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 Han Gao, et al.

ByHan Gao

Jan 14, 2022 1 Introduction

In 1995, Cranston and Le Jan  introduced a linear self-attracting diffusion

$Xt=Bt−θ∫0t∫0sXs−Xududs+νt,t≥0$

with θ > 0 and X0 = 0, where B is a 1-dimensional standard Brownian motion. They showed that the process Xt 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  introduced in 1992 as a model for the shape of a growing polymer (Brownian polymer) as follows

$Xt=X0+Bt+∫0t∫0sfXs−Xududs,$

where B is a d-dimensional standard Brownian motion and f is Lipschitz continuous. 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 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, Xt is a continuous analogue of a process introduced by Diaconis and studied by Pemantle . Let

$LX(t,x)$

be the local time of the solution process X. Then, we have

$Xt=X0+Bt+∫0tds∫Rf−xLXs,Xs+xdx$

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

$x∈Rd$

, we call it self-repelling (resp. self-attracting). In 2002, Benaïm et al  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. , Cranston and Mountford , Gauthier , Herrmann and Roynette , Herrmann and Scheutzow , Mountford and Tarr , Shen et al , Sun and Yan  and the references therein.

On the other hand, starting from the application of fractional Brownian motion in polymer modeling, Yan et al  considered an analogue of the linear self-interacting diffusion:

$XtH=BtH−θ∫0t∫0sXsH−XuHduds+νt,t≥0$

with θ ≠ 0 and

$X0H=0$

, where BH is a fractional Brownian motion (fBm, in short) with Hurst parameter

$12≤H<1$

. The solution of (1.3) is a Gaussian process. When θ > 0, Yan et al  showed that the solution XH of (1.3) converges in L2 and almost surely, to the random variable

$X∞H=∫0∞hθsdBsH+ν∫0∞hθsds$

where the function is defined ar follows

$hθs=1−θse12θs2∫s∞e−12θu2du,s≥0$

with θ > 0. Recently, Sun and Yan  considered the related parameter estimations with θ > 0 and

$12≤H<1$

, and Gan and Yan  considered the parameter estimations with θ < 0 and

$12≤H<1$

.

Motivated by these results, as a natural extension one can consider the following stochastic differential equation:

$Xt=Gt−θ∫0t∫0sXs−Xududs+νt,t≥0$

with θ > 0 and X0 = 0, where G = {Gt, 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

$SH={StH,t≥0}$

with

$S0H=0$

and the covariance

$RHt,s≡EStHSsH=s2H+t2H−12s+t2H+|t−s|2H$

for 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 (see, for example, Alós et al ). The sub-fBm has properties analogous to those of fBm and satisfies the following estimates:

$2−22H−1∧1t−s2H≤EStH−SsH2≤2−22H−1∨1t−s2H.$

More works for sub-fBm and related processes can be found in Bojdecki et al. , Li , Shen and Yan [25, 26], Sun and Yan , Tudor , Ciprian A. Tudor  Yan et al  and the references therein.

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

$XtH=StH−θ∫0t∫0sXsH−XuHduds+νt,t≥0$

with θ < 0 and

$X0H=0$

, where SH is a sub-fBm with Hurst parameter

$12≤H<1$

. 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

$\frac{1}{2}

, the random variable

$ξ∞H=∫0∞se12θs2dSsH$

exists as an element in L2.

(II) For θ < 0 and

$\frac{1}{2}

, as t, we have

$J0Ht;θ,ν≔te12θt2XtH→ξ∞H−νθ$

in L2 and almost surely.

(III) For θ < 0 and

$\frac{1}{2}

, define the processes

${J}^{H}\left(n,\theta ,\nu \right)=\left\{{J}_{t}^{H}\left(n,\theta ,\nu \right),t\ge 0\right\},n\ge 1$

by

$JnHt;θ,ν≔θt2Jn−1Ht;θ,ν−2n−3‼ξ∞H−νθ,n=1,2,…,$

for all t ≥ 0, where (−1)!! = 1. We then have

$JnHt;θ,ν→2n−1‼ξ∞H−νθ$

holds in L2 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.

