DIFFUSION IN LONG-RANGE CORRELATED ORNSTEIN-UHLENBECK FLOWS

: We study a diﬁusion d x ( t ) = V ( t; x ( t )) dt + p 2 D 0 d w ( t ) with a random Markovian and Gaussian drift for which the usual (spatial) P¶eclet number is inﬂnite. We introduce a temporal P¶eclet number and we prove that, under the ﬂniteness of the temporal P¶eclet number, the laws of scaled diﬁusions † x ( t=† 2 ), t ‚ 0 converge weakly, as † ! 0, to the law of a Brownian motion. We also show that the eﬁective diﬁusivity has a ﬂnite, nonzero limit as D 0 ! 0.


Introduction
The motion of a passive scalar in a random velocity is described by Itô's stochastic differential equation where V = (V 1 , · · · , V d ) : R × R d × Ω → R d is a d-dimensional random vector field with incompressible (∇ x · V(t, x) ≡ 0) realizations and w(t), t ≥ 0 is a d-dimensional standard Brownian motion, independent of V. The coefficient D 0 > 0 is called the molecular diffusivity.
We are particularly interested in the following class of velocity fields: V is a time-space stationary, centered (EV(0, 0) = 0) Gaussian, Markovian field with the co-variance matrix given by with Γ(k) := I − k ⊗ k|k| −2 and the power-law energy spectrum where a(k) is a (ultraviolet or infrared) cut-off function to ensure the finiteness of the integral (1.2). This class of velocity fields plays an important role in statistical hydrodynamics because the particular member with β = 1/3, α = 4/3 satisfies Kolmogorov-Obukhov's self-similarity hypothesis for the developed turbulence.
The main object of interest here is the large-scale diffusive scaling x ε (t) := εx(t/ε 2 ), ε ↓ 0. (1.4) In the case of α ≥ 1 the infrared cutoff is necessary. Once the infrared cutoff is in place, the velocity field is spatially homogeneous and temporally strongly mixing. Then, with an additional arbitrary ultraviolet cutoff to ensure regularity, the motion (D 0 ≥ 0) on the large (integral) scale is diffusive by the results of [5].
For α < 1 the infrared cutoff is optional (thus long-range correlation is possible) but an ultraviolet cutoff is necessary. However, we will only assume that sup k≥1 k n a(k) < +∞, ∀n ≥ 1. (1.5) Our main objective is to prove a sharp convergence theorem for the diffusive limit in flows with long-range correlation.
It is well known (see [1], Corollary of Theorem 3.4.1) that under these assumptions almost all realizations of the field are jointly continuous in (t, x) and of C ∞ -class in x for any t fixed. One can further prove the global existence and uniqueness of solutions of (1.1). We denote by Q ε the laws of the scaled trajectories x ε (t) = εx(t/ε 2 ), t ≥ 0 in C([0, +∞); R d ). The main theorem of this paper is formulated as follows.
The following questions arise naturally: Does the diffusion limit hold when D 0 = 0? Do the diffusion coefficients D (called the effective diffusivity) established in Theorem 1 have a non-zero limit as D 0 tends to zero? We don't know the answers. However, we can prove the following.
Theorem 2 Let D(D 0 ) be half the covariance matrix of the limiting Brownian motion as a function of the molecular diffusivity D 0 . We have lim sup Beside the framework and techniques developed in the paper, the main interest of the theorems is that they establish a new regime for the diffusive limit. Previous diffusive limit theorems have been proved either for random flows that have finite Péclet number ii (0, k)|k| −2 dk < ∞ [2,8] or for Markovian flows that are strongly mixing in time [5,11]. For the flows considered here, finite Péclet number means α < 0 while temporal mixing means β = 0. In the regime α + β < 1, α > 0, β > 0 the velocity neither has finite Péclet number nor is temporally mixing.
our results suggest the introduction of the temporal Péclet number defined as R ii (t, 0) dt whose convergence may be an alternative general condition for long-time diffusive behavior. The condition α + β < 1 is believed and partially shown to be sharp (see [6] for more discussion).
Without loss of generality we will set D 0 = 1 till we turn to the proof of Theorem 2.

