Convergence Theorems for Operators Sequences on Functionals of Discrete-Time Normal Martingales
We aim to investigate the convergence of operators sequences acting on functionals of discrete-time normal martingales . We first apply the 2D-Fock transform for operators from the testing functional space to the generalized functional space and obtain a necessary and sufficient condition for such operators sequences to be strongly convergent. We then discuss the integration of these operator-valued functions. Finally, we apply the results obtained here and establish the existence and uniqueness of solution to quantum stochastic differential equations in terms of operators acting on functionals of discrete-time normal martingales . And also we prove the continuity and continuous dependence on initial values of the solution.
White noise analysis created by Hida  is essentially a branch of infinite-dimensional calculus on generalized functionals of Brownian motion, which connected with the applications to the study of random processes and stochastic differential equations. Ito’s theory is based on the notion of the Wiener measure in the space of continuous functions, interpreted as the trajectories of Brownian motion. And considering the space of slowly growing Schwartz distributions as the space of trajectories of white noise, Hida has constructed a new theory of white noise functionals [2–4], which proved to be useful in the studying of stochastic differential equations and was further developed in [5, 6] and other papers. On the other hand, many problems in mathematical physics and financial mathematics lead to the necessity of studying stochastic differential equations with operator coefficients in infinite-dimensional spaces. In , the authors generalize the classical Ito theory to this case.
In 1998, Ito introduced his analysis of generalized Poisson functionals in paper , which can be viewed as a theory of infinite-dimensional calculus on generalized functionals of Poisson martingale. As we know, both Brownian motion and Poisson martingale are continuous-time normal martingales. And other continuous-time processes theories of white noise analysis appeared in [9–12].
Discrete-time normal martingales  also play an important role in many theoretical and applied fields. For example, the classical random walk is just such a discrete-time normal martingale [14, 15]. It would then be interesting to develop a theory of infinite-dimensional calculus on generalized functionals of discrete-time normal martingales.
Let be a discrete-time normal martingale satisfying some mild conditions. In paper , the authors constructed the testing functional space and generalized functional space of by using a specific orthonormal basis for square integrable functionals of and also characterized these functionals via a transform acting on them.
It is well known that operators on functional spaces play a fundamental role in quantum mechanics. In paper , the authors introduce a transform, called 2D-Fock transform, for operators from the testing functional space to the generalized functional space of and characterize continuous linear operators from to . In this paper, we apply the 2D-Fock transform to investigate the convergence of operators sequences acting on functionals of discrete-time normal martingales . We obtain a necessary and sufficient condition for such sequence to be strongly convergent. And we give a criterion for checking whether such operator-valued functions are Bochner-integrable.
Finally, as application of the results obtained here, we investigate the following nonlinear quantum stochastic differential equation:where are two maps, is the initial value, and stands for the convolution of operators. is a given -valued quantum stochastic process which plays the role of quantum noises, and the solution will be a -valued quantum stochastic process.
Equation (1) can describe the evolution of a quantum system at a level of operators in the presence of quantum noise. We will establish the existence and uniqueness of solution to equation (1). And also we prove the continuity and continuous dependence on initial values of the solution.
The paper is organized as follows. In Section 2, we briefly recall the construction and characterization of continuous linear operators from to , which were made in recent paper . In Section 3, we apply the 2D-Fock transform to investigate the convergence of operators sequences acting on functionals of discrete-time normal martingales . We first obtain a necessary and sufficient condition for such sequence to be strongly convergent. And then we give a criterion for checking whether such operator-valued functions are Bochner-integrable. In Section 4, we apply the results obtained here and establish the existence and uniqueness of solution to quantum stochastic differential equations (1) and prove the continuity and continuous dependence on initial values of the solution.
Notation and Conventions. Throughout the paper, designates the set of all nonnegative integers and denotes the finite power set of ; namely,where means the cardinality of as a set. In addition, we always assume that is a given probability space with denoting the expectation with respect to . We denote by the usual Hilbert space of square integrable complex-valued functions on and use and to mean its inner product and norm, respectively. By convention, is conjugate-linear in its first argument and linear in its second argument.
2. Generalized Functionals of Discrete-Time Normal Martingales
Definition 1. A stochastic process on is called a discrete-time normal martingale if it is square integrable and satisfies(i) and for ;(ii) and for , where , for , and means the conditional expectation.
Let be a discrete-time normal martingale on . Then one can construct from a process asIt can be verified that admits the following properties:Thus can be viewed as a discrete-time noise, which we call the discrete-time normal noise associated with .
Let ; the -field over is generated by . In the literature, -measurable functions on are also known as functionals of . Thus elements of can be called square integrable functionals of .
Definition 3. The discrete-time normal martingale is said to have the chaotic representation property if the system defined by (5) is total in .
