# Dirichlet Forms Constructed from Annihilation Operators on Bernoulli Functionals

Authors: Caishi Wang; Beiping Wang
Journal: Advances in Mathematical Physics
 (2017)

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## 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. Letw be a nonnegative function on N. By using the Bernoulli annihilators, we first define in a dense subspace of L2-space of Bernoulli functionals a positive, symmetric, bilinear form Ew associated with w. And then we prove that Ew is closed and has the contraction property; hence, it is a Dirichlet form. Finally, we consider an interesting semigroup of operators associated with w on L2-space of Bernoulli functionals, which we call the w-Ornstein-Uhlenbeck semigroup, and, by using the Dirichlet form, Ew we show that the w-Ornstein-Uhlenbeck semigroup is a Markov semigroup.


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## Body

## 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 L2-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 L2-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 C∗-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{∂k}k≥0 be the Bernoulli annihilators. For a nonnegative function w on N, we first define in the L2-space of Bernoulli functionals a positive, symmetric, densely defined bilinear form Ew in the following manner: (1)Ewξ,η=∑k=0∞wk∂kξ,∂kη,where 〈·,·〉 denotes the inner product in the L2-space of Bernoulli functionals. We then prove that Ew is closed and has the contraction property; hence, it is a Dirichlet form. We also obtain an operator representation of Ew. Finally, we consider an interesting semigroup of operators associated with w on the L2-space of Bernoulli functionals, which we call the w-Ornstein-Uhlenbeck semigroup, and, by using the Dirichlet form Ew, we show that the w-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, N always denotes the set of all nonnegative integers. We denote by Γ the finite power set of N; namely, (2)Γ=σ∣σ⊂N,#σ<∞,where #(σ) means the cardinality of σ as a set. Unless otherwise stated, letters like j, k, and n stand for nonnegative integers, namely, elements of N.


## 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].LetΩ={-1,1}N be the set of all mappings ω:N↦{-1,1} and (ζn)n≥0 the sequence of canonical projections on Ω given by (3)ζnω=ωn,ω∈Ω.Denote by F the σ-field on Ω generated by the sequence (ζn)n≥0. Let (pn)n≥0 be a given sequence of positive numbers with the property that 0<pn<1 for all n≥0. It is known [2] that there exists a unique probability measure P on F such that (4)P∘ζn1,ζn2,…,ζnk-1ϵ1,ϵ2,…,ϵk=∏j=1kpj1+ϵj/21-pj1-ϵj/2.For nj∈N, ϵj∈{-1,1} (1≤j≤k) with ni≠nj when i≠j and k∈N with k≥1. Thus we have a probability measure space (Ω,F,P).LetZ=(Zn)n≥0 be the sequence of random variables on (Ω,F,P) defined by (5)Zn=ζn+qn-pn2pnqn,n≥0,where qn=1-pn. Clearly Z=(Zn)n≥0 is an independent sequence of random variables on (Ω,F,P), and, for each n≥0, Zn has a distribution: (6)PZn=θn=pn,PZn=-1θn=qn,n≥0with θn=qn/pn. To be convenient, we set F-1={∅,Ω} and (7)Fn=σZk;0≤k≤n,the σ-field generated by (Zk)0≤k≤n for n≥0. By convention, E will denote the expectation with respect to P.LetL2(Ω) be the space of square integrable random variables on (Ω,F,P), namely, (8)L2Ω=L2Ω,F,P.We denote by 〈·,·〉 the usual inner product of the space L2(Ω) and by · the corresponding norm. It is known [2] that Z has the chaotic representation property. Thus L2(Ω) has {Zσ∣σ∈Γ} as its orthonormal basis, where Z∅=1 and (9)Zσ=∏j∈σZj,σ∈Γ,σ≠∅,which shows that L2(Ω) is an infinite dimensional real Hilbert space.Remark 1.
It is easy to see thatF=(Zn;n≥0). Thus F-measurable functions on Ω are usually known as functionals of Z or Bernoulli functionals simply. In particular, functions in L2(Ω) are called square integrable Bernoulli functionals.Lemma 2 (see [1]).
For eachk∈N, there exists a bounded operator ∂k on L2(Ω) such that (10)∂kZσ=1σkZσ∖k,σ∈Γ,where σ∖k=σ∖{k} and 1σ(k) is the indicator of σ as a subset of  N.Lemma 3 (see [1]).
Letk∈N. Then ∂k∗, the adjoint of operator ∂k, has following property: (11)∂k∗Zσ=1-1σkZσ∪k,σ∈Γ,where σ∪k=σ∪{k}.The operator∂k and its adjoint ∂k∗ are referred to as the annihilation operator and creation operator at site k, respectively. The next lemma shows that these operators satisfy a canonical anticommutation relations (CAR) in equal-time.Lemma 4 (see [1]).
Letk,l∈N. Then it holds true that (12)∂k∂l=∂l∂k,∂k∗∂l∗=∂l∗∂k∗,∂k∗∂l=∂l∂k∗,k≠l,∂k∂k=∂k∗∂k∗=0,∂k∂k∗+∂k∗∂k=I,where I is the identity operator on L2(Ω).


