Hypercontractivity for Functional Stochastic Partial Differential Equations

Explicit sufficient conditions on the hypercontractivity are presented for two classes of functional stochastic partial differential equations driven by, respectively, non-degenerate and degenerate Gaussian noises. Consequently, these conditions imply that the associated Markov semigroup is $L^2$-compact and exponentially convergent to the stationary distribution in entropy, variance and total variational norm. As the log-Sobolev inequality is invalid under the framework, we apply a criterion presented in the recent paper \cite{Wang14} using Harnack inequality, coupling property and Gaussian concentration property of the stationary distribution. To verify the concentration property, we prove a Fernique type inequality for infinite-dimensional Gaussian processes which might be interesting by itself.


Introduction
The hypercontractivity was introduced in 1973 by Nelson [10] for the Ornstein-Ulenbeck semigroup. As applications, it implies the exponential convergence of the Markov semigroup in entropy (and hence, also in variance) to the associated stationary distribution, and it also implies the L 2 -compactness of the semigroup subject to the existence of a density with respect to the stationary distribution, see [15] for more details. In the setting of symmetric Markov processes, Gross [9] proved that the hypercontractivity of the semigroup is equivalent to the log-Sobolev inequality for the associated Dirichlet form. This leads to an intensive study of the log-Sobolev inequality.
However, as explained in [3] that the log-Sobolev inequality does not hold for the segment solution to a stochastic delay differential equation (SDDE). As the segment solution is a process on a functional space, the equation is also called a functional stochastic differential equation (FSDE). In this case, an efficient tool to prove the hypercontractivity is the dimension-free Harnack inequality introduced in [11], where diffusion semigroups on Riemannian manifolds are concerned. By using the coupling by change of measures, this type Harnack inequality has been established for various stochastic equations, see the recent monograph [14] and references within. The aim of the present paper is to prove the hypercontractivity for functional stochastic partial differential equations (FSPDEs) in Hilbert spaces. We will consider non-degenerate noise and degenerate noise, respectively, so that the corresponding results derived in [3] for finite-dimensional FSDEs as well as in [15] for degenerate SPDEs are extended.
In the recent paper [15], the second named author presented a general criterion on the hypercontractivity by using the Harnack inequality of the semigroup, the concentration property of the underlying probability measure, and a coupling property. In general, let P t be a Markov semigroup on L 2 (µ) for a probability space (E, F , µ) such that µ is P tinvariant. By definition, P t is hypercontractive if P t 2→4 = 1 holds for large enough t > 0, where · 2→4 is the operator norm form L 2 (µ) to L 4 (µ). For any (x, y) ∈ E × E, a process (X t , Y t ) on E × E is called a coupling for the Markov semigroup with initial point (x, y) if where B b (·) stands for the set of all bounded measurable functions on a measurable space. The criterion is stated as follows. 14]). Assume that the following three conditions hold for some measurable functions ρ : E × E → (0, ∞) and φ : [0, ∞) → (0, ∞) such that lim t→∞ φ(t) = 0 : (i) (Harnack Inequality) There exist constants t 0 , c 0 > 0 such that Then P t is hypercontractive and compact in L 2 (µ) for large enough t > 0, and hold for some constants c, α > 0.
We will apply this result to non-degenerate and degenerate FSPDEs, respectively. To state our main results, we first introduce some notation.
For two separable Hilbert spaces H 1 , H 2 , let L (H 1 , H 2 ) (respectively, L HS (H 1 , H 2 )) be the set of all bounded (respectively, Hilbert-Schmidt) linear operators from H 1 to H 2 . We will use | · | and ·, · to denote the norm and inner product on a Hilbert space, and let · and · HS stand for the operator norm and the Hilbert-Schmidt norm for a linear operator. Below we introduce our main results for non-degenerate FSPDEs and degenerate FSPDEs, respectively.

Non-Degenerate FSPDEs
Let H be a separable Hilbert space. For a fixed constant r 0 > 0, let Let W (t) be a cylindrical Brownian motion on H under a complete filtered probability space (Ω, F , {F t } t≥0 , P); that is, for an orthonormal basis {e i } i≥1 on H and a sequence of independent one-dimensional Brownian motions {B i (t)} i≥1 on (Ω, F , {F t } t≥0 , P).
Consider the following FSPDE on H: where (A, D(A)) is a densely defined closed operator on H generating a C 0 -contraction semigroup e tA , b : C → H is measurable, (σ, D(σ)) is a densely defined linear operator on H. We assume that A, b and σ satisfy the following conditions.
We first observe that assumptions (A1) and (A2) imply the existence and uniqueness of continuous mild solutions to (1.2); that is, for any F 0 -measurable random variable X 0 on C , there exists a unique continuous adapted process X t on H such that P-a.s.
This implies that µ j := |σ * e j | 2 λ 1−δ j (j ≥ 1) gives rise to a finite measure on N, so that by Hölder's inequality, To emphasize the initial datum X 0 = ξ ∈ C , we denote the solution and the segment solution by {X ξ (t)} t≥−r 0 and {X ξ t } t≥0 , respectively. Then the Markov semigroup for the segment solution is defined as We are ready to state the main result in this part.
(3) For any t 0 > r 0 there exists a constant c > 0 such that where · var is the total variational norm and µ ξ t stands for the law of X ξ t for (t, ξ) ∈ [0, ∞) × C .

