.sub.t], [U.sub.0] = F, (3.6)

has a unique solution in the space [F'.sub.[theta]](N') given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

where T is the generalized function given by Equation (3.3).

Proof. Use Equation (3.3) to rewrite Equation (3.6) as

[partial derivative][U.sub.t]/[partial derivative]t = [(-1).sup.p+1] 1/2 [T.sup.*p] * [U.sub.t] + [V.sub.t], [U.sub.0] = F.

Then we can apply Theorem 3.1 to this equation to get the unique solution in Equation (3.7).

4. Concluding remarks

1. We can further rewrite the solution in Equation (3.7) in another form. For t > 0, define the distribution [[mu].sub.t,p] by its Laplace transform

[??] [xi]) = exp [[(-1).sup.p+1]t/2 <[xi], [xi]>.sup.p]], [xi] [member of] N. (4.1)

Recall the duality theorem [4] which states that the Laplace transform is a topological isomorphism from [F'.sub.[theta]](N') onto [G.sub.[theta]*](N). Hence Equation (4.1) implies that [[mu].sub.t,p], t > 0; are generalized functions in the space [F'.sub.[theta]](N') with the Young function given by

[theta](x) = [x.sup.2p/2p-1], x [greater than or equal to] 0.

Therefore, the solution [U.sub.t] in equation (3.7) can be rewritten as

[U.sub.t] = F * [[mu].sub.t,p] + [[integral].sup.t.sub.0] [[mu].sub.t-s,p] * [V.sub.s] ds.

In particular, when [V.sub.t] = 0, we have the evolution equation

[partial derivative][U.sub.t]/[partial derivative]t = [(-1).sup.p+1] 1/2 [[DELTA].sup.p.sub.G][U.sub.t], [U.sub.0] = F, (4.2)

which has a unique solution given by

[U.sub.t] = F * [[mu].sub.t,p]. (4.3)

Hochberg [7] has studied the one-dimensional case of Equation (4.2) and showed that the fundamental solution defines a finitely additive measure with unbounded total variation. Using the white noise theory, we can now interpret this "finitely additive measure with unbounded total variation" as a generalized function in the space [F'.sub.[theta]](N'), which is given by Equation (4.3). This phenomenon is very much like the case of Feynman integral, which had been regarded as a finitely additive measure with unbounded total variation before the theory of white noise was introduced by T. Hida in 1975. It is a well-known fact that the Feynman integral is a generalized function [6] [9].

2. When p = 1, Equation (4.1) gives the equality

[??]([xi]) = exp [- t/2 [|[xi]|.sup.2.sub.0], [xi] [member of] X,

which shows that [[mu].sub.t,1] is the standard Gaussian measure on X0 with variance t, i.e., [[mu].sub.t,1] = [[gamma].sub.t] with [gamma]t defined by

[[gamma].sub.t](*) = [gamma] (*/[[square root of t]).

Note that the probability measure [[mu].sub.t,1] induces a positive distribution in the space [F'.sub.[theta]](N') given by

<<[[mu].sub.t,1][psi]>> = [[integral].sub.X'] [psi](x) d[[mu].sub.t,1](x) = [[integral].sub.X'] [psi]([square root of t] x) d[gamma](x), [psi] [member of] [F.sub.[theta]](N').

For more details, see the book [9]. Moreover, if the potential function is given by [V.sub.t] = [alpha] [W.sub.t] with [alpha] [member of] R and [W.sub.t] a white noise, then the solution in Equation (3.7) reduces to the one obtained in [2].

References

[1] N. Asai, I. Kubo, and H.-H. Kuo, General characterization theorems and intrinsic topologies in white noise analysis; Hiroshima Math. J. 31 (2001) 299-330.

[2] A. Barhoumi, H. H. Kuo, and H. Ouerdiane, Generalized Gross Heat equation with noises; Soochow J. of Math. 32 (2006) 113-125.

[3] M. Ben Chrouda, M. El Oued, and H. Ouerdiane, Convolution calculus and applications to stochastic differentntial equations; Soochow J. of Math. 28 (2002) 375-388.

[4] R. Gannoun, R. Hachaichi, H. Ouerdiane, and A. Rezgui, Un theoreme de dualite entre espace de fonctions Holomorphes a croissance exponentielle; J. Functional Analysis 171 (2000) 1-14.

[5] L. Gross, Potential theory on Hilbert space; J. Functional Analysis 1 (1967) 123-181.

[6] T. Hida, H.-H. Kuo, J. Potthoff, and L. Streit, White Noise: An Infinite Dimensional Calculus. Kluwer Academic Publishers, Dordrecht, 1993.

[7] K. J. Hochberg, A signed measure on path space related to Wiener measure; The Annals of Probability 6 (1978) 433-458.

[8] H.-H. Kuo, Gaussian Measures in Banach Spaces. Lecture Notes in Math., Vol. 463, Springer-Verlag, 1975 (reprinted by BookSurge, 2006)

[9] H.-H. Kuo, White Noise Distribution Theory. CRC Press, Boca Raton, 1996.

[10] N. Obata, White Noise Calculus and Fock Space. Lecture Notes in Math. 1577, Springer-Verlag, 1994.

Soumaya Gheryani

Department of Mathematics, Faculty of Sciences of Tunis,

University of Tunis El Manar, Tunis, Tunisia

E-mail: soumaye_gheryani@yahoo.fr

Hui-Hsiung Kuo

Department of Mathematics, Luisiana State University,

Baton Rouge, LA 70803, USA

E-mail: kuo@math.lsu.edu

Habib Ouerdiane

Department of Mathematics, Faculty of Sciences of Tunis,

University of Tunis El-Manar, Tunis, Tunisia

E-mail: habib.ouerdiane@fst.rnu.tn
COPYRIGHT 2007 Research India Publications
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2007 Gale, Cengage Learning. All rights reserved.

Terms of use | Privacy policy | Copyright © 2019 Farlex, Inc. | Feedback | For webmasters