Dirichlet Forms Constructed from Annihilation Operators on Bernoulli Functionals

Abstract

The annihilation operators on Bernoulli functionals (Bernoulli annihilators, for short) and their adjoint operators satisfy a canonical anticommutation relation (CAR) in equal-time. As a mathematical structure, Dirichlet forms play an important role in many fields in mathematical physics. In this paper, we apply the Bernoulli annihilators to constructing Dirichlet forms on Bernoulli functionals. Let be a nonnegative function on . By using the Bernoulli annihilators, we first define in a dense subspace of -space of Bernoulli functionals a positive, symmetric, bilinear form associated with . And then we prove that is closed and has the contraction property; hence, it is a Dirichlet form. Finally, we consider an interesting semigroup of operators associated with on -space of Bernoulli functionals, which we call the -Ornstein-Uhlenbeck semigroup, and, by using the Dirichlet form, we show that the -Ornstein-Uhlenbeck semigroup is a Markov semigroup.

1. Introduction

The annihilation operators on Bernoulli functionals (Bernoulli annihilators, for short) admit much good operation properties with physical meanings. For example, they together with their adjoint operators satisfy a canonical anticommutation relation (CAR) in equal-time [1]. In recent years, these operators have begun to find applications in developing a discrete-time stochastic calculus in infinite dimensions. Privault [2] used the Bernoulli annihilators to define the gradients for Bernoulli functionals in 2008. In 2010, Nourdin et al. [3] investigated normal approximation of Rademacher functionals (a special case of Bernoulli functionals) with the help of the Bernoulli annihilators. Recently [4], it has been shown that a wide class of quantum Markov semigroups can be constructed from the Bernoulli annihilators and their adjoint operators. As is known, quantum Markov semigroups are quantum analogues of the classical Markov semigroups in probability theory, which provide a mathematical model for describing the irreversible evolution of quantum systems interacting with the environment (see, e.g., [5–7]).

A Dirichlet form [8] in an -space is a closed, positive, symmetric, densely defined bilinear form that has the contraction property (also known as the Markov property). As a mathematical structure, Dirichlet forms have close connections with many objects in probability theory, quantum mechanics, and quantum field theory (see [8] and references therein). One typical example in this respect is that, under some mild conditions, a Dirichlet form determines a Markov process through the Markov semigroup associated with it [9]. Thus, Dirichlet forms play an important role in construction of Markov processes [9]. The classical Dirichlet forms [9, 10] are usually defined in the -spaces of functions on locally compact metric spaces. In the past three decades, however, Dirichlet forms on noncommutative and even infinite dimensional settings have also appeared successively. For example, Albeverio and Hoegh-Krohn [11] constructed Dirichlet forms on -algebras, while Hida et al. [12] gave a Dirichlet form on white noise functionals by using Hida’s differential operators.

In this paper, motivated by the work of Hida et al. [12], we would like to apply the Bernoulli annihilators to constructing Dirichlet forms on Bernoulli functionals. Our main work is as follows.

Let be the Bernoulli annihilators. For a nonnegative function on , we first define in the -space of Bernoulli functionals a positive, symmetric, densely defined bilinear form in the following manner: where denotes the inner product in the -space of Bernoulli functionals. We then prove that is closed and has the contraction property; hence, it is a Dirichlet form. We also obtain an operator representation of . Finally, we consider an interesting semigroup of operators associated with on the -space of Bernoulli functionals, which we call the -Ornstein-Uhlenbeck semigroup, and, by using the Dirichlet form , we show that the -Ornstein-Uhlenbeck semigroup is a Markov semigroup.

We mention that the whole family of the annihilation and creation operators acting on Bernoulli functionals is interpreted as a type of quantum Bernoulli noises [1, 13]. As is seen, however, we only make use of the annihilation operators in the construction of our Dirichlet form. Thus, our work here does not mean that one can construct a Dirichlet form from quantum Bernoulli noises. In fact, from a physical point of view, any Dirichlet forms cannot be constructed from any quantum Bernoulli noises.

