Quantum Stochastic Cable Equation Acting on Functionals of Discrete-Time Normal Martingales

Abstract

Let be a discrete-time normal martingale satisfying some mild conditions. Then Gel’fand triple can be constructed of functionals of , where elements of are called testing functionals of , while elements of are called generalized functionals of . In this paper, we consider a quantum stochastic cable equation in terms of operators from to . Mainly with the 2D-Fock transform as the tool, we establish the existence and uniqueness of a solution to the equation. We also examine the continuity of the solution and its continuous dependence on initial values.

1. Introduction

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. It was originally proposed for developing mathematical models of signal decay in submarine (underwater) telegraphic cables. However, its utility for neuroscience was not recognized until many years later, after nerve axons came to be regarded as “core conductors.” Neurons are the basic unit in the brain; it transmits electrical signals through axons and receives electrical signals through dendrites. Axons and dendrites can be described as cables with special properties. Thus, the cable equation has been useful to explain different phenomena in dendrites and axons (see, e.g., [1–3]).

The potential of the responses of nerve cells may evolve through a combination of diffusion and random fluctuation. To describe the change of such potential, Walsh [4, 5] introduced a stochastic cable equation, which is a second-order, parabolic, partial differential equation with a certain stochastic term. In 1998, Applebaum [6] proposed a quantum stochastic cable equation in the framework of Fock space, which can be viewed as a quantum version of Walsh’s stochastic cable equation and was intended to describe the microtubular potential as an observable that evolves in space and time through diffusion and quantum effects.

Hida’s white noise analysis is essentially a theory of infinite dimensional calculus on generalized functionals of Brownian motion (also known as Gaussian white noise generalized functionals). In 2000, a quantum stochastic cable equation was considered in the framework of white noise analysis [7]. The main features of the equation considered in [7] lie in that the role of quantum noises was played by a family of generalized operators, which are continuous operators defined on Gaussian white noise testing functionals and valued in Gaussian white noise generalized functionals, and can be regarded as quantum Gaussian noises.

Functionals of discrete-time normal martingales have attracted much attention in recent years (see, e.g., [8–12]). Let be a discrete-time normal martingale satisfying some mild conditions. Then, by using a specific orthonormal basis for the space of square integrable functionals of , Gel’fand triple can be constructed [11], where elements of are called testing functionals of , while elements of are called generalized functionals of . Recently, a transform, called 2D-Fock transform, has been introduced [13] for operators from to . It has been shown [13] that continuous linear operators from to are completely determined by a growth condition on their 2D-Fock transforms. Let denote the set of all continuous linear operators from to . Then one can define the convolution in by means of the 2D-Fock transform, which makes form a commutative algebra.

In this paper, we consider the following quantum stochastic cable equation in terms of operators from to :where is a given constant, stands for the convolution of operators, is a given map, and are -valued quantum stochastic processes, and the solution , if it exists, will be a -valued quantum stochastic process. Compared to the equation considered in [7], (2) might be used to describe quantum stochastic evolution in space and time of a quantum system in the presence of quantum Bernoulli noises.

The rest of the paper is organized as follows. In Section 2, we briefly recall some notions and results on continuous linear operators from to , which will be used in our later discussion. Sections 3 and 4 are our main work. Here, by using the 2D-Fock transform, we establish the existence and uniqueness of a solution to (2). We also examine the continuity and continuous dependence on initial values of the solution to (2).

Throughout this paper, designates the set of all nonnegative integers and the finite power set of , namely, where means the cardinality of as a set.

2. Preliminaries

In this section, we briefly recall some notions and results on continuous linear operators from to . For details, see [10–14] and references therein.

Let be a given probability space and the usual Hilbert space of square integrable complex-valued functions on with inner product and norm , respectively. By convention, is conjugate-linear in its first argument and linear in its second argument.

Let be a discrete-time normal martingale on that has the chaotic representation property. Denote by the Hilbert space of square integrable functionals of , which shares the same inner product and norm with , namely, and . It is known that has an orthonormal basis defined by and where is the discrete-time normal noise associated with (see [11] for details).

By using the orthonormal basis , one can construct [11] a dense linear subspace of , which itself is a countable Hilbert nuclear space and continuously contained in . Let be the dual of endowed with the strong topology. Then, by identifying with its dual, one comes to Gel’fand triplewhich is the framework where we will work. Elements of are called testing functionals of , while elements of are called generalized functionals of . As mentioned above, we denote by the set of all continuous linear operators from to .

Lemma 1 (see [11]). Let be the -valued function on given byThen, for , the positive term series converges and moreover

Definition 2 (see [13]). For an operator , its 2D-Fock transform is the function on defined aswhere 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 3 (see [13]). Let and 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 4 (see [13]). 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 has and in particular takes values in , where where .

Definition 5 (see [13]). Let and and then their convolution is defined by

It can be verified that forms a commutative algebra with a unit. The next definition describes convergence of sequences in .