## 3. Forms Constructed from Bernoulli Annihilators

In the present section, we show how to use the annihilation operators{∂k}k≥0 to construct a closed, positive, symmetric, and densely defined bilinear form in L2(Ω).We first prove a convergence result concerning{∂k}k≥0, which will play a key role in our later discussions.For a nonnegative functionw:N→R, we define the w-counting measure #w(·) as(13)#wσ=∑j∈σwj,σ∈Γ,where #w(σ)=0 if σ=∅. Clearly, 0≤#w(σ)<∞ for all σ∈Γ.Theorem 5.
Letw:N→R be a nonnegative function. Define Dw as the linear subspace of L2(Ω) given by(14)Dw=ξ∈L2Ω∣∑σ∈Γ#wσZσ,ξ2<∞.Then, Dw is a dense linear subspace of L2(Ω), and, moreover, for all ξ, η∈Dw, the following series, is absolutely convergent:(15)∑k=0∞wk∂kξ,∂kη.Proof.
Clearly,Dw is a linear subspace of L2(Ω). On the other hand, for each τ∈Γ, in view of the fact that #w(τ)<∞, we have (16)∑σ∈Γ#wσZσ,Zτ2=#wτ<∞.Thus, {Zτ∣τ∈Γ}⊂Dw, which implies that Dw is dense in L2(Ω).
Now, letξ,η∈Dw. Then, for each k≥0, it follows from Lemma 2 as well as the expansion ξ=∑σ∈Γ‍〈Zσ,ξ〉Zσ that(17)∂kξ=∑σ∈Γ1σkZσ,ξZσ∖k.Thus, by a direct calculation, we have (18)∑k=0∞wk∂kξ2=∑k=0∞∑σ∈Γ1σkwkZσ,ξ2=∑σ∈Γ#wσZσ,ξ2<∞.With the same argument, we have (19)∑k=0∞wk∂kη2<∞.It then follows from these two equalities that (20)∑k=0∞wk∂kξ,∂kη≤∑k=0∞wk∂kξ21/2∑k=0∞wk∂kη21/2<∞,which implies that the series ∑k=0∞‍w(k)〈∂kξ,∂kη〉 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 functionw:N→R, we define Ew as(21)Ewξ,η=∑k=0∞wk∂kξ,∂kη,ξ,η∈Dw,and call (Ew,Dw) the w-energy form.It is easy to see that(Ew,Dw) is a positive, symmetric, and densely defined bilinear form in L2(Ω). Note that if we take w(k)≡1, then we get (22)Ewξ,ξ=∫Ω∇ξωl2N2dPω,ξ∈Dw,where ∇ denotes the gradient operator, which is defined on Dw and valued in L2(Ω×N), the space of jointly square integrable stochastic processes on (Ω,F,P) (see [14] for details). This justifies the name of (Ew,Dw).In order to examine basic properties of thew-energy form (Ew,Dw), we introduce another bilinear form E^w on Dw as(23)E^wξ,η=Ewξ,η+ξ,η,ξ,η∈Dw,where 〈·,·〉 is the inner product of L2(Ω). It is then easy to see that E^w is again an inner product on Dw. We denote by ·E^w the norm induced by E^w. A direct calculation gives that(24)ξE^w2=∑σ∈Γ#wσ+1Zσ,ξ2,ξ∈Dw,where #w(·) is the w-counting measure as defined by (13).Theorem 7.
Letw:N→R be a nonnegative function. Then the w-energy form (Ew,Dw) is closed; namely, (Dw,E^w) is a Hilbert space.Proof.
We need only to show thatDw is complete with respect to norm ·E^w.
Let{ξn}n≥1⊂Dw be a Cauchy sequence with respect to norm ·E^w. Then it is also a Cauchy sequence with respect to norm · since ·≤·E^w. Thus there exists ξ∈L2(Ω) such that ξn-ξ→0 as n→∞. Now we show that ξ∈Dw and ξn-ξE^w→0 as n→∞.
Letϵ>0. Then, by the property that {ξn}n≥1 is a Cauchy sequence with respect to ·E^w, we know that there exists a positive integer K≥1 such that(25)∑σ∈Γ#wσ+1Zσ,ξm-ξn2=ξm-ξnE^w2<ϵ2,∀m,n>K.For all n>K, since (26)limm→∞#wσ+1Zσ,ξm-ξn2=#wσ+1Zσ,ξ-ξn2,∀σ∈Γ,it follows by applying the well-known Fatou’s Lemma [15] to (25) that(27)∑σ∈Γ#wσ+1Zσ,ξ-ξn2≤ϵ2,which implies that ξ-ξn∈Dw; hence ξ∈Dw and (28)ξ-ξnE^w=∑σ∈Γ#wσ+1Zσ,ξ-ξn21/2≤ϵ.Therefore, ξn-ξE^w→0 as n→∞; namely, {ξn} converges in Dw with respect to norm ·E^w.


