Harnack Inequalities for Stochastic (Functional) Differential Equations with Non-Lipschitzian Coefficients

By using coupling arguments, Harnack type inequalities are established for a class of stochastic (functional) differential equations with multiplicative noises and non-Lipschitzian coefficients. To construct the required couplings, two results on existence and uniqueness of solutions on an open domain are presented.


Introduction
Consider the following stochastic differential equation (SDE): (1.1) dX(t) = σ(t, X(t))dB(t) + b(t, X(t))dt, where (B(t)) t≥0 is the d-dimensional Brownian motion on a complete filtered probability space (Ω, (F t ) t≥0 , F , P), σ : [0, ∞)×R d → R d ⊗R d and b : [0, ∞)×R d → R d are measurable, locally bounded in the first variable and continuous in the second variable. This timedependent stochastic differential equation has intrinsic links to non-linear PDEs (cf. [19]) as well as geometry with time-dependent metric (cf. [8]). When the equation has a unique solution for any initial data x, we denote the solution by X x (t). In this paper we aim to investigate Harnack inequalities for the associated family of Markov operators (P (t)) t≥0 : where B b (R d ) is the set of all bounded measurable functions on R d .
In the recent work [23] the second named author established some Harnack-type inequalities for P (t) under certain ellipticity and semi-Lipschitz conditions. Precisely, if there exists an increasing function K : [0, ∞) → R such that σ(t, x) − σ(t, y) 2 HS + 2 b(t, x) − b(t, y), x − y ≤ K(t)|x − y| 2 , x, y ∈ R d , t ≥ 0, and there exists a decreasing function λ : [0, ∞) → (0, ∞) such that then for each T > 0, the log-Harnack inequality , x, y ∈ R d holds for all strictly positive f ∈ B b (R d ). If, in addition, there exists an increasing function δ : [0, ∞) → (0, ∞) such that almost surely σ(t, x) − σ(t, y) * (x − y) ≤ δ(t)|x − y|, x, y ∈ R d , t ≥ 0, then for p > (1 + δ(T ) λ(T ) ) 2 there exists a positive constant C(T ) (see [23,Theorem 1.1(2)] for expression of this constant) such that the following Harnack inequality with power p holds: This type Harnack inequality is first introduced in [20] for diffusions on Riemannian manifolds, while the log-Harnack inequality is firstly studied in [14,22] for semi-linear SPDEs and reflecting diffusion process on Riemannian manifolds respectively. Both inequalities have been extended and applied in the study of various finite-and infinite-dimensional models, see [1,2,4,5,7,12,13,21,23] and references within. In particular, these inequalities have been studied in [24] for the stochastic functional differential equations (SFDE) (1.4) dX(t) = Z(t, X(t)) + a(t, X t ) dt + σ(t, X(t))dB(t), X 0 ∈ C , where C = C([−r 0 , 0]; R d ) for a fixed constant r 0 > 0 is equipped with the uniform norm · ∞ ; X t ∈ C is given by X t (u) = X(t + u), u ∈ [−r 0 , 0]; σ : [0, ∞) × R d → R d ⊗ R d , Z : [0, ∞) × R d → R d , and a : [0, ∞) × C → R d are measurable, locally bounded in the first variable and continuous in the second variable. Let X φ t be the solution to this equation with X 0 = φ ∈ C . In [24] the log-Harnack inequality of type (1.2) and the Harnack inequality of type (1.3) were established for provided σ is invertible and for any T > 0 there exist constants K 1 , K 2 ≥ 0, K 3 > 0 and The aim of this paper is to extend the above mentioned results to SDEs and SFDEs with less regular coefficients as considered in Fang and Zhang [6] (see also [11]), where the existence and uniqueness of solutions were investigated. In section 2, we consider the SDE case; and in section 3, we consider the SFDE case. Finally, in section 4 we present two results for the existence and uniqueness of solutions on open domains of SDEs and SFDEs with non-Lipschitz coefficients, which are crucial for constructions of couplings in the proof of Harnack-type inequalities.

