Large Time Behavior on the Linear Self-Interacting Diffusion Driven by Sub-Fractional Brownian Motion With Hurst Index Large Than 0.5 I: Self-Repelling Case

Gao, Han; Guo, Rui; Jin, Yang; Yan, Litan

doi:10.3389/fphy.2021.795210

ORIGINAL RESEARCH article

Front. Phys., 14 January 2022

Sec. Interdisciplinary Physics

Volume 9 - 2021 | https://doi.org/10.3389/fphy.2021.795210

Large Time Behavior on the Linear Self-Interacting Diffusion Driven by Sub-Fractional Brownian Motion With Hurst Index Large Than 0.5 I: Self-Repelling Case

HG
Han Gao ¹^*
RG
Rui Guo ²
YJ
Yang Jin ³
LY
Litan Yan ³

1. College of Fashion and Art Design, Donghua University, Shanghai, China
2. College of Information Science and Technology, Donghua University, Shanghai, China
3. Department of Statistics, College of Science, Donghua University, Shanghai, China

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Abstract

Let S^H be a sub-fractional Brownian motion with index . In this paper, we consider the linear self-interacting diffusion driven by S^H, which is the solution to the equation

where θ < 0 and are two parameters. Such process X^H is called self-repelling and it is an analogue of the linear self-attracting diffusion [Cranston and Le Jan, Math. Ann. 303 (1995), 87–93]. Our main aim is to study the large time behaviors. We show the solution X^H diverges to infinity, as t tends to infinity, and obtain the speed at which the process X^H diverges to infinity as t tends to infinity.

1 Introduction

In 1995, Cranston and Le Jan [1] introduced a linear self-attracting diffusionwith θ > 0 and X₀ = 0, where B is a 1-dimensional standard Brownian motion. They showed that the process X_t converges in L² and almost surely, as t tends infinity. This is a special case of path dependent stochastic differential equations. Such path dependent stochastic differential equation was first developed by Durrett and Rogers [2] introduced in 1992 as a model for the shape of a growing polymer (Brownian polymer) as followswhere B is a d-dimensional standard Brownian motion and f is Lipschitz continuous. X_t corresponds to the location of the end of the polymer at time t. Under some conditions, they established asymptotic behavior of the solution of stochastic differential equation and gave some conjectures and questions. The model is a continuous analogue of the notion of edge (resp. vertex) self-interacting random walk. If f(x) = g(x)x/‖x‖ and g(x) ≥ 0, X_t is a continuous analogue of a process introduced by Diaconis and studied by Pemantle [3]. Let be the local time of the solution process X. Then, we havefor all t ≥ 0. This formulation makes it clear how the process X interacts with its own occupation density. We may call this solution a Brownian motion interacting with its own passed trajectory, i.e., a self-interacting motion. In general, the Eq. 1.2 defines a self-interacting diffusion without any assumption on f. Iffor all , we call it self-repelling (resp. self-attracting). In 2002, Benaïm et al [4] also introduced a self-interacting diffusion with dependence on the (convolved) empirical measure. A great difference between these diffusions and Brownian polymers is that the drift term is divided by t. It is noteworthy that the interaction potential is attractive enough to compare the diffusion (a bit modified) to an Ornstein-Uhlenbeck process, in many case of f, which points out an access to its asymptotic behavior. More works can be found in Benaïm et al. [5], Cranston and Mountford [6], Gauthier [7], Herrmann and Roynette [8], Herrmann and Scheutzow [9], Mountford and Tarr [10], Shen et al [11], Sun and Yan [12] and the references therein.

On the other hand, starting from the application of fractional Brownian motion in polymer modeling, Yan et al [13] considered an analogue of the linear self-interacting diffusion:with θ ≠ 0 and , where B^H is a fractional Brownian motion (fBm, in short) with Hurst parameter . The solution of (1.3) is a Gaussian process. When θ > 0, Yan et al [13] showed that the solution X^H of (1.3) converges in L² and almost surely, to the random variablewhere the function is defined ar followswith θ > 0. Recently, Sun and Yan [14] considered the related parameter estimations with θ > 0 and , and Gan and Yan [15] considered the parameter estimations with θ < 0 and .

