Harnack Inequalities for SDEs Driven by Time-Changed Fractional Brownian Motions

We establish Harnack inequalities for stochastic differential equations (SDEs) driven by a time-changed fractional Brownian motion with Hurst parameter $H\in(0,1/2)$. The Harnack inequality is dimension-free if the SDE has a drift which satisfies a one-sided Lipschitz condition, otherwise we still get Harnack-type estimates, but the constants will, in general, depend on the space dimension. Our proof is based on a coupling argument and a regularization argument for the time-change.


Introduction
Throughout this paper, (Ω, A , P) is a probability space. Consider the following d-dimensional SDE (1.1) where b : [0, ∞) × R d → R d is measurable, locally bounded in the time variable t ≥ 0 and continuous in the space variable x ∈ R d ; the driving noise U = (U t ) t≥0 is a locally bounded measurable process on R d starting at zero U 0 = 0. Let us assume, for the time being, that this SDE has a unique non-explosive solution.
In this paper, we want to establish for the solution to the SDE (1.1) a dimensionfree Harnack inequality with power, first introduced by Wang [19] for diffusions on Riemannian manifolds, and a log-Harnack inequality, considered in [16] for semi-linear SDEs. These two Harnack-type inequalities have many applications, for example when studying the strong Feller property, heat kernel estimates, contractivity properties, entropy-cost inequalities, and many more; for an in-depth explanation we refer to the monograph by Wang [20,Subsection 1.4.1] and the references given there. Both, the power-Harnack and log-Harnack inequalities have been thoroughly investigated for various finite-and infinite-dimensional SDEs and SPDEs driven by Brownian noise; the main tool was a coupling method and the Girsanov transformation, see [20] and the references mentioned there. If the noise is a jump process, it is usually very difficult to construct a successful coupling, and the methods from diffusion processes cannot be directly applied. One notable exception are driving noises which are subordinate to a diffusion process.
Let Σ : [0, ∞) → R d ⊗ R d be a measurable and locally bounded deterministic function, and assume that U is of the following form: where W = (W t ) t≥0 is a standard d-dimensional Brownian motion, S = (S(t)) t≥0 is a subordinator (i.e. a non-decreasing process on [0, ∞) with stationary and independent increments a.k.a. increasing Lévy process) and V = (V t ) t≥0 is a locally bounded (B[0, ∞)⊗A /B(R d )-)measurable process on R d with V 0 = 0; we will, in addition, assume that the processes W, S and V are stochastically independent.
In this setting, Wang & Wang [21] were able to obtain Harnack and log-Harnack inequalities, using an approximation of the subordinator (as in [23]) and a coupling argument. The following assumptions turned out to be crucial: The coefficient b has to satisfy a so-called one-sided Lipschitz condition, i.e. there exists a locally bounded (H) moreover, the inverse Σ −1 t exists for each t ≥ 0, and there exists a non-decreasing The first-named author used in [6] the same approximation argument and a gradient estimate approach, in order to improve the Harnack inequalities derived in [21]. Recently, in [22] the approximation argument was also used to establish Harnack-type inequalities for SDEs with non-Lipschitz drift and anisotropic subordinated Brownian noise, i.e. with U having the form is a standard Brownian motion in R d , and S (1) , . . . , S (d) is an independent d-dimensional Lévy process such that each coordinate process S (i) is a subordinator. Unfortunately, this gives only dimension-dependent Harnack inequalities. Note that the techniques of [21,6,22] do not really need that the time-change is a subordinator; we may, as we do here, assume that the time-change is any non-decreasing process on [0, ∞) starting from zero and which is independent of the original process.
It is a natural question to ask whether one can still get Harnack-type inequalities if the driving noise U is a more general, maybe non-Markovian, process. As far as we know, Harnack inequalities were established in [8,10,9] for SDEs driven by fractional Brownian motions. Inspired by these papers as well as [21,6], we will combine general time-change and coupling arguments to obtain Harnack inequalities for SDEs driven by time-changed fractional Brownian motions. Brownian motions delayed by inverse α-stable subordinators. Assume that U = W H Z + V where V is a locally bounded measurable process on R d starting from zero V 0 = 0. In this paper, we restrict ourselves to the case H ∈ (0, 1/2).
In order to ensure the existence and uniqueness of the solution to the SDE (1.1) and to construct a successful coupling, we assume that the coefficient b satisfies the one-sided Lipschitz condition (H). As a direct consequence of the log-Harnack inequality, we obtain a gradient estimate for the associated Markov operator.
As in [22], we can also deal with the anisotropic case, i.e.
where, for each i = 1, . . . , d, W Hi,(i) = (W Hi,(i) t ) t≥0 is a real-valued fractional Brownian motion with Hurst index H i ∈ (0, 1/2), Z (i) = (Z (i) (t)) t≥0 is a one-dimensional non-decreasing process such that Z (i) (0) = 0, and V = (V t ) t≥0 is a locally bounded measurable process with values in R d and V 0 = 0; moreover, we assume that these processes are independent. As in [22], we replace the Lipschitz condition for the drift coefficient b by a Yamada-Watanabe-type condition; in general, however, this condition cannot be compared with the one-sided Lipschitz condition.
The remaining part of this paper is organized as follows. We collect some basics on fractional Brownian motions in Section 2. In Section 3 we establish the Harnack inequalities for SDEs driven by a time-changed fractional Brownian motion and with drift coefficient satisfying the one-sided Lipschitz condition (H). More explicit expressions in the Harnack and log-Harnack inequalities are obtained if the time-change Z is (the inverse of) a subordinator; this is a consequence of our moment estimates from [7]; if Z is the inverse of a subordinator, only the log-Harnack inequality holds, since the exponential moment of Z(t) −θ is usually infinite for θ > 0. The last section is devoted to the case of an anisotropic driving noise; as one would expect from [22], the Harnack inequalities turn out to be dimension-dependent.