Notation and Conventions. Throughout, always denotes the set of all nonnegative integers. We denote by the finite power set of ; namely, where means the cardinality of as a set. Unless otherwise stated, letters like , , and stand for nonnegative integers, namely, elements of .

2. Bernoulli Annihilators

In this section, we briefly recall some necessary notions and facts about Bernoulli functionals and the annihilation operators on them. For details, we refer to [1].

Let be the set of all mappings and the sequence of canonical projections on given by Denote by the -field on generated by the sequence . Let be a given sequence of positive numbers with the property that for all . It is known [2] that there exists a unique probability measure on such that For , () with when and with . Thus we have a probability measure space .

Let be the sequence of random variables on defined by where . Clearly is an independent sequence of random variables on , and, for each , has a distribution: with . To be convenient, we set and the -field generated by for . By convention, will denote the expectation with respect to .

Let be the space of square integrable random variables on , namely, We denote by the usual inner product of the space and by the corresponding norm. It is known [2] that has the chaotic representation property. Thus has as its orthonormal basis, where and which shows that is an infinite dimensional real Hilbert space.

Remark 1. It is easy to see that . Thus -measurable functions on are usually known as functionals of or Bernoulli functionals simply. In particular, functions in are called square integrable Bernoulli functionals.

Lemma 2 (see [1]). For each , there exists a bounded operator on such that where and is the indicator of as a subset of  .

Lemma 3 (see [1]). Let . Then , the adjoint of operator , has following property: where .

The operator and its adjoint are referred to as the annihilation operator and creation operator at site , respectively. The next lemma shows that these operators satisfy a canonical anticommutation relations (CAR) in equal-time.

Lemma 4 (see [1]). Let . Then it holds true that where is the identity operator on .

3. Forms Constructed from Bernoulli Annihilators

In the present section, we show how to use the annihilation operators to construct a closed, positive, symmetric, and densely defined bilinear form in .

We first prove a convergence result concerning , which will play a key role in our later discussions.

For a nonnegative function , we define the -counting measure aswhere if . Clearly, for all .

Theorem 5. Let be a nonnegative function. Define as the linear subspace of given byThen, is a dense linear subspace of , and, moreover, for all , , the following series, is absolutely convergent:

Proof. Clearly, is a linear subspace of . On the other hand, for each , in view of the fact that , we have Thus, , which implies that is dense in .
Now, let . Then, for each , it follows from Lemma 2 as well as the expansion thatThus, by a direct calculation, we have With the same argument, we have It then follows from these two equalities that which implies that the series is absolutely convergent.

In view of the above theorem, we come naturally to the next definition, which introduces our main object of study.

Definition 6. For a nonnegative function , we define asand call the -energy form.

It is easy to see that is a positive, symmetric, and densely defined bilinear form in . Note that if we take , then we get where denotes the gradient operator, which is defined on and valued in , the space of jointly square integrable stochastic processes on (see [14] for details). This justifies the name of .

In order to examine basic properties of the -energy form , we introduce another bilinear form on aswhere is the inner product of . It is then easy to see that is again an inner product on . We denote by the norm induced by . A direct calculation gives thatwhere is the -counting measure as defined by (13).

Theorem 7. Let be a nonnegative function. Then the -energy form is closed; namely, is a Hilbert space.

Proof. We need only to show that is complete with respect to norm .
Let be a Cauchy sequence with respect to norm . Then it is also a Cauchy sequence with respect to norm since . Thus there exists such that as . Now we show that and as .
Let . Then, by the property that is a Cauchy sequence with respect to , we know that there exists a positive integer such thatFor all , since it follows by applying the well-known Fatou’s Lemma [15] to (25) thatwhich implies that ; hence and Therefore, as ; namely, converges in with respect to norm .

4. Contraction Property

Let be a given nonnegative function. In this section, we further prove that the -energy form has the contraction property; hence, it is a Dirichlet form.

We first make some necessary preparations. Let , the linear subspace of spanned by the system , which is obviously a dense linear subspace of . For , we denote by the linear subspace of all -measurable random variables in . It can be verified that is a -dimensional closed subspace of and has an orthonormal basis , where . And, moreover, these subspaces have the following relation:

Recall that elements of are mappings . For and , we can naturally define two mappings , aswhich, of course, remain elements of .