SDE with non-Lipschitzian coefficients
To characterize the non-Lipschitz regularity of coefficients, we introduce the class Here, the restriction that u ≥ 1 is more technical than essential, since in applications one may usually replace u by u ∨ 1 (see condition (H1) below). To ensure the existence and uniqueness of the solution and to establish the log-Harnack inequality, we shall need the following assumptions: (H1) There exist u,ũ ∈ U with u ′ ≤ 0 and increasing functions K,K ∈ C([0, ∞); (0, ∞)) such that for all t ≥ 0 and x, y ∈ R d , (H2) There exists a decreasing function λ ∈ C([0, ∞); (0, ∞)) such that The log-Harnack inequality we are establishing depends only on functions u, K and λ,K andũ will be only used to ensure the existence of coupling constructed in the proof. As in [23], in order to derive the Harnack inequality with a power, we need the following additional assumption: Theorem 2.1. Assume that (H1) holds.
(1) For any initial data X(0), the equation (1.1) has a unique solution, and the solution is non-explosive.
Although the main idea of the proof is based on [23], due to the non-Lipschitzian coefficients we have to overcome additional difficulties for the construction of coupling. In fact, to show that the coupling we are going to construct is well defined, a new result concerning existence and uniqueness of solutions to SDEs on a domain is addressed in section 4.

Construction of the coupling and some estimates
It is easy to see fromm Theorem 4.1 that the equation (1.1) has a unique strong solution which is non-explosive (see the beginning of the next subsection). To establish the desired log-Harnack inequality, we modify the coupling constructed in [23]. For fixed T > 0 and θ ∈ (0, 2), let then ξ is a smooth and strictly positive on [0, T ) so that For any x, y ∈ R d , we construct the coupling processes (X(t), Y (t)) t≥0 as follows: We intend to show that the Y (t) (hence, the coupling process) is well defined up to time τ and τ ≤ T , where τ := inf{t ≥ 0 : is the coupling time. To this end, we apply Theorem 4.1 to It is easy to verify (4.2) from (H1). Then Y (t) is well defined up to time ζ ∧ τ , where ζ := lim n→∞ ζ n , and ζ n := inf{t ∈ [0, T ); |Y (t)| ≥ n}.
Here and in what follows, we set inf ∅ = ∞. As in [23], to derive Harnack-type inequalities, we need to prove that the coupling is successful before ζ ∧ T under the weighted probability Q := R(T ∧τ ∧ζ)P, where for s ∈ [0, T ∧ζ ∧τ ). To ensure the existence of the density R(T ∧τ ∧ζ) , letting we verify that (R(s∧ζ n ∧τ n )) s∈[0,T ),n≥1 is uniformly integrable, so that is a well defined probability density due to the martingale convergence theorem. Then we prove that ζ ∧T ≥ τ a.s.-Q, so that Q = R(τ )P. Both assertions are ensured by the following lemma. .
exists as a probability density function of P, and . .
Then, before time T ∧ τ ∧ ζ, (2.4) can be reformulated as For fixed s ∈ [0, T ) and n ≥ 1, let ϑ n,s = s ∧ τ n ∧ ζ n and Q n,s = R(ϑ n,s )P. Then by the Girsanov theorem, ( . By the Itô formula and condition (H1), we obtain Applying the Itô formula to ϕ(|Z(t)| 2 ) and noting that ϕ ′′ = u ′ ≤ 0, we derive Taking the expectation w.r.t. the probability measure Q n,s and noting ( B(t)) t∈[0,ϑn,s] is a Brownian motion under Q n,s , we get On the other hand, it follows from (H2) that Combining with (2.9), we arrive at This implies the desired inequality in (1), and the consequence then follows from the martingale convergence theorem.
If moreover (H3) holds, then we have the following moment estimate on R(τ ), which will be used to prove the Harnack inequality with power.
Proof. By (2.8) and (H3), for any r > 0 we have where in the last step we use the inequality for a continuous exponentially integrable martingale M(t), and M (t) denotes the quadratic variational process corresponding to M(t). Putting r = θ 2 8δ(T ) 2 such that r = Due to Lemma 2.2, we have τ ≤ T ∧ ζ, Q-a.s. By taking s = T − n −1 and letting n → ∞ in the above inequality, we arrive at Since for any continuous Q-martingale M(t) Taking q = 1 + 1 + p −1 which minimizes q(pq + 1)/(q − 1), and using the definition of p, we have Combining this with (2.13), we complete the proof.

