Brownian particle in the curl of 2-d stochastic heat equations

We study the long time behaviour of a Brownian particle evolving in a dynamic random environment. Recently, [G. Cannizzaro, L. Haunschmid-Sibitz, F. Toninelli, preprint arXiv:2106.06264] proved sharp $\sqrt{log}$-super diffusive bounds for a Brownian particle in the curl of (a regularisation of) the 2-d Gaussian Free Field (GFF) $\underline{\omega}$. We consider a one parameter family of Markovian and Gaussian dynamic environments which are reversible with respect to the law of $\underline{\omega}$. Adapting their method, we show that if $s\ge1$, with $s=1$ corresponding to the standard stochastic heat equation, then the particle stays $\sqrt{log}$-super diffusive, whereas if $s<1$, corresponding to a fractional heat equation, then the particle becomes diffusive. In fact, for $s<1$, we show that this is a particular case of [T. Komorowski, S. Olla, J. Func. Anal., 2003], which yields an invariance principle through a Sector Condition result. Our main results agree with the Alder-Wainwright scaling argument (see [B. Alder, T. Wainright, Phys. Rev. Lett. 1967]) used originally in [B. T\'oth, B. Valk\'o, J. Stat. Phys., 2012] to predict the $\log$-corrections to diffusivity. We also provide examples which display $log^a$-super diffusive behaviour for $a\in(0,1/2]$.


Introduction and main result
We study the motion of a Brownian particle in R 2 , evolving in a dynamic random environment (DRE), given by the solution to the Itô SDE # dXptq " ω t pXptqq dt`?2 dBptq, t ě 0, where pBptqq tě0 is a standard two-dimensional Brownian motion and pω t pxqq tě0,xPR 2 is a time-dependent random field which is independent from pBptqq tě0 . We take pω t pxqq tě0,xPR 2 to be a regularised version of the curl of the solution to the (fractional) stochastic heat equation with additive noise in R 2 and initial condition given by the curl of the regularised Gaussian Free Field (GFF) ω. The coordinates of ω t " pω 1 t , ω 2 t q satisfy # dω k t "´p´∆q s ω k t dt`?2B K k p´∆q s´1 2 dW t , t ě 0 , k " 1, 2 , ω 0 " ω , where s P r0, 8q and pB K 1 , B K 2 q :" pB x2 ,´B x1 q. Here, W is a mollified (in space) space-time white noise, with covariance ErW r pxqW t pyqs " mintr, tuV px´yq, and ω is distributed according to the law of the curl of a mollified GFF. More precisely, for every k, l " 1, 2, r, t ě 0 and x, y P R 2 , W s t pxq :" p´∆q s´1 2 W t pxq and ω have mean zero and covariance ErB K k W s r pxqB K l W s t pyqs " mintr, tuB K k B K l V˙g 1´s px´yq " mintr, tupB K k p´∆q s´1 δ x , B K l δ y q V if s ď 1 , ErB K k W s r pxqB K l W s t pyqs " mintr, tuB K k B K l p´∆q s´1 V px´yq " mintr, tupB K k p´∆q s´1 δ x , B K l δ y q V if s ą 1 , (4) Erω k pxqω l pyqs "´B K k B K l V˙g 1 px´yq " pB K k p´∆q´1δ x , B K l δ y q V , where˙denotes the convolution over R 2 and, for every ϕ 1 , ϕ 2 P SpR 2 q, the space of Schwartz functions over R 2 , The smooth function V is given by V :" U˙U , for a U P C 8 pR 2 q, radially symmetric, decaying exponentially fast at infinity and with ş R 2 U pxq dx " 1. To simplify some computations, we may also assume that U has Fourier transform supported in ball of radius 1. Also, the kernel g r : R 2 zt0u Ñ R in (3) and (5)  if r P p0, 1q ; where Γ denotes the Gamma function. I.e., g r is the Green's function of p´∆q r in R 2 , for r P r0, 1s. Also, the fractional Laplacian p´∆q s´1 for s ą 1 can be defined in terms of its Fourier multiplier, as { p´∆q s´1 f ppq " |p| 2ps´1q p f ppq.