## 4. Contraction Property

Letw:N→R be a given nonnegative function. In this section, we further prove that the w-energy form (Ew,Dw) has the contraction property; hence, it is a Dirichlet form.We first make some necessary preparations. LetS=Span{Zσ∣σ∈Γ}, the linear subspace of L2(Ω) spanned by the system {Zσ∣σ∈Γ}, which is obviously a dense linear subspace of L2(Ω). For n≥0, we denote by Sn the linear subspace of all Fn-measurable random variables in L2(Ω). It can be verified that Sn is a 2n+1-dimensional closed subspace of L2(Ω) and has an orthonormal basis {Zσ∣σ∈Γn]}, where Γn]={σ∈Γ∣max⁡σ≤n}. And, moreover, these subspaces have the following relation: (29)⋃n=0∞Sn=S.Recall that elements ofΩ are mappings ω:N→{-1,1}. For k≥0 and ω∈Ω, we can naturally define two mappings ωk+, ωk-:N→{-1,1} as(30)ωk+n=1,n=k;ωn,n≠k,ωk-n=-1,n=k;ωn,n≠k,which, of course, remain elements of Ω.The following proposition is a slight variant of a result given in [2], which shows that the annihilation operator ∂k acts just like a difference operator.Lemma 8 (see [2]).
Letk≥0 and ξ∈S. Then,(31)∂kξω=pkqkξωk+-ξωk-,ω∈Ω.Proof.
We need only to show that (31) holds for each ξ=Zσ with σ∈Γ. Let σ∈Γ. Then, it follows from (3), (5), and (30) that(32)Zjω=Zjωk+=Zjωk-,ω∈Ω,j≠k,(33)Zkωk+=qkpk,Zkωk-=-pkqk.Thus, in case of k∉σ, Zσ(ω)=Zσ(ωk+)=Zσ(ωk-) for ω∈Ω, which implies (34)pkqkZσωk+-Zσωk-=0,ω∈Ω,which, together with the formula ∂kZσ=1σ(k)Zσ∖k, gives (35)∂kZσω=pkqkZσωk+-Zσωk-,ω∈Ω.Now, if k∈σ, then Zσ∖k(ω)=Zσ∖k(ωk+)=Zσ∖k(ωk-) for ω∈Ω, which together with (33) yields (36)Zσ∖kω=pkqkZσ∖kωk+qkpk+Zσ∖kωk-pkqk=pkqkZσ∖kωk+Zkωk+-Zσ∖kωk-Zkωk-=pkqkZσωk+-Zσωk-,ω∈Ω,which, together with the formula ∂kZσ=1σ(k)Zσ∖k, still leads to (37)∂kZσω=pkqkZσωk+-Zσωk-,ω∈Ω.This completes the proof.Definition 9.