So, if the discrete-time normal martingale has the chaotic representation property, then the system defined by (5) is actually an orthonormal basis for , which is a closed subspace of as is known.
And, for brevity, we use to mean the space of square integrable functionals of ; namely,which shares the same inner product and norm with , namely, and . We note that forms a countable orthonormal basis for .
Lemma 4 (see ). Let be the -valued function on given by Then, for , the positive term series converges and moreover
Using the -valued function defined by (7), we can construct a chain of Hilbert spaces consisting of functionals of as follows. For , we define a norm on throughand put It is not hard to check that is a Hilbert norm and becomes a Hilbert space with .
Lemma 5 (see ). For , one has and moreover the system forms an orthonormal basis for .
It is easy to see that for all . This implies that and whenever . Thus we actually get a chain of Hilbert spaces of functionals of :We now putand endow it with the topology generated by the norm sequence . Note that, for each , is just the completion of with respect to . Thus is a countably-Hilbert space . The next lemma, however, shows that even has a much better property.
Lemma 6 (see ). The space is a nuclear space; namely, for any , there exists such that the inclusion mapping defined by is a Hilbert-Schmidt operator.
For , we denote by the dual of and by the norm of . Then and whenever . The lemma below is then an immediate consequence of the general theory of countably-Hilbert spaces (see ).
Lemma 7 (see ). Let be the dual of and endow it with the strong topology. Thenand moreover the inductive limit topology on given by space sequence coincides with the strong topology.
We mention that, by identifying with its dual, one comes to a Gel’fand triple,which we refer to as the Gel’fand triple associated with .
Definition 8 (see ). Elements of are called generalized functionals of , while elements of are called testing functionals of .
Throughout this paper, we denote by the set of all continuous linear operators from to ; that is, .
Definition 9 (see ). For an operator , its 2D-Fock transform is the function on given by where is the canonical bilinear form on .
Much like generalized functionals of , continuous linear operators in are also completely determined by their 2D-Fock transforms.
Lemma 10 (see ). Let , be continuous linear operators. Then if and only if .
The following lemma is known as the characterization theorem of operators in through their 2D-Fock transforms.
Lemma 11 (see ). Let be a function on . Then is the 2D-Fock transform of an element in if and only if it satisfiesfor some constants and . In that case, for , one hasand in particular takes values in , where where .
For two operators , , their usual product may not make sense. However, one can introduce a product of other type for them.
Definition 12 (see ). Let , ; then their convolution is defined as
It can be verified that forms a commutative algebra with an involution and a unit.
3. Convergence Theorems for Operators in
Let be the same discrete-time normal martingale as described in Section 2. In the present section, we apply the 2D-Fock transform to establish convergence theorems for operators in . Furthermore, we discuss the integration of these operator-valued functions.
Definition 13. A sequence is called strongly convergent to , if, for any , one has (in the strong topology of ).
The following theorem offers a necessary condition for operators sequences in to be strongly convergent.
Theorem 14. Let be an operator sequence in , and let be operator in . Then the sequence converges strongly to if and only if(1) for all ;(2)there exist constants and such that
Proof. For the “only if” part, let converge strongly to ; then it is easy to see thatOn the other hand, by Lemma 11, we know that, for each , there exist and such thatLet , according to the principle of uniform boundedness, , which implies thatFor the “if” part, let satisfy conditions (1) and (2); by taking and using Lemma 11, we getwhich means that is an operator sequence from to and it is uniformly bounded. On the other hand, is total in , and condition (1) equivalent to converges in for any , so there exists a linear operator form to such that, for any , we havethus is a linear operator form to , and, for any , converges weakly to in . By Proposition in paper , converges strongly to in . So converges strongly to .
In the following, we discuss the integration of -valued mapping.
Definition 15. Let be a measure space and let be a -valued function; then is said to be strongly Bochner-integrable with respect to if generalized functional valued function is Bochner-integrable with respect to for any . In that case, a linear operator from to ,is defined. We called it the Bochner integral of with respect to and it is denoted by .
We note that the strong Bochner integral may not be an operator in . The next theorem, however, provides sufficient conditions for to be strongly Bochner-integrable and its strong Bochner integral to be an operator in .
Theorem 16. Let be a -valued mapping satisfying the following conditions:(1)For any , the function is measurable.(2)There exist and a nonnegative function , such that, for -a.e., , it holds thatThen is strongly Bochner-integrable with respect to and its strong Bochner integral satisfies the following norm inequality: where . In particular, .
Proof. We take . Then, by Lemma 11, for -a.e. , we havewhich means that is an operator from to . On the other hand, according to (1), for any , is measurable. And is a separable space, so is strongly measurable for any , where, together with (27), for any , the mapis Bochner-integrable with respect to . Thus, is strongly Bochner-integrable with respect to . Finally, (28) easily follows from (29) and we find .
