Harnack inequalities for SDEs driven by subordinator fractional Brownian motion

Accepted Manuscript Harnack inequalities for SDEs driven by subordinator fractional Brownian motion Zhi Li, Litan Yan

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S0167-7152(17)30329-2 https://doi.org/10.1016/j.spl.2017.10.015 STAPRO 8056

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Statistics and Probability Letters

Received date : 11 July 2017 Revised date : 14 September 2017 Accepted date : 23 October 2017 Please cite this article as: Li Z., Yan L., Harnack inequalities for SDEs driven by subordinator fractional Brownian motion. Statistics and Probability Letters (2017), https://doi.org/10.1016/j.spl.2017.10.015 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Harnack Inequalities for SDEs Driven by Subordinator Fractional Brownian Motion∗ Zhi Li a,b) † Litan Yan a) a)

College of Information Science and Technology, Donghua University, 2999 North Renmin Rd, Songjiang, Shanghai 201620, P.R. China. b) School of Information and Mathematics, Yangtze University, Jingzhou 434023, P.R. China.

Abstract By using a transformation formulas for fractional Brownian motion, the Harnack inequalities for stochastic differential equations driven by subordinator fractional Brownian motion with Hurst parameter H ∈ (1/2, 1) are established. 2000 Mathematics Subject Classification(s): 60H15, 60G15, 60H05. Keywords: Harnack inequality; Fractioanl Brownian motion; Subordinator.

1. Introduction In this paper, we consider the following stochastic differential equation on Rd Z t H Xt (x) = x + bs (Xs )ds + WS(t) + Vt , t ≥ 0,

(1.1)

0

where b : [0, ∞) × Rd → Rd , (t, x) → bt (x) is measurable, locally bounded as a function of t ≥ 0, continuous in x and satisfying a so-called one-sided Lipschitz condition, i.e. there exists a locally bounded measurable function K on [0, ∞) such that hbt (x) − bt (y), x − yi ≤ K(t)|x − y|2 ,

∀x, y ∈ Rd , t ≥ 0.

(1.2)

The processes W := (Wt )t≥0 , S := (S(t))t≥0 , and V := (Vt )t≥0 are independent stochastic processes and satisfy  H  W is a fractional Brownian motion on Rd with Hurst parameter H ∈ (1/2, 1); S is a time-changed, .i.e. a non-decreasing process on [0, ∞) with S(0) = 0; (1.3)  V is a locally bounded measurable process on Rd with V0 = 0. I∗

This research is partially supported by the NNSF of China (No.11571071), Natural Science Foundation of Hubei Province (No.2016CFB479) and China postdoctoral fund (No. 2017M610216). II† Corresponding author. E-mail:[email protected]. Preprint submitted to Statistics and Probability Letters

September 14, 2017

Very recently, by using the coupling argument, the Girsanov transformations and the regularization argument for the time-change, Deng and Schilling [9] established Harnack inequalities for the equation (1.1) driven by a time-changed fractional Brownian motion with H ∈ (0, 1/2) under the condition (1.2). As far as I know that the fBm becomes the standard Brownian motion when H = 1/2, and the fBm B H neither is a semimartingale nor a Markov process if H 6= 1/2. However, the fBm B H , H > 1/2 is a long-memory process and presents an aggregation behavior. The long-memory property make fBm as a potential candidate to model noise in mathematical finance (see [5]); in biology (see [4, 6]); in communication networks (see, for instance [17]); the analysis of global temperature anomaly [13] electricity markets [15] etc. −1 But, in virtue of irregularity of the operator KH , it is very difficult to obtain the Harnack inequality by using directly Girsanov transformations when the Hurst parameter 1/2 < H < 1. To overcome this difficulty, motivating mainly by [11], we will transform the fractional Brownian motion with Hurst parameter 1/2 < H < 1 into some integral with respect to the fractional Brownian motion with Hurst parameter 0 < H < 1/2 by using of the transformation formula for fractional Brownian motion.