Function spaces and random fields
In the sequel, we shall denote by T 0 := (Ω, V, P) the probability space of random vector fields and by T 1 := (Σ, W, Q) the probability space of the (molecular) Brownian motion. Their respective expectations are denoted by E and M. The trajectory x(t), t ≥ 0 is then a stochastic process over the probability space T 0 ⊗ T 1 . The space T 0 is detailed in the rest of this section.

Spatially homogeneous Gaussian measures
Let H m ρ be the Hilbert space of d-dimensional incompressible vector fields that is the completion for any positive integer m and the weight function ϑ ρ (x) := (1 + |x| 2 ) −ρ , where ρ > d/2. When m, ρ = 0 we shall write L 2 div := H 0 0 . Let the Gaussian measure µ be the probability law of V(0, ·) ∈ H m ρ as given by the correlation functions (1.2) with (1.3) and (1.5). Let We will suppress writing the measure µ when there is no danger of confusion. Both here and in the sequel B(M) is the Borel σ-algebra of subsets of a metric space M. On H m ρ we define a group of µ-preserving transformations τ This group is ergodic and stochastically continuous, s., with derivatives of all orders bounded by deterministic constants (i.e. constants independent of f ). For any 1 ≤ p < +∞ and a positive integer m let W p,m be the Sobolev space as the closure of C ∞ b in the norm This definition can be extended in an obvious way to include the case of p = +∞.
Let (·, ·) L 2 denote the generalized pairing between tempered distributions and the Schwartz test functions. Let Obviously F ϕ ∈ (H m ρ ) * and B(H m ρ ) is the smallest σ-algebra with respect to which all F ϕ , ϕ ∈ S div are measurable.
We denote by P 0 the space of all polynomials over H m ρ , i.e.
denotes the conditional expectation operator with respect to the the σ-algebra Σ(l) generated by all polynomials from P 0 (l) in the probability space T 2 . Hence we can extend Q(l) to a positivity preserving contraction operator Q(l) : L p → L p for any 1 ≤ p ≤ +∞. Notice that hence the following proposition holds.
Proof. Observe that for any integer n ≥ 1, a bounded, Borel measurable function G : R 2n → R and ϕ 1 , · · · , ϕ n ∈ S d we have ¿From (2.5) we infer that U x (H(l)) = H(l) and the conclusion of the proposition follows.

Ornstein-Uhlenbeck Velocity Field
Let W (t), t ≥ 0 be a cylindrical Wiener process over the probability space T 0 = (Ω, V, P) so that dW (t) is a divergence-free, space-time-white-noise Gaussian field. Let B : L 2 div → H m ρ be the continuous extension of the operator defined on S div as where E(k) is given by (1.3)-(1.5). Here and belowψ denotes the Fourier transform of ψ. It can be shown (see [7], Proposition 2) that B is a Hilbert-Schmidt operator. Let It can be shown (see [7], Proposition 2) that S(t), t ≥ 0 extends to a C 0 -semigroup of operators on H m ρ , provided that β is an integer. For a non-integral β the above is still true provided that d/2 < ρ < d/2 + β. S div is a core of the generator −A of the semigroup and We also introduce the operator C : L 2 div → H m ρ defined as the continuous extension of The same argument as in the proof of Part 1) of Proposition 2 of [7] yields that C is Hilbert-Schmidt. Let By V µ (t), t ≥ 0 we denote the process V f (t), t ≥ 0 over T 0 ⊗ T 2 with the random initial condition f , distributed according to µ and independent of the cylindrical Wiener process and in what follows we shall identify those two processes. The measure µ is stationary (see Section 2.3 of [7]). Its ergodicity follows from Lemma 1 below (Corollary 1). A direct calculation shows that for any bounded and measurable G, H : therefore µ is reversible. We denote by R t , t ≥ 0, L and E L (·, ·) respectively the L 2 -semigroup, generator and Dirichlet form corresponding to the process V (t), t ≥ 0.
A useful formula for the Dirichlet form of linear functionals is given by the following.
Proposition 2 For any ϕ ∈ S div , the linear functional F ϕ as defined by (2.1) is in the domain D(L) of L and Then, a direct calculation shows that F ϕ ∈ D(L) and LF ϕ (f ) = (Aϕ, f ) L 2 and (2.11) follows. On the other hand for an arbitrary ϕ ∈ S div one can choose a sequence of ϕ n , n ≥ 1 such that their respective Fourier transforms ϕ n satisfy (2.12) and This implies that F ϕ ∈ D(L). We obtain (2.11) by passing to the limit in the expression for E L (F ϕn , F ϕn ).
The following proposition holds.
Proposition 3 Operators R t and Q(l) commute i.e.
Proof. Suppose that n ≥ 1, ϕ 1 , · · · , ϕ n ∈ S div (l) and G(f ) : with V f given by (2.10). By Theorem 1.36 p. 16 of [9] there exists a polynomial P (x 1 , · · · , x n ), (x 1 , · · · , x n ) ∈ R n such that the right hand side of (2.15) is given by Since R t and Q(l) are self-adjoint, by taking the adjoint of both sides of this equality we arrive at R t Q(l) = Q(l)R t . Hence, (2.16) follows.
As a consequence of the above proposition, we have

