Stochastic functional Hamiltonian system with singular coefficients

By Zvonkin type transforms, the existence and uniqueness of the strong solutions for a class of stochastic functional Hamiltonian systems are obtained, where the drift contains a Holder-Dini continuous perturbation. Moreover, under some reasonable conditions, the non-explosion of the solution is proved. In addition, as applications, the Harnack and shift Harnack inequalities are derived by method of coupling by change of measure. These inequalities are new even in the case without delay and the shift Harnack inequality is also new even in the non-degenerate functional SDEs with singular drifts.


1.
Introduction. As a typical model of degenerate diffusion system, the stochastic Hamiltonian system has been investigated in [10,23,24,26], see [4] for the functional version of this model with regular coefficients. Recently, Zvonkin type transforms ( [29]) have been used to prove existence and uniqueness of the strong solutions fo SDEs with singular drift, see e.g. [1,11,14,16,21,27]. In [13,15], the first author and his co-author have investigated the non-degenerate functional SPDEs with Dini continuous drift, see also [2,12] for the finite dimensional non-degenerate functional SDEs with integrable drifts. So far, there is few results on the degenerate functional SDEs with singular coefficients. The purpose of this paper is to deal with this problem. We will adopt the Zvonkin type transforms considered in [25] for SDEs without delay which enable us to regularize a singular perturbation without time delay. The main difficulty is to treat the delay part, which is a function on an infinite dimension space. To this end, we construct a family of homeomorphisms on C d1+d2 (see (17) below) besides the homeomorphisms on R d1+d2 .
On the other hand, Harnack and shift Harnack inequalities have many applications. For instance, Harnack inequalities can yield strong Feller property and the uniqueness of invariant probability measure, shift Harnack inequalities implies the regularity of heat kernel with respect to Lebesgue measure, see [22, Chapter 1] for more details. For the stochastic Hamiltonian system with multiplicative noise, the coupling by change of measure is so hard to establish that there is no any result on the Harnack inequalities for SDEs of this type. Unfortunately, the Zvonkin type transforms make a stochastic Hamiltonian system even with additive noise into a new one with multiplicative noise. Thus, we can not obtain the Harnack inequalities for the original SDEs from the ones after Zvonkin type transforms as in the non-degenerate case, see [13,15]. Instead, we adopt a new idea, i.e. directly construct the coupling by change of measure for the original SDE with singular drift. Compared with the result in [4], Σ(T, h, r) in Harnack inequality in Theorem 3.2 contains two additional terms, i.e. the second and third term in Σ(T, h, r), which comes from the singularity of the drift Z in (27).
The paper is organized as follows: In Section 2, we prove the existence and uniqueness of the solution for the stochastic functional Hamiltonian system; In Section 3, we investigate the Harnack and shift Harnack inequalities and their applications.
Throughout the paper, the letter C or c will denote a positive constant, and C(θ) or c(θ) stands for a constant depending on θ. The value of the constants may change from one appearance to another.
2. Existence and uniqueness. Fix a constant r > 0. For any d ∈ N + , let which is called the segment process.
Consider the following stochastic functional Hamiltonian system on R d1+d2 : where W = (W (t)) t≥0 is a d 2 -dimensional standard Brownian motion with respect to a complete filtered probability space (Ω, are measurable and locally bounded (bounded on bounded sets). When B = 0 and b = ∇V for some potential V , (1) is called stochastic Hamiltonian system, which includes the kinetic Fokker-Planck equation as a typical example (see [20]). If d 1 = 0, then equation (1) reduces to the non-degenerate case.
In the following, we will use ∇ (1) and ∇ (2) to denote the gradient operators on the first space R d1 and the second space R d2 respectively. For simplicity, we denote is called a strong solution to (1) with life time ζ, if the segment process X t is F t -measurable, and ζ > 0 is a stopping time such that P-a.s lim sup t↑ζ |X(t)| = ∞ holds on {ζ < ∞}, and P-a.s When B = 0, the infinitesimal generator associated to (1) is where Σ(t, ·) := 1 2 σ(t, ·)σ * (t·), tr(·) denotes the trace of a matrix, and ∇ b(t,·) f := ∇f, b(t, ·) is the directional derivative of f along b(t, ·) for a differentiable function on R d1+d2 .
A function f defined on the Euclidean space is called Hölder − Dini continuous of order α ∈ [0, 1) if holds for some Dini function φ, and is called Dini − continuous if this condition holds for α = 0.
A measurable function φ : [0, ∞) → [0, ∞) is called a slowly varying function at zero (see [3]) if for any λ > 0, Let D 0 be the set of all Dini functions, and T 0 the set of all slowly varying functions at zero that are bounded away from 0 and ∞ on [ε, ∞) for any ε > 0.