The rest of this paper is organized as follows. In Section 2, we introduce some necessary notations and preliminaries. In Section 3, we devote ourselves to establish the Harnack inequalities for SDEs driven by time-changed fractional Brownian motion with Hurst parameter H ∈ (1/2, 1). 2. Preliminaries In this section, we recall briefly some basic facts on fractional Brownian motion (fBm) which will be used later on. Let W H = {WtH , t ≥ 0} on Rd be a fractional Brownian motion with Hurst parameter H ∈ (0, 1) defined on the probability space (Ω, F, P), i.e., W H is a centered Gauss process with covariance function 1 RH (t, s) = E(WtH,i WsH,j ) = (t2H + s2H − |t − s|2H )δij , 2

t, s ≥ 0, 1 ≤ i, j ≤ d,

where δij denotes Kronecker’s delta. In particular, if H = 21 , W H is a Brownian motion. It is well known that if H 6= 12 , W H does not have independent increments and has α-order H¨older continuous path for all α ∈ (0, H). For more details of fBm and proofs we refer the readers, for instance, to [3, 8]. On the other hand, from [8], we know the covariance kernel RH (t, s) can be written as Z t∧s RH (t, s) = KH (t, r)KH (s, r)dr, 0

where KH is a square integrable kernel given by 1 1 1 1 1 t KH (t, s) = Γ(H + )−1 (t − s)H− 2 F (H − , − H, H + , 1 − ), 2 2 2 2 s

2

in which F (·, ·, ·, ·) is the Gauss hypergeometric function and Γ(·) denotes the Euler Gamma function. Moreover, according to [1], fractional Brownian motion has the following integral representation with respect to the usual d-dimensional standard Brownian motion W = (Wt )t≥0 Z t

WtH =

KH (t, s)dWs .

0

H+ 1

By [8], the operator KH : L2 ([0, T ]; Rd ) → I0+ 2 (L2 [0, T ]; Rd ) associated with the square integrable kernel KH (·, ·) is defined as follows Z t KH (t, s)f (s)ds, f ∈ L2 ([0, T ]; Rd ). (KH f )(t) := 0

H+ 21

where I0+ is the H + 21 -order left fractional Riemann-Liouville operator on [0, T ], one can see [14]. It is an isomorphism and for each f ∈ L2 ([0, T ]; Rd ), 1 1 2H 12 −H 2 −H H− 21 (KH f )(s) = I0+ s I0+ s f, H ≤ , 2 1 1 1 H− 12 H− 2 12 −H I0+ s f, H ≥ . (KH f )(s) = I0+ s 2

H+ 1

−1 As a consequence, for every h ∈ I0+ 2 (L2 [0, T ]; Rd ), the inverse operator KH is of the following form 1 1 H− 1 1 −1 (KH h)(s) = sH− 2 D0+ 2 s 2 −H h0 , H > , 2 1 1 1 −H H− 1 2H −1 2 (KH h)(s) = s 2 −H D0+ s 2 D0+ h, H < , 2 H− 1

1

−H

2 where D0+ 2 (D0+ ) is H − 21 ( 12 − H)-order left-sided Riemann-Liouville derivative, one can see [14]. In particular, if h is absolutely continuous, we have 1 1 1 −H 1 −H 0 −1 2 s2 h , H < . (KH h)(s) = sH− 2 I0+ 2

(2.1)

3. Main results Throughout this section, we write |x| :=



(1) 2

(d) 2

|x | + · · · + |x |

1/2

for the Euclidean

norm of x = (x , · · · , x ) ∈ R . It is easy to see that under (1.2), there exists a unique non-explosive solution to the SDE (1.1). We will consider the Harnack inequality for the solution to the equation (1.1). To this end, we define (1)

(d)

d

Pt f (x) := Ef (Xt (x)), t ≥ 0, f ∈ Bb (Rd ), x ∈ Rd .

(3.1)

where Bb (Rd ) denotes the set of all bounded measurable functions on Rd . We also denote for any f : Rd → R the local Lipschitz constant at the point x by | 5 f |(x) := lim sup y→x

|f (y) − f (x)| ∈ [0, ∞], |y − x| 3

x ∈ Rd .

