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International Journal of Analysis
Volume 2016 (2016), Article ID 5963420, 11 pages
http://dx.doi.org/10.1155/2016/5963420
Research Article

Some Nonlinear Stochastic Cauchy Problems with Generalized Stochastic Processes

Equipe Analyse Algébrique Non Linéaire, Université des Antilles, Campus de Schoelcher, BP 7209, 97275 Schoelcher Cedex, Martinique

Received 4 July 2016; Accepted 25 July 2016

Academic Editor: Ying Hu

Copyright © 2016 Victor Dévoué. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study some nonlinear stochastic Cauchy problems in the framework of the -algebras. We adapt the definitions to this framework. By means of suitable regularizations, we define associated generalized problems. We use our previous results about the wave equation in canonical form to obtain generalized solutions. We compare the generalized solutions with the classical ones when they exist.

1. Introduction

A possibility in studying stochastic differential equations is to make use of the theory of Colombeau-type generalized functions spaces to overcome the multiplication problem in distribution space. Here, to study some nonlinear stochastic Cauchy problems, we choose to reformulate them correctly in the framework of the -algebras of Marti [13] in order to show that, following the example of the theory of Colombeau [4, 5], these algebras may serve as a tool for treating singular processes in stochastic analysis. Until now, similar studies were made only in Colombeau-type algebras.

A -algebra, where is overgenerated by a finite set , is always isomorphic to a Colombeau-type algebra but the asymptotic scale of this last algebra, which can be obtained from the generators of , is not explicit and difficult to work with. So, by using the overgeneration, we are able to work with asymptotics which are explicit. Therefore, we consider this framework more convenient.

We adapt the definitions to the framework of the -algebras. We interpret generalized stochastic processes on as measurable maps with values in a -algebra. Using our previous results about our study on the wave equation in canonical form [6, 7], we can prove that some nonlinear non-Lipschitz stochastic Cauchy problems have a unique solution. So this paper completes our research. We find some results similar to those of [8, 9] about the solutions in theory of Colombeau algebra.

The paper is organized as follows. In Section 2, we give some definitions and references for stochastic analysis, -algebras, and algebras of generalized stochastic processes. In Section 3, we study the following Cauchy problems formally written as where is a monotonic curve of equation , is not a characteristic curve, and are generalized stochastic processes on , and is a generalized process on . That is, , , and are weakly measurable maps of some probability space with values in the Schwartz distribution space , respectively, . The function is smooth; it can be non-Lipschitz (in ) but and all derivatives have polynomial growth.

For fixed, , we replace problem (resp., ) by a generalized one well-formulated (resp., ) in a convenient algebra. To do this, we use regularizations and cutoff techniques. We use two parameters. The first parameter regularizes the data and the second one replaces the problem by a family of Lipschitz problems. We show that problem (resp., ) has a unique solution in some algebras of generalized stochastic processes. Moreover, if problem (resp., ) admits a solution satisfying appropriate growth estimates on some open subset of , then this solution and the generalized one are equal in a meaning given in Theorem 16.

In Section 4, we are interested in a nonlinear stochastic Cauchy problem with the white noise as initial data: where is the curve of equation , is not a characteristic curve, and is the white noise on . The function is smooth; it can be non-Lipschitz but and all derivatives have polynomial growth. We treat problem in the same way as the previous ones and we study the limiting behavior of the generalized solution.

2. Algebra of Generalized Stochastic Processes

2.1. Stochastic Analysis

We refer the reader to [8, 10, 11], for some basic facts from stochastic analysis as construction of white noise and the relation between the white noise and Wiener process on .

Let be a probability space. A weakly measurable map is called a generalized stochastic process on .

For each fixed test function , the map ; is a random variable. The space of generalized stochastic processes is denoted by .

White noise on is constructed as follows.

The probability space is the space of tempered distributions and is the Borel -algebra generated by the weak topology. According to Bochner-Minlos theorem, there is a unique probability measure on such that for .

The white noise can be define as the identity mapping , . Remark that is a generalized Gaussian process with mean zero and variance , where denotes mathematical expectation. Its covariance is The white noise on is realized as the -fold distributional derivative of the Wiener process.

2.2. Algebras of Generalized Functions
2.2.1. The Presheaves of -Algebras

We recall briefly some notions that form the basis of our study [7, 12]. We refer the reader to the references. Take or . Consider

(1) a set of indices, a solid subring of the ring (that is to say, for any , with , if (i.e., for any , ), then ), and a solid ideal of ;

(2) a sheaf of -topological algebras on a topological space , such that, for any open set in , the algebra is endowed with a family of seminorms satisfying

Assume that, for any open subsets , of such that , we have and if is the restriction operator , then, for each , the seminorm extends to .

Assume that, for any family of open subsets of if , then, for each , , there exists a finite subfamily of and corresponding seminorms , , such that, for any , .

