1 Introduction

In 1995, Cranston and Le Jan [1] introduced a linear self-attracting diffusion

Xt=Btθ0t0sXsXududs+νt,t0

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 [2] introduced in 1992 as a model for the shape of a growing polymer (Brownian polymer) as follows

Xt=X0+Bt+0t0sfXsXududs,

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 [3]. Let

LX(t,x)

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

Xt=X0+Bt+0tdsRfxLXs,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

xRd

, 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:

XtH=BtHθ0t0sXsHXuHduds+νt,t0

with θ ≠ 0 and

X0H=0

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

12H<1

. The solution of (1.3) is a Gaussian process. When θ > 0, Yan et al [13] showed that the solution XH of (1.3) converges in L2 and almost surely, to the random variable

XH=0hθsdBsH+ν0hθsds

where the function is defined ar follows

hθs=1θse12θs2se12θu2du,s0

with θ > 0. Recently, Sun and Yan [14] considered the related parameter estimations with θ > 0 and

12H<1

, and Gan and Yan [15] considered the parameter estimations with θ < 0 and

12H<1

.

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

Xt=Gtθ0t0sXsXududs+νt,t0

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,t0}

with

S0H=0

and the covariance

RHt,sEStHSsH=s2H+t2H12s+t2H+|ts|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 [16]). The sub-fBm has properties analogous to those of fBm and satisfies the following estimates:

222H11ts2HEStHSsH2222H11ts2H.

More works for sub-fBm and related processes can be found in Bojdecki et al. [1720], Li [2124], Shen and Yan [25, 26], Sun and Yan [27], Tudor [2831], Ciprian A. Tudor [32] Yan et al [3335] and the references therein.

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

XtH=StHθ0t0sXsHXuHduds+νt,t0

with θ < 0 and

X0H=0

, where SH is a sub-fBm with Hurst parameter

12H<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

12<H<1

, the random variable

ξH=0se12θs2dSsH

exists as an element in L2.

(II) For θ < 0 and

12<H<1

, as t, we have

J0Ht;θ,νte12θt2XtHξHνθ

in L2 and almost surely.

(III) For θ < 0 and

12<H<1

, define the processes

JH(n,θ,ν)={JtH(n,θ,ν),t0},n1

by

JnHt;θ,νθt2Jn1Ht;θ,ν2n3ξHνθ,n=1,2,,

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

JnHt;θ,ν2n1ξHνθ

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.

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FIGURE 1. A path of XH with θ = − 1 and H = 0.7.

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FIGURE 2. A path of XH with θ = − 10 and H = 0.7.

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FIGURE 3. A path of XH with θ = − 100 and H = 0.7.

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FIGURE 4. A path of XH with θ = − 1 and H = 0.5.

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FIGURE 5. A path of XH with θ = − 10 and H = 0.5.

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FIGURE 6. A path of XH with θ = − 100 and H = 0.5.

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TABLE 1. The data of

XtH

with θ = − 1 and H = 0.7.

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TABLE 2. The data of

XtH

with θ = − 10 and H = 0.7.

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TABLE 3. The data of

XtH

with θ = − 100 and H = 0.7.

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TABLE 4. The data of

XtH

with θ = − 1 and H = 0.5.

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TABLE 5. The data of

XtH

with θ = − 10 and H = 0.5.

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TABLE 6. The data of

XtH

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

SH=StH,t0

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,sEStHSsH=s2H+t2H12s+t2H+|ts|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 [17] 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 [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

tStH

admits almost surely a bounded

1Hϑ

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

tStH

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 [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

for s, t ∈ [0, T]. When

12<H<1

, we can show that

φH2=0T0Tφtφs2tsRHt,sdsdt=0T0TφtφsψHt,sdsdt,φH,

where

ψHt,s=2tsRa,bt,s=H2H1|ts|2H2|t+s|2H2

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

EφSH(φ)

by

10,tSH10,t=0T10,tsdSsHStH

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

φH2=E0TφsdSsH2

for any

φH

.

For simplicity, in this paper we assume that

12<H<1

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

exists in L2 and

00φtφsψHt,sdsdt<,

we can define the integral

and

E0φsdSsH2=00φtφsψHt,sdsdt.

Denote by

S

the set of smooth functionals of the form

F=fSHφ1,SHφ2,,SHφn,

where

fCb(Rn)

and

φiH

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

DF=j=1nfxjSHφ1,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

F1,2E|F|2+EDFH2.