The following proposition is a slight variant of a result given in [2], which shows that the annihilation operator acts just like a difference operator.

Lemma 8 (see [2]). Let and . Then,

Proof. We need only to show that (31) holds for each with . Let . Then, it follows from (3), (5), and (30) thatThus, in case of , for , which implies which, together with the formula , gives Now, if , then for , which together with (33) yields which, together with the formula , still leads to This completes the proof.

Definition 9. A contraction function is a function with , and for all .

As usual, we use to mean the composition of a contraction function and a random variable . The next theorem then shows that is invariant under the action of contraction functions.

Theorem 10. Let be a contraction function. Then for all .

Proof. Let . Then there exists some such that , which implies since is a continuous function. Thus .

Theorem 11. Let and be a contraction function. Then, for all , it holds that

Proof. Let . Then, we can take a sequence in such that as . It follows from Theorem 10 that . Thus, for each , by using Lemma 8 and the contraction property of , we have with , which implies It is easy to see that as . Thus, by letting in the above inequality, we finally come to (38).

Theorem 12. The -energy form has the contraction property; namely, for all contraction function and all , it holds that and

Proof. Let and be a contraction function. Then, by Theorem 11, we have which, together with the assumption , implies . It then follows from the definition of that

In the literature, a closed, positive, symmetric, densely defined bilinear form in an -space is called a Dirichlet form if it additionally has the contraction property (see, e.g., [8, 9, 12]). Summing up our discussions above, we actually arrive at the next important conclusion.

Corollary 13. The -energy form is a Dirichlet form in .

5. Application

Markov semigroups are semigroups of contraction operators that are closely related to Markov processes in probability theory. In the final section, we consider an interesting class of semigroups of contraction operators on . As an application of our results in the previous sections, we will prove that this class of semigroups are actually Markov ones.

Let be a nonnegative function. For each , one can define a contraction operator on aswhere is the -counting measure defined by (13). One can verify that the family forms a strongly continuous semigroup of contraction operators on .

Definition 14. The semigroup defined by (44) is called the -Ornstein-Uhlenbeck semigroup on .

We note that, with , the -Ornstein-Uhlenbeck semigroup becomes the usual Ornstein-Uhlenbeck semigroup considered in [2, 14].

Theorem 15. Let be a nonnegative function. Then the -Ornstein-Uhlenbeck semigroup is a Markov semigroup on ; namely, for all , it holds thatwhenever with -a.e.

Proof. Consider the operator in given bywhere denotes the domain of the operator , which is defined as where is the -counting measure defined by (13). It is easy to show that is a positive, self-adjoint, densely defined operator in . And, moreover, by a direct calculation, we find thatwhere means the spectral integral of the function with respect to the spectral measure of . Thus, is exactly the infinitesimal generator of the semigroup .
We now verify that and share the following relationship:with . In fact, if , then, by the well-known Cauchy inequality, we have which implies . Thus, . Let and . Then, by (17) and (21), we have Here, we take use of the absolute convergence of the following series: which can be verified straightforwardly. On the other hand, by the definition of and the expansion , we have Thus, .
Combining (48), (49), and Corollary 13 with the general theory of Dirichlet forms [8], we finally know that is a Markov semigroup.

6. Discussion

Let be a locally compact separable metric space and be a positive Radon measure on . For a Dirichlet form in , one thing researchers are interested in is whether it is regular [9]. Indeed, if is a regular Dirichlet form in , then there is a symmetric Hunt process with as its Dirichlet form. If, additionally, possesses the local property, then the process is even a diffusion process [9]. In that case, the process is called the diffusion process associated with the Dirichlet form . It is well known that diffusion processes play an important role in many problems in mathematical physics.

As is shown, in our case, is a Dirichlet form in for a positive function . On the other hand, being endowed with an appropriate metric, can become a compact separable metric space and can become a positive Radon measure on . It is then natural to consider whether or not is regular and whether or not possesses the local property. Such problems, however, are far from being simple and might be related to deeper topological properties of .

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant no. 11461061).