Motivated by these results, as a natural extension one can consider the following stochastic differential equation:with θ > 0 and X₀ = 0, where G = {G_t, t ≥ 0} is a Gaussian process with some suitable conditions which includes fractional Brownian motion and some related processes. However, for a (general) abstract Gaussian process it is difficult to find some interesting fine estimates associated with the calculations. So, in this paper we consider the linear self-attracting diffusion driven by a sub-fractional Brownian motion (sub-fBm, in short). We choose this kind of Gaussian process because it is only the generalization of Brownian motion rather than the generalization of fractional Brownian motion. It only has some similar properties of fractional Brownian motion, such as long memory and self similarity, but it has no stationary increment. The so-called sub-fBm with index H ∈ (0, 1) is a mean zero Gaussian process with and the covariancefor all s, t ≥ 0. For H = 1/2, S^H coincides with the standard Brownian motion B. S^H 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 S^H. As a Gaussian process, it is possible to construct a stochastic calculus of variations with respect to S^H (see, for example, Alós et al [16]). The sub-fBm has properties analogous to those of fBm and satisfies the following estimates:

More works for sub-fBm and related processes can be found in Bojdecki et al. [17–20], Li [21–24], Shen and Yan [25, 26], Sun and Yan [27], Tudor [28–31], Ciprian A. Tudor [32] Yan et al [33–35] and the references therein.

In this present paper, we consider the linear self-interacting diffusion

with

< 0 and

, where

S^H

is a sub-fBm with Hurst parameter

. Our main aim is to show that the solution of

(1.7)

diverges to infinity and obtain the speed diverging to infinity, as

tends to infinity. The object of this paper is to expound and prove the following statements:

(I) For θ < 0 and , the random variable

exists as an element in

L²

(II) For θ < 0 and , as t → ∞, we have

L²

and almost surely.

(III) For θ < 0 and , define the processes by

for all

≥ 0, where (−1)!! = 1. We then have

holds in

L²

and almost surely for every

≥ 1, as

→

∞

This paper 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 result. In Section 5 we give some numerical results.

FIGURE 1

FIGURE 2

FIGURE 3

FIGURE 4

FIGURE 5

FIGURE 6

TABLE 1

t		T		t
0.0000	0.0000	0.3438	−0.1077	0.6875	−0.0995
0.0156	−0.0167	0.3594	−0.1190	0.7031	−0.1091
0.0313	−0.0178	0.3750	−0.1153	0.7188	−0.1163
0.0469	−0.0320	0.3906	−0.1116	0.7344	−0.1165
0.0625	−0.0338	0.4063	−0.0965	0.7500	−0.1122
0.0781	−0.0420	0.4219	−0.0937	0.7656	−0.1205
0.0938	−0.0492	0.4375	−0.0971	0.7813	−0.1170
0.1094	−0.0496	0.4531	−0.0974	0.7969	−0.1192
0.1250	−0.0564	0.4688	−0.0997	0.8125	−0.1180
0.1406	−0.0590	0.4844	−0.0976	0.8281	−0.1316
0.1563	−0.0682	0.5000	−0.0956	0.8438	−0.1245
0.1719	−0.0692	0.5156	−0.0983	0.8594	−0.1202
0.1875	−0.0834	0.5313	−0.0959	0.8750	−0.1241
0.2031	−0.0886	0.5469	−0.0877	0.8906	−0.1212
0.2188	−0.0969	0.5625	−0.0919	0.9063	−0.1250
0.2344	−0.0983	0.5781	−0.0818	0.9219	−0.1219
0.2500	−0.0961	0.5938	−0.0757	0.9375	−0.1199
0.2656	−0.1022	0.6094	−0.0717	0.9531	−0.1191
0.2813	−0.1120	0.6250	−0.0834	0.9688	−0.1223
0.2969	−0.1182	0.6406	−0.0894	0.9844	−0.1089
0.3125	−0.1094	0.6563	−0.0923	1.0000	−0.1023
0.3281	−0.1042	0.6719	−0.0996	—	—

The data of with θ = − 1 and H = 0.7.

