In a previous study (I) (see [12]), as an extension to classical result, we considered the linear self-interacting diffusion as follows: X t H = S t H − θ ∫ 0 t ∫ 0 s ( X s H − X u H ) d u d s + ν t , t ≥ 0 , (1) with θ ≠ 0, where θ and ν are two real numbers, and S ^H is a sub-fBm with the Hurst parameter 1 2 ≤ H < 1 . 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 X ^H diverges to infinity as t tends to infinity and J 0 H ( t ; θ , ν ) ≔ t e 1 2 θ t 2 X t H → ξ ∞ H − ν θ and J n H ( t ; θ , ν ) ≔ θ t 2 J n − 1 H ( t ; θ , ν ) − ( 2 n − 3 ) ‼ ξ ∞ H − ν θ → ( 2 n − 1 ) ‼ ξ ∞ H − ν θ in L ² and almost surely, for all n = 1, 2, …, where ( − 1)!! = 1 and ξ ∞ H = ∫ 0 ∞ s e 1 2 θ s 2 d S s H .

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 H = 1 2 , 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 X _t converges in L ² 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: X t = X 0 + B t + ∫ 0 t ∫ 0 s f ( X s − X u ) d u d s , (2) where B is a d-dimensional standard Brownian motion and f is Lipschitz continuity. 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 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 x ∈ R d , x ⋅ f ( x ) ≥ 0 (respectively, ≤ 0 ). 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 1 2 < H < 1 , the random variable

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.

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 S ^H is a rather special class of self-similar Gaussian processes such that S 0 H = 0 and R H ( t , s ) ≔ E S t H S s H = s 2 H + t 2 H − 1 2 ( s + t ) 2 H + | t − s | 2 H (3) 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 [4] in a limit of occupation time fluctuations of a system of independent particles moving in R d 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 S ^H imply that t ↦ S t H 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 a sub-fBm S ^H ∫ 0 t u s d S s H as the limit in probability of a Riemann sum. Clearly, when u is of bounded q _H variation on any finite interval with q _H > 1 and 1 p H + 1 q H > 1 , the integral is well-defined and u t S t H = ∫ 0 t u s d S s H + ∫ 0 t S s H d u s for all t ≥ 0.

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: ⟨ 1 [ 0 , s ] , 1 [ 0 , t ] ⟩ H = R H ( t , s ) for s, t ∈ [0, T]. For every φ ∈ H , we can define the Wiener integral with respect to S ^H , denoted by S H ( φ ) = ∫ 0 T φ ( s ) d S s H as a linear (isometric) mapping from H onto S H by using the limit in probability of a Riemann sum, where S H is the Gaussian Hilbert space generating by S H and ‖ φ ‖ H 2 = E ∫ 0 T φ ( s ) d S s H 2 (4) for any φ ∈ H . In particular, when 1 2 < H < 1 , we can show that ‖ φ ‖ H 2 = ∫ 0 T ∫ 0 T φ ( t ) φ ( s ) ψ H ( t , s ) d s d t , ∀ φ ∈ H , where ψ H ( t , s ) = ∂ 2 ∂ t ∂ s R H ( t , s ) = H ( 2 H − 1 ) | t − s | 2 H − 2 − | t + s | 2 H − 2 for s, t ∈ [0, T]. Thus, when 1 2 < H < 1 if for every T > 0, the integral ∫ 0 T φ ( s ) d S s H exists in L ² and ∫ 0 ∞ ∫ 0 ∞ φ ( t ) φ ( s ) ψ H ( t , s ) d s d t < ∞ , we can define the integral as follows: ∫ 0 ∞ φ ( s ) d S s H and E ∫ 0 ∞ φ ( s ) d S s H 2 = ∫ 0 ∞ ∫ 0 ∞ φ ( t ) φ ( s ) ψ H ( t , s ) d s d t .