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 , Nualart , and Tudor  for a complete description of stochastic calculus with respect to Gaussian processes. Throughout this paper we assume that

$SH=StH,t≥0$

denotes a sub-fBm defined on the probability space

$(Ω,F,P)$

with index H. As we pointed out before, the sub-fBm SH is a rather special class of self-similar Gaussian processes such that

$S0H=0$

,

$E[StH]=0$

and

$RHt,s≔EStHSsH=s2H+t2H−12s+t2H+|t−s|2H$

for 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  in a limit of occupation time fluctuations of a system of independent particles moving in

$Rd$

according a symmetric α-stable Lévy process, and it also appears in Bojdecki et al  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

$t↦StH$

$1H−ϑ$

-variation on any finite interval for any sufficiently small ϑ ∈ (0, H). That is, the paths of

$t↦StH$

admits a bounded pH-variation on any finite interval with

$pH>1H$

. As an immediate result, one can define the Young integral of a process u = {ut, t ≥ 0} with respect to sub-fBm Ba,b

as the limit in probability of a Riemann sum. Clearly, the integral is well-defined and

$utStH=∫0tusdSsH+∫0tSsHdus$

for all t ≥ 0, provided u is of bounded qH-variation on any finite interval with qH > 1 and

$1pH+1qH>1$

(see, for examples, Bertoin  and FöIllmer ).

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

for s, t ∈ [0, T]. When

$12

, we can show that

$‖φ‖H2=∫0T∫0Tφtφs∂2∂t∂sRHt,sdsdt=∫0T∫0TφtφsψHt,sdsdt,∀φ∈H,$

where

$ψHt,s=∂2∂t∂sRa,bt,s=H2H−1|t−s|2H−2−|t+s|2H−2$

for s, t ∈ [0, T]. Define the linear mapping

$E∋φ↦SH(φ)$

by

$10,t↦SH10,t=∫0T10,tsdSsH≡StH$

for all t ∈ [0, T] and it can be continuously extended to

$H$

and we call the mapping Φ is called the Wiener integral with respect to SH, denoted by

and

$‖φ‖H2=E∫0TφsdSsH2$

for any

$φ∈H$

.

For simplicity, in this paper we assume that

$12

. Thus, if for every T > 0, the integral

exists in L2 and

$∫0∞∫0∞φtφsψHt,sdsdt<∞,$

we can define the integral

and

$E∫0∞φsdSsH2=∫0∞∫0∞φtφsψHt,sdsdt.$

Denote by

$S$

the set of smooth functionals of the form

$F=fSHφ1,SHφ2,…,SHφn,$

where

$f∈Cb∞(Rn)$

and

$φi∈H$

. The Malliavin derivative D of a functional F as above is given by

$DF=∑j=1n∂f∂xjSHφ1,SHφ2,…,SHφnφj.$

The derivative operator D is then a closable operator from L2(Ω) into

$L2(Ω;H)$

. We denote by

$D1,2$

the closure of

$S$

with respect to the norm

$‖F‖1,2≔E|F|2+E‖DF‖H2.$

The divergence integral δ is the adjoint of derivative operator DH. That is, we say that a random variable u in

$L2(Ω;H)$

belongs to the domain of the divergence operator δ, denoted by Dom(δS), if

$E⟨DF,u⟩H≤c‖F‖L2Ω$

for every

$F∈D1,2$

, where c is a constant depending only on u. In this case δ(u) is defined by the duality relationship

for any

$F∈D1,2$

. We have

$D1,2⊂Dom(δ)$

and for any

$u∈D1,2$

$Eδu2=E‖u‖H2+E⟨Du,Du∗⟩H⊗H=E‖u‖H2+E∫0,T4DξurDηusϕHη,rϕHξ,sdsdrdξdη,$

where

$(DSu)∗$

is the adjoint of Du in the Hilbert space

$H⊗H$

. We will denote

for an adapted process u, and it is called Skorohod integral. Alós et al , we can obtain the relationship between the Skorohod and Young integral as follows

$∫0TusdSsH=∫0TusδSsH+∫0T∫0TDsutψt,sdsdt,$

provided u has a bounded q-variation with

$1≤q<1H$

and

$u∈D1,2(H)$

such that

$∫0T∫0TDsutψt,sdsdt<∞.$

Theorem 2.1 (Alós et al ). Let 0 < H < 1 and let

$f∈C2(R)$

such that

$max|fx|,|f′x|,|f″x|≤κeβx2,$

where κ and β are two positive constants with

$β<14T−2H$

. Then we have

$fStH=f0+∫0tf′SsHdSsH+H2−22H−1∫0tf″SsHs2H−1ds$

for all t ∈ [0, T].