TABLE 2

t		t		t
0.0000	0.0000	0.3438	−0.1597	0.6875	−0.5552
0.0156	0.0087	0.3594	−0.1729	0.7031	−0.5943
0.0313	0.0113	0.3750	−0.1912	0.7188	−0.6439
0.0469	0.0040	0.3906	−0.2051	0.7344	−0.7019
0.0625	−0.0153	0.4063	−0.2130	0.7500	−0.7595
0.0781	−0.0239	0.4219	−0.2342	0.7656	−0.8345
0.0938	−0.0234	0.4375	−0.2494	0.7813	−0.9066
0.1094	−0.0279	0.4531	−0.2654	0.7969	−0.9868
0.1250	−0.0348	0.4688	−0.2820	0.8125	−1.0919
0.1406	−0.0372	0.4844	−0.2980	0.8281	−1.2177
0.1563	−0.0395	0.5000	−0.3156	0.8438	−1.3507
0.1719	−0.0530	0.5156	−0.3363	0.8594	−1.5050
0.1875	−0.0587	0.5313	−0.3543	0.8750	−1.6776
0.2031	−0.0648	0.5469	−0.3694	0.8906	−1.8811
0.2188	−0.0835	0.5625	−0.3865	0.9063	−2.1081
0.2344	−0.0942	0.5781	−0.4093	0.9219	−2.3699
0.2500	−0.1100	0.5938	−0.4204	0.9375	−2.6701
0.2656	−0.1213	0.6094	−0.4368	0.9531	−3.0170
0.2813	−0.1317	0.6250	−0.4620	0.9688	−3.4144
0.2969	−0.1365	0.6406	−0.4810	0.9844	−3.8989
0.3125	−0.1418	0.6563	−0.5086	1.0000	−4.4510
0.3281	−0.1541	0.6719	−0.5258	—	—

The data of with θ = − 10 and H = 0.7.

TABLE 3

t		t		t
0.0000	0.0000	0.3438	−1.0056	0.6875	−2.29E+07
0.0156	0.0132	0.3594	−1.6439	0.7031	−6.63E+07
0.0313	0.0093	0.3750	−2.7733	0.7188	−1.97E+08
0.0469	0.0070	0.3906	−4.8028	0.7344	−5.99E+08
0.0625	0.0103	0.4063	−8.5377	0.7500	−1.87E+09
0.0781	0.0116	0.4219	−15.5941	0.7656	−5.98E+09
0.0938	0.0092	0.4375	−29.2598	0.7813	−1.96E+10
0.1094	0.0066	0.4531	−56.3669	0.7969	−6.59E+10
0.1250	0.0081	0.4688	−111.4786	0.8125	−2.27E+11
0.1406	0.0049	0.4844	−226.2866	0.8281	−8.02E+11
0.1563	0.0094	0.5000	−471.3711	0.8438	−2.91E+12
0.1719	−0.0029	0.5156	−1.01E+03	0.8594	−1.08E+13
0.1875	−0.0114	0.5313	−2.21E+03	0.8750	−4.10E+13
0.2031	−0.0279	0.5469	−4.97E+03	0.8906	−1.60E+14
0.2188	−0.0484	0.5625	−1.15E+04	0.9063	−6.40E+14
0.2344	−0.0557	0.5781	−2.72E+04	0.9219	−2.62E+15
0.2500	−0.0837	0.5938	−6.59E+04	0.9375	−1.10E+16
0.2656	−0.1240	0.6094	−1.64E+05	0.9531	−4.75E+16
0.2813	−0.1834	0.6250	−4.19E+05	0.9688	−2.10E+17
0.2969	−0.2706	0.6406	−1.10E+06	0.9844	−9.48E+17
0.3125	−0.4085	0.6563	−2.95E+06	1.0000	−4.40E+18
0.3281	−0.6332	0.6719	−8.12E+06	—	—

The data of with θ = − 100 and H = 0.7.

