ORIGINAL RESEARCH article

Front. Phys., 25 January 2022

Sec. Interdisciplinary Physics

Volume 9 - 2021 | https://doi.org/10.3389/fphy.2021.791858

Large Time Behavior on the Linear Self-Interacting Diffusion Driven by Sub-Fractional Brownian Motion II: Self-Attracting Case

  • 1. College of Information Science and Technology, Donghua University, Shanghai, China

  • 2. College of Fashion and Art Design, Donghua University, Shanghai, China

  • 3. Department of Statistics, College of Science, Donghua University, Shanghai, China

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Abstract

In this study, as a continuation to the studies of the self-interaction diffusion driven by subfractional Brownian motion SH, we analyze the asymptotic behavior of the linear self-attracting diffusion:

where θ > 0 and are two parameters. When θ < 0, the solution of this equation is called self-repelling. Our main aim is to show the solution XH converges to a normal random variable with mean zero as t tends to infinity and obtain the speed at which the process XH converges to as t tends to infinity.

1 Introduction

In a previous study (I) (see [12]), as an extension to classical result, we considered the linear self-interacting diffusion as follows:with θ ≠ 0, where θ and ν are two real numbers, and SH is a sub-fBm with the Hurst parameter . The solution of Eq. 1 is called self-repelling if θ < 0 and is called self-attracting if θ > 0. When θ < 0, in a previous study (I), we showed that the solution XH diverges to infinity as t tends to infinity andandin L2 and almost surely, for all n = 1, 2, …, where ( − 1)!! = 1 and

In the present study, we consider the case θ > 0 and study its large time behaviors.

Let us recall the main results concerning the system (Eq. 1). When , as a special case of path-dependent stochastic differential equations, in 1995, Cranston and Le Jan [8] introduced a linear self-attracting diffusion (Eq. 1) with θ > 0. They showed that the process Xt converges in L2 and almost surely as t tends infinity. This path-dependent stochastic differential equation was first developed by Durrett and Rogers [10] introduced in 1992 as a model for the shape of a growing polymer (Brownian polymer). The general form of this kind of model can be expressed as follows:where B is a d-dimensional standard Brownian motion and f is Lipschitz continuity. Xt corresponds to the location of the end of the polymer at time t. Under some conditions, they established asymptotic behavior of the solution of the stochastic differential equation. The model is a continuous analog of the notion of edge (respectively, vertex) self-interacting random walk (see, e.g., Pemantle [22]). By using the local time of the solution process X, we can make it clear how the process X interacts with its own occupation density. In general, Eq. 2 defines a self-interacting diffusion without any assumption on f. We call it self-repelling (respectively, self-attracting) if, for all (respectively, ). More examples can be found in Benaïm et al. [2, 3], Cranston and Mountford [9], Gan and Yan [11], Gauthier [13], Herrmann and Roynette [14], Herrmann and Scheutzow [15], Mountford and Tarr [20], Sun and Yan [26, 27], Yan et al [34], and the references therein.

In this present study, our main aim is to expound and prove the following statements:

  • (I) For θ > 0 and , the random variable

exists as an element in

L2

, where the function is defined as follows:

with

θ

> 0.

  • (II) For θ > 0 and , we have

in

L2

and almost surely as

t

→

∞

.

  • (III) For θ > 0 and , we have

in distribution as

t

→

∞

, where

  • (IV) For θ > 0 and , we have

Then the convergence

holds in

L2

as

T

tends to infinity.

This article is organized as follows. In Section 2, we present some preliminaries for sub-fBm and Malliavin calculus. In Section 3, we obtain some lemmas. In Section 4, we prove the main results given as before. In Section 5, we give some numerical results.

2 Preliminaries

In this section, we briefly recall the definition and properties of stochastic integral with respect to sub-fBm. We refer to Alós et al [1], Nualart [21], and Tudor [31] for a complete description of stochastic calculus with respect to Gaussian processes.