3 Some Basic Estimates

Throughout this paper we assume that θ < 0 and

$12

. Recall that the linear self-interacting diffusion with sub-fBm SH defined by the stochastic differential equation

$XtH=StH−θ∫0t∫0sXsH−XuHduds+νt,t≥0$

with θ < 0. Define the kernel (t, s)↦hθ(t, s) as follows

$hθt,s=1−θse12θs2∫ste−12θu2du,t≥s,0,t

for s, t ≥ 0. By the variation of constants method (see, Cranston and Le Jan ) or Itô’s formula we may introduce the following representation:

$XtH=∫0thθt,sdSsH+ν∫0thθt,sds$

for t ≥ 0.

The kernel function (t, s)↦hθ(t, s) with θ < 0 admits the following properties (these properties are proved partly in Sun and Yan ):

• For all s ≥ 0, the limit

$limt→∞te12θt2hθt,s=se12θs2$

for all s ≥ 0.

• For all ts ≥ 0, we have

$1≤hθt,s≤e−12θt2−s2.$

• For all ts, r ≥ 0, we have

$hθt,0=hθt,t=1,∫sthθt,udu=e12θs2∫ste−12θu2du.$

Lemma 3.1 Let θ < 0 and define function

$Iθt=−θte12θt2∫0te−12θu2du−1.$

We then have

$limt→∞t2Iθ(t)=−1θ$

and

$limt→∞t21+θte−12θt2∫t∞e12θu2du=−1θ$

Proof This is simple calculus exercise.

Lemma 3.2 (Sun and Yan ). Let θ < 0 and define the functions tIθ(t, n), n = 1, 2, … as follows

$Iθt,1=−θt2Iθt,Iθt,n+1=−θt2Iθt,n−2n−1‼.$

Then we have

$limt→∞Iθt,n=2n−1‼.$

for every n ≥ 0, where (−1)! = 1.

Lemma 3.3 Let θ < 0. Then the integral

$ΔH=∫0∞∫0∞xye12θx2+y2ψHx,ydxdy$

converges and as t,

$limt→∞t2e−θt2EXtH2=ΔH.$

Proof An elementary may show that (3.6) converges for all θ < 0. It follows from L’Hôspital’s rule that

$limt→∞t2eθt2EXtH2=limt→∞t2eθt2∫0t∫0thθt,xhθt,yψHx,ydxdy=limt→∞θ2t−2e−θt2∫0tdx∫0txye12θx2+y2ψHx,ydy∫xtdu∫yte−12θu2+v2dv=2limt→∞θ2t−2e−θt2∫0tdu∫0udx∫0udv∫0vdyxye12θx2+y2−u2−v2ψHx,y=limt→∞−θt−1e−12θt2∫0tdx∫0te−12θv2dv∫0vxye12θx2+y2ψHx,ydy=limt→∞−θt−1e−12θt2∫0te−12θv2dv∫0tdx∫0vxye12θx2+y2ψHx,ydy=limt→∞−θt−1e−12θt2∫0te−12θv2dv∫0vdx∫0vxye12θx2+y2ψHx,ydy=∫0∞dx∫0∞xye12θx2+y2ψHx,ydy,$

where we have used the following fact:

$limt→∞1t−1e−12θt2∫0te−12θv2dv∫vtdx∫0vxye12θx2+y2ψHx,ydy=limt→∞1t−1e−12θt2∫0tdx∫0xe−12θv2dv∫0vxye12θx2+y2ψHx,ydy=0.$

This completes the proof.

Lemma 3.4 Let θ < 0. Then, convergence

$limt→∞1t2−2He−θt2∫t∞∫s∞sre12θs2+r2ψHs,rdsdr=14−θ−2HΓ2H+1.$

holds.

Proof It follows from L’Hôspital’s rule that

$limt→∞1t2−2Heθt2∫t∞ue12θu2∫u∞ve12θv2ψHu,vdvdu=−12θlimt→∞1t2−2He12θt2∫t∞ve12θv2ψHt,vdv=−limt→∞H2H−12θt2−2H∫t∞ve12θv2−t2v−t2H−2−v+t2H−2dv$

for all θ < 0 and

$12

. By making the change of variable

$12θ(v2−t2)=x$

, we see that

$limt→∞12θt2−2H∫t∞ve12θv2−t2v−t2H−2−v+t2H−2dv=limt→∞12θ2t2−2H∫0∞e−x{t2+2x−θ−t2H−2−t2+x+t2H−2}dx=limt→∞12θ2t2−2H∫0∞e−x2x−θ2H−2t2+2x−θ+t2−2Hdx−limt→∞12θ2t2−2H∫0∞e−xt2+x+t2H−2dx=12−θ−2H−1Γ2H−1$

for all θ < 0 and

$12

. This completes the proof.