TABLE 4

t		t		t
0.0000	0.0000	0.3438	0.2713	0.6875	0.6225
0.0156	0.0711	0.3594	0.3234	0.7031	0.7483
0.0313	0.0168	0.3750	0.2698	0.7188	0.9047
0.0469	−0.1326	0.3906	0.3765	0.7344	0.7963
0.0625	−0.1887	0.4063	0.4725	0.7500	0.8221
0.0781	−0.1911	0.4219	0.2156	0.7656	0.7416
0.0938	−0.0792	0.4375	0.1224	0.7813	0.6743
0.1094	−0.0320	0.4531	0.0691	0.7969	1.0655
0.1250	−0.1853	0.4688	0.0377	0.8125	1.0480
0.1406	−0.0827	0.4844	0.1668	0.8281	0.9146
0.1563	−0.0861	0.5000	0.3344	0.8438	0.9478
0.1719	0.0616	0.5156	0.2866	0.8594	1.0125
0.1875	0.1014	0.5313	0.1759	0.8750	1.0931
0.2031	0.1542	0.5469	0.0739	0.8906	1.2403
0.2188	0.2224	0.5625	0.2168	0.9063	0.9036
0.2344	0.2205	0.5781	0.3676	0.9219	0.8949
0.2500	0.3345	0.5938	0.3904	0.9375	0.8626
0.2656	0.3581	0.6094	0.3878	0.9531	0.9140
0.2813	0.2635	0.6250	0.3985	0.9688	1.0247
0.2969	0.4084	0.6406	0.4900	0.9844	1.1976
0.3125	0.2820	0.6563	0.4769	1.0000	1.0780
0.3281	0.4043	0.6719	0.6713	—	—

The data of with θ = − 1 and H = 0.5.

TABLE 5

t		t		t
0.0000	0.0000	0.3438	0.3643	0.6875	1.0084
0.0156	0.1112	0.3594	0.3489	0.7031	1.0312
0.0313	0.1668	0.3750	0.2532	0.7188	1.1722
0.0469	0.1353	0.3906	0.2453	0.7344	1.2474
0.0625	0.2259	0.4063	0.4297	0.7500	1.1783
0.0781	0.0764	0.4219	0.3837	0.7656	1.1997
0.0938	0.0025	0.4375	0.4639	0.7813	1.3114
0.1094	0.2166	0.4531	0.3663	0.7969	1.5335
0.1250	0.2593	0.4688	0.5287	0.8125	1.3820
0.1406	0.2412	0.4844	0.5164	0.8281	1.5679
0.1563	0.5773	0.5000	0.4502	0.8438	1.4858
0.1719	0.4322	0.5156	0.4488	0.8594	1.6145
0.1875	0.4384	0.5313	0.4538	0.8750	1.6282
0.2031	0.2872	0.5469	0.2729	0.8906	1.7043
0.2188	0.3078	0.5625	0.5069	0.9063	1.9432
0.2344	0.3761	0.5781	0.6164	0.9219	1.8384
0.2500	0.1896	0.5938	0.9359	0.9375	2.1171
0.2656	0.1558	0.6094	0.8222	0.9531	2.3878
0.2813	0.3807	0.6250	0.7422	0.9688	2.5204
0.2969	0.3637	0.6406	0.9326	0.9844	2.7823
0.3125	0.3641	0.6563	1.0095	1.0000	3.1237
0.3281	0.3580	0.6719	1.0371	—	—

The data of with θ = − 10 and H = 0.5.