3.1. Statement of the main result Theorem 3.1. If b satisfies the assumption (1.2). Then, for any T > 0, max{0, 2H − 32 } < σ1 < H − 12 , x, y ∈ Rd , the operator PT satisfies that

1 2

< H < 1,

(i) For any bounded Borel function f : Rd → [1, ∞)

i h S(T )2−2H e |x − y|2 . PT logf (y) ≤ logPT f (x) + C(H)E R T Rt ( 0 exp[− 0 K(s)ds]dS(t))2

(ii) For any bounded Borel function f : Rd → R

h i S(T )2−2H e | 5 PT f |2 (x) ≤ {PT f 2 (x) − (PT f (x))2 }2C(H)E . RT Rt ( 0 exp[− 0 K(s)ds]dS(t))2

(iii) For any bounded Borel function f : Rd → [0, ∞) and for any p > 1  h 1 PT f (y) ≤ (PT f p (x)) p E where e C(H) =

i1− p1 e − y|2 pS(T )2−2H C(H)|x RT Rt (p − 1)2 ( 0 exp[− 0 K(s)ds]dS(t))2 min

max{0,2H− 32 }< σ1
C ∗ (H, σ),

 2 2 i + 2B 2H − , 3 − 4H + , 1 σ σ σ   σ−1 3 σ−1 + 1, (− + H) + 1 B σ (H − 12 ) σ−1 σ 2 σ C(H, σ) = , 1 1 c˜H Γ(H − 2 )|σ(1 − 2H) + 1| σ

C ∗ (H, σ) =C(H, σ) ·

h

1 H−

and B(·, ·) and Γ(·) are the standard Beta and Gamma functions. 3.2. Proof of Theorem 3.1

We first consider the following regularization of S: Z 1 t+ε S(s)ds + εt, t ≥ 0, ε > 0. Sε (t) := ε t Then Sε is strictly increasing, absolutely continuous and Sε ↓ S as ε ↓ 0. For each ε > 0, we consider the approximation equation Z t ε Xt (x) = x + bs (Xsε (x))ds + WSHε (t) + Vt , t ≥ 0. (3.2) 0

Since Sε is absolutely continuous, this equation is indeed driven by the fractional Brownian motion so that we are able to establish the Harnack inequalities for the associated operator 4

Ptε by using coupling. Finally, by proving Ptε → Pt as ε → 0, we derive the corresponding Harnack inequalities for Pt . Let ` be an absolutely continuous and strictly increasing function on [0, +∞) with `(0) = 0 and v : [0, +∞) → Rd be measurable and locally bounded with v0 = 0. We consider the following equation Z t `,v H Xt (x) = x + bs (Xs`,v (x))ds + W`(t) + vt , t ≥ 0. (3.3) 0

Under our general assumptions, this equation has an unique solution. Let Pt`,v f (x) := Ef (Xt`,v (x)), t ≥ 0, f ∈ Bb (Rd )

(3.4)

where Xt`,v (x) is the solution to (3.3) for X0`,v = x. Proposition 3.1. If b satisfies the assumption (1.2). Then, for any T > 0, max{0, 2H − 32 } < σ1 < H − 12 , x, y ∈ Rd , the operator PT`,v satisfies that

1 2

< H < 1,

(i) For any bounded Borel function f : Rd → [1, ∞)

PT`,v logf (y) ≤ logPT`,v f (x) + C(H, σ) · ρ(T, H, x, y),

(3.5)

(ii) For any bounded Borel function f : Rd → [0, ∞) and for any p > 1 h pC(H, σ) · ρ(T, H, x, y) i (PT`,v f (y))p ≤ (PT`,v f p (x))exp , (p − 1) where B

σ−1 σ

C(H, σ) =



(H − 12 ) σ−1 + 1, (− 32 + H) σ−1 +1 σ σ 1

c˜H Γ(H − 21 )|σ(1 − 2H) + 1| σ



(3.6)

,

ρ(T, H, x, y) is given in (3.10) and B(·, ·) and Γ(·) are the standard Beta and Gamma functions. Proof. Now, for fixed T > 0 and y ∈ Rd , we intend to construct a coupling to derive the Harnack inequalities of PT`,v . To this end, let {Yt }t≥0 solve the equation Z t Z t X `,v (x) − Ys H Yt = y + bs (Ys )ds + W`(t) + vt + γI[0,τ ) (s) · s`,v d`(s), t ≥ 0, (3.7) |Xs (x) − Ys | 0 0 where γ := R T 0

and

|x − y| , Rt exp[− 0 K(s)ds]d`(t)

t≥0

τ := inf{t ≥ 0 : Xt`,v = Yt }.