Lemma 3.5 Let θ < 0 and 0 ≤ s < tT. We then have

$ct−s2H≤EXtH−XsH2≤Ct−s2H$

Proof Given 0 ≤ s < tT and denote

$X̂tH=∫0thθt,rdSrH,t≥0.$

It follows that

$EX̂tH−X̂sH2=E∫0shθt,x−hθs,xdSxH2+E∫sthθt,xdSxH2+2E∫sthθt,ydSyH∫0shθt,x−hθs,xdSxH.$

Now, we estimate the three terms. For the first term, we have

$0≤E∫0shθt,x−hθs,xdSxH2=∫0s∫0shθt,x−hθs,xhθt,y−hθs,yψHx,ydxdy=θ2∫ste−12θu2du2∫0s∫0sxye12θx2+y2ψHx,ydxdy≤θ2s2t−s2e−θt2∫0s∫0sψHx,ydxdy=θ2s2t−s2e−θt2ESsH2≤CH,Tt−s2$

for all θ < 0 and 0 < s < tT. For the second term, we have

$E∫sthθt,xdSxH2=∫st∫sthθt,xhθt,yψHx,ydxdy≤e−θt2∫st∫stxye12θx2+y2ψHx,ydxdy≤t2e−θt2∫st∫stψHx,ydxdy≤CH,Tt−s2H.$

for all θ < 0 and 0 < s < tT. Similarly, for the third term, we also prove

$0≤E∫sthθt,ydSyH∫0shθt,x−hθs,xdSxH=∫st∫0shθt,yhθt,x−hθs,xψHx,ydxdy≤θ2e−12θt2∫ste−12θu2du∫stye12θy2dy∫0sxe12θx2ψHx,ydx≤θ2e−θt2t−s∫stye12θy2dy∫0sxe12θx2ψHx,ydx≤CH,Tt−s2$

for all θ < 0 and 0 < s < tT. Thus, we have obtained the following estimate:

$EX̂tH−X̂sH2≤CH,T|t−s|2H$

for all θ < 0 and 0 < s < tT.On the other hand, elementary calculations may show that

$∫0shθt,r−hθs,rdr=θ∫ste−12θu2du∫0sre12θr2dr≤CH,Tt−s$

and

$∫sthθt,rdr=e−12θs2∫ste12θr2dr≤CH,Tt−s$

for all θ < 0 and 0 < s < tT. It follows that

$∫0thθt,rdr−∫0shθs,rdr2=∫0shθt,r−hθs,rdr2+∫sthθt,rdr2+2∫sthθt,rdr∫0shθt,r−hθs,rdr≤CH,Tt−s2$

for all θ < 0 and 0 < s < tT, which implies that

$EXta,b−Xsa,b2=EX̂ta,b−X̂sa,b2+ν2∫0thθt,rdr−∫0shθs,rdr2≤CH,Tt−s2H$

for all θ < 0 and 0 < s < tT. Noting that the above calculations are invertible for all θ < 0 and 0 < s < tT, 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 XH to Eq. 3.1. From Lemma 3.5 and Guassianness, we find that the self-repelling diffusion

${XtH,t≥0}$

is H-Hölder continuous. So, the integral

exists with t ≥ 0 as a Young integral and

$tXtH=∫0tsdXsH+∫0tXsHds$

for all t ≥ 0. Define the process Y = {Yt, t ≥ 0} by

$Yt:=∫0tXtH−XsHds=tXtH−∫0tXsHds=∫0tsdXsH=∫0tsdSsH−∫0tθsYsds+12νt2.$

By the variation of constants method, one can prove

$Yt=e−12θt2∫0tse12θs2dSsH−νθe−12θt2−1$

for all t ≥ 0. Define Gaussian process

$ξH={ξtH,t≥0}$

as follows

$ξtH≔∫0tse12θs2dSsH,t≥0.$

Lemma 4.1 Let θ < 0 and

$12

. Then, the random variable

$ξ∞H≔∫0∞se12θs2dSsH$

exists as an element in L2. Moreover, ξH is H-Hölder continuous and

$ξtH→ξ∞H$

in L2 and almost surely, as t tends to infinity.