Basics of fractional Brownian motion
In this section, we recall briefly some basic facts on fractional Brownian motion (fBM) which will be used later on. For further details of fBM and proofs we refer the readers, for instance, to [2,5] or [14].
Denote by Γ(·), resp., B(·, ·), the Euler Gamma and Beta functions, and write 2 F 1 for Gauss' hypergeometric function. Let a, b ∈ R with a < b. For f ∈ L 1 [a, b] and α > 0, be a fractional Brownian motion on R d with Hurst parameter H ∈ (0, 1/2) ∪ (1/2, 1) and define for 0 < s < t the kernel It is known that the operator K H , associated with the kernel K H (·, ·) [17, p. 187] or [5]. Moreover, fractional Brownian motion has the following integral representation with respect to a standard d-dimensional Brownian motion W = (W t ) t≥0 : In particular, if 0 < H < 1 2 and h ∈ I the inverse operator is given by see also [15, p. 108].

SDEs driven by delayed fractional Brownian motions
Consider the following SDE on R d are stochastically independent and satisfy Moreover, we assume that the coefficient b satisfies the one-sided Lipschitz condition (H).
Remark 3.1. The one-sided Lipschitz condition (H) ensures, in particular, the existence, uniqueness and non-explosion of the solution to the SDE (3.1). Indeed, it is well known that the following ordinary differential equation we conclude that the the SDE (3.1) has a unique non-explosive solution.
Throughout this section, we write |x|

Statement of the main result
In order to state our main result, we need the following notation: where k(s) is the constant appearing in (H), and we denote for any function f : Theorem 3.2. We assume that (3.2) and (H) hold for the SDE (3.1) and we denote its unique solution by X t (x).
i) For T > 0, x, y ∈ R d and any bounded Borel function f : ii) For T > 0, x ∈ R d and any bounded Borel function f :

Proof of Theorem 3.2
For the proof of Theorem 3.2, we need a few preparations. Let : [0, ∞) → [0, ∞) be a non-decreasing and càdlàg function with (0) = 0, and v : [0, ∞) → R a locally bounded measurable function with v(0) = 0. By Remark 3.1 the following SDE has a unique non-explosive solution We want to transform the equation (3.6) into an SDE driven by a fractional Brownian motion which will allow us to establish Harnack inequalities using a combination of coupling and the Girsanov transformation, cf. [8,10,9]. First, however, we have to approximate the (deterministic) time-change by an absolutely continuous function. Consider the following regularization of : By construction, for each ∈ (0, 1) the function is absolutely continuous, strictly increasing and satisfies for any t ≥ 0 and define P ,v by (3.7) with instead of .

Lemma 3.3.
We assume that (3.2) and (H) hold for the SDE (3.1) and we denote its unique solution by X t (x). Fix ∈ (0, 1) and let and X ,v t (x) be as above.
i) For T > 0, x, y ∈ R d and any bounded Borel function f : ii) For T > 0, x, y ∈ R d , p > 1 and any bounded Borel function f : Proof. Fix T > 0, x, y ∈ R d and denote by (Y t ) t≥0 a solution of the equation is locally Lipschitz continuous off the diagonal, the system of coupled equations (3.9) and (3.10) has a unique solution for t ∈ [0, τ ). If τ < ∞, we set Y t = X ,v t (x) for all t ≥ τ . In this way, we can construct a unique solution (Y t ) t≥0 to (3.10). Let us show that the coupling time satisfies τ ≤ T . Let t < τ , write ζ t for the difference of the solutions to the SDEs (3.9) and (3.10), and observe that ζ t admits a classic differential satisfying d|ζ t | = 1 {ζt =0} |ζ t | −1 ζ t , dζ t ; therefore, (H) yields Now assume that τ (ω) > T for some ω ∈ Ω. Taking t = T in the above inequality, we get By definition, (γ (t)) = t for t ≥ (0), γ ( (t)) = t for t ≥ 0, and t → γ (t) is absolutely continuous and strictly increasing. Let A simple calculation shows |x − y| s respectively. Thus, the distribution of (X ,v T (y)) 0≤t≤T under P coincides with the law of (Y t ) 0≤t≤T under RP; in particular, we get for all bounded Borel functions f : By the Jensen inequality, we get for any random variable F ≥ 1, Combining this with (3.12) and the observation that we get for all bounded Borel functions f : This completes the proof of the log-Harnack inequality.
Using (3.11) we get

is a martingale with mean 1 -this is due to Novikov's criterion -we get
Inserting this expression into (3.13), completes the proof of the power-Harnack inequality.