TABLE 6

t		t		t
0.0000	0.0000	0.3438	2.1870	0.6875	5.26E+07
0.0156	−0.1749	0.3594	3.5867	0.7031	1.52E+08
0.0313	−0.3397	0.3750	6.3084	0.7188	4.52E+08
0.0469	−0.4106	0.3906	11.0159	0.7344	1.37E+09
0.0625	−0.3348	0.4063	19.5047	0.7500	4.29E+09
0.0781	−0.3567	0.4219	35.6469	0.7656	1.37E+10
0.0938	−0.3936	0.4375	66.9024	0.7813	4.50E+10
0.1094	−0.3411	0.4531	129.1499	0.7969	1.51E+11
0.1250	−0.2522	0.4688	255.5964	0.8125	5.21E+11
0.1406	−0.1583	0.4844	518.9528	0.8281	1.84E+12
0.1563	−0.1543	0.5000	1.08E+03	0.8438	6.66E+12
0.1719	0.0877	0.5156	2.31E+03	0.8594	2.47E+13
0.1875	−0.1242	0.5313	5.07E+03	0.8750	9.42E+13
0.2031	−0.0522	0.5469	1.14E+04	0.8906	3.67E+14
0.2188	0.1336	0.5625	2.63E+04	0.9063	1.47E+15
0.2344	0.0243	0.5781	6.23E+04	0.9219	6.02E+15
0.2500	0.1665	0.5938	1.51E+05	0.9375	2.53E+16
0.2656	0.2096	0.6094	3.77E+05	0.9531	1.09E+17
0.2813	0.4085	0.6250	9.62E+05	0.9688	4.81E+17
0.2969	0.5852	0.6406	2.52E+06	0.9844	2.18E+18
0.3125	0.8397	0.6563	6.76E+06	1.0000	1.01E+19
0.3281	1.3366	0.6719	1.86E+07	—	—

The data of with θ = − 100 and H = 0.5.

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 [16], Nualart [36], and Tudor [31] for a complete description of stochastic calculus with respect to Gaussian processes. Throughout this paper we assume that denotes a sub-fBm defined on the probability space with index H. As we pointed out before, the sub-fBm S^H is a rather special class of self-similar Gaussian processes such that , andfor all s, t ≥ 0. For H = 1/2, S^H coincides with the standard Brownian motion B. S^H 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 S^H. As a Gaussian process, it is possible to construct a stochastic calculus of variations with respect to S^H. The sub-fBm appeared in Bojdecki et al [17] in a limit of occupation time fluctuations of a system of independent particles moving in according a symmetric α-stable Lévy process, and it also appears in Bojdecki et al [18] in a high-density limit of occupation time fluctuations of the above mentioned particle system, where the initial Poisson configuration has finite intensity measure.

The estimate (1.6) and normality imply that the sub-fBm admits almost surely a bounded -variation on any finite interval for any sufficiently small ϑ ∈ (0, H). That is, the paths of admits a bounded p_H-variation on any finite interval with . As an immediate result, one can define the Young integral of a process u = {u_t, t ≥ 0} with respect to sub-fBm B^a,bas the limit in probability of a Riemann sum. Clearly, the integral is well-defined andfor all t ≥ 0, provided u is of bounded q_H-variation on any finite interval with q_H > 1 and (see, for examples, Bertoin [37] and FöIllmer [38]).

Let be the completion of the linear space generated by the indicator functions 1_[0,t], t ∈ [0, T] with respect to the inner productfor s, t ∈ [0, T]. When , we can show thatwherefor s, t ∈ [0, T]. Define the linear mapping byfor all t ∈ [0, T] and it can be continuously extended to and we call the mapping Φ is called the Wiener integral with respect to S^H, denoted byandfor any .

For simplicity, in this paper we assume that . Thus, if for every T > 0, the integralexists in L² andwe can define the integraland

Denote by the set of smooth functionals of the formwhere and . The Malliavin derivativeD of a functional F as above is given by

The derivative operator D is then a closable operator from L²(Ω) into . We denote by the closure of with respect to the norm

The divergence integralδ is the adjoint of derivative operator D^H. That is, we say that a random variable u in belongs to the domain of the divergence operator δ, denoted by Dom(δ^S), iffor every , where c is a constant depending only on u. In this case δ(u) is defined by the duality relationshipfor any . We have and for any where is the adjoint of Du in the Hilbert space . We will denotefor an adapted process u, and it is called Skorohod integral. Alós et al [16], we can obtain the relationship between the Skorohod and Young integral as followsprovided u has a bounded q-variation with and such that

Theorem 2.1

(Alós et al [16]). Let 0 < H < 1 and letsuch thatwhereκandβare two positive constants with. Then we havefor allt ∈ [0, T].