Proof This is simple calculus exercise. In fact, we have

$E∫0∞xe12θx2dSxH2=∫0∞∫0∞xye12θx2+y2ψHx,ydxdy=2∫0∞xe12θx2dx∫0xye12θy2ψHx,ydy=2H2H−1∫0∞xe12θx2dx∫0xye12θy2x−y2H−2−x+y2H−2dy≤2H2H−1∫0∞xe12θx2dx∫0xx−y2H−2−x+y2H−2ydy=2H2H−1CH∫0∞x2H+1e12θx2dx=Cθ,HΓ2H+2$

for all θ < 0 and

$12

, which shows that the random variable

$ξ∞H$

exists as an element in L2.

Now, we show that the process ξa,b is Hölder continuous. For all 0 < s < t by the inequality

$e−x2x≤C$

for all x ≥ 0, we have

$EξtH−ξsH2=E∫stxe12θx2dSxH2=∫st∫stxye12θx2+y2ψHx,ydxdy=2∫stxe12θx2dx∫sxye12θy2ψHx,ydy=2H2H−1∫stxe12θx2dx∫sxye12θy2x−y2H−2−x+y2H−2dy≤2HCθ2H−1∫stdx∫sxx−y2H−2dy=Cθ,Ht−s2H.$

Thus, the normality of ξH implies that

$EξtH−ξsH2n≤Cθ,H,nt−s2nH$

for all 0 ≤ s < t,

$12

and integer numbers n ≥ 1, and the Hölder continuity follows.

Nextly, we check the

$ξta,b$

converges to

$ξ∞H$

in L2. This follows from the next estimate:

$EξtH−ξ∞H2=∫t∞∫t∞xye12θx2+y2ψHx,ydxdy=2∫t∞∫txxye12θx2+y2ψHx,ydxdy≤2e12θt2∫t∞xe12θx2dx∫txyψHx,ydy≤2e12θt2∫t∞xe12θx2dx∫0xyψHx,ydy≤2H2H−1e12θt2⋅∫t∞xe12θx2dx∫0xyx−y2H−2−x+y2H−2dy≤2H2H−1e12θt2⋅∫t∞xe12θx2dx∫0xyx−y2H−2dy=2H2H−1∫01u1−u2H−2due12θt2∫t∞x2H+1e12θx2dx→0,$

as t tends to infinity.

Finally, we check the

$ξta,b$

converges to

$ξ∞H$

almost surely. By integration by parts we see that

$ξtH−ξ∞H=∫t∞se12θs2dSsH=−te12θt2StH−∫t∞1+θs2e12θs2SsHds$

for all t ≥ 0. Elementary may check that the convergence

$ηtH≔∫t∞1+θs2e12θs2SsHds→a.s0$

holds almost surely, as t tends to infinity. In fact, by inequality

$∫t∞sαe12θs2ds≤Ctα−1e12θt2,α>−1,$

with t ≥ 0, we may show that

$Esupn≤t

for all integer numbers n ≥ 1, and hence

$∑n=0∞Psupn≤t

Thus, Borel-Cantelli’s lemma implies that

$ηtH$

converges to zero almost surely as t tends to infinity, and the lemma follows from (4.2).

Corollary 4.1 For all γ > 0, we have

$tγξtH−ξ∞H=tγ∫t∞se12θs2dSsH→0,$

in L2 and almost surely, as t tends to infinity.

Lemma 4.2 Let θ < 0 and

$12

. Then, we have

$Λγt,θ≔tγ+1e12θt2∫0te−12θu2ξ∞a,b−ξua,bdu→0$

in L2 and almost surely for every γ ≥ 0, as t tends to infinity.

Proof Given 0 < st, θ < 0 and denote

$ϒθs,t≔∫0te−12θv2dv∫v∞re12θr2ψHs,rdr=∫0tre12θr2ψHs,rdr∫0re−12θv2dv+∫0te−12θv2dv∫t∞re12θr2ψHs,rdr≤C∫0trψHs,rdr+Cte−12θt2∫t∞re12θr2ψHs,rdr≤C∫0trψHs,rdr+t−s2H−2t−1,$

where we have used the fact

$∫0xe−12θv2dv≤Cxe−12θx2,∀x≥0$

and estimates

$∫t∞re12θr2ψHs,rdr=H2H−1∫t∞rr−s2H−2−s+r2H−2e12θr2dr≤H2H−1∫t∞rr−s2H−2e12θr2dr≤H2H−1t−s2H−2∫t∞re12θr2dr=H2H−1−θt−s2H−2e12θt2.$