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

EDF,uHcFL2Ω

for every

FD1,2

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

for any

FD1,2

. We have

D1,2Dom(δ)

and for any

uD1,2

Eδu2=EuH2+EDu,DuHH=EuH2+E0,T4DξurDηusϕHη,rϕHξ,sdsdrdξdη,

where

(DSu)

is the adjoint of Du in the Hilbert space

HH

. We will denote

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

0TusdSsH=0TusδSsH+0T0TDsutψt,sdsdt,

provided u has a bounded q-variation with

1q<1H

and

uD1,2(H)

such that

0T0TDsutψt,sdsdt<.

Theorem 2.1 (Alós et al [16]). Let 0 < H < 1 and let

fC2(R)

such that

max|fx|,|fx|,|fx|κeβx2,

where κ and β are two positive constants with

β<14T2H

. Then we have

fStH=f0+0tfSsHdSsH+H222H10tfSsHs2H1ds

for all t ∈ [0, T].

3 Some Basic Estimates

Throughout this paper we assume that θ < 0 and

12<H<1

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

XtH=StHθ0t0sXsHXuHduds+νt,t0

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

hθt,s=1θse12θs2ste12θu2du,ts,0,t<s

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:

XtH=0thθt,sdSsH+ν0thθt,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 [12]):

• For all s ≥ 0, the limit

limtte12θt2hθt,s=se12θs2

for all s ≥ 0.

• For all ts ≥ 0, we have

1hθt,se12θt2s2.

• For all ts, r ≥ 0, we have

hθt,0=hθt,t=1,sthθt,udu=e12θs2ste12θu2du.

Lemma 3.1 Let θ < 0 and define function

Iθt=θte12θt20te12θu2du1.

We then have

limtt2Iθ(t)=1θ

and

limtt21+θte12θt2te12θu2du=1θ

Proof This is simple calculus exercise.

Lemma 3.2 (Sun and Yan [12]). Let θ < 0 and define the functions tIθ(t, n), n = 1, 2, … as follows

Iθt,1=θt2Iθt,Iθt,n+1=θt2Iθt,n2n1.

Then we have

limtIθt,n=2n1.

for every n ≥ 0, where (−1)! = 1.

Lemma 3.3 Let θ < 0. Then the integral

ΔH=00xye12θx2+y2ψHx,ydxdy

converges and as t,

limtt2eθt2EXtH2=ΔH.

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

limtt2eθt2EXtH2=limtt2eθt20t0thθt,xhθt,yψHx,ydxdy=limtθ2t2eθt20tdx0txye12θx2+y2ψHx,ydyxtduyte12θu2+v2dv=2limtθ2t2eθt20tdu0udx0udv0vdyxye12θx2+y2u2v2ψHx,y=limtθt1e12θt20tdx0te12θv2dv0vxye12θx2+y2ψHx,ydy=limtθt1e12θt20te12θv2dv0tdx0vxye12θx2+y2ψHx,ydy=limtθt1e12θt20te12θv2dv0vdx0vxye12θx2+y2ψHx,ydy=0dx0xye12θx2+y2ψHx,ydy,

where we have used the following fact:

limt1t1e12θt20te12θv2dvvtdx0vxye12θx2+y2ψHx,ydy=limt1t1e12θt20tdx0xe12θv2dv0vxye12θx2+y2ψHx,ydy=0.

This completes the proof.

Lemma 3.4 Let θ < 0. Then, convergence

limt1t22Heθt2tssre12θs2+r2ψHs,rdsdr=14θ2HΓ2H+1.

holds.

Proof It follows from L’Hôspital’s rule that

limt1t22Heθt2tue12θu2uve12θv2ψHu,vdvdu=12θlimt1t22He12θt2tve12θv2ψHt,vdv=limtH2H12θt22Htve12θv2t2vt2H2v+t2H2dv

for all θ < 0 and

12<H<1

. By making the change of variable

12θ(v2t2)=x

, we see that

limt12θt22Htve12θv2t2vt2H2v+t2H2dv=limt12θ2t22H0ex{t2+2xθt2H2t2+x+t2H2}dx=limt12θ2t22H0ex2xθ2H2t2+2xθ+t22Hdxlimt12θ2t22H0ext2+x+t2H2dx=12θ2H1Γ2H1

for all θ < 0 and

12<H<1

. This completes the proof.