Definition 6 (see [14]). A sequence in is called to converge strongly to , if for any , we have in the strong topology of .

Lemma 7 (see [14]). Let be a sequence of operators in and an operator in . Then the sequence converges strongly to if and only if:
(1) for all ;
(2) there exist constants and , such that

Lemma 8 (see [14]). Let be a measure space and be a -valued function satisfying the following conditions:
(1) for any , the function is measurable;
(2) there exist and a positive function , such that for -a.e., it holds that Then is Bochner integrable with respect to on and its Bochner integral satisfies the following norm inequality: where . In particular, .

Definition 9 (see [14]). Let be a compact metric space. A map is said to be continuous, if for each sequence in that converges to , one has converges to .

Lemma 10 (see [14]). Let be a compact metric space and is a -valued function. Then is continuous if and only if
(1) is continuous for any ;
(2) there exist and , such thatfor each and .

3. Existence and Uniqueness of Solution

In the present section, we establish the existence and uniqueness of a solution to quantum (2). Recall that is the space of all continuous linear operators from to , which forms a commutative algebra with the convolution .

Let be the Green’s function associated with the classical cable equation. Then has following properties (see[5] for details):

(1) is continuous;

(2) ;

(3) ;

(4) for each , there exists , such thatAnd we put Then, obviously, , , .

Let be a -valued function. Then, is said to be a solution to Equation(2) if it satisfies the following integral equation: where , the integral is the Bochner integral of operator valued function in .

Definition 11. Let be a map. If there exists a constant , such that
(1) for any and , we have (2) for any , we have .

Then is said to satisfy Lipschitz and linear growth conditions by means of 2D-Fock transforms.

Theorem 12. Let and satisfy
(1) for any , the functions and are continuous;
(2) there exist constants associated with , and , such that In addition, let satisfy Lipschitz and linear growth conditions. Then there exists a sequence of -valued functions , satisfying
(1) for any and , the function is continuous;
(2) for , it holds that (3) for , we have

Proof. By properties of the Green’s function and the conditions of Theorem 12, the integral exists and belongs to for . Put Then, for any , the function is continuous. For , , let Then, we have , which implies that the function is continuous. Furthermore, we have where and . By Lemma 8, the integral exists and belongs to . Now we define as Then, by 2D-Fock transform, we have , which implies that the function is continuous and satisfies Assume that, for some , has been defined and the function is continuous and satisfies For , define Similarly, for any , the function is continuous, and by computation we have where and . Hence, the integral exists and belongs to . So we can define as . For , the function is continuous and satisfiesNote that Thus from (38), we get . Hence, by induction we come to our conclusion.

Theorem 13. Let the noise process , initial value process , and the map in (2) satisfy the conditions in Theorem 12. Then there exists a unique continuous -valued function which satisfies (2). Moreover, the solution has following estimate:

Proof (existence). Consider the sequence of -valued functions given in Theorem 12. We assert that, for any , . In fact, this is obviously true for . Now, we assume that the inequality is satisfied for some , then, for , we have . By induction, we come to the assertion.
For , consider the sequence ; we have for any . Hence, the sequence converges in . On the other hand, from the Theorem 12, we have , which together with Lemma 7 implies that converges in .
Let ; then, it is easy to see that and Thus, for any , the integral exists and belongs to .
Moreover, by the dominated convergence theorem, for each , On the other hand, we have , so , which implies that is a solution to (2).
For the continuity of , we should only show that for any is continuous on . In fact, for any , we have So which means that the function is continuous. In addition, we have . Then by Lemma 10, the function is continuous.
Finally, we prove the uniqueness of the solution. Assume that is a another continuous solution to (2). Then for any , we have where . Hence, By Gronwall inequality, we obtain that By Lemma 3, and ; namely, the solution is unique.

4. Stability of Solution

In this section, we will prove that the solution to (2) continuously depends on the initial values.

Let be the set of continuous functions from to , namely, . For any , we define seminorm on as and endow with the topology generated by the family of seminorms . Then becomes a Hausdorff topological linear space. In the following, we always assume that the map satisfies Lipschitz and linear growth conditions in terms of 2D-Fock transform, and the noise process satisfies conditions:

(1) For any , the function is continuous;

(2) there exist constants and associated with , such that

Theorem 14. For each , there exists a unique continuous valued function satisfying (2); moreover, for each .

Proof. By Theorem 13, we should only prove that there exist constants , , and , such thatandIn fact, because , so by Lemma 10, there exist , such that Set ; then for any , we have ; hence, and

For , we denote by the solution to (2) with J as its initial value. Then, we have the following estimate:In fact, because . Therefore, . Then, by Gronwall inequatily, we come to the estimate (64).

Using estimate (64) and 2D-Fock transform and together with the theory of classical cable equation, we obtain the next theorem, which shows that the solution to (2) depends continuously on the initial value.

Theorem 15. For each , define a map as ; then is a family of continuous injective maps and .

Data Availability

We declare that no data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant no. 11461061 and Grant no. 11861057).