A contraction function is a functionC:R→R with C(0)=0, and |C(s)-C(t)|≤|s-t| for all s,t∈R.As usual, we useC∘ξ to mean the composition of a contraction function C and a random variable ξ. The next theorem then shows that S is invariant under the action of contraction functions.Theorem 10.
LetC be a contraction function. Then C∘ξ∈S for all ξ∈S.Proof.
Letξ∈S. Then there exists some n≥0 such that ξ∈Sn, which implies C∘ξ∈Sn since C is a continuous function. Thus C∘ξ∈S.Theorem 11.
Letξ∈L2(Ω) and C be a contraction function. Then, for all k≥0, it holds that(38)∂kC∘ξ≤∂kξ.Proof.
Letk≥0. Then, we can take a sequence (ξn)n≥1 in S such that ξn-ξ→0 as n→∞. It follows from Theorem 10 that {C∘ξn}n≥1⊂S. Thus, for each n≥1, by using Lemma 8 and the contraction property of C, we have (39)∂kC∘ξnω=pkqkC∘ξnωk+-C∘ξnωk-≤pkqkξnωk+-ξnωk-=∂kξnωwith ω∈Ω, which implies (40)∂kC∘ξn≤∂kξn,n≥1.It is easy to see that C∘ξn-C∘ξ→0 as n→∞. Thus, by letting n→∞ in the above inequality, we finally come to (38).Theorem 12.
Thew-energy form (Ew,Dw) has the contraction property; namely, for all contraction function C:R→R and all ξ∈Dw, it holds that C∘ξ∈Dw and(41)EwC∘ξ,C∘ξ≤Ewξ,ξ.Proof.
Letξ∈Dw and C:R→R be a contraction function. Then, by Theorem 11, we have (42)∑σ∈Γ#wσZσ,C∘ξ2=∑k=0∞wk∂kC∘ξ2≤∑k=0∞wk∂kξ2=∑σ∈Γ#wσZσ,ξ2,which, together with the assumption ξ∈Dw, implies C∘ξ∈Dw. It then follows from the definition of Ew that (43)EwC∘ξ,C∘ξ=∑k=0∞wk∂kC∘ξ2≤∑k=0∞wk∂kξ2=Ewξ,ξ.In the literature, a closed, positive, symmetric, densely defined bilinear form in anL2-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.
Thew-energy form (Ew,Dw) is a Dirichlet form in L2(Ω).