Definition 17. Let ; the map is said to be sequentially continuous; if is a sequence in and converges to , we have that converges to .
Definition 17 is the notion of sequentially continuous, because -valued map is defined on interval, so it is equivalent to continuous. The next theorem offers a necessary and sufficient condition for -valued functions to be continuous.
Theorem 18. Let ; is a -valued function; then is continuous if and only if(1) is continuous for any ;(2)there exist and such that for each and .
Proof. We just prove the “if” part. Let and converges to ; then the sequence and satisfies the following conditions:
(1) For every , .
(2) There exist and , such thatwhich implies that the operator sequence satisfies the conditions in Theorem 14; thus, there exists such that , andThen , which implies thatand is continuous.
4. Application to Quantum SDEs
In the present section, we show applications of our main results in Section 3. We establish the existence and uniqueness of solution to (1). Equation (1) can describe quantum stochastic evolutions in the presence of quantum noise. As usual, by a solution of (1) we mean a map which satisfies the following integral equation:where the integral is the Bochner integral of operator function in .
Definition 19. Let be a map. is said to satisfy a locally uniform Lipschitz condition and a locally uniform linear growth condition, if, for any , there exists a constant , such that, for any and , one has
Theorem 20. Let be continuous and satisfy a locally uniform Lipschitz condition and locally uniform linear growth condition, and let the noise process satisfy the following:(1) is continuous.(2)For any , there exists a constant , such that, for any ,Then, for any , there exists a unique continuous map , solving (1).
Proof. We just prove that, for any , the integral equation (35) has a unique continuous solution on .
First, using iteration, we will prove that, for each , there exists a sequence , which has the following properties:
(1) For any , the function is continuous.
(2) For any and , there exists a constant , such that(3)where .
In fact, from the assumptions and Theorem 16, for any , the integral and exist and belong to . Now we define as follows:It is easy to see that, for any , the function is continuous. On the other hand, from the assumptions, there exist constants , , and , such thatwhere and .
Then we assume that, for some and , has been defined and has the properties shown above. Then, by Theorem 18, we find that is continuous; thusis also continuous, which means that the function , , is measurable. On the other hand, we havewhere . By Theorem 16, for any , the integral exists and belongs to . Similarly, for any , the integral also exists and belongs to .
Now we define a map as follows:Obviously, for any , the function is continuous. Furthermore, because and satisfy a locally uniform Lipschitz condition and locally uniform linear growth condition, we haveNote thatSo we obtain that, for any ,Therefore, by induction, we come to our result.
Next, we show that, for any , converges in . Indeed, by induction, for any and , we haveSo, for any and , the sequence converges in . On the other hand, from the properties of , we havewhere . Thus, it follows from Theorem 14 that for any , converges in .
We letThen, it is easy to see thatMoreover, for , , and , we haveSowhich means that the function is continuous. Then, by Theorem 18, the function is continuous.
In the same manner as above, we can show that, for each , the integrals and exist and belong to . Further, by the dominated convergence theorem, we find that, for any and ,Note that, for any and , we haveThus, by taking limit, we getwhich means that, for any , we haveSo the map is a continuous solution to (35).
Finally, we prove the uniqueness. Assume that is another continuous solution to (35). Then together with and satisfying a locally uniform Lipschitz condition and locally uniform linear growth condition, for each and , we haveBy Gronwall inequality, we obtain that By Lemma 10, , ; that is, the solution is unique.
In the following, we always assume that the map and the noise process are the same as in Theorem 20. For , we denote by the solutions of (1) with being its initial value. The following theorem shows that the solution to (1) continuously depends on the initial value.
Theorem 21. Let , be given. Then, for any , there exists constant , such that, for any and , one has
Proof. For , let , , and be such that, for all and , we have(1);(2), ;(3). It is easy to see that, for any ,which implies that Similarly, we can prove that Then, using estimates (62) and (63), we have
Conflicts of Interest
The author declares that there are no conflicts of interest.
This work is supported by the National Natural Science Foundation of China (Grants nos. 11461061 and 11761047).
T. Hida, Analysis of Brownian Functionals, Carleton Math, Carleton Univ., Ottawa, Canada, 1975.View at: MathSciNet
T. Hida, H.-H. Kuo, J. Potthoff, and L. Streit, White Noise: An Infinite Dimensional Calculus, Kluwer Academic, Dordrecht, Netherlands, 1993.
Z. Y. Huang and J. A. Yan, Introduction to Infinite Dimensional Stochastic Analysis, Kluwer Academic, Dordrecht, Netherlands, 1999.View at: MathSciNet
H. H. Kuo, White Noise Distribution Theory, CRC Press, Boca Raton, Fla, USA, 1996.View at: MathSciNet
I. M. Gel'fand and N. Y. Vilenkin, Generalized Functions, vol. 4, Academic Press, New York, NY, USA, 1964.View at: MathSciNet