Proof of Theorem 2.1
According to Theorem 4.1 below for D = R d , (H1) implies that (1.1) has a unique solution.
Since u is decreasing, the first inequality in (H1) with y = 0 implies that for |x| ≥ 1, Moreover, the second inequality in (H1) with y = 0 implies that for |x| ≥ 1, where [|x|] stands for the integer part of |x|. Combining this with (2.14) we may find a function h ∈ C([0, ∞); (0, ∞)) such that which implies the non-explosion of X(t) as is well known. Thus, the proof of (1) is finished. Next, by Lemma 2.2 and the Girsanov theorem, is a d-dimensional Brownian motion under the probability measure Q. Then, according to Theorem 2.1(1), the equation has a unique solution for all t ≥ 0. Moreover, it is easy to see that (X(t)) t≥0 solves the equation Thus, we have extended equation (2.7) to all t ≥ 0, which has a global solution (X(t), Y (t)) t≥0 under the probability measure Q, and τ := inf{t ≥ 0 : Moreover, since the equations (2.15) and (2.16) coincide for t ≥ τ , by the uniqueness of the solution and X(τ ) = Y (τ ), we conclude that X(T ) = Y (T ), Q-a.s. Now, by Lemma 2.2 and the Young inequality we obtain Taking θ = 1, we derive the desired log-Harnack inequality. Moreover, by the Hölder inequality, for any q > 1 we have Setting q = 1 + 4δ(T ) 2 + 4θλ(T )δ(T ) λ(T ) 2 θ 2 such that it then follows from Lemma 2.3 that It is easy to see that for any q > 1 + δ(T ) 2 +2λ(T )δ(T ) Therefore, the desired Harnack inequality with power q follows. Let T > r 0 be fixed, for

SFDEs with non-Lipschitzian coefficients
Consider the following type of stochastic functional differential equation holds for all t ≥ 0 and φ, ψ ∈ C . Since su(s) is increasing and concave in s, we have su(s) ≤ c(1 + s) for some constant c > 0. Therefore, it is easy to see that the above conditions also imply the non-explosion of the solution. Let X φ t be the segment solution to (3.1) for X 0 = φ. We aim to establish the Harnack inequality for the associated Markov operators (P t ) t≥0 : As already known in [5,24], to establish a Harnack inequality using coupling method, one has to assume thatσ(·, φ) depends only on φ(0); that is,σ(t, φ) = σ(t, φ(0)) holds for some Therefore, below we will consider the equation locally bounded in the first variable and continuous in the second variable. We shall make use of the following assumption, which is weaker than (1)-(4) introduced in the end of Section 1 since u might be unbounded.
The proof is modified from Section 2. But in the present setting we are not able to derive the Harnack inequality with power as in Theorem 2.1(3). The reason is that according to the proof of Lemma 3.3 below, to estimate ER(τ ) q for q > 0 one needs upper bounds of the exponential moments of Z t 2 ∞ u( Z t 2 ∞ ), which is however not available.
Let T > 0 and φ, ψ ∈ C be fixed. Combining the construction of coupling in Section 2 for the SDE case with non-Lipschitz coefficients and that in [24] for the SFDE case with Lipschitz coefficients, we construct the coupling process (X(t), Y (t)) as follows: As In particular,τ ≤ T implies that X T +r 0 = X T +r 0 . To show thatτ ≤ T , we make use of the Girsanov theorem as in Section 2. Let Z(t) = X(t) − Y (t) and We intend to show that For t < T ∧ζ n ∧τ n , rewrite (3.4) as We have Z 0 = φ − ψ and for t < T ∧τ n ∧ζ n .
Since su(s) is concave in s so that E s,n [ℓ n (r)u(ℓ n (r))] ≤ E s,n ℓ n (r)u(E s,n ℓ n (r)), this implies that t 0 E s,n ℓ n (r)u(E s,n ℓ n (r))dr, t ≤ s.
Therefore, the desired estimate follows from the Bihari's inequality. E R(s∧ζ n ∧τ n ) log R(s∧ζ n ∧τ n ) Proof. By the first inequality in (A2), (3.7) and using the Itô formula, we obtain for t ≤ s ∧τ n ∧ζ n . So, as in the proof of Lemma 2.2, there exists a Q s,n -martingale M(t) such that for t ≤ s ∧τ n ∧ζ n , where in the last step we have used u ≥ 1 andξ ′ (t) = − 1 2γ . Therefore, Proof of Theorem 3.1. As discussed in Section 2 that Lemma 3.3 and (3.9) imply thatτ ≤ T ∧ζ Q-a.s., where Q := R(τ ∧ T ∧ζ)P = R(τ )P. Since by the construction we have X(t) = Y (t) for t ≥τ , this implies that X T +r 0 = Y T +r 0 . Applying the Young inequality and Lemma 3.3, we obtain