The main lemma
The purpose of this section is to prove the following estimate.
Lemma 1 There exists an absolute constant C 0 > 0 such that where Q(l) is the projection onto H(l) -the L 2 -closure of P 0 (l) as defined in (2.3).
Since, by Lemma 1, E L (F, F ) = 0 implies that Q(l)(F ) = 0, ∀l, and hence F is a constant, we have the following.
The proof of Lemma 1 (Section 3.3) uses the general scheme of periodization and periodic approximation (Section 3.1 and 3.2) which is valid for general stationary Markov fields.

Periodization of the Ornstein-Uhlenbeck flow
For an arbitrary integer n ≥ 1 let Λ n : We define considered as the subspace of the Sobolev space of all divergence free vector fields f : [12] p. 5). We set π n : H m ρ → H m n by the formula The images of Wiener process CW(t), t ≥ 0 under π n are finite dimensional Brownian motions given by We define an Ornstein-Uhlenbeck process in H m n as Let µ n := µπ −1 n and Π n : L 2 (µ n ) → L 2 , J n : P 0 → L 2 (µ n ) be linear maps given by The process V (n) µn (t), t ≥ 0, is a stationary process defined over the probability space with the initial condition f distributed according to µ n . This process gives rise to a random, time-space stationary and spatially 2 n+1 π-periodic vector field V (n) We denote by R t n , L n , E Ln (·, ·) the L 2 (µ n )-semigroup, generator and Dirichlet form corresponding to the process V (n) µn (t), t ≥ 0. Let M n be the cardinality of Λ n and c l the cardinality of those j-s for which |k j | ≤ l. We denote by ν, ν l the Gaussian measures on (R d ) 2Mn , (R d ) 2c l with the corresponding characteristic functions and The operator U extends to a unitary map between L 2 (ν) and L 2 (µ n ). As in Section 2.2 let H n (l) be the L 2 -closure of the polynomials J n (P 0 (l)) and let Q n (l) be the corresponding orthogonal projection. Notice that U (K n (l)) = H n (l), (3.11) where K n (l) is the L 2 -closure of polynomials depending only on variables a j , b j corresponding to those j-s for which |k j | > l.

Periodic approximation
In the next proposition we show that V (n) µn (t), t ≥ 0 is an approximation of V µ (t), t ≥ 0.

Proposition 4
i) For any F ∈ P 0 we have ii) For any F ∈ P 0 we have J n F ∈ D(E Ln ) and The class of polynomials P 0 is a core of E L . where and for any F ∈ P 0 lim Proof. Part i). It suffices to verify that (3.12) holds for the polynomials. Let ϕ 1 , · · · , ϕ N ∈ S div and F (·) = (ϕ 1 , ·) L 2 div · · · (ϕ N , ·) L 2 div , (3.17) πn(f ) (t))) L 2 div (3.18) and the right hand side of (3.18) can be expressed as a finite sum of certain products made of expressions of the form Taking into account the definitions of X we conclude that as n ↑ +∞ the right hand side of (3.18) tends to Part ii). Note that R t (P 0 ) ⊆ P 0 so P 0 is a core of L and for F as in (3.17) Likewise, (3.18) implies that R t n J n (P 0 ) ⊆ J n (P 0 ), ∀ t ≥ 0, so J n (P 0 ) is a core of L n and L n J n F (f ) = N k=1 (ϕ 1 , j n (f )) L 2 div · · · (Aϕ k , j n (f )) L 2 div · · · (ϕ N , j n (f )) L 2 div , Π n J n F (f ) = (ϕ 1 , π n j n (f )) L 2 div · · · (ϕ N , π n j n (f )) L 2 div , for F (·) = (ϕ 1 , ·) L 2 div · · · (ϕ N , ·) L 2 div ∈ P 0 . Thus, and (3.13) follows for all F ∈ P 0 .
Part iii) Notice that the orthogonal projection Q n (l) onto K n (l) in L 2 ((R d ) 2Mn ) is the conditional expectation with respect to the σ-algebra generated by the functions in variables a j , b j , |k j | > l only. Thus, for G := G(a j , b j ; j ∈ Λ n ) we have which in turn implies (3.14), thanks to (3.11). (3.16) follows from the fact that the co-variance matrices of the fields V (n) approximate, as n ↑ +∞, the co-variance matrix of the field V.