Set , is a sheaf of subalgebras of the sheaf and is a sheaf of ideals of [13]. The constant sheaf is the sheaf . We call presheaf of -algebra the factor presheaf of algebras .

We write the class in of . We simplify the notations by writing (resp., ) instead of (resp., ).

Overgenerated Rings. Let and be the subset of obtained as rational functions with coefficients in , of elements in as variables. Define We say that is overgenerated by ( is a solid subring of ). If is some solid ideal of , we also say that is overgenerated by .

Relationship with Distribution Theory. Let be an open subset of . If is a family of mollifiers , , and if , the convolution product family is a family of smooth functions slowly increasing in . So, the space of distributions can be embedded into . Taking as component of the multi-index , we shall choose the subring overgenerated by some of containing the family .

Association Process. Consider an open subset of , a sheaf of topological -vector spaces containing as a subsheaf, and a map from to such that is an element of . Assume that We say that and are - associated if, for each neighborhood of for the -topology, there exists such that . We write if . We define an association process between and by .

-Singular Support. Assume that Let be the set of all having a neighborhood on which is associated with a distribution. The -singular support of is the set

2.2.2. Generalized Operator and Generalized Restriction Mapping

We denote by an element of , .

Let be an open subset of and . We say that the algebra is stable under the family if for all and we have ,

If is stable under , for , is a well defined element of (i.e., not depending on the representative of ).

The stability condition is verified if the family is smoothly tempered, that is to say, if the following two conditions are satisfied:

(i) For each , , and , there is a positive finite sequence , such that .

(ii) For each , , , and , there is a positive finite sequence such that Let and ; we define If is stable under , the operator is called the generalized operator associated with via the family .

Consider . We say that is compatible with second-side restriction if Clearly, if , then is a well defined element of .

If the function is compatible with second-side restriction, the mapping is called the generalized second-side restriction mapping associated with .

2.3. The Algebras and

Set and for and and set as a set of indices, . For any open set , in , is endowed with the topology defined by the family of the seminorms: and for , , , and means that is a compact subset of . Let be a subring of the ring . We consider a solid ideal of . Put The generalized derivation provides with a differential algebraic structure.

Take . For any open set , in , is endowed with the topology defined by the family of the seminorms and , . Put The generalized derivation provides with a differential algebraic structure.

Remark 1. The () norms are bounded by the norms. We have .

2.4. Algebras of Generalized Stochastic Processes

Let be an open set in .

Definition 2. A -generalized stochastic process on a probability space is a mapping such that there is a representing function with the following properties:(i)For fixed , the map is jointly measurable on .(ii)Almost surely in , the map belongs to and it is a representative of ; that is, almost surely in , . The algebra of generalized stochastic processes is denoted by .

Definition 3. A -generalized stochastic process on a probability space is a map such that there is a representing function with the following properties:(i)For fixed , the map is jointly measurable on .(ii)Almost surely in , the map belongs to and it is a representative of ; that is, almost surely in , . The algebra of generalized stochastic processes is denoted by .

Remark 4. Let having the propertiesdefine by Let be a generalized stochastic process. If , then is measurable with respect to and smooth with respect to , hence jointly measurable. Also, belongs to . Thus, qualifies as a representative for a random generalized function. We obtain an imbedding .

3. Nonlinear Stochastic Problems

3.1. A Nonlinear Stochastic Problem with Additive Generalized Stochastic Process

We consider the Cauchy problem formally written aswhere is a monotonic curve of equation , is not a characteristic curve, , , and is a -generalized stochastic process on a probability space . The function is smooth; it can be non-Lipschitz but and all derivatives have polynomial growth. Assume that . We look for a solution (e.g., we can take or ).

Then, is a solution to problem if and only if, for any , is solution to the problem :

3.2. A Nonlinear Stochastic Problem with Multiplicative Generalized Stochastic Process

We consider the Cauchy problem formally written aswhere is a monotonic curve of equation , is not a characteristic curve, , , and is a -generalized stochastic process on a probability space . The function is smooth; it can be non-Lipschitz but and all derivatives have polynomial growth. Assume that . We look for a solution (e.g., we can take ).

Then, is a solution to problem if and only if, for any , is solution to the problem :

3.3. Cutoff Procedure

Let be in such that and . Set .

Consider a family of smooth one-variable functions such that Assume that is bounded on for any integer , . Set Let . We approximate the function by the family of functions defined by

Assume that . Here, function is smooth; it can be non-Lipschitz but and all derivatives have polynomial growth. More precisely, we assume the existence of such that Thus, So, according to [7, 12], is stable under the family .

3.4. Construction of

Put and Define by with having the propertiesConsider a family of mollifiers such that ; then, . Put and .

We make the following assumptions to generate a convenient -algebra adapted to our problem:, , , , , , is overgenerated by the following elements of :   is built on with is built on with

3.5. Generalized Differential Problems Associated with the Formal Ones

Our goal is to give a meaning to the problems formally written as and .