3 Some Basic Estimates

Throughout this paper we assume that θ < 0 and . Recall that the linear self-interacting diffusion with sub-fBm S^H defined by the stochastic differential equationwith θ < 0. Define the kernel (t, s)↦h_θ(t, s) as followsfor s, t ≥ 0. By the variation of constants method (see, Cranston and Le Jan [1]) or Itô’s formula we may introduce the following representation:for t ≥ 0.

The kernel function (

)↦

h_θ

(

) with

< 0 admits the following properties (these properties are proved partly in Sun and Yan [

]):

• For all s ≥ 0, the limit

for all

≥ 0.

• For all t ≥ s ≥ 0, we have

• For all t ≥ s, r ≥ 0, we have

Lemma 3.1Letθ < 0 and define functionWe then haveand

ProofThis is simple calculus exercise.

Lemma 3.2(Sun and Yan [12]). Letθ < 0 and define the functionst↦I_θ(t, n), n = 1, 2, … as followsThen we havefor everyn ≥ 0, where (−1)! = 1.

Lemma 3.3Letθ < 0. Then the integralconverges and ast → ∞,

ProofAn elementary may show that (3.6) converges for all θ < 0. It follows from L’Hôspital’s rule thatwhere we have used the following fact:This completes the proof.

Lemma 3.4Letθ < 0. Then, convergenceholds.

ProofIt follows from L’Hôspital’s rule thatfor all θ < 0 and . By making the change of variable , we see thatfor all θ < 0 and . This completes the proof.

Lemma 3.5Letθ < 0 and 0 ≤ s < t ≤ T. We then have

ProofGiven 0 ≤ s < t ≤ T and denoteIt follows thatNow, we estimate the three terms. For the first term, we havefor all θ < 0 and 0 < s < t ≤ T. For the second term, we havefor all θ < 0 and 0 < s < t ≤ T. Similarly, for the third term, we also provefor all θ < 0 and 0 < s < t ≤ T. Thus, we have obtained the following estimate:for all θ < 0 and 0 < s < t ≤ T.On the other hand, elementary calculations may show thatandfor all θ < 0 and 0 < s < t ≤ T. It follows thatfor all θ < 0 and 0 < s < t ≤ T, which implies thatfor all θ < 0 and 0 < s < t ≤ T. Noting that the above calculations are invertible for all θ < 0 and 0 < s < t ≤ T, one can obtain the left hand side in (3.8) and the lemma follows.

4 Convergence

In this section, we obtain the large time behaviors associated with the solution X^H to Eq. 3.1. From Lemma 3.5 and Guassianness, we find that the self-repelling diffusion is H-Hölder continuous. So, the integralexists with t ≥ 0 as a Young integral andfor all t ≥ 0. Define the process Y = {Y_t, t ≥ 0} by

By the variation of constants method, one can provefor all t ≥ 0. Define Gaussian process as follows

Lemma 4.1Letθ < 0 and. Then, the random variableexists as an element inL². Moreover,ξ^HisH-Hölder continuous andinL²and almost surely, asttends to infinity.

ProofThis is simple calculus exercise. In fact, we havefor all θ < 0 and , which shows that the random variable exists as an element in L².Now, we show that the process ξ^a,b is Hölder continuous. For all 0 < s < t by the inequality for all x ≥ 0, we haveThus, the normality of ξ^H implies thatfor all 0 ≤ s < t, and integer numbers n ≥ 1, and the Hölder continuity follows.Nextly, we check the converges to in L². This follows from the next estimate:as t tends to infinity.Finally, we check the converges to almost surely. By integration by parts we see thatfor all t ≥ 0. Elementary may check that the convergenceholds almost surely, as t tends to infinity. In fact, by inequalitywith t ≥ 0, we may show thatfor all integer numbers n ≥ 1, and henceThus, Borel-Cantelli’s lemma implies that converges to zero almost surely as t tends to infinity, and the lemma follows from (4.2).