Proof of Lemma 1
The conclusion of Lemma 1 follows from Proposition 4 and the following.
Lemma 2 There exists an absolute constant C > 0, independent of n, such that for all F ∈ P 0 such that F dµ = 0.
Proof. Let F be the polynomial given by (3.17). We have J n F = U G, for a certain polynomial G(a j , b j ; j ∈ Λ n ) (cf. (3.5), (3.9)), i.e. Consequently, The above argument generalizes to any polynomial F ∈ P 0 . We shall denote by G the corresponding polynomial in a j , b j , j ∈ Λ n . For any integer m ≥ 1 we can write that A simple calculation shows that the right side of (3.20) is greater than or equal to Moreover, by Jensen's inequality, where G(l)(a j , b j : |k j | > l) := · · · a j , b j :|k j |≤l G(a j , b j : j ∈ Λ n ) dν l .
Because J n F has zero mean, so does G(l) and the coercivity of E j implies with an absolute constant C independent of n. Here we have used the fact that The conclusion of the lemma follows upon the passage to the limit with m ↑ +∞.

Lagrangian velocity process
In what follows we introduce the so-called Lagrangian canonical process over the probability space T 0 ⊗ T 1 , with the state space H m ρ that, informally speaking, describes the random environment viewed from the moving particle. Let x f (t), t ≥ 0 be the trajectory of (1.1) with the drift replaced by V f (t, t ≥ 0. η(t), t ≥ 0 is a continuous, Markov process (see e.g. [10] Theorem 1 p. 424) i.e. there exists Q t , t ≥ 0 a C 0 -semigroup Q t , t ≥ 0 of Markovian operators on L 2 satisfying where V t , t ≥ 0 is the natural filtration corresponding to the Lagrangian process. Moreover thanks to incompressibility of V the measure µ is stationary, i.e. Q t F dµ = F dµ, t ≥ 0. Ergodicity of the measure for the semigroup Q t , t ≥ 0 follows from the ergodicity for R t , t ≥ 0 (see Theorem 1 p. 424 of [10]). The generator of the process is given by In order to make sense of (4.3) we need to assume that m > [d/2] + 1. The set C L is dense in L 2 . In what follows we shall also consider a family of approximate Lagrangian processes obtained as follows. Let We define a random field V (n) (t, x) := V (n) (τ x (V (t))), (t, x) ∈ R × R d and set η (n) (t) := τ x (n) (t) (V (t)), t ≥ 0 where x (n) (t; ω, σ), t ≥ 0 is a solution of (1.1) with V (n) as the drift. One can choose V (n) , n ≥ 1, (for details on this point see the remark before formula (11) in [10]) in such a way that lim n↑+∞ ME sup 0≤t≤T |x(t; ω, σ) − x n (t; ω, σ)| 2 = 0. (4.4) As before we introduce also the process η All the facts stated for η(t), t ≥ 0 hold also for η (n) (t), t ≥ 0. In particular these processes are Markovian with the respective semigroups Q t n , t ≥ 0. These semigroups satisfy lim for any t ≥ 0 and f ∈ L p . The generator of the approximate process is given by C Ln is a core of M n (see [10] Theorem 2 p. 424).
On C L × C L we define a non-negative definite bilinear form The form is closable and we denote by H + the completion of C L,0 = C L ∩ L 2 0 under the norm · + := (·, ·) 1/2 + . It is easy to observe that H + = W 2,1 ∩ D(E L ). The scalar product (·, ·) + over H + is the Dirichlet form associated with the Markovian process ξ t := τ w(t) V(t), t ≥ 0 with w(t), t ≥ 0 a standard d-dimensional Brownian motion independent of V(t), t ≥ 0. We denote also by H 0 − the space of all F ∈ L 2 0 for which The completion of H 0 − in the · − norm shall be denoted by H − .
5 Proof of Theorem 1