## 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 onL2(Ω). As an application of our results in the previous sections, we will prove that this class of semigroups are actually Markov ones.Letw:N→R be a nonnegative function. For each t≥0, one can define a contraction operator Ptw on L2(Ω) as(44)Ptwξ=∑σ∈Γe-t#wσZσ,ξZσ,ξ∈L2Ω,where #w(·) is the w-counting measure defined by (13). One can verify that the family Pw=(Ptw)t≥0 forms a strongly continuous semigroup of contraction operators on L2(Ω).Definition 14.
The semigroupPw=(Ptw)t≥0 defined by (44) is called the w-Ornstein-Uhlenbeck semigroup on L2(Ω).We note that, withw(k)≡1, the w-Ornstein-Uhlenbeck semigroup becomes the usual Ornstein-Uhlenbeck semigroup considered in [2, 14].Theorem 15.
Letw:N→R be a nonnegative function. Then the w-Ornstein-Uhlenbeck semigroup Pw=(Ptw)t≥0 is a Markov semigroup on L2(Ω); namely, for all t≥0, it holds that(45)0≤Ptwξ≤1P-a.e.whenever ξ∈L2(Ω) with 0≤ξ≤1P-a.e.Proof.
Consider the operatorNw in L2(Ω) given by(46)Nwξ=∑σ∈Γ#wσZσ,ξZσ,ξ∈DomNw,where DomNw denotes the domain of the operator Nw, which is defined as (47)DomNw=ξ∈L2Ω∣∑σ∈Γ#wσ2Zσ,ξ2<∞,where #w(·) is the w-counting measure defined by (13). It is easy to show that Nw is a positive, self-adjoint, densely defined operator in L2(Ω). And, moreover, by a direct calculation, we find that(48)Ptw=e-tNw,t≥0,where e-tNw means the spectral integral of the function s↦e-ts with respect to the spectral measure of Nw. Thus, -Nw is exactly the infinitesimal generator of the semigroup Pw=(Ptw)t≥0.
We now verify thatNw and Ew share the following relationship:(49)Ewξ,η=ξ,Nwη,ξ∈Dw,η∈DomNwwith DomNw⊂Dw. In fact, if η∈DomNw, then, by the well-known Cauchy inequality, we have (50)∑σ∈Γ#wσZσ,η2≤∑σ∈Γ#wσ2Zσ,η21/2∑σ∈ΓZσ,η21/2<∞,which implies η∈Dw. Thus, DomNw⊂Dw. Let ξ∈Dw and η∈DomNw. Then, by (17) and (21), we have (51)Ewξ,η=∑k=0∞wk∂kξ,∂kη=∑k=0∞wk∑σ∈Γ1σkZσ,ξZσ,η=∑σ∈Γ#wσZσ,ξZσ,η.Here, we take use of the absolute convergence of the following series: (52)∑σ∈Γ∑k=0∞1σkwkZσ,ξZσ,η,which can be verified straightforwardly. On the other hand, by the definition of Nw and the expansion ξ=∑σ∈Γ‍〈Zσ,ξ〉Zσ, we have (53)ξ,Nwη=∑σ∈Γ#wσZσ,ξZσ,η.Thus, Ew(ξ,η)=〈ξ,Nwη〉.
Combining (48), (49), and Corollary 13 with the general theory of Dirichlet forms [8], we finally know that Pw=(Ptw)t≥0 is a Markov semigroup.


## 6. Discussion

LetX be a locally compact separable metric space and μ be a positive Radon measure on X. For a Dirichlet form E in L2(X,μ), one thing researchers are interested in is whether it is regular [9]. Indeed, if E is a regular Dirichlet form in L2(X,μ), then there is a symmetric Hunt process M with E as its Dirichlet form. If, additionally, E possesses the local property, then the process M is even a diffusion process [9]. In that case, the process M is called the diffusion process associated with the Dirichlet form E. It is well known that diffusion processes play an important role in many problems in mathematical physics.As is shown, in our case,Ew is a Dirichlet form in L2(Ω)≡L2(Ω,P) for a positive function w:N→R. On the other hand, being endowed with an appropriate metric, Ω can become a compact separable metric space and P can become a positive Radon measure on Ω. It is then natural to consider whether or not Ew is regular and whether or not Ew possesses the local property. Such problems, however, are far from being simple and might be related to deeper topological properties of Ω.



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