Remark 1. Note that expressions (3)-(5) make sense due to the presence of the smooth function V . Plugging, e.g., the right-hand side of (5) into (6), we get which is equal to the expression in the middle of (5). Furthermore, even though the GFF in the full space is only defined up to a constant (i.e. inverting the Laplacian ∆), taking the derivatives B K k B K l of its regularisation makes it rigorous without ambiguity. The same reasoning holds to define the noise B K k p´∆q s´1 2 W t when s " 0.
Remark 2. As we show in Proposition 8, the dynamics (2) leave the law of ω invariant. The case s " 1 corresponds to the standard stochastic heat equation (SHE), whereas s " 0 is the infinite dimensional Ornstein Uhlenbeck process, as defined, for example in [17,Chapter 1.4]. The parameter s P r0, 8q controls the speed of the environment on different scales: smaller values of s correspond to faster movement of the larger scales.
By definition, the drift field ω t pxq in (1) is divergence-free. Brownian particles evolving in stationary divergence-free random fields have been considered as a toy model for anomalous diffusions in inhomogeneous media, such as the motion of a tracer particle in an incompressible turbulent flow. See e.g. the surveys [11,Chapter 11] and [14]. Depending on the decay of the spatial correlations of the drift field, the particle could behave either diffusively or superdiffusively, meaning that the mean square displacement satisfies for large t Dptq :" Here, E denotes the expectation under the joint law of B and ω, see Section 2. If the correlations of the environment decay fast enough (see e.g. [11,Chapter 11]), one gets diffusive behaviour, and if the decay is too slow (see [12]), one gets superdiffusive behaviour. There is, however, an intermediate regime for which the correlations decay in such a way that Dptq diverges only as plog tq γ , for γ ą 0. These logarithmic corrections are expected to be present in two-dimensional Brownian particles evolving in isotropic random drift fields. Indeed, by the Alder-Wainwright scaling argument (see [1,2,3,8]), in 2d, if the displacement of the particle scales faster than the correlations of the environment field, then the (only) expected behaviour for the mean square displacement of the particle is to be of order t ? log t. We briefly elaborate on this, following the Appendix of [21]. Let Kpt, xq :" Erω 0 p0qω t pxqs. Now, assume that PpXptq P dxq « αptq´2ϕpαptq´1xq dx, where ϕ is a density and αptq " t ν plog tq γ for some ν, γ ě 0. If we also assume that Kpt, xq « βptq´2ψpβptq´1xq, for another density ψ, then if we must have ν " 1{2 and γ " 1{4, which yields Xptq « t ? log t. We emphasise here that this argument, even though instructive, it is not mathematically rigorous. Indeed, the ? log correction was rigorously established recently by Toninelli et al. [7]. They showed that for a time-independent drift field ω distributed according to the law of ω, one has Dptq « a log t as t Ñ 8 , up to log log t corrections, confirming a conjecture made by Tóth and Valkó [21] based on this scaling argument. The result was obtained in the Tauberian sense 1 , i.e., in terms of the Laplace transform of the mean square displacement Note that in the case considered by [21,7], the correlations of the drift field do not scale in time since the drift field is time-independent, so (9) is trivially satisfied. Moving to the time-dependent case treated in the present work, if we take s ě 1 in (2), we still have that the correlations of the drift field do not scale fast enought 1 2s for ω vs. t 1 2 plog tq 1 4 for X. Therefore, we should still expect for the particle X to behave ?
log-superdiffusively, since condition (9) remains true. However, if we move to the case where s ă 1 in (2), then the picture changes substantially and condition (9) is no longer satisfied, since t 1 2 plog tq 1 4 ăă t 1 2s for t ąą 1. Theorem 3 and Theorem 4 below rigorously establish the expected abrupt difference between super diffusive and diffusive behaviours depending on the exponent s, agreeing with the scaling argument.
Theorem 3. If s ě 1 in (2), then, for every ε ą 0, there exist constants A ε , B ε ą 0, depending only on ε and s, such that, for λ P p0, 1q, we have For the case s P r0, 1q we can apply a sector condition result of Komorowski and Olla [13] to obtain the following invariance principle.