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

cts2HEXtHXsH2Cts2H

Proof Given 0 ≤ s < tT and denote

X̂tH=0thθt,rdSrH,t0.

It follows that

EX̂tHX̂sH2=E0shθt,xhθs,xdSxH2+Esthθt,xdSxH2+2Esthθt,ydSyH0shθt,xhθs,xdSxH.

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

0E0shθt,xhθs,xdSxH2=0s0shθt,xhθs,xhθt,yhθs,yψHx,ydxdy=θ2ste12θu2du20s0sxye12θx2+y2ψHx,ydxdyθ2s2ts2eθt20s0sψHx,ydxdy=θ2s2ts2eθt2ESsH2CH,Tts2

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

Esthθt,xdSxH2=ststhθt,xhθt,yψHx,ydxdyeθt2ststxye12θx2+y2ψHx,ydxdyt2eθt2ststψHx,ydxdyCH,Tts2H.

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

0Esthθt,ydSyH0shθt,xhθs,xdSxH=st0shθt,yhθt,xhθs,xψHx,ydxdyθ2e12θt2ste12θu2dustye12θy2dy0sxe12θx2ψHx,ydxθ2eθt2tsstye12θy2dy0sxe12θx2ψHx,ydxCH,Tts2

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

EX̂tHX̂sH2CH,T|ts|2H

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

0shθt,rhθs,rdr=θste12θu2du0sre12θr2drCH,Tts

and

sthθt,rdr=e12θs2ste12θr2drCH,Tts

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

0thθt,rdr0shθs,rdr2=0shθt,rhθs,rdr2+sthθt,rdr2+2sthθt,rdr0shθt,rhθs,rdrCH,Tts2

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

EXta,bXsa,b2=EX̂ta,bX̂sa,b2+ν20thθt,rdr0shθs,rdr2CH,Tts2H

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,t0}

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:=0tXtHXsHds=tXtH0tXsHds=0tsdXsH=0tsdSsH0tθsYsds+12νt2.

By the variation of constants method, one can prove

Yt=e12θt20tse12θs2dSsHνθe12θt21

for all t ≥ 0. Define Gaussian process

ξH={ξtH,t0}

as follows

ξtH0tse12θs2dSsH,t0.

Lemma 4.1 Let θ < 0 and

12<H<1

. Then, the random variable

ξH0se12θs2dSsH

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

ξtHξH

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

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

E0xe12θx2dSxH2=00xye12θx2+y2ψHx,ydxdy=20xe12θx2dx0xye12θy2ψHx,ydy=2H2H10xe12θx2dx0xye12θy2xy2H2x+y2H2dy2H2H10xe12θx2dx0xxy2H2x+y2H2ydy=2H2H1CH0x2H+1e12θx2dx=Cθ,HΓ2H+2

for all θ < 0 and

12<H<1

, which shows that the random variable

ξH

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

ex2xC

for all x ≥ 0, we have

EξtHξsH2=Estxe12θx2dSxH2=ststxye12θx2+y2ψHx,ydxdy=2stxe12θx2dxsxye12θy2ψHx,ydy=2H2H1stxe12θx2dxsxye12θy2xy2H2x+y2H2dy2HCθ2H1stdxsxxy2H2dy=Cθ,Hts2H.

Thus, the normality of ξH implies that

EξtHξsH2nCθ,H,nts2nH

for all 0 ≤ s < t,

12<H<1

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

Nextly, we check the

ξta,b

converges to

ξH

in L2. This follows from the next estimate:

EξtHξH2=ttxye12θx2+y2ψHx,ydxdy=2ttxxye12θx2+y2ψHx,ydxdy2e12θt2txe12θx2dxtxyψHx,ydy2e12θt2txe12θx2dx0xyψHx,ydy2H2H1e12θt2txe12θx2dx0xyxy2H2x+y2H2dy2H2H1e12θt2txe12θx2dx0xyxy2H2dy=2H2H101u1u2H2due12θt2tx2H+1e12θx2dx0,

as t tends to infinity.

Finally, we check the

ξta,b

converges to

ξH

almost surely. By integration by parts we see that

ξtHξH=tse12θs2dSsH=te12θt2StHt1+θs2e12θs2SsHds

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

ηtHt1+θs2e12θs2SsHdsa.s0

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

tsαe12θs2dsCtα1e12θt2,α>1,

with t ≥ 0, we may show that

Esupnt<n+1ηtH2nn1+θs21+θr2e12θs2+r2E|SsHSrH|drdsCns2+He12θs2ds2Cn2+2Heθn2,

for all integer numbers n ≥ 1, and hence

n=0Psupnt<n+1ηtH2εCε2n=0n2+2Heθn2<.