Degenerate FSPDEs
Let H := H 1 × H 2 for two separable Hilbert spaces H 1 and H 2 , and let C = C([−r 0 , 0]; H) as in Subsection 1.1. Consider the following degenerate FSPDE on H: ) is a densely defined closed operator on H 2 , and W (t) is the cylindrical Brownian motion on H 2 . Corresponding to (A1)-(A3) in the non-degenerate case, we make the following assumptions (see [15] for the case without delay, i.e. b(X t , Y t ) depends only on X(t) and Y (t)).
Similarly to the case without delay considered in [15], assumptions (B3) and (B4) will be used to prove the Harnack inequality. Moreover, as explained in Subsection 1.1 for the non-degenerate case, from [14, Theorem 4.1.3] we conclude that assumptions (B1) and (B2) imply the existence, uniqueness and non-explosion of the continuous mild solution (X ξ,η (t), Y ξ,η (t)) for any initial point (ξ, η) ∈ C . Let P t be the Markov semigroup for the segment solution. We have then all assertions in Theorem 1.1 hold with λ := sup s∈(0,λ 1 ] s − e sr 0 λ ′ . The remainder of this paper is organized as follows. In Section 2 we present a Fernique inequality for infinite-dimensional Gaussian processes, which will be used to prove the concentration condition required in Theorem 1.1(3). Theorems 1.2 and 1.3 are proved in Sections 3 and 4, respectively.

Infinite-dimensional Fernique inequality
In [8], Fernique introduced an inequality for the distribution of the maximum of Gaussian processes. To prove the exponential integrability of X t ∞ for FSPDEs, one needs an infinitedimensional version of this inequality. However, as the dimension goes to infinity, known Fernique inequality for multi-dimensional Gaussian processes becomes invalid. So, we modify the inequality such that it holds also in infinite-dimensions. To this end, we first recall the inequality for one-dimensional Gaussian processes (see e.g. [4, page 49] for the multidimensional case).
Now, we call a process {γ(t)} t∈[0,1] on the Hilbert space H a cylindrical continuous Gaussian process, if, for an orthonormal basis {e i } i≥1 , every one-dimensional process γ i (t) := γ(t), e i is a continuous Gaussian process. For a cylindrical continuous Gaussian process γ(t) with zero mean, let Theorem 2.2. Let γ(t) be a cylindrical continuous Gaussian process on H with zero mean such that Then, for any positive constant λ < min i≥1 , there exists a constant c > 0 such that . Obviously, (2.1) implies lim i→∞ δ i = 0 so that λ > 0. For any λ ∈ (0, λ), it suffices to prove (2.2) for some constant c > 0 and large enough r > 0. Below, we assume that In this case, (2.4) Since, by (2.3) and the definition of λ, we have Combining this with (2.4), we finish the proof.
Proof. The proof is similar to that of [3,Lemma 2.4]. Let µ ξ t be the law of X ξ t . It is easy to see that if µ ξ t converges weakly to a probability measure µ ξ as t → ∞, then µ ξ is an invariant probability measure of P t . Let P(C ) be the set of all probability measures on C . Consider the L 1 -Wasserstein distance induced by ρ(ξ, η) := 1 ∧ ξ − η ∞ : where C (µ 1 , µ 2 ) is the set of all couplings for µ 1 and µ 2 . It is well known that P(C ) is a complete metric space with respect to the distance W (see, e.g., [5, Lemma 5.3 and Lemma 5.4]), and the topology induced by W coincides with the weak topology (see, e.g., [5,Theorem 5.6]). So, to show existence of an invariant measure, it is sufficient to prove that µ ξ t is a W -Cauchy sequence as t → ∞, i.e., For any t 2 > t 1 > 0, consider the following SPDEs Then, the laws of X t 2 (ξ) and Y t 2 (ξ) are µ ξ t 2 and µ ξ t 1 , respectively. Also, following the argument leading to derive (3.2), we obtain for some constant c 1 > 0. By Gronwall's inequality and λ = λ 1 − Le λ 1 r 0 as assumed above, this implies Combining with (3.4) yields 2 . Therefore, (3.11) holds, and, by the completeness of W , there exists µ ξ ∈ P(C ) such that (3.12) lim To prove the uniqueness, it suffices to show that µ ξ is independent of ξ ∈ C . This follows since, by the triangle inequality, (3.3) and (4.2), Thus, the proof is finished.
With the above preparations, we present below a proof of Theorem

By [2, Proposition 2.3], this implies
Combining this with the Markov property we obtain Therefore, the last assertion follows from (3.3).

Proof of Theorem 1.3
According to what we have done in the last section for the proof of Theorem 1.2, it suffices to verify the existence and uniqueness of the invariant probability measure, as well as conditions (i)-(iii) in Theorem 1.1. In the present setting we have to pay more attention on the degenerate part. In particular, the known Harnack inequality (see [14,Corollary 4.4.4]) does not meet our requirement as the exponential term in the upper bound is not integrable with respect to the invariant probability measure. So, we first establish the following Harnack inequality which extends the corresponding one in [15] for the case without delay. The proof is modified from [15] using the coupling by change measures. This method was introduced in [1] on manifolds and further developed in [12] for SPDEs and in [7] for SDDEs, see [14] for a self-contained account on coupling by change of measures and applications.
Moreover, corresponding to Lemma 3.1 in the non-degenerate case, we have the following result on the exponential integrability of the solution. Proof. By Lemma 4.2, it suffices to prove for (ξ, η) = (0, 0). Simply denote (X t , Y t ) = (X 0,0 t , Y 0,0 t ). We have for some constant c 1 > 0. By Gronwall's inequality and λ ′′ = λ 1 − λ ′ e λ 1 r 0 as assumed above, this yields for some constant c 2 > 0. Hence, by using Hölder's and Jensen's inequalities as in (3.10) and applying (3.5) for the present Z t , we finish the proof.
Finally, the following lemma ensures the existence and uniqueness of invariant probability measure and verifies condition (iii) in Theorem 1.1, so that the proof of Theorem 1.3 is finished.