It follows that

$EΛγt,θ2=t2γ+2eθt2∫0t∫0te−12θu2+v2⋅E∫u∞se12θs2dSsH∫v∞re12θr2dSrHdudv=t2γ+2eθt2∫0t∫0te−12θu2+v2dudv∫u∞∫v∞rse12θr2+s2ψHs,rdrds=t2γ+2eθt2∫0te−12θu2du∫u∞se12θs2ϒθs,tds=t2γ+2eθt2∫0tse12θs2ψHs,θds∫0se−12θu2du+t2γ+2eθt2∫t∞se12θs2ϒθs,tds∫0te−12θu2du≤t2γ+2eθt2∫0ts2ϒθs,tds+t2γ+1e12θt2∫t∞se12θs2ϒθs,tds→0t→∞,$

which shows that Λγ(t, θ) converges to zero in L2.Now, we obtain the convergence with probability one. Noting that

$ξ∞H−ξuH=∫u∞se12θs2dSsH$

for all u ≥ 0, we get

$|Λγt,θ|≤tγ+1e12θt2∫0te−12θu2∫u∞se12θs2dSsHdu≤tγ+1e12θt2∫0te−12θu2u|SuH|e12θu2+∫u∞|SsH1−θs2|e12θs2dsdu=tγ+1e12θt2∫0tu|SuH|du+tγ+1e12θt2∫0te−12θu2du∫u∞|SsH1−θs2|e12θs2ds=tγ+1e12θt2∫0tu|SuH|du+tγ+1e12θt2∫0t|SsH1−θs2|e12θs2ds∫0se−12θu2du+tγ+1e12θt2∫t∞|SsH1−θs2|e12θs2ds∫0te−12θu2du≤tγ+1e12θt2∫0tu|SuH|du+tγ+1e12θt2∫0t|SsH1−θs2|sds+Cθtγ∫t∞|SsH1−θs2|e12θs2ds→0$

almost surely for all γ ≥ 0, θ < 0 and

$12

, 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 XH with

$12

.

Theorem 4.1 Let θ < 0 and

$12

. Then, as t∞, the convergence

$J0Ht;θ,ν≔te12θt2XtH→ξ∞H−νθ$

holds in L2 and almost surely.

Proof Given t > 0 and θ < 0. Simple calculations may prove

$J0Ht;θ,ν=te12θt2XtH=te12θt2∫0thθt,sdSsH+νte12θt2∫0thθt,sds=te12θt2StH−θt2e12θt2∫0tse12θs2∫ste−12θu2dudSsH+νte12θt2∫0te−12θs2ds=te−12θt2StH−θte12θt2∫0te−12θu2∫0use12θs2dSsHdu+νte12θt2∫0te−12θs2ds=te12θt2StH−θte12θt2∫0te−12θu2ξuHdu+νte12θt2∫0te−12θs2ds.$

It follows from Lemma 4.1, Corollary 4.1, and Lemma 4.2 that

$J0Ht;θ,ν−ξ∞H−νθ=te12θt2XtH−ξ∞H−νθ=te12θt2StH−θte12θt2∫0te−12θu2ξuH−ξ∞Hdu+ξ∞H−νθ−θte12θt2∫0te−12θu2du−1→0t→∞$

in L2 and almost surely for all θ < 0 and

$12

, as t tends to infinity.

Theorem 4.2 Define the processes

$JH(n,θ,ν)={JtH(n,θ,ν),t≥0},n≥1$

by

$JnHt;θ,ν≔θt2Jn−1Ht;θ,ν−2n−3‼ξ∞H−νθ,n=1,2,…,$

for all t ≥ 0, where (−1)!! = 1. Then, the convergence

$JnHt;θ,ν→2n−1‼ξ∞H−νθ$

holds in L2 and almost surely for every n ≥ 1, as t∞.

Proof From the proof of Theorem 4.1, we find that the identities

$J0Ht;θ,ν−ξ∞H−νθ=te12θt2StH+θte12θt2∫0te−12θu2ξuH−ξ∞Hdu+ξ∞H−νθθte12θt2∫0te−12θu2du−1,JnHt;θ,ν=ξ∞H−νθInt,θ+tθt2ne−12θt2StH+θtθt2ne12θt2∫0te12θu2ξuH−ξ∞Hdu.$

holds for all t > 0, n ≥ 1 and θ < 0, where In(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 SH with

$12

:

$dXtH=dStH−θ∫0tXtH−XsHdsdt+νdt,X0H=0,$

where θ < 0 and

$ν∈R$

are two parameters. We will simulate the process with ν = 0 in the following cases:

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);

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.