Corrector field and energy identity
Proposition 5 Under the assumptions of Theorem 1, we have V p ∈ H − , for any p = 1, · · · , d.
Let χ (p) * ,>l := Q(l)χ (p) * ∈ L 2 0 ∩ H + . Thanks to Proposition 1 we have Let g a := f a (χ (p) * ,>l ), (5.13) where f a (r) := −a γ ∨ (r ∧ a γ ), a > 0, γ > 0 and F a is the integral of f a satisfying F a (0) = 0. Substituting into (5.10) g a for the test function we conclude that From the contraction property for Dirichlet forms we have g a + ≤ χ (p) * ,>l + , so the set g a , a > 0 is H + -weakly pre-compact. To prove that g a H + -weakly converges as a ↑ +∞ we show that for any F ∈ W 1,∞ lim a↑+∞ ∇g a · ∇F dµ = ∇χ and |f a (χ (p) * ,>l ) − 1| ≤ min{a −γ |χ (p) * ,>l |, 1}. We infer therefore that The right hand side of (5.17) tends to 0, as a ↑ +∞. Hence we have (5.16) and g a χ (p) * ,>l , as a ↑ +∞, H + -weakly. As a consequence, we have that The same argument shows also that lim a↑+∞ G p (g a ) = G p (χ (p) * ,>l ). Notice that V ≤l is L 2 -orthogonal to any F ϕ , with ϕ ∈ S div (l) Hence by virtue of the classical Kolmogorov-Rozanov Theorem (see e.g. Theorem 10.1 p. 181 in [13]) V ≤l is independent of Σ(l). Since g a L 2 ≤ χ (p) * ,>l L 2 , a > 0 we obtain, using independence of V ≤l and any g a , a > 0, that The last term on the left hand side of (5.15) equals with H a given by (5.14). Here we used the fact that ∇ · V >l = 0. Thus, we have proved that . Observe that the integrand appearing in the last term on the left hand side of (5.19) belongs to L 1 and Here we have used the fact that V ≤l and Q(l)χ  λ converges H + -strongly, as λ ↓ 0.

Convergence of finite dimensional distributions
weakly, as ε ↓ 0, to those of a Brownian Motion whose co-variance equals 2D(ε 0 ) : Passing to the limit ε 0 ↓ 0 we conclude that the f.d.d. of x ε (t), t ≥ 0, as ε ↓ 0, converges to the Wiener measure with the co-variance matrix given by 2D = 2[D p,q ], where Proof of (5.27). Using (4.5) we conclude that For an arbitrary λ > 0 the correctors χ (p) n,λ converge both strongly in L 2 and weakly in H + , as n ↑ +∞. In consequence χ (p) λ satisfies (5.5) with V in place of V n and test functions Φ ∈ H + ∩ L ∞ . Choosing f a (χ (p) λ ) as the test functions and letting a ↑ +∞ we conclude χ we conclude, thanks to (5.32) that the right hand side of (5.33) equals That, in consequence, validates our claim (5.27)

Proof of Theorem 2
Now we turn to the proof of Theorem 2 for the limit of vanishing molecular diffusivity D 0 → 0.
For the lower bound, we turn to Eq. (5.10), which after some elementary approximation argument can be written in the form We want to show that, if the infimum of D is zero as D 0 tends to zero, then the entire left side of (6.4) drops out in the limit while the right side equals V p 2 L 2 > 0, thus leading to contradiction. Let us assume the infimum of D as D 0 → 0 is zero and take an infimum-achieving subsequence of χ The right side of (6.6) can be explicitly calculated using Feynman diagrams and the result is for α + β < 1. The third term on the left hand side of (6.4) can be therefore bounded by ) * (D 0 )), p = 1, · · · , d, which vanishes by (6.5). Theorem 2 is proved.