As we pointed out in the previous study (I) (see [12]), the sub-fBm SH is a rather special class of self-similar Gaussian processes such that andfor all s, t ≥ 0. For H = 1/2, SH coincides with the standard Brownian motion B. SH is neither a semimartingale nor a Markov process unless H = 1/2, so many of the powerful techniques from stochastic analysis are not available when dealing with SH. As a Gaussian process, it is possible to construct a stochastic calculus of variations with respect to SH. The sub-fBm appeared in Bojdecki et al [4] in a limit of occupation time fluctuations of a system of independent particles moving in according a symmetric α-stable Lévy process. More examples for sub-fBm and related processes can be found in Bojdecki et al. [4–7], Li [16–19], Shen and Yan [23, 24], Sun and Yan [25], C. A. Tudor [32], Tudor [28–31], C. A. Tudor [33], Yan et al [33, 35, 36], and the references therein.

The normality and Hölder continuity of the sub-fBm SH imply that admits a bounded pH variation on any finite interval with . As an immediate result, one can define the Young integral of a process u = {ut, t ≥ 0} with respect to a sub-fBm SHas the limit in probability of a Riemann sum. Clearly, when u is of bounded qH variation on any finite interval with qH > 1 and , the integral is well-defined andfor all t ≥ 0.

Let be the completion of the linear space generated by the indicator functions 1[0,t], t ∈ [0, T] with respect to the inner product:for s, t ∈ [0, T]. For every , we can define the Wiener integral with respect to SH, denoted byas a linear (isometric) mapping from onto by using the limit in probability of a Riemann sum, where is the Gaussian Hilbert space generating by andfor any . In particular, when , we can show thatwherefor s, t ∈ [0, T]. Thus, when if for every T > 0, the integral exists in L2 andwe can define the integral as follows:and

Let now D and δ be the (Malliavin) derivative and divergence operators associated with the sub-fBm SH. And let denote the Hilbert space with respect to the norm as follows:Then the duality relationshipholds for any and . Moreover, for any , we havewhere (Du)∗ is the adjoint of Du in the Hilbert space given as follows: . We denotefor an adapted process u, and it is called the Skorohod integral. By using Alós et al [1], we can obtain the relationship between the Skorohod and the Young integral as follows:provided u has a bounded q variation with and such that

3 Some Basic Estimates

For simplicity, we throughout let C stand for a positive constant which depends only on its superscripts, and its value may be different in different appearances, and this assumption is also suitable to c. Recall that the linear self-attracting diffusion with sub-fBm SH is defined by the following stochastic differential equation:with θ > 0. The kernel (t, s)↦hθ(t, s) is defined as follows:for s, t ≥ 0. By the variation of constants method (see, Cranston and Le Jan [8]) or Itô’s formula, we may introduce the following representation:for t ≥ 0.

The kernel function (

t

,

s

)↦

hθ

(

t

,

s

) with

θ

> 0 admits the following properties (these properties are proved partly in Cranston and Le Jan [

8

]):

  • • For all s ≥ 0, the limit

exists.

  • • For all t ≥ s ≥ 0, we have hθ(s) ≤ hθ(t, s), and

  • • For all t ≥ s, r ≥ 0 and θ ≠ 0, we have

and

  • • For all t > 0, we have

Lemma 3.1Letandθ > 0. Then the random variableexists as an element in L2.

ProofThis is a simple calculus exercise. In fact, we havefor all θ > 0 and . Clearly, Eq. 10 implies that and andfor all θ > 0 and . These show that the random variable exists as an element in L2.

Lemma 3.2Letθ > 0. We then have

ProofThis is a simple calculus exercise. In fact, we havefor all t ≥ 0 and θ > 0. Noting thatandwe see thatby L’Hopital’s rule.

Lemma 3.3Letθ > 0. We then havefor all t ≥ 0.

Lemma 3.4Let θ > 0 and . We then have

ProofBy L’Hopital’s rule and the change of variable , it follows thatwhere we have used the equationThis completes the proof.

Lemma 3.5Let θ > 0 and . We then havefor all 0 ≤ s < t ≤ T, where C and c are two positive constants depending only on H, θ, ν and T.

ProofThe lemma is similar to Lemma 3.5 in the previous study (I).

Lemma 3.6Let θ > 0 and . Then the convergenceholds in L2 and almost surely as t tends to infinity.

ProofConvergence (18) in L2 follows from Lemma (3.1). In fact, by Eq. 10, we haveas t tends to infinity.On the other hand, by Lemma (3.5), 3.3 and the equation almost surely as t tends to infinity, we find thatas t tends to infinity. It follows from the integration by parts thatalmost surely as t tends to infinity.