Let now D and δ be the (Malliavin) derivative and divergence operators associated with the sub-fBm S ^H . And let D 1,2 denote the Hilbert space with respect to the norm as follows: ‖ F ‖ 1,2 ≔ E | F | 2 + E ‖ D F ‖ H 2 . Then the duality relationship E F δ ( u ) = E ⟨ D F , u ⟩ H (5) holds for any F ∈ D 1,2 and D 1,2 ⊂ D o m ( δ ) . Moreover, for any u ∈ D 1,2 , we have E δ ( u ) 2 = E ‖ u ‖ H 2 + E ⟨ D u , ( D u ) * ⟩ H ⊗ H = E ‖ u ‖ H 2 + E ∫ [ 0 , T ] 4 D ξ u r D η u s ψ H ( η , r ) ψ H ( ξ , s ) d s d r d ξ d η , where (Du)^∗ is the adjoint of Du in the Hilbert space given as follows: H ⊗ H . We denote δ ( u ) = ∫ 0 T u s δ S s H for 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: ∫ 0 T u s d S s H = ∫ 0 T u s δ S s H + ∫ 0 T ∫ 0 T D s u t ψ H ( t , s ) d s d t , provided u has a bounded q variation with 1 ≤ q < 1 H and u ∈ D 1,2 such that ∫ 0 T ∫ 0 T D s u t ψ H ( t , s ) d s d t < ∞ .

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 S ^H is defined by the following stochastic differential equation: X t H = S t H − θ ∫ 0 t ∫ 0 s ( X s H − X u H ) d u d s + ν t , t ≥ 0 (6) with θ > 0. The kernel (t, s)↦h _θ (t, s) is defined as follows: h θ ( t , s ) = 1 − θ s e 1 2 θ s 2 ∫ s t e − 1 2 θ u 2 d u , t ≥ s , 0 , t < s (7) 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: X t H = ∫ 0 t h θ ( t , s ) d S s H + ν ∫ 0 t h θ ( t , s ) d s (8) for t ≥ 0.

The kernel function (t, s)↦h _θ (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

Lemma 3.1

Let 1 2 < H < 1 and θ > 0. Then the random variable X ∞ H = ∫ 0 ∞ h θ ( s ) d S s H + ν ∫ 0 ∞ h θ ( s ) d s exists as an element in L² .

Proof

This is a simple calculus exercise. In fact, we have E ∫ 0 ∞ h θ ( s ) d S s H 2 = ∫ 0 ∞ ∫ 0 ∞ h θ ( s ) h θ ( r ) ψ H ( s , r ) d s d r = 2 H ( 2 H − 1 ) ∫ 0 ∞ ∫ 0 s h θ ( s ) h θ ( r ) ( s − r ) − s | 2 H − 2 − ( r + s ) 2 H − 2 d r d s = 2 H ( 2 H − 1 ) ∫ 0 1 ∫ 0 s h θ ( s ) h θ ( r ) ( s − r ) 2 H − 2 − ( r + s ) 2 H − 2 d r d s + 2 H ( 2 H − 1 ) ∫ 1 ∞ ∫ 0 1 h θ ( s ) h θ ( r ) ( s − r ) 2 H − 2 − ( r + s ) 2 H − 2 d r d s + 2 H ( 2 H − 1 ) ∫ 1 ∞ ∫ 1 s h θ ( s ) h θ ( r ) ( s − r ) 2 H − 2 − ( r + s ) 2 H − 2 d r d s for all θ > 0 and 1 2 < H < 1 . Clearly, Eq. 10 implies that ∫ 0 1 ∫ 0 s h θ ( s ) h θ ( r ) ( s − r ) 2 H − 2 − ( r + s ) 2 H − 2 d r d s ≤ ( C θ ) 2 ∫ 0 1 ∫ 0 s ( s − r ) 2 H − 2 − ( r + s ) 2 H − 2 d r d s = ( C θ ) 2 ∫ 0 1 ∫ 0 1 s 2 H − 1 ( 1 − x ) 2 H − 2 − ( 1 + x ) 2 H − 2 d x d s < ∞ , and ∫ 1 ∞ ∫ 0 1 h θ ( s ) h θ ( r ) ( s − r ) 2 H − 2 − ( r + s ) 2 H − 2 d r d s ≤ ( C θ ) 2 ∫ 1 ∞ ∫ 0 1 s − 2 ( s − r ) 2 H − 2 − ( r + s ) 2 H − 2 d r d s ≤ ( C θ ) 2 ∫ 1 ∞ s − 2 ( s − 1 ) 2 H − 2 − s 2 H − 2 d s < ∞ . and ∫ 1 ∞ ∫ 1 s h θ ( s ) h θ ( r ) ( s − r ) 2 H − 2 − ( r + s ) 2 H − 2 d r d s ≤ ( C θ ) 2 ∫ 1 ∞ ∫ 1 s ( r s ) − 2 ( s − r ) 2 H − 2 − ( r + s ) 2 H − 2 d r d s ≤ ( C θ ) 2 ∫ 1 ∞ ∫ r ∞ ( r s ) − 2 ( s − r ) 2 H − 2 − ( r + s ) 2 H − 2 d r d s = ( C θ ) 2 ∫ 1 ∞ ∫ 1 ∞ r 2 H − 5 x − 2 ( x − 1 ) 2 H − 2 − ( 1 + x ) 2 H − 2 d x d r < ∞ for all θ > 0 and 1 2 < H < 1 . These show that the random variable X ∞ H exists as an element in L ².