Then, by virtue of (3.3) and (3.7) we have d(Xt`,v (x)

− Yt ) =

(bt (Xt`,v (x))

− bt (Yt ))dt + γ · 5

Xt`,v (x) − Yt

|Xt`,v (x) − Yt |

d`(t),

t < τ.

Then, applying the Tanaka’s formula to |Xt`,v (x) − Y (t)|, we have for t < τ d|Xt`,v (x) − Yt | =sgn(Xt`,v (x) − Yt )d(Xt`,v (x) − Yt )

=sgn(Xt`,v (x) − Yt )(bt (Xt`,v (x)) − bt (Yt ))dt − γd`(t).

Then, by virtue of the above inequality and the chain rule, we have that  Rt  d e− 0 K(s)ds |Xt`,v (x) − Yt | = − K(t)e−

Rt 0

Rt

K(s)ds

|Xt`,v (x) − Yt |dt + e−

Rt 0

K(s)ds

Rt

d|Xt`,v (x) − Yt |

|Xt`,v (x) − Yt |dt − γe− 0 K(s)ds d`(t) Rt 1 + e− 0 K(s)ds `,v hXt`,v (x) − Yt , bt (Xt`,v (x)) − bt (Yt )idt |Xt (x) − Yt |

= − K(t)e−

≤ − γe−

Rt 0

0

K(s)ds

K(s)ds

d`(t).

Therefore, if T < τ then 0<

|XT`,v (x)

−

− YT |e

RT 0

K(s)ds

≤ |x − y| −

Z

T

γe−

0

Rt 0

K(s)ds

d`(t) = 0,

which is a contradiction to the assumption T < τ . Therefore, we have T ≥ τ and XT`,v (x) = YT . X `,v −1

(x)−Y`−1 (t)

Let u(t) := γ{I[0,`−1 (τ )) (t)} |X``,v (t) (x)−Y `−1 (t)

Z

0

`(T ) 1−2H

(`(T ) − v)

`−1 (t) |

1 u(v)dv = 3 Γ( 2 − H) 3 −H 2

Z

. Then, we have `(T )

0

=(I0+ u)(`(T )) where

 3 1 1 − H (`(T ) − v) 2 −H u(v)dv (`(T ) − v) 2 −H Γ 2

3  1 u(v) = Γ − H (`(T ) − v) 2 −H u(v). 2

Since 1 − 2H > −1 and u(v) is bound on [0, `(T )], we know u(v) ∈ L2 ([0, `(T )]; Rd ). Thus 3 R· −H 1−2H 2 we have that 0 c˜−1 u(v)dv ∈ I0+ (L2 ([0, `(T )]; R)). Note that by means of H (`(T ) − v) the integral representation of fractional Brownian motion, the definition of the operator KH and transformation formulas for fractional Brownian motion (see [11]), we get for any 0 ≤ `(t) ≤ `(T ), Z `(t) H H f W`(t) = u(v)dv + W`(t) 0 Z `(t) Z `(t) (3.8) = u(v)dv + c˜H (`(t) − v)2H−1 dWv1−H 0 0 Z `(t) 1−2H = c˜H (`(t) − v)2H−1 [˜ c−1 u(v)dv + dWv1−H ]. H (`(t) − v) 0

6

For given `(t) in (3.8) and any 0 ≤ v ≤ `(t), let Z v 1−2H 1−H f c˜−1 u(s)ds + Wv1−H = Wv H (`(t) − s) Z v Z0 v −1 1−2H c˜H (`(t) − s) u(s)dv + K1−H (v, s)dWs = 0 0 Z · Z v h  i −1 −1 1−2H K1−H (v, s) K1−H c˜H (`(t) − z) u(z)dz (s)ds + dWs = 0

where c˜H = Now, let



0

2H Γ(2H)Γ(3−2H)

 21

.