Proof of Theorem 3.2. By [1, Proposition 2.3], ii) is a direct consequence of i).
Fix T > 0. By a standard approximation argument, it is enough to prove the formulae in i) and iii) for f ∈ C b (R d ).
Step 1: Assume that b t : R d → R d is, uniformly for t in compact intervals, a global Lipschitz function, i.e. for any t > 0 there is some C t > 0 such that (3.14) This implies that for all x ∈ R d and ∈ (0, 1) Since X ,v t (x) and X ,v t (x) are non-explosive, the integral in the above expression is finite. Therefore, we can apply Gronwall's inequality with g( , t) and find Since is of bounded variation, the limit ↓ also holds for the integrals We can now use Lemma 3.3 i) and let ↓ 0 to get Since the processes W , V and Z are independent, v=V holds for all bounded Borel functions f : R d → R. Thus, the Jensen inequality yields for all x, y ∈ R d and f ∈ C b (R d ) with f ≥ 1 For the power-Harnack inequality we use Hölder's inequality to find for all x, y ∈ R d and non-negative f ∈ C b (R d ) Step 2: For the general case, we use the approximation argument proposed in [21, part (c) are, uniformly for t in compact intervals, globally Lipschitz continuous, see [4]. Denote by (X (n) t (x)) t≥0 the solution of (3.1) with b replaced by b (n) , and define P Therefore, the claim follows if we let n → ∞.
We will call S −1 = (S −1 (t)) t≥0 an inverse subordinator associated with the Bernstein function φ. Since we assume that the subordinator S is strictly increasing, we know that almost all paths of S −1 are continuous and non-decreasing. We will frequently use the following identity: P (S(r) ≥ t) = P S −1 (t) ≤ r , r, t > 0. If lim sup r↓0 φ(r)r −σ < ∞ and lim sup r→∞ φ(r)r −σ < ∞ for some σ > 0, then the following assertions hold.
i) For any T > 0, x, y ∈ R d and all bounded Borel functions f : ii) For any T > 0, x ∈ R d and all bounded Borel functions f :   Proof. By our assumption, there exists a constant c = c(σ) > 0 such that φ(r) ≤ c r σ for all r > 0. Combining this with and Tonelli's theorem, we get that for all s, t > 0 This yields for all t > 0 Remark 3.7. Let α ∈ (0, 1). For an α-stable subordinator S, the corresponding Bernstein function φ(r) = r α satisfies the conditions of Corollary 3.5 with σ = α. Because of (3.16) and the well known two-sided estimate P (S(r) ≥ t) 1 ∧ rt −α , r, t > 0, (f g means that c −1 f (t) ≤ g(t) ≤ cf (t) for some c ≥ 1 and all t) we have for any t > 0 This shows that Lemma 3.6 is sharp for α-stable subordinators.
This means that we cannot expect, in the setting of Corollary 3.5, to get a power-Harnack inequality as we did in Corollary 3.4 iii).

SDEs with non-Lipschitz drift and anisotropic noise
W Hi,(i) are fBMs on R with Hurst parameter H i ∈ (0, 1/2); V is a locally bounded measurable process on R d with V 0 = 0. We consider the following stochastic equation on R d : In this section, we will use the 1 -norm on R d , i.e. x 1 := |x (1) | + · · · + |x (d) |, x ∈ R d , and we replace the one-sided Lipschitz condition (H) by the following Yamada-Watanabetype condition   

 
There exists some u ∈ U and a locally bounded As in Section 3, it is easy to see that (A) guarantees the existence, uniqueness and non-explosion of the solution to (4.2). We define for bounded Borel functions f :

Statement of the main result
Let k(t) be the constant appearing in (A), and denote by K(t) = t 0 k(s) ds its primitive. For i ∈ {1, . . . , d} we define Θ Hi by (3.5) with H i instead of H. Finally, we set for u ∈ U  G −1 u is the inverse function of G u . Since u ∈ U , it is easy to see that G u is strictly increasing with G u (0) = −∞ and lim r↑∞ G u (r) = ∞, so that Φ u,k is well-defined. If, in particular, u(s) = cs for some constant c > 0, then Φ u,k (t, r) = 1 + c t 0 k(s) e cK(s) ds r, t, r ≥ 0.  i) For T > 0, x, y ∈ R d , and any bounded Borel function f : ii) For T > 0, x, y ∈ R d , p > 1 and any bounded Borel function f : R d → [0, ∞) iii) If (A) holds with u(s) = cs for some constant c > 0, then we have for T > 0, x ∈ R d and any bounded Borel function f : Θ Hi (Z (i) (T )) 2Hi .

Deterministic time-changes
The proof of Theorem 4.2 uses the same strategy as the proof of Theorem 3.