Lemma 3.2

Let θ > 0. We then have lim t → ∞ t e 1 2 θ t 2 ∫ 0 t h θ ( t , s ) d s − ∫ 0 ∞ h θ ( s ) d s = − 1 θ . (13)

Proof

This is a simple calculus exercise. In fact, we have ∫ 0 t h θ ( t , s ) d s − ∫ 0 ∞ h θ ( s ) d s = ∫ 0 t h θ ( t , s ) − h θ ( s ) d s − ∫ t ∞ h θ ( s ) d s = ∫ 0 t θ s e 1 2 θ s 2 ∫ s ∞ e − 1 2 θ u 2 d u − ∫ s t e − 1 2 θ u 2 d u d s − ∫ t ∞ h θ ( s ) d s = e 1 2 θ t 2 − 1 ∫ t ∞ e − 1 2 θ u 2 d u − ∫ t ∞ h θ ( s ) d s . for all t ≥ 0 and θ > 0. Noting that lim t → ∞ t e 1 2 θ t 2 − 1 ∫ t ∞ e − 1 2 θ u 2 d u = lim t → ∞ 1 t − 1 e − 1 2 θ t 2 ∫ t ∞ e − 1 2 θ u 2 d u = 1 θ and lim t → ∞ t ∫ t ∞ h θ ( s ) d s = lim t → ∞ 1 t − 1 ∫ t ∞ h θ ( s ) d s = lim t → ∞ t 2 h θ ( t ) = lim t → ∞ t 2 1 − θ t e 1 2 θ t 2 ∫ t ∞ e − 1 2 θ u 2 d u = 1 θ , (14) we see that lim t → ∞ t e 1 2 θ t 2 ∫ 0 t h θ ( t , s ) d s − ∫ 0 ∞ h θ ( s ) d s = lim t → ∞ 1 t − 1 e − 1 2 θ t 2 e 1 2 θ t 2 − 1 ∫ t ∞ e − 1 2 θ u 2 d u − ∫ t ∞ h θ ( s ) d s = − 1 θ . by L’Hopital’s rule.

Lemma 3.3

Let θ > 0. We then have d d t h θ ( t ) ≤ C θ min 1 , 1 t 3 (15) for all t ≥ 0.

Lemma 3.4

Let θ > 0 and 1 2 < H < 1 . We then have lim t → ∞ 1 t 2 − 2 H e − θ t 2 ∫ 0 t ∫ 0 s s r e 1 2 θ ( s 2 + r 2 ) ψ H ( s , r ) d s d r = 1 4 θ − 2 H Γ ( 2 H + 1 ) . (16)

