Malliavin Derivatives in Spaces with Variable Exponents

Abstract

Spaces with variable exponents and are introduced. After discussing some approximation results of , Sobolev spaces on H with variable exponents are introduced. At last, we define Malliavin derivatives in and discuss some properties of Malliavin derivatives in .

1. Introduction

Variable exponent spaces play an important role in the study of some nonlinear problems in natural science and engineering. In decades, there is a rapid development on the subject of variable exponent function spaces. Basic properties of the spaces have been discussed by Kováik and Rákosník in [1]. Some theories of variable exponent spaces can also be found in [2, 3]. Harjulehto et al. present an overview of applications to differential equations with nonstandard growth in [4]. Diening et al. [5] summarize most of the existing literature of theory of function spaces with variable exponents and applications to partial differential equations. In [6], Aoyama discusses the properties of Lebesgue spaces with variable exponents on a probability space.

Malliavin derivatives have many applications on mathematical finance (see Malliavin and Thalmaier [7] and di Nunno et al. [8]), and we are interested in the behavior of Malliavin derivatives in spaces with variable exponents. In this paper, in Section 2, motivated by [6, 9, 10], we will first introduce and and give some approximation results of , which are useful in the definition on Malliavin derivatives in the following parts. In Section 3 we discuss gradients in and give properties of the linear operator . At the end of this section, we define variable exponent Sobolev spaces on . After the above preparation, we give Malliavin derivatives in and discuss some properties of the Malliavin derivative operator in in the last section.

2. Preliminaries

Let be a separable Hilbert space. A Borel probability measure in with mean and compact covariance operator is called Gaussian measure if the Fourier transform satisfies is called nondegenerate if  . We are given a nondegenerate Gaussian measure in the Hilbert space . Since the operator is compact, there exists a complete orthonormal system in and a sequence of positive numbers such that We denote by the space of all mappings , which are both continuous and bounded. is a Banach space with the norm . And denote by , , the space of all mappings which are continuous and bounded together with their derivatives of order not bigger than .

Given a variable exponent , it is assumed to be a Borel measurable function. On the set of all Borel measurable functions, the moduli and are defined, respectively, by where ,  , and .

Definition 1. The space is the set of Borel measurable functions such that , and it is endowed with the Luxemburg norm:

Define the equivalence relation if and only if . Denote by the quotient with respect to the equivalence relation .

Definition 2. The space is the set of Borel measurable functions such that , and it is endowed with the Luxemburg norm:

Define the equivalence relation if and only if . Denote by the quotient with respect to the equivalence relation .

Proposition 3 (see Lemma in [5]). Let variable exponents be such that ,  -a.e.; then the inequality holds for every and , where in the case we use the convention .

Proposition 4. The variable exponent satisfies ,  -a.e. Let and set ; . Then for all and where the series is convergent in .

Proof. For , , so . Since is a complete orthonormal system in and for , we have Set , and we have so by Lebesgue’s Dominated Convergence Theorem, we have Thus, the series are convergent in . The proof is completed.

Denote the linear span of all real and imaginary parts functions , , and , by .

Proposition 5 (see Lemma 2.2 in [9]). For all there exists a two-index sequence such that , , and

is the space of all mappings which are continuous and have compact support in . We have the following proposition about .

Proposition 6. If the variable exponent satisfies ,  -a.e., then is dense in .

Proof. Note that is the set of all the simple functions. It is easy to check that . Let with . Since is a Borel measurable function, there exist with ,  -a.e., and , so by Lebesgue's Dominated Convergence Theorem, we have in . Thus, is in the closure of . If we drop the assumption , then we split into positive and negative parts which belong to the closure of . Thus is dense in . For any and , there exists such that .
On the other hand, for , is dense in . So there exists such that . By Proposition 3, we have Thus, and is dense in . The proof is completed.

Proposition 7. If the variable exponent satisfies ,  -a.e., then is dense in .

Proof. Since , by Proposition 6   is dense in . So for any and , there exists such that By Proposition 5 and diagonal rule, there exists a sequence such that , , and By Lebesgue's Dominated Convergence Theorem, we have And as we have in . Suppose that , for , and we have for . Thus, is dense in .

Proposition 8. Suppose that , , is a white noise function on . If the variable exponent satisfies ,  -a.e., then where and .

Proof. Since is a Gaussian random variable with mean 0 and covariance , by Proposition 3, we have The proof is completed.

3. Variable Exponent Sobolev Spaces on Separable Hilbert Space

For any and , we denote by its derivative in the direction of ; that is, Denote the gradient of by .

We will consider the following linear mapping: where ,  -a.e.

Lemma 9 (see Lemma 2.6 in [9]). Suppose , ; then where and .

Lemma 10 (see Corollary 2.7 in [9]). Suppose and ; then

Given a linear operator , not necessarily closed, if the closure of its graph happens to be the graph of some operator, that operator is called the closure of , and we say that is closable. Denote the closure of by . And is closable if and only if for any sequence such that one has .

Theorem 11. The mapping is a closable linear operator.

Proof. Let such that By the definition of closable operators, we only need to prove that .
For any and , by Lemma 10, we have By Proposition 3, we have where ; . Since is bounded, by Propositions 3 and 8, we have Thus, , as . Since , for a fixed , suppose such that in . We have that is, . Thus, we have ,  -a.e., and  , -a.e. The proof is completed.

We denote by the closable operator of and by the domain of .

Proposition 12. For any , the linear operator is closable and its closure satisfies for .

Proof. For any , let such that By the definition of closable operators, we only need to prove that . For any and , by Lemma 9, we have Similar to the proof of Theorem 11, we have as and ,  -a.e. Thus is closable. And The proof is completed.

4. Malliavin Derivatives in

For any , define the linear operator: where ,  -a.e. And where and .

Proposition 13 (see Corollary 2.10 in [9]). Suppose and ; then the following identity holds: In a similar way to Theorem 11, one has Theorem 14.

Theorem 14. The mapping is a closable linear operator.

Proof. Let such that We only need to prove that .
For any and , by Proposition 13, we have And we have , as . Similar to Theorem 11, we have ,  -a.e. The proof is completed.

We denote by the closable operator of and by the domain of . For any , we call the Malliavin derivative of .

Proposition 15. If satisfies ,  -a.e., then and for any .

Proof. For any , there exists such that in . As we have and .

To prove Proposition 17, we need the following lemma.

Lemma 16 (see Lemma 2.3 in [9]). For all there exists a two-index sequence such that , , and

Proposition 17. If satisfies ,  -a.e., then one has and for any and .

Proof. First, we prove that for any . By Lemma 16, for , there exists a two-index sequence such that , , , and By diagonal rule and Lebesgue’s Dominated Convergence Theorem, in . Since is closed, we have and .
Now we will prove the proposition. For , there exists such that By the first part of this proof, . And Since and , we have As is closed, . The proof is completed.

Proposition 18. Assume that satisfies ,  -a.e. For , ; suppose and are bounded; then and .

Proof. First, for , as , there exists such that in . And since we have As , in , and in , we have .
Secondly, for , there exists such that in . By the first part of the proof, Since and are bounded, we have and . The proof is completed.

5. Conclusion

In this work some new results on Malliavin derivatives are given. Malliavin derivatives are extended from to variable exponent spaces and some properties of Malliavin derivatives in variable exponent spaces are obtained. We introduce and , and this work generalizes classical variable exponents spaces based on Kováik and Rákosník [1]. Then, we prove that is dense in and by this approximation result we give extension of Malliavin derivatives which are given by P. Malliavin [11]. Some properties of Malliavin derivatives in are also discussed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant no. 11371110).