  Z · −1 −1 1−2H K1−H c˜H (`(T ) − z) u(z)dz (v)dW (v) R(`(T )) = exp − 0 0  2 Z Z · i 1 `(T ) −1 −1 1−2H − K1−H c˜H (`(T ) − z) u(z)dz (v)dv . 2 0 0 h

Z

`(T )

f H )0≤t≤`(T ) is an F B H -fractional Using Corollary 5.2 of [11], we immediately know that (W `(t) `(t) Brownian motion with Hurst parameter H ∈ ( 12 , 1) under the new probability Q(dω) = f 1−H )0≤t≤`(T ) is an F W 1−H -fractional Brownian motion with Hurst paR(`(T ))P(dω) if (W `(t) `(t) rameter 1 − H under the new probability Q(dω) = R(`(T ))P(dω).

ft1−H )0≤t≤`(T ) is an F B 1−H -fractional Brownian motion with Next we want to show (W `(t) Hurst parameter H ∈ ( 21 , 1) under the new probability R(`(T ))P(dω). Due to R · −1 Q(dω) = 1−2H P [12], it suffices to show that E R(`(T )) = 1. Since 0 c˜H (`(T ) − z) u(z)dz is absolutely continuous, we have by (2.1) that Z ·   1 1 −1 −H H− 2 H− 21 −1 1−2H 2 K1−H I c˜H (`(T )−v)1−2H u(v), v ∈ [0, `(T )]. c˜−1 (`(T )−z) u(z)dz (v) = v 0+ v H 0

Hence, we have further that for v ∈ [0, `(T )], Z ·   −1 1−2H c˜−1 (`(T ) − z) u(z)dz (v) K1−H H 0 Z v 1 −1 12 −H H− 21 1−2H − 32 +H v z (`(T ) − z) u(z)(v − z) dz c ˜ = H 1 Γ(H − 2 ) 0 Z 1 σ  σ−1 o σ−1 nZ v o σ1 −H 2 3 c˜−1 γ n v  H− 1 σ − +H σ(1−2H) H v 2 (v − z) 2 (`(T ) − z) dz ≤ · z dz · Γ(H − 21 ) 0 0   −1 H− 12 − σ1 1−2H+ σ1 1−2H+ σ1 σ−1 γ `(T ) + (`(T ) − v) 1 σ−1 3 σ−1 c˜ v σ ≤ H B (H − ) + 1, (− + H) + 1 1 2 σ 2 σ Γ(H − 12 ) |σ(1 − 2H) + 1| σ h i 1 1 1 1 ≤C(H, σ)γv H− 2 − σ `(T )1−2H+ σ + (`(T ) − v)1−2H+ σ . (3.9)

7

Then, we can further obtain that for the fixed T > 0, Z `(T )  Z ·  2 1 −1 −1 1−2H K c ˜ (`(T ) − z) u(z)dz (v) dv 1−H H C(H, σ) 0 0 Z `(T ) h i n 2 1 1 2 v 2H−1− σ `(T )1−2H+ σ + (`(T ) − v)1−2H+ σ dv · R T ≤ 0

≤ `(T )2−2H

h

1 H−

=: 2ρ(T, H, x, y).

1 σ

o2 |x − y| Rt exp[− 0 K(s)ds]d`(t) 0 o2  i n 2 2 |x − y| + 2B 2H − , 3 − 4H + · RT Rt σ σ exp[− 0 K(s)ds]d`(t) 0

(3.10)

As a consequence, we get Z · h 1 Z `(T )  2 i −1 −1 c˜H K1−H (`(T ) − z)1−2H u(z)dz (v)dv ≤ exp[C 2 (H, σ)ρ(T, H, x, y)]. Eexp 2 0 0 (3.11) Using the well-known Novikov criterion, one can have EP R(`(T )) = 1. Then we can rewrite (3.7) as Z t Z t X `,v − Ys H bs (Ys )dt + W`(t) + vt + γI[0,τ ) (s) · s`,v d`(s) Yt =y + |Xs − Ys | 0 0 Z t Z t Z `(t) `,v − Y X s s H f + vt + =y + bs (Ys )dt + W γI[0,τ ) (s) · `,v d`(s) − u(s)ds `(t) |Xs − Ys | 0 0 0 Z t Z t Z t Xs`,v − Ys H f =y + bs (Ys )dt + W`(t) + vt + γI[0,τ ) (s) · `,v d`(s) − u(`(s))d`(s) |Xs − Ys | 0 0 0 Z t Z t f H + vt . =y + bs (Ys )dt + σ s dW `(s) 0