Proof

By L’Hopital’s rule and the change of variable 1 2 θ ( t 2 − r 2 ) = x , it follows that lim t → ∞ 1 t 2 − 2 H e θ t 2 ∫ 0 t ∫ 0 s s r e 1 2 θ ( s 2 + r 2 ) ψ H ( s , r ) d s d r = lim t → ∞ 1 2 θ t 2 − 2 H e 1 2 θ t 2 ∫ 0 t e 1 2 θ r 2 ψ H ( t , r ) r d r = lim t → ∞ H ( 2 H − 1 ) 2 θ t 2 − 2 H ∫ 0 t e − 1 2 θ ( t 2 − r 2 ) ( t − r ) 2 H − 2 − ( t + r ) 2 H − 2 r d r = lim t → ∞ H ( 2 H − 1 ) 2 θ 2 t 2 − 2 H ∫ 0 1 2 θ t 2 e − x t − t 2 − 2 x θ 2 H − 2 d x = lim t → ∞ H ( 2 H − 1 ) 2 θ 2 t 2 − 2 H ∫ 0 1 2 θ t 2 e − x 2 x θ 2 H − 2 t + t 2 − 2 x θ 2 − 2 H d x = 1 2 θ − 2 H H ( 2 H − 1 ) Γ ( 2 H − 1 ) = 1 4 θ − 2 H Γ ( 2 H + 1 ) , where we have used the equation lim t → ∞ 1 t 2 − 2 H e 1 2 θ t 2 ∫ 0 t e 1 2 θ r 2 ( t + r ) 2 H − 2 r d r = 0 . This completes the proof.

Lemma 3.5

Let θ > 0 and 1 2 < H < 1 . We then have c ( t − s ) 2 H ≤ E X t H − X s H 2 ≤ C ( t − s ) 2 H (17) for all 0 ≤ s < t ≤ T, where C and c are two positive constants depending only on H, θ, ν and T .

Proof

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

Lemma 3.6

Let θ > 0 and 1 2 ≤ H < 1 . Then the convergence ∫ t ∞ h θ ( s ) d S s H → 0 (18) holds in L ² and almost surely as t tends to infinity.

Proof

Convergence (18) in L ² follows from Lemma (3.1). In fact, by Eq. 10, we have E ∫ t ∞ h θ ( s ) d S s H 2 ≤ ∫ t ∞ ∫ t ∞ | h θ ( s ) ‖ h θ ( r ) ‖ ψ ( s , r ) | d s d r ≤ C ∫ t ∞ ∫ t ∞ min 1 , 1 s 2 min 1 , 1 r 2 | ψ ( s , r ) | d s d r = C H ( 2 H − 1 ) ∫ t ∞ ∫ t ∞ | s − r | 2 H − 2 − | s + r | 2 H − 2 d s d r ( s r ) 2 → 0 , as t tends to infinity.

On the other hand, by Lemma (3.5), 3.3 and the equation S t H t → 0 almost surely as t tends to infinity, we find that ∫ t ∞ S s H d h θ ( s ) ≤ C θ ∫ t ∞ | S s H | d s s 3 → 0 , as t tends to infinity. It follows from the integration by parts that ∫ t ∞ h θ ( s ) d S s H = − h θ ( t ) S t H − ∫ t ∞ S s H d h θ ( s ) → 0 almost surely as t tends to infinity.

In this section, we consider the long time behaviors for X ^H with 1 2 < H < 1 and θ > and our objects are to prove the statements given in Section 1.

Theorem 4.1

Let θ > 0 and 1 2 ≤ H < 1 . Then the convergence X t H → a . s X ∞ H (19) holds in L ² and almost surely as t tends to infinity.

Proof

When H = 1 2 , the convergence is obtained in Cranston-Le Jan [8]. Consider the decomposition X t H − X ∞ H = ∫ 0 t [ h θ ( t , s ) − h θ ( s ) ] d S s H + ∫ t ∞ h θ ( s ) d S s H + ν ∫ 0 t h θ ( t , s ) d s − ∫ 0 ∞ h θ ( s ) d s ≡ ϒ t H + ∫ t ∞ h θ ( s ) d S s H + ν Δ t H ( θ ) (20) for all t ≥ 0.