Notice that the typical examples for functions contained in
To characterise the non-Lipschitz condition of B and σ, we introduce the class Typical functions in D 1 contain We will need the following local condition (see [25, (A)] for details).
holds for some positive increasing function Φ, then the solution to (1) is nonexplosive and for any p, T > 0, there exists a constant C 1 (p, T, Φ) depending on p, T, Φ such that Remark 2.2. We should remark that Zvonkin's transform can not regularize the functional drift with singular condition. That is why we assume the functional part is regular. In fact, the finite dimensional noise W can not remove the drift B which is a function on the infinite dimensional space C d1+d2 . More precisely, if we adopt the same trick as in Zvonkin's transform, we need to deal with the equation like STOCHASTIC FUNCTIONAL HAMILTONIAN SYSTEM ...

1261
Proof of Theorem 2.1. We first assume that (A) holds for some C n = C, φ n = φ and γ n = γ independent of n ≥ 1.
(2) Next, if σ and b do not depend on x (1) , then so does u n . In this case, if (H1), (H4) and (7) hold with C n , φ n and γ n uniformly in n ≥ 1, then by [25, (3.24)], we may repeat the above argument to prove the pathwise uniqueness.
(4) Finally, we prove the non-explosion. [22,Lemma 4.4.6] gives the proof for the time-homegeneous SDEs with additive noise. For reader's convenience, we give the proof in detail.
Let X ξ t be the solution to (1) up to life time ζ(ξ). For simplicity, we denote X t = X ξ t . Define τ n = inf{t ≥ 0, |X(t)| ≥ n}. By Itô's formula, and using (8), it is easy to see that s H(X(s))ds for some constant C > 0. Combining (25) with (26), there exists a constant H p (X(t)) It follows from Gronwall's lemma that Letting n → ∞ in the above inequality, we obtain Since lim |x|→∞ H(x) = ∞, it is easy to see that the solution is non-explosive.

Harnack and shift Harnack inequalities.
In this section, consider the following stochastic functional Hamiltonian system on R m+d : where W = (W (t)) t≥0 is an d-dimensional standard Brownian motion with respect to a complete filtered probability space (Ω, When m = 0, we let M = 0 and C m+d = C d := C([−r, 0]; R d ), and then (27) reduces to non-degenerate functional SDEs. Throughout this section, we make the following assumptions: (A1) (Hypoellipticity) σ is invertible and M M * is invertible if m > 0.
(A2) (Regularity and growth of Z) There exists φ ∈ D 0 ∩ T 0 such that for any z,z ∈ R m+d , Moreover, there exists a constant C > 0 such that (A3) (Regularity of B) There exists a constant C > 0 such that for any ξ, η ∈ C m+d ,  (1) and (3), (A1)-(A3) implies that (27) has a unique non-explosive strong solution X ξ t with X 0 = ξ ∈ C m+d . Let P t be the associated Markov semigroup, i.e.
is used to construct the coupling by change of measure for the Harnack inequalities and shift Harnack inequalities.
We use the coupling constructed in [4] to derive the Harnack inequalities.
and C > 0 is a constant. If m = 0 then the assertion holds for M = 0. In addition, since lim h ∞→0 Σ(h, T, r) = 0, P T is strong Feller for any T > r.
Proof of Theorem 3.2. By the semigroup property and Jensen's inequality, we only need to consider T − r ∈ (0, 1]. For any η ∈ C m+d , let (X η (t), Y η (t)) solve (27) with (X 0 , Y 0 ) = η. As in [4], for h = (h 1 , h 2 ) ∈ C m+d , definẽ which implies that the distribution of (X T ,Ỹ T ) under Q T coincides with that of (X ξ+h T , Y ξ+h T ) under P. On the other hand, by Young's inequality, P T log f (ξ + h) = E Q T log f ((X T ,Ỹ T )) = E Q T log f ((X ξ T , Y ξ T )) ≤ log P T f (ξ) + ER(T ) log R(T ), and by Hölder inequality, SinceW is a Brownian motion under Q T , by the definition of R(T ), it is easy to see that  and any positive f ∈ B b (C m+d ),