4 Some Large Time Behaviors

In this section, we consider the long time behaviors for XH with and θ > and our objects are to prove the statements given in Section 1.

Theorem 4.1

Let θ > 0 and . Then the convergenceholds in L2 and almost surely as t tends to infinity.

ProofWhen , the convergence is obtained in Cranston-Le Jan [8]. Consider the decompositionfor all t ≥ 0.We first check that Eq. 19 holds in L2. By Lemma 3.6 and Lemma 3.2, we only need to prove converges to zero in L2. It follows from the equationfor all θ > 0 as t tends to infinity and Lemma 3.4 thatfor all θ > 0 and as t tends to infinity, which implies that Eq. 19 holds in L2.We now check that Eq. 19 holds almost surely as t tends to infinity. By Lemma 3.6, we only need check that converges to zero almost surely as t tends to infinity. We havefor all θ > 0 and as t tends to infinity. To obtain the convergence, we define the random sequencefor every integer n ≥ 1. Then {Zn,k, k = 0, 1, 2, …, n} is Gaussian for every integer n ≥ 1. It follows from Lemma 3.4 thatfor every integer n ≥ 1 and 0 ≤ k ≤ n, which implies thatfor any ɛ > 0, every integer n ≥ 1 and 0 ≤ k ≤ n.On the other hand, for every s ∈ (0, 1), we denoteThen also is Gaussian for every integer n ≥ 1 and 0 ≤ k ≤ n. It follows thatfor all s, s′ ∈ [0, 1]. Thus, for any ɛ > 0, by Slepian’s theorem and Markov’s inequality, one can getfor every integer n ≥ 1 and 0 ≤ k ≤ n. Combining this with the Borel–Cantelli lemma and the relationshipwe show that almost surely as t tends to infinity. This completes the proof.

Theorem 4.2

Let θ > 0 and . Then the convergenceholds in distribution, where is a central normal random variable with its variance

ProofWhen , this result also is unknown. We only consider the case and similarly one can prove the convergence for . By Eq. 20, Slutsky’s theorem, and Lemma 3.2, we only need to show thatin probability andin distribution.First, Eq. 22 follows from Eq. 10 andfor all θ > 0 and as t tends to infinity.We now obtain convergence (23). By the equationas t tends to infinity and Lemma 3.4, we getfor all θ > 0 and as t tends to infinity. Thus, convergence (23) follows from the normality of for all and the theorem follows.At the end of this section, we obtain a law of large numbers. Consider the process YH defined byThen the self-attracting diffusion XH satisfiesandby integration by parts. It follows thatfor all and t ≥ 0. By the variation of constant method, we can give the explicit representation of YH as follows:

Lemma 4.1Let and θ > 0. Then we havealmost surely and in L2 as T tends to infinity.

ProofThis lemma follows from Eq. 24 and the estimatesas T tends to infinity.

Theorem 4.3

Let and θ > 0. Then we havein L2 as T tends to infinity.

ProofGiven and θ > 0,for all t ≥ 0. Thenfor all t ≥ 0. We now prove the lemma in three steps.

Step IWe claim thatas t tends to infinity. Clearly, we haveThus, 29 is equivalent toBy L’Hôspital’s rule and Lemma 3.4, it follows thatfor all .

Step IIWe claim thatas T tends to infinity. We have thatfor all t > s > 0. An elementary calculation may show thatfor all t > s > 0. It follows from the equation with x ≥ 0 and β > − 1 thatfor all t > s > 0 and 0 ≤ α ≤ 1. For the term Λ2(H; t, s), by the proof of Lemma 3.4, we find thatfor all . Combining this with the equationand the equation with x > 0 and 0 < ϱ < 1, we getfor all t > s > 0, and 0 ≤ γ ≤ 2 − 2H. Thus, we have showed that the estimateholds for all t > s ≥ 0. In particular, we havefor all t, s ≥ 0. As a corollary, we getas T tends to infinity.

Step IIIWe claim thatas t tends to infinity. By steps I and II, we find that Eq. 37 is equivalent toas t tends to infinity. Noting that the equationfor all t, s > 0, we further find that convergence (38) also is equivalent toas T tends to infinity. We now check that convergence (40) in two cases.