We first check that Eq. 19 holds in L ². By Lemma 3.6 and Lemma 3.2, we only need to prove ϒ t H converges to zero in L ². It follows from the equation ∫ t ∞ e − 1 2 θ u 2 d u ∼ 1 θ t e − 1 2 θ t 2 for all θ > 0 as t tends to infinity and Lemma 3.4 that E ϒ t H 2 = ∫ 0 t ∫ 0 t | h θ ( t , s ) − h θ ( s ) ‖ h θ ( t , r ) − h θ ( r ) | ψ H ( s , r ) d s d r = ∫ t ∞ e − 1 2 θ u 2 d u 2 ∫ 0 t ∫ 0 t θ 2 s r e θ 2 ( s 2 + r 2 ) ψ H ( s , r ) d s d r ∼ 1 t 2 e − θ t 2 ∫ 0 t ∫ 0 s s r e θ 2 ( s 2 + r 2 ) ψ H ( s , r ) d s d r = H ( 2 H − 1 ) t 2 e − θ t 2 ∫ 0 t ∫ 0 s s r e θ 2 ( s 2 + r 2 ) | s − r | 2 H − 2 − | s + r | 2 H − 2 d s d r → 0 for all θ > 0 and 1 2 < H < 1 as t tends to infinity, which implies that Eq. 19 holds in L ².

We now check that Eq. 19 holds almost surely as t tends to infinity. By Lemma 3.6, we only need check that ϒ t H converges to zero almost surely as t tends to infinity. We have ϒ t H = ∫ 0 t [ h θ ( t , s ) − h θ ( s ) ] d S s H = ∫ t ∞ e − 1 2 θ u 2 d u ∫ 0 t θ s e 1 2 θ s 2 d S s H ∼ 1 t e − 1 2 θ t 2 ∫ 0 t s e 1 2 θ s 2 d S s H for all θ > 0 and 1 2 < H < 1 as t tends to infinity. To obtain the convergence, we define the random sequence Z n , k = ϒ n + k n H , k = 0,1,2 , … , n for every integer n ≥ 1. Then {Z _n,k , k = 0, 1, 2, …, n} is Gaussian for every integer n ≥ 1. It follows from Lemma 3.4 that σ 2 ( n ) ≔ E ( Z n , k ) 2 ∼ 1 n + k n 2 e − θ n + k n 2 E ∫ 0 n + k n s e 1 2 θ s 2 d S s H 2 ≤ 1 n + k n 2 e − θ n + k n 2 ∫ 0 n + k n ∫ 0 n + k n s r e 1 2 θ ( s 2 + r 2 ) | ψ H ( s , r ) | d s d r ∼ C n 2 H for every integer n ≥ 1 and 0 ≤ k ≤ n, which implies that P ( | Z n , k | > ε ) = ∫ ε ∞ 1 2 π σ ( n ) e − x 2 2 σ 2 ( n ) d x ≤ 1 ε ∫ ε ∞ x 2 π σ ( n ) e − x 2 2 σ 2 ( n ) d x = σ ( n ) ε ∫ ε / σ ( n ) ∞ y 2 π e − y 2 2 d y ≤ σ ( n ) ε e − ε 2 4 σ 2 ( n ) ∫ ε / σ ( n ) ∞ y 2 π e − y 2 4 d y ≤ C ε n H exp − C 1 ε 2 n 2 H for any ɛ > 0, every integer n ≥ 1 and 0 ≤ k ≤ n.

On the other hand, for every s ∈ (0, 1), we denote R s n , k = ϒ n + k + s n H − ϒ n + k n H .

Then { R s n , k , 0 ≤ s ≤ 1 } also is Gaussian for every integer n ≥ 1 and 0 ≤ k ≤ n. It follows that E ( R s n , k − R s ′ n , k ) 2 ≤ C n 2 H E ( S s H − S s ′ H ) 2 for all s, s′ ∈ [0, 1]. Thus, for any ɛ > 0, by Slepian’s theorem and Markov’s inequality, one can get P sup 0 ≤ s ≤ 1 | R s n , k | > ε ≤ P C n H sup 0 ≤ s ≤ 1 | S s H | > ε ≤ C ε 6 n 6 H E sup 0 ≤ s ≤ 1 | S s H | 6 ≤ C ε 6 n 6 H for every integer n ≥ 1 and 0 ≤ k ≤ n. Combining this with the Borel–Cantelli lemma and the relationship { sup n + k n < t < n + k + 1 n | ϒ t H | > ε } ⊆ { | Z n , k | > ε / 2 } ∪ sup 0 ≤ s ≤ 1 | R s n , k | > ε / 2 , we show that ϒ t H → 0 almost surely as t tends to infinity. This completes the proof.