0

Thus, the distribution of (Xt`,v (y))0≤t≤T under P coincides with the law of (Yt )0≤t≤T under R(`(T ))P. Therefore, we conclude from the definition of Pt`,v that for all bounded Borel functions f : Rd → R PT`,v f (y) = EQ f (YT (y)) = EQ f (XT`,v (x)) = EP R(`(T ))f (XT`,v (x)). By Jensen’s inequality, we obtain for any random variable F ≥ 1, h E R(`(T ))log

Then we have

(3.12)

i h i h F i F F = ER(`(T ))P log ≤ logER(`(T ))P = logE[F ]. R(`(T )) R(`(T )) R(`(T )) h i E R(`(T ))logF ≤ logE[F ] + E[R(`(T ))logR(`(T ))]. 8

  R · −1 −1 1−2H Let η(s) = K1−H c ˜ (`(T ) − z) u(z)dz (s). By the definition of R(`(T )), combining 0 H with (3.12), we have Z

Z 1 `(T ) hη(s), dWs i − logR(`(T )) = − |η(s)|2 ds 2 0 0 Z `(T ) Z 1 `(T ) f =− hη(s), dWs i + |η(s)|2 ds 2 0 0 Z `(T ) fs i + C(H, σ) · ρ(T, H, x, y). hη(s), dW ≤− `(T )

0

Thus, for all bounded Borel functions f ≥ 1 on Rd , we have

PT`,v logf (y) =E[logf (XT`,v (y))] = E[R(`(T ))logf (XT`,v (x))] ≤logE[f (XT`,v (x))] + E[R(`(T ))logR(`(T ))] =logPT`,v f (x) + ER(`(T ))P [logR(`(T ))]

≤logPT`,v f (x) + C(H, σ) · ρ(T, H, x, y). The proof of the log-Harnack inequality is complete. Next, we prove the (3.6). For all bounded Borel functions f ≥ 1 on Rd , by (3.12) and the H¨older’s inequality, we can obtain p

(PT`,v f (y))p = (Ef (XT`,v (y)))p = (ERf (XT`,v (x)))p ≤ (PT`,v f p (x))(ER p−1 (`(T )))p−1 . Let Mt := −

(3.13)

Rt

hη(s), dWs i, t ≥ 0. By using (3.12) we get h p i p p M`(T ) − hM i`(T ) R p−1 (`(T )) =exp p−1 2(p − 1) h i h p i p2 p =exp hM i`(T ) × exp M`(T ) − hM i`(T ) 2(p − 1)2 p−1 2(p − 1)2 h pC(H, σ) · ρ(T, H, x, y) i h p i p2 ≤exp × exp M`(T ) − hM i`(T ) . (p − 1)2 p−1 2(p − 1)2 h i p p2 Notice that exp p−1 M`(T ) − 2(p−1) hM i is a martingale with mean 1. Then, using 2 `(T ) Novikov’s criterion we get 0

E[R

p p−1

h pC(H, σ) · ρ(T, H, x, y) i (`(T ))] ≤ exp . (p − 1)2

Thus, we get (3.6) by plugging the above expression into (3.13). The proof is complete. The Proof of the Theorem 3.1. According to [2], ii) follows from i). By a standard approximation argument, it is enough to prove the i) and iii) for f ∈ Cb (Rd ). To prove the Theorem 3.1 using the Proposition 3.1, we need to ensure that Xtε (x) → Xt (x) as ε → 0. To 9

this end, we first assume that bt : Rd → Rd is, uniformly for t in compact intervals, a global Lipschitz function, i.e. for any t > 0 there is some Ct > 0 such that |bs (x) − bs (y)| ≤ Ct |x − y|,

0 ≤ t, x, y ∈ Rd .