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

tγξtHξH=tγtse12θs2dSsH0,

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

Lemma 4.2 Let θ < 0 and

12<H<1

. Then, we have

Λγt,θtγ+1e12θt20te12θu2ξa,bξua,bdu0

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

Proof Given 0 < st, θ < 0 and denote

ϒθs,t0te12θv2dvvre12θr2ψHs,rdr=0tre12θr2ψHs,rdr0re12θv2dv+0te12θv2dvtre12θr2ψHs,rdrC0trψHs,rdr+Cte12θt2tre12θr2ψHs,rdrC0trψHs,rdr+ts2H2t1,

where we have used the fact

0xe12θv2dvCxe12θx2,x0

and estimates

tre12θr2ψHs,rdr=H2H1trrs2H2s+r2H2e12θr2drH2H1trrs2H2e12θr2drH2H1ts2H2tre12θr2dr=H2H1θts2H2e12θt2.

It follows that

EΛγt,θ2=t2γ+2eθt20t0te12θu2+v2Euse12θs2dSsHvre12θr2dSrHdudv=t2γ+2eθt20t0te12θu2+v2dudvuvrse12θr2+s2ψHs,rdrds=t2γ+2eθt20te12θu2duuse12θs2ϒθs,tds=t2γ+2eθt20tse12θs2ψHs,θds0se12θu2du+t2γ+2eθt2tse12θs2ϒθs,tds0te12θu2dut2γ+2eθt20ts2ϒθs,tds+t2γ+1e12θt2tse12θs2ϒθs,tds0t,

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

ξHξuH=use12θs2dSsH

for all u ≥ 0, we get

|Λγt,θ|tγ+1e12θt20te12θu2use12θs2dSsHdutγ+1e12θt20te12θu2u|SuH|e12θu2+u|SsH1θs2|e12θs2dsdu=tγ+1e12θt20tu|SuH|du+tγ+1e12θt20te12θu2duu|SsH1θs2|e12θs2ds=tγ+1e12θt20tu|SuH|du+tγ+1e12θt20t|SsH1θs2|e12θs2ds0se12θu2du+tγ+1e12θt2t|SsH1θs2|e12θs2ds0te12θu2dutγ+1e12θt20tu|SuH|du+tγ+1e12θt20t|SsH1θs2|sds+Cθtγt|SsH1θs2|e12θs2ds0

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

12<H<1

, 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<H<1

.

Theorem 4.1 Let θ < 0 and

12<H<1

. Then, as t∞, the convergence

J0Ht;θ,νte12θt2XtHξHνθ

holds in L2 and almost surely.

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

J0Ht;θ,ν=te12θt2XtH=te12θt20thθt,sdSsH+νte12θt20thθt,sds=te12θt2StHθt2e12θt20tse12θs2ste12θu2dudSsH+νte12θt20te12θs2ds=te12θt2StHθte12θt20te12θu20use12θs2dSsHdu+νte12θt20te12θs2ds=te12θt2StHθte12θt20te12θu2ξuHdu+νte12θt20te12θs2ds.

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

J0Ht;θ,νξHνθ=te12θt2XtHξHνθ=te12θt2StHθte12θt20te12θu2ξuHξHdu+ξHνθθte12θt20te12θu2du10t

in L2 and almost surely for all θ < 0 and

12<H<1

, as t tends to infinity.

Theorem 4.2 Define the processes

JH(n,θ,ν)={JtH(n,θ,ν),t0},n1

by

JnHt;θ,νθt2Jn1Ht;θ,ν2n3ξHνθ,n=1,2,,

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

JnHt;θ,ν2n1ξHνθ

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,νξHνθ=te12θt2StH+θte12θt20te12θu2ξuHξHdu+ξHνθθte12θt20te12θu2du1,JnHt;θ,ν=ξHνθInt,θ+tθt2ne12θt2StH+θtθt2ne12θt20te12θu2ξuHξHdu.

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<H<1

:

dXtH=dStHθ0tXtHXsHdsdt+νdt,X0H=0,

where θ < 0 and

νR

are two parameters. We will simulate the process with ν = 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);

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.

References


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By Han Gao