Corollary 4.1For allγ > 0, we haveinL²and almost surely, asttends to infinity.

Lemma 4.2Letθ < 0 and. Then, we haveinL²and almost surely for everyγ ≥ 0, asttends to infinity.

ProofGiven 0 < s ≤ t, θ < 0 and denotewhere we have used the factand estimatesIt follows thatwhich shows that Λ_γ(t, θ) converges to zero in L².Now, we obtain the convergence with probability one. Noting thatfor all u ≥ 0, we getalmost surely for all γ ≥ 0, θ < 0 and , as t tends to infinity. This completes the proof.The objects of this paper are to prove the following theorems which give the long time behaviors for X^H with .

Theorem 4.1

Letθ < 0 and. Then, ast → ∞, the convergenceholds inL²and almost surely.

ProofGiven t > 0 and θ < 0. Simple calculations may proveIt follows from Lemma 4.1, Corollary 4.1, and Lemma 4.2 thatin L² and almost surely for all θ < 0 and , as t tends to infinity.

Theorem 4.2

Define the processesbyfor allt ≥ 0, where (−1)!! = 1. Then, the convergenceholds inL²and almost surely for everyn ≥ 1, ast → ∞.

ProofFrom the proof of Theorem 4.1, we find that the identitiesholds for all t > 0, n ≥ 1 and θ < 0, where I_n(t, θ) is given in Lemma 3.2. Thus, the theorem follows from Lemma 4.1, Corollary 4.1, Lemma 4.2 and Theorem 4.1.

5 Simulation

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

S^H

with

where

< 0 and

are two parameters. We will simulate the process with

= 0 in the following cases:

• H = 0.7 and θ = − 1, θ = − 10, and θ = − 100, respectively (see, Figure 1, Figure 2, Figure 3, and Table 1, Table 2, Table 3);
• H = 0.5 and θ = − 1, θ = − 10, and θ = − 100, respectively (see, Figure 4, Figure 5, Figure 6, and Table 4, Table 5, Table 6);

Remark 1From 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 author.

Author contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

Funding

This study was funded by the National Natural Science Foundation of China (NSFC), grant no. 11971101.

Conflict of interest

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.

Publisher’s note

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

the self-repelling diffusion, asymptotic distribution, convergence, sub-fractional Brownian motion, stochastic integral

Citation

Gao H, Guo R, Jin Y and Yan L (2022) Large Time Behavior on the Linear Self-Interacting Diffusion Driven by Sub-Fractional Brownian Motion With Hurst Index Large Than 0.5 I: Self-Repelling Case. Front. Phys. 9:795210. doi: 10.3389/fphy.2021.795210

Received

14 October 2021

Accepted

05 November 2021

Published

14 January 2022

Volume

9 - 2021

Edited by

Ming Li, Zhejiang University, China

Reviewed by

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

Zhenxia Liu, Linköping University, Sweden

Updates

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, 1061760802@qq.com

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.

Interdisciplinary Physics

ORIGINAL RESEARCH article

Large Time Behavior on the Linear Self-Interacting Diffusion Driven by Sub-Fractional Brownian Motion With Hurst Index Large Than 0.5 I: Self-Repelling Case

Abstract

1 Introduction

2 Preliminaries

3 Some Basic Estimates

4 Convergence

5 Simulation

Statements

Data availability statement

Author contributions

Funding

Conflict of interest

Publisher’s note

References

Summary

Outline

Figures

Cite article

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ORIGINAL RESEARCH article

Large Time Behavior on the Linear Self-Interacting Diffusion Driven by Sub-Fractional Brownian Motion With Hurst Index Large Than 0.5 I: Self-Repelling Case

Abstract

1 Introduction

2 Preliminaries

3 Some Basic Estimates

4 Convergence

5 Simulation

Statements

Data availability statement

Author contributions

Funding

Conflict of interest

Publisher’s note

References

Summary

Outline

Figures

Cite article

Share article

Article metrics