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

Let S H be a sub-fractional Brownian motion with index 1 2 < H < 1 . In this paper, we consider the linear self-interacting diffusion driven by S H , which is the solution to the equation d X t H = d S t H − θ ( ∫ 0 t X t H − X s H d s ) d t + ν d t , X 0 H = 0 , where θ < 0 and ν ∈ R 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.


INTRODUCTION
In 1995, Cranston and Le Jan [1] introduced a linear self-attracting diffusion with θ > 0 and X 0 0, where B is a 1-dimensional standard Brownian motion. They showed that the process X t converges in L 2 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 follows where 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 L X (t, x) be the local time of the solution process X. Then, we have for 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. If for all x ∈ R d , 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 X H 0 0, where B H is a fractional Brownian motion (fBm, in short) with Hurst parameter 1 2 ≤ H < 1. 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 L 2 and almost surely, to the random variable where the function is defined ar follows Recently, Sun and Yan [14] considered the related parameter estimations with θ > 0 and 1 2 ≤ H < 1, and Gan and Yan [15] considered the parameter estimations with θ < 0 and 1 2 ≤ H < 1.
Motivated by these results, as a natural extension one can consider the following stochastic differential equation: with θ > 0 and X 0 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 S H {S H t , t ≥ 0} with S H 0 0 and the covariance for 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][18][19][20], Li [21][22][23][24], Shen and Yan [25,26], Sun and Yan [27], Tudor [28][29][30][31], Ciprian A. Tudor [32] Yan et al [33][34][35] and the references therein.
In this present paper, we consider the linear self-interacting diffusion with θ < 0 and X H 0 0, where S H is a sub-fBm with Hurst parameter 1 2 ≤ H < 1. Our main aim is to show that the solution of (1.7) diverges to infinity and obtain the speed diverging to infinity, as t tends to infinity. The object of this paper is to expound and prove the following statements: (I) For θ < 0 and 1 2 < H < 1, the random variable (II) For θ < 0 and 1 2 < H < 1, as t → ∞, we have in L 2 and almost surely.
(III) For θ < 0 and for all t ≥ 0, where (−1)!! 1. We then have holds in L 2 and almost surely for every n ≥ 1, as t → ∞. 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.

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 S H S H t , t ≥ 0 denotes a sub-fBm defined on the probability space (Ω, F , P) 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 S H for 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 R d 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 t1S H t admits almost surely a bounded 1 H−ϑ -variation on any finite interval for any sufficiently small ϑ ∈ (0, H). That is, the paths of t1S H t admits a bounded p H -variation on any finite interval with p H > 1 H . 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,b for all t ≥ 0, provided u is of bounded q H -variation on any finite interval with q H > 1 and 1 p H + 1 q H > 1 (see, for examples, Bertoin [37] and FöIllmer [38]).
Let H be the completion of the linear space E generated by the indicator functions 1 [0,t] , t ∈ [0, T] with respect to the inner product and it can be continuously extended to H and we call the mapping Φ is called the Wiener integral with respect to S H , denoted by for any φ ∈ H. For simplicity, in this paper we assume that 1 Denote by S the set of smooth functionals of the form The Malliavin derivative D of a functional F as above is given by Frontiers in Physics | www.frontiersin.org January 2022 | Volume 9 | Article 795210 3 Gao et al.
Self-Interacting Diffusion Driven by SubfBm I The derivative operator D is then a closable operator from L 2 (Ω) into L 2 (Ω; H). We denote by D 1,2 the closure of S with respect to the norm The divergence integral δ is the adjoint of derivative operator D H . That is, we say that a random variable u in L 2 (Ω; H) belongs to the domain of the divergence operator δ, denoted by Dom(δ S ), if for every F ∈ D 1,2 , where c is a constant depending only on u. In this case δ(u) is defined by the duality relationship    for any F ∈ D 1,2 . We have D 1,2 ⊂ Dom(δ) and for any u ∈ D 1,2 where (D S u) p is the adjoint of Du in the Hilbert space H ⊗ H. We will denote δ u ( ) T 0 u s δS H s for 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 follows provided u has a bounded q-variation with 1 ≤ q < 1 H and u ∈ D 1,2 (H) such that Theorem 2.1. (Alós et al [16]). Let 0 < H < 1 and let f ∈ C 2 (R) such that

SOME BASIC ESTIMATES
Throughout this paper we assume that θ < 0 and 1 2 < H < 1. Recall that the linear self-interacting diffusion with sub-fBm S H defined by the stochastic differential equation with θ < 0. Define the kernel (t, s)1h θ (t, s) as follows for 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 (t, s)1h θ (t, s) with θ < 0 admits the following properties (these properties are proved partly in Sun and Yan [12]): • For all s ≥ 0, the limit lim t→∞ te 1 2 θt 2 h θ t, s ( ) se for all s ≥ 0.  • For all t ≥ s ≥ 0, we have • For all t ≥ s, r ≥ 0, we have We then have lim t→∞ t 2 I θ (t) − 1 θ and Proof. This is simple calculus exercise.
Proof. It follows from L'Hôspital's rule that for all θ < 0 and 1 2 < H < 1. By making the change of variable for all θ < 0 and 1 2 < H < 1. This completes the proof.
Lemma 3.5. Let θ < 0 and 0 ≤ s < t ≤ T. We then have Proof. Given 0 ≤ s < t ≤ T and denotê Frontiers in Physics | www.frontiersin.org January 2022 | Volume 9 | Article 795210 It follows that (3.9) Now, we estimate the three terms. For the first term, we have for all θ < 0 and 0 < s < t ≤ T. For the second term, we have for all θ < 0 and 0 < s < t ≤ T. It follows that for all θ < 0 and 0 < s < t ≤ T, which implies that for 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.

CONVERGENCE
In this section, we obtain the large time behaviors associated with the solution X H to Eq. Proof. This is simple calculus exercise. In fact, we have for all θ < 0 and 1 2 < H < 1, which shows that the random variable ξ H ∞ exists as an element in L 2 . Now, we show that the process ξ a,b is Hölder continuous. For all 0 < s < t by the inequality e −x 2 x ≤ C for all x ≥ 0, we have x − y Thus, the normality of ξ H implies that for all 0 ≤ s < t, 1 2 < H < 1 and integer numbers n ≥ 1, and the Hölder continuity follows.
Nextly, we check the ξ a,b t converges to ξ H ∞ in L 2 . This follows from the next estimate: yψ H x,y dy in L 2 and almost surely for every c ≥ 0, as t tends to infinity.

DATA AVAILABILITY STATEMENT
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