Case 1Let . Clearly, by Eq. 36, we have to

Case 2Let . By Eq. 36, we have thatwith andwith for all T > 1. Similarly, by Eq. 35, we also havefor all T > 1 and since 0 < t2 − s2 < 1 for . Thus, we have shown thatwith andas T tends to infinity. This shows that convergence (40) holds for all . Similarly, we can also show the theorem holds for and the theorem follows.

Remark 1By using the Borel–Cantelli lemma and Theorem 4.3, we can check that convergence (28) holds almost surely.

5 Simulation

We have applied our results to the following linear self-attracting diffusion driven by a sub-fBm

SH

with

as follows:

where

θ

> 0 and

are two parameters. We will simulate the process with

ν

= 0 in the following cases:

FIGURE 1

FIGURE 1

Path of XH with θ = 1 and H = 0.7.

FIGURE 2

FIGURE 2

Path of XH with θ = 10 and H = 0.7.

FIGURE 3

FIGURE 3

Path of XH with θ = 100 and H = 0.7.

TABLE 1

ttt
0.000 00.000 00.343 8−0.121 60.687 5−0.097 9
0.015 60.008 70.359 4−0.129 00.703 1−0.096 8
0.031 30.011 30.375 0−0.140 60.718 8−0.101 7
0.046 90.003 90.390 6−0.146 70.734 4−0.109 0
0.062 5−0.015 30.406 3−0.145 90.750 0−0.108 8
0.078 1−0.023 80.421 9−0.157 90.765 6−0.118 8
0.093 8−0.022 90.437 5−0.162 40.781 3−0.116 3
0.109 4−0.027 00.453 1−0.166 60.796 9−0.112 5
0.125 0−0.033 50.468 8−0.170 10.812 5−0.123 1
0.140 6−0.035 30.484 4−0.171 70.828 1−0.140 0
0.156 3−0.037 00.500 0−0.173 80.843 8−0.146 5
0.171 9−0.049 80.515 6−0.177 40.859 4−0.155 4
0.187 5−0.054 40.531 3−0.176 60.875 0−0.160 4
0.203 1−0.059 30.546 9−0.171 30.890 6−0.170 9
0.218 8−0.076 50.562 5-0.166 70.906 3−0.174 3
0.234 4−0.085 00.578 1−0.166 40.921 9−0.178 1
0.250 0−0.098 10.593 8−0.152 10.937 5−0.179 4
0.265 6−0.106 20.609 4−0.142 20.953 1−0.180 3
0.281 3−0.112 70.625 0−0.139 50.968 8−0.175 8
0.296 9−0.113 20.640 6−0.128 20.984 4−0.193 5
0.312 5−0.114 00.656 3−0.123 41.000 0−0.198 0
0.328 1−0.121 40.671 9−0.105 4

Data of with θ = 1 and H = 0.7

TABLE 2

ttt
0.000 00.000 00.343 8−0.098 30.687 5−0.110 9
0.015 6−0.006 40.359 4−0.110 40.703 1−0.112 1
0.031 3−0.010 40.375 0−0.110 80.718 8−0.112 6
0.046 9−0.010 10.390 6−0.109 80.734 4−0.103 4
0.062 5−0.017 90.406 3−0.111 90.750 0−0.099 1
0.078 1−0.017 70.421 9−0.110 60.765 6−0.090 1
0.093 8−0.024 20.437 5−0.112 60.781 3−0.089 0
0.109 4−0.031 90.453 1−0.117 00.796 9−0.089 4
0.125 0−0.030 60.468 8−0.118 50.812 5−0.090 9
0.140 6−0.041 60.484 4−0.120 50.828 1−0.085 7
0.156 3−0.052 30.500 0−0.113 10.843 8−0.085 1
0.171 9−0.057 70.515 6−0.106 80.859 4−0.095 1
0.187 5−0.063 70.531 3−0.106 70.875 0−0.090 9
0.203 1−0.069 00.546 9−0.113 70.890 6−0.089 0
0.218 8−0.070 80.562 5−0.110 50.906 3−0.094 0
0.234 4−0.067 00.578 1−0.110 10.921 9−0.097 6
0.250 0−0.063 00.593 8−0.107 80.937 5−0.100 6
0.265 6−0.074 40.609 4−0.107 80.953 1−0.099 8
0.281 3−0.083 10.625 0−0.106 90.968 8−0.094 1
0.296 9−0.086 50.640 6−0.105 90.984 4−0.093 3
0.312 5−0.088 10.656 3−0.108 51.000 0−0.092 8
0.328 1−0.096 20.671 9−0.110 7