This implies that for all x ∈ Rd Z T H ε |bs (Xsε (x)) − bs (Xs (x))|ds + |WSH1/n (T ) − WS(T |XT (x) − XT (x)| ≤ )| 0 Z T H ≤CT |Xsε (x) − Xs (x)|ds + |WSHε (T ) − WS(T ) |.

(3.14)

0

(n)

Since XT (x) and XT (x) are non-explosive, the integral in the above expression is finite. H Therefore, we can apply Gronwall’s inequality with g(T, n) := |WSHε (T ) − WS(T ) | and find |XTε (x)

− XT (x)| ≤ g(T, n) + CT

Z

T

g(s, n)e(T −s)CT ds.

0

Notice that limε↓0 g(s, n) = 0 for all s ≥ 0. Then we get from the dominated convergence theorem that lim XTε (x) = XT (x), x ∈ Rd ; (3.15) ε↓0

then lim PTε f (x) = PT f (x), ε↓0

f ∈ Cb (Rd ).

Now, we consider the following equation Z t ε,v Xt (x) = x + bs (Xsε,v (x))ds + WSHε (t) + vt , 0

(3.16)

t ≥ 0.

(3.17)

Under our general assumptions, this equation has an unique solution. Let Ptε,v f (x) := Ef (Xtε,v (x)), t ≥ 0, f ∈ Bb (Rd )

(3.18)

where Xtε,v (x) is the solution to (3.17) for X0ε,v = x. We can now use the Proposition 3.1 to get for all x, y ∈ Rd and all f ∈ Cb (Rd ) with f ≥ 1 PTε,v logf (y) ≤ logPTε,v f (x) + C(H, σ) · ρ(T, H, x, y),

(3.19)

and for all x, y ∈ Rd and all non-negative f ∈ Cb (Rd )

h pC(H, σ) · ρ(T, H, x, y) i ≤ . (p − 1) h Since the processes W , V and S are independent, PTε f = E PTε,v f (·) (PTε,v f (y))p

(PTε,v f p (x))exp

v=V,`=Sε

(3.20) i

holds for all

bounded Borel functions f : Rd → R. Thus, the Jensen inequality yields for all x, y ∈ Rd 10

and all f ∈ Cb (Rd ) with f ≥ 1 h i ε,v ε PT logf (y) =E PT logf (y) v=V,`=Sε i h i h ε,v + E C(H, σ) · ρ(T, H, x, y) ≤E logPT f (x) v=V,`=Sε v=V,`=Sε h i 2−2H Sε (T ) ε ∗ ≤logPT f (x) + C (H, σ)E R T |x − y|2 , Rt 2 ( 0 exp[− 0 K(s)ds]dSε (t)) where

C ∗ (H, σ) =C(H, σ) ·

h

1 H−

1 σ

 2 i 2 . + 2B 2H − , 3 − 4H + σ σ

For the power-Harnack inequality we use H¨older’s inequality to find for all x, y ∈ Rd and all non-negative f ∈ Cb (Rd ) i h ε,v ε PT f (y) =E PT f (y) v=V,`=Sε h i h C(H, σ) · ρ(T, H, x, y) i 1 ≤E (PTε,v f p (x)) p exp (p − 1) v=V,`=Sε i1− p1  h 2−2H ∗ 1 pSε (T ) C (H, σ)|x − y|2 ε p p ≤(PT f (x)) E . RT Rt (p − 1)2 ( 0 exp[− 0 K(s)ds]dSε (t))2 Since Sε is of bounded variation, the limit Sε ↓ S also holds for the integrals Z T Z t Z T Z t lim exp[− K(s)ds]dSε (t) = exp[− K(s)ds]dS(t). ε↓0

0

0

0

(3.21)

0

Thus, combining (3.15), (3.16) and with (3.21) and letting ε ↓ 0, we get for all x, y ∈ Rd and all f ∈ Cb (Rd ) with f ≥ 1 h i S(T )2−2H PT logf (y) ≤ logPT f (x) + C (H, σ)E R T |x − y|2 Rt ( 0 exp[− 0 K(s)ds]dS(t))2 ∗

and for all x, y ∈ Rd and all non-negative f ∈ Cb (Rd )  h PT f (y) ≤ (PT f (x)) E p