3.5.1. Generalized Differential Problem Associated with

For fixed, the problem associated with can be written as the well-formulated problem then, In terms of representatives, and thanks to the stability and restriction hypothesis, if we can find verifying and if we can prove that , then is a solution of .

Let be another solution to . The uniqueness of the solution to will be the consequence of .

Remark 5. Dependence from some regularizing family. The problem itself, and so a solution of it, a priori depends on the family of cutoff functions and, in the case of irregular data, on the family of mollifiers [7].

Remark 6. is such that Moreover, with .

3.5.2. Generalized Differential Problem Associated with

For fixed, the problem associated with can be written as the well-formulated problem then, In terms of representatives, and thanks to the stability and restriction hypothesis, if we can find verifying and if we can prove that , then is a solution of .

3.6. Generalized Problems
3.6.1. Solution to the Parametrized Regular Problems

For fixed, we consider the families of regularized problems and . We are going to prove that and have a unique smooth solution under assumptions and and the assumption(a),  ,  ,(b),(c) Following [6], one can prove that is equivalent to the integral formulation and is equivalent to the integral formulation where and denotes a primitive of , with

Theorem 7. Under Assumptions , , and , problem resp., has a unique solution, , in .

See [6, 12] for a detailed proof; replace by and by (resp., ).

3.6.2. Solution to the Problems

Theorem 8. Assume that is the solution to problem resp., ; then, problem resp., has a unique solution in .

is the solution to (resp., ) if ; that is, The proof follows the same steps as the existence results which can be found in [12], replacing by and by (resp., ). An induction process on the order of the successive derivatives shows that belongs to . The Gronwall lemma is essential tool for the uniqueness.

Theorem 9. Assume that is the solution to problem resp., ; then, problem resp., has a unique solution in .

Proof. is the solution to if . We shall prove that But and, as , we have . Thus, So and it is the solution to problem in . Set Then,

Theorem 10. The mapping is the solution to problem resp., and it is almost surely unique in .

Proof. Since is the unique solution to problem in , then almost surely in , the map , (), belongs to and it is a representative of (i.e., ). For fixed , the map is jointly measurable on . So is the solution to problem almost surely unique in .

3.7. The Regularized and the Nonregularized Problems

Remark 11. The generalized function represented by the family of solutions to the regularized problems (resp., ) is defined from the integral representation. Thus, we are going to study the relationship between this generalized function and the classical solutions to (resp., ), when they exist, on a domain such that , . This justifies choosing when with .
If problem (resp., ) has a smooth solution on , then, necessarily, we have .

Let us recall that there exists a canonical sheaf embedding of into , through the morphism of algebra where is any open subset of and . The presheaf allows defining restriction and, as usually, we denote by the restriction on of .

Theorem 12. Let be an open subset of such that Assume that when is an increasing family of open subsets of such that when with . Assume that the nonregularized problem resp., has a smooth solution on such that for any . Let be the generalized function represented by the family of solutions to resp., . Then, .

We refer the reader to [12] for a detailed proof.

We can say that the solution to the regularized problem coincides with the solution to the nonregularized equation on .

3.8. A Special Case

We consider the Cauchy problem formally written aswhere is a monotonic curve of equation , is not a characteristic curve, , , and is a -generalized stochastic process on a probability space .

This problem coincides with problem for and with problem for . We obtain the solution to problem . in is defined, with the previous notations, by

4. A Nonlinear Stochastic Cauchy Problem with the White Noise as Data

We consider the Cauchy problems formally written aswhere is a monotonic curve of equation , is not a characteristic curve, and is the white noise on . The function is smooth; it can be non-Lipschitz but and all derivatives have polynomial growth. Assume that . We look for a solution and .

is a solution to problem if and only if, for any , is a solution to the formally problem is a solution to problem if and only if, for any , is a solution to the formally problem We consider the same hypotheses and we take the same spaces and built for problems and .

4.1. A Generalized Differential Problem Associated with the Formal One

Our goal is to give a meaning to the problem formally written as . For fixed, the problem associated with can be written as the well-formulated problem then, The problem associated with can be written as the well-formulated problem then, In terms of representatives, and thanks to the stability and restriction hypothesis, if we can find verifying and if we can prove that , then is a solution of .

We have ; then, .

4.2. Generalized Problem
4.2.1. Solution to the Parametrized Regular Problem

For fixed, we consider the family of regularized problems . We are going to prove that has a unique smooth solution under the following assumptions:(a),  ,  ,(b),(c),  ,  , .Following [6], one can prove that is equivalent to the integral formulation where , with

Theorem 13. Under Assumptions and , problem has a unique solution, , in .

We refer the reader to [6, 12] for a detailed proof (replace by and by ).

4.2.2. Solution to

Theorem 14. Assume that is the solution to problem ; then, problem has a unique solution