Data of with θ = 10 and H = 0.7

TABLE 3

ttt
0.000 00.000 00.343 80.015 30.687 50.001 5
0.015 6−0.004 70.359 40.013 50.703 10.011 6
0.031 3−0.021 00.375 00.000 50.718 80.014 8
0.046 9−0.024 10.390 6−0.002 00.734 40.002 7
0.062 5−0.029 00.406 30.002 50.750 0-0.000 8
0.078 1−0.020 00.421 90.002 30.765 6-0.008 6
0.093 8−0.014 30.437 50.011 60.781 3-0.006 9
0.109 4−0.012 90.453 10.003 80.796 90.000 1
0.125 0−0.020 60.468 8−0.007 40.812 50.006 0
0.140 6−0.015 70.484 4−0.010 50.828 10.016 0
0.156 30.004 70.500 0−0.012 40.843 80.006 2
0.171 90.013 60.515 6−0.009 00.859 40.011 1
0.187 50.011 00.531 3−0.009 40.875 00.009 0
0.203 10.006 70.546 9−0.016 00.890 60.000 3
0.218 80.020 00.562 5−0.011 40.906 3-0.003 2
0.234 40.016 20.578 1−0.006 70.921 90.010 5
0.250 00.002 40.593 8−0.002 80.937 5-0.001 1
0.265 60.002 50.609 40.002 80.953 10.001 0
0.281 30.008 20.625 00.000 90.968 80.005 6
0.296 90.007 60.640 6−0.006 20.984 40.001 9
0.312 50.008 30.656 3−0.015 81.000 0-0.004 6
0.328 10.008 60.671 9−0.005 1

Data of with θ = 100 and H = 0.7

FIGURE 4

FIGURE 4

Path of XH with θ = 1 and H = 0.5.

FIGURE 5

FIGURE 5

Path of XH with θ = 10 and H = 0.5.

FIGURE 6

FIGURE 6

Path of XH with θ = 100 and H = 0.5.

TABLE 4

ttt
0.000 00.000 00.343 80.939 30.687 51.188 3
0.015 6−0.176 10.359 40.991 30.703 10.992 1
0.031 30.009 90.375 01.036 30.718 80.956 4
0.046 9−0.040 00.390 61.218 00.734 40.994 3
0.062 50.019 00.406 31.204 20.750 00.885 2
0.078 10.088 30.421 91.122 90.765 60.861 1
0.093 80.020 00.437 51.111 00.781 30.688 6
0.109 40.274 40.453 11.021 10.796 90.653 8
0.125 00.231 70.468 81.066 00.812 50.731 2
0.140 60.246 10.484 41.007 00.828 10.750 8
0.156 30.200 40.500 01.099 50.843 80.866 3
0.171 90.172 30.515 61.149 70.859 40.746 9
0.187 50.233 20.531 31.162 00.875 00.608 0
0.203 10.485 90.546 91.222 90.890 60.618 4
0.218 80.697 40.562 51.435 00.906 30.655 0
0.234 40.684 80.578 11.447 40.921 90.632 1
0.250 00.627 50.593 81.453 50.937 50.610 1
0.265 60.777 40.609 41.479 40.953 10.623 8
0.281 30.825 00.625 01.276 40.968 80.406 6
0.296 90.775 40.640 61.281 40.984 40.389 3
0.312 50.878 30.656 31.284 81.000 00.234 5
0.328 10.838 00.671 91.168 9