1 p

i1− p1 pS(T )2−2H C ∗ (H, σ)|x − y|2 . RT Rt (p − 1)2 ( 0 exp[− 0 K(s)ds]dS(t))2

Secondly, for the general case, we use the approximation argument proposed in [10, 16, 18]. Let ˜bt (x) := bt (x) − K(t)x, t ≥ 0, x ∈ Rd . Using (1.2), it is not difficult to see that the mapping id − n−1˜bt : Rd → Rd is injective for any n ∈ N and t ≥ 0. The maps h i (n) bt := n (id − n−1˜bt )−1 (x) − x + K(t)x, n ∈ N, t ≥ 0, x ∈ Rd , 11

are, uniformly for t in compact intervals, globally Lipschitz continuous, see [7]. Denote by (n) (n) (Xt (x))t≥0 the solution of (1.1) with b replaced by b(n) , and define Pt by (3.1) with Xt (x) (n) replaced by Xt (x). Because of the first part of the proof, the statements of the Theorem (n) 3.1 hold with PT replaced by PT . On the other hand, we see as in [16], that (n)

(n)

lim PT f = PT f for all f ∈ Cb (Rd ).

lim XT (x) = XT (x), a.s., hence,

n→∞

n→∞

Therefore, the claim follows if we let n → ∞. References

[1] E. Al`os, O. Mazet, D. Nualart, Stochastic calculus with respect to Gaussian processes, Ann. Probab. 29 (2001) 766-801. [2] M. Arnaudon, A. Thalmaier, F.Y. Wang, Equivalent Harnack and gradient inequalities for pointwise curvature lower bound, Bulletin des Sciences Math´ematiques. 38 (2014) 643-655. [3] F. Biagini, Y. Hu, B. Øksendal, T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Spring, London 2008. [4] S. Boudrahem, P.R. Rougier, Relation between postural control assessment with eyes open and centre of pressure visual feed back effects in healthy individuals, Exp Brain Res. 195 (2009) 145-152. [5] F. Comte, E. Renault, Long memory continuous time models, J Econometrics 73(1)(1996) 101-149. [6] F. De La, AL. Perez-Samartin, L. Matnez, MA. Garcia, A. Vera-Lopez, Long-range correlations in rabbit brain neural activity, Ann Biomed Eng February. 34(2) (2006) 295-299. [7] G. Da Prato, M. R¨ockner, F.Y. Wang, Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups, J. Funct. Anal. 257 (2009) 9921017. ¨ unel, Stochastic analysis of the fractional Brownian motion, [8] L. Decreusefond, A.S. Ust¨ Potential Anal. 10 (1998) 177-214. [9] C.S. Deng, R. Schilling, Harnack inequalities for SDEs driven by time-changed fractional Brownian motions, arXiv:1609.00885v1. [10] C.S. Deng, Harnack inequalities for SDEs driven by subordinate Brownian motions, J. Math. Anal. Appl. 417 (2014) 970-978. 12

[11] C. Jost, Transformation formulas for fractional Brownian motion, Stochastic Processes and their Applications. 116 (2006) 1341-1357. [12] D. Nualart, Y. Ouknine, Regularization of differential equations by fractional noise, Stochastic Processes and their Applications. 102 (2002) 103-116. [13] M. Rypdal, K. Rypdal, Testing hypotheses about sun-climate complexity linking, Phys Rev Lett. 104(12) (2010) 128-151. [14] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yvendon. 1993. [15] I. Simonsen, Measuring anti-correlations in the nordic electricity spot market by wavelets, Physica A. 322(1) (2003) 597-606. [16] F.Y. Wang, J. Wang, Harnack inequalities for stochastic equations driven by L´evy noise, J. Math. Anal. Appl. 410 (2014) 513-523. [17] W. Willinger, W. Leland, M. Taqqu, D. Wilson, On self-similar nature of ethernet traffic, IEEE/ACM Trans Networking. 2(1) (1994) 1-15. [18] X.C. Zhang, Derivative formulas and gradient estimates for SDEs driven by α-stable processes, Stochastic Processes and their Applications. 123 (2013) 1213-1228.

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