Existence and uniqueness of solutions
There are a lot of literature on the existence and uniqueness of SDEs and SFDEs under non-Lipschitz condition, see e.g. Taniguchi [17,18] and references therein. In the following two subsections, for the construction of couplings given in the previous sections, we present below two results in this direction for SDEs and SFDEs on open domains respectively.

Stochastic differential equations
Let D be a non-empty open domain in R d , and let T > 0 be fixed. Consider the following SDE: where (B(t)) t≥0 is the m-dimensional Brownian motion on a complete filtered probability space (Ω, locally bounded in the first variable and continuous in the second variable. Theorem 4.1. If there exist u ∈ U , a sequence of compact sets K n ↑ D and functions {Θ n } n≥1 ∈ C([0, T ); (0, ∞)) such that for every n ≥ 1, Then for any initial data X(0) ∈ D, the equation (4.1) has a unique solution X(t) up to life time where inf ∅ := ∞.
Proof. For each n ≥ 1, we may find h n ∈ C ∞ (R d ) with compact support contained in D such that h n | Kn = 1. Let Then for any n ≥ 1, b n and σ n are bounded on [0, nT n+1 ] × R d and continuous in the second variable. According to the Skorokhod theorem [15] (see also [9, Theorem 0.1]), the equation dX n (t) = σ n (t, X n (t))dB(t) + b n (t, X n (t))dt, X n (0) = X 0 has a weak solution for t ∈ [0, nT n+1 ]. So, by Yamada-Watanabe principle [25], to prove the existence and uniqueness of the (strong) solution, we only need to verify the pathwise uniqueness.
Therefore, (4.1) has a unique solution up to the life time ζ = T ∧ lim n→∞ ζ n .

Stochastic functional differential equations
T ) and compact set K ⊂ D, and continuous in the second variable.
Theorem 4.2. Assume that there exists a sequence of compact sets K n ↑ D such that for every n ≥ 1, hold for some u n ∈Ū and all φ, ψ ∈ K C n , t ≤ nT n+1 . Then for any initial data X 0 ∈ D C , the equation Applying [3, Corollary 1.3] for σ = 1 n I d×d , m = 0, Z = b = 0 and T = 1 + r 0 , we see that for every n = 1,σ n andb n are Lipschitz continuous in the second variable uniformly in the first variable. Therefore, the equation dX (n) (t) =b n (t, X |h(t + s)) − h(t)| s ε for a fixed number ε ∈ (0, 1 2 ). It is well known that g ε is a compact function on C([0, T ′ ]; R d ), i.e. {g ε ≤ r} is compact under the uniform norm for any r > 0. Sinceb n andσ n are uniformly bounded and ε ∈ (0, 1 2 ), we have sup n≥1 Eg ε (X (n) ) < ∞.
Sinceσ n →σ andb n →b uniformly and P (n) → P weakly, by letting n → ∞ we conclude that is a P-martingale with According to [10, Theorem II.7.1], this implies for some m-dimensional Brownian motion B on the filtered probability space (Ω, F t , P). Therefore, the equation has a weak solution up to time T ′ .
Noting that su n (s) is increasing in s, we have Z t 2 ∞ u n ( Z t 2 ∞ ) ≤ ℓ n (t)u n (ℓ n (t)), t ≥ 0.
Since s → su n (s) is concave, due to Jensen's inequality this implies that Eℓ n (s)u n Eℓ n (s) ds.