Data of with θ = 1 and H = 0.5

TABLE 5

ttt
0.000 00.000 00.343 8−0.024 70.687 5−0.492 7
0.015 6−0.054 80.359 4−0.366 60.703 1−0.589 4
0.031 30.122 70.375 0−0.452 20.718 8−0.689 0
0.046 90.167 90.390 6−0.690 70.734 4−0.507 9
0.062 50.151 50.406 3−0.915 40.750 0−0.370 3
0.078 1−0.217 70.421 9−0.954 10.765 6−0.283 2
0.093 80.041 10.437 5−1.020 50.781 3−0.445 5
0.109 4−0.061 70.453 1−0.906 90.796 9−0.551 5
0.125 0−0.069 70.468 8−0.855 30.812 5−0.579 9
0.140 6−0.359 20.484 4−0.820 10.828 1−0.509 3
0.156 3−0.348 90.500 0−0.735 70.843 8−0.556 1
0.171 9−0.481 80.515 6−0.822 00.859 4−0.589 2
0.187 5−0.296 60.531 3−0.785 20.875 0−0.501 7
0.203 1−0.471 70.546 9−0.814 60.890 6−0.458 0
0.218 8−0.417 50.562 5−0.823 90.906 3−0.689 5
0.234 4−0.169 30.578 1−0.833 70.921 9−0.784 6
0.250 0−0.126 50.593 8−0.735 30.937 5−0.825 7
0.265 6−0.017 80.609 4−0.539 70.953 1−0.903 4
0.281 3−0.053 60.625 0−0.515 20.968 8−0.736 4
0.296 9−0.071 40.640 6−0.524 50.984 4−0.669 2
0.312 5−0.115 80.656 3−0.489 91.000 0−0.506 1
0.328 1−0.132 20.671 9−0.525 8

Data of with θ = 10 and H = 0.5

TABLE 6

ttt
0.000 00.000 00.343 8−0.207 40.687 5-0.149 3
0.015 6−0.012 90.359 4−0.373 20.703 1−0.230 8
0.031 3−0.134 80.375 0−0.464 90.718 80.164 4
0.046 90.069 70.390 6−0.292 50.734 4−0.050 0
0.062 50.111 50.406 3−0.244 50.750 0−0.131 7
0.078 10.002 90.421 9−0.246 70.765 6−0.218 2
0.093 8−0.058 90.437 50.062 80.781 3−0.313 7
0.109 4−0.288 80.453 1−0.091 70.796 9−0.069 1
0.125 0−0.195 60.468 8−0.307 20.812 5−0.239 1
0.140 6−0.046 90.484 4−0.216 20.828 1−0.306 2
0.156 3−0.139 10.500 0−0.241 80.843 8−0.147 8
0.171 9−0.183 30.515 6−0.159 30.859 4−0.203 4
0.187 5−0.117 50.531 3−0.250 90.875 0−0.219 3
0.203 1−0.261 60.546 9−0.344 20.890 6−0.376 9
0.218 8−0.156 80.562 5−0.129 50.906 30.051 5
0.234 4−0.221 50.578 1−0.113 00.921 9−0.107 6
0.250 0−0.173 60.593 8−0.191 50.937 5−0.117 3
0.265 6−0.198 50.609 4−0.131 30.953 1−0.274 6
0.281 30.067 40.625 0−0.175 80.968 8−0.155 6
0.296 9−0.163 30.640 6−0.100 80.984 4−0.223 2
0.312 5−0.121 90.656 3−0.104 91.000 0−0.232 0
0.328 1−0.161 00.671 9−0.270 3

Data of with θ = 100 and H = 0.5

Remark 2From the following numerical results, we can find that it is important to study the estimates of parameters θ and ν.

Statements

Data availability statement

The original contributions presented in the study are included in the article/supplementary material; further inquiries can be directed to the corresponding authors.

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The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Summary

Keywords

subfractional Brownian motion, self-attracting diffusion, law of large numbers, Malliavin calculus, asymptotic distribution

Citation

Guo R, Gao H, Jin Y and Yan L (2022) Large Time Behavior on the Linear Self-Interacting Diffusion Driven by Sub-Fractional Brownian Motion II: Self-Attracting Case. Front. Phys. 9:791858. doi: 10.3389/fphy.2021.791858

Received

09 October 2021

Accepted

19 November 2021

Published

25 January 2022

Volume

9 - 2021

Edited by

Ming Li, Zhejiang University, China

Reviewed by

Xiangfeng Yang, Linköping University, Sweden

Yu Sun, Our Lady of the Lake University, United States

Updates

Copyright

© 2022 Guo, Gao, Jin and Yan.

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Han Gao,

This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics

Disclaimer

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.

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