Theorem 4. If s P r0, 1q in (2), then there exist constants A, B ą 0, such that, for all t ě 0, we have Furthermore, let pQ ω ε q εPp0,1s denote the laws of pεXp t ε 2 qq tě0 , over Cr0, 8q, for ε P p0, 1s, given the initial configuration ω 0 " ω. Then pQ ω ε q εPp0,1s converge weakly, with respect to the law of ω, as ε Ó 0, to the law of a Brownian motion with deterministic covariance matrix D, which only depends on s. The covariance matrix D is defined in (66).
The asymptotic behaviour of D T pλq in (12) is a reflection of the fact that the dynamics provided by the SHE (with the full Laplacian) does not mix the environment fast enough to produce a scaling of the correlations which is faster than the scaling of the displacement of the particle, as discussed above. On the other hand, the result in (13) confirms that the fractional dynamics on the environment changes dramatically the behaviour of the particle. Moreover, the estimates in (12) are exactly the same as the ones obtained in [7], and our proof is an adaptation of theirs, which is based on Yau's method [22] of recursive estimates of iterative truncations of the resolvent equation in (21). Indeed, when s ě 1, the dominant terms in the estimates are the ones coming from the stationary drift field, which are the same as for the static case. What we show is that we can remove the additional terms coming from the dynamics of the environment in the estimates, maintaining the same asymptotic behaviour. However, when s ă 1, the dominant terms are now precisely the ones coming from the dynamics of the environment. The effect can be seen already in the first upper bound obtained by the first truncation of (21), and it is enough to show (13) in Theorem 4, see Remark 11.
If now we consider intermediate regimes between s " 1 and s ă 1, only adding a logarithmic divergence to the operator ∆ in (2), we obtain something which was not predicted by the Alder-Wainwright scaling argument. Namely, for any given a P p0, 1 2 s, we can find an interpolation between the regimes s " 1 and s ă 1 such that we prove corrections to diffusivity of order plog tq a . More precisely, if we consider that the coordinates of ω t " pω 1 t , ω 2 t q satisfy # dω k t " plogpe`p´∆q´1qq γ ∆ω k t dt`?2plogpe`p´∆q´1qq for a parameter γ ą 0. Then, we can show the following Theorem 5. If pXptqq tě0 is the solution to (1) with pω t q tě0 solution to (14), then, for every γ P r 1 2 , 8q, there exist constants A, B ą 0, only depending on γ, such that: If γ " 1, then for λ P p0, 1q,

Structure of the paper
In Section 2 we define the environment seen from the particle process as a technical tool. In Section 3 we derive the action of the infinitesimal generator of the environment seen from the particle on Fock space, and show that the law of ω is invariant under the family of dynamics given by (2). Section 4 contains the proof of the main recursive estimates through an iterative analysis of the resolvent equation in (21) and a proof of (13) in Theorem 4 using only the first truncation of the resolvent equation. In Section 5 we prove Theorem 3 by using the recursive estimates obtained in Section 4. In Section 6, we present a general overview of the method in [13] of homogenisation of diffusions in divergence-free, Gaussian and Markovian fields and show that for s ă 1 we may apply their results to get Theorem 4. In Section 7 we prove Theorem 5. Appendices A and B gather important ingredients from Toninelli et al. [7], and some generalisations to the present setting, necessary in Sections 4 and 5 and Appendix C presents the final argument for the proof of Theorem 3, taken from [7].

Setting and preliminaries
Let T 0 :" pΩ, B, Pq be a probability space supporting ω and an independent Wiener process W as defined between displays (2) and (3). Let T 1 :" pΣ, F , Qq be another probability space supporting a standard 2d Brownian motion B. We consider solutions to the system (1), (2) on ΩˆΣ equipped with the product measure P " P b Q. The law of pXptqq tě0 under P is called the annealed law. Note, that under P, the process pXptqq tě0 alone is not Markovian. Notwithstanding, we may define a different Markovian process, the so-called environment seen from the particle, which takes values on the larger space of functions over R 2 [10]. It evolves by spatially shifting the environment by the position of the walker, at any given time t ě 0. Precisely, we set η t :" ω t p¨`Xptqq , t ě 0 .
The law of X is rotationally invariant, and therefore we have that Er|Xptq| 2 s " ErX 1 ptq 2`X 2 ptq 2 s " 2ErX 1 ptq 2 s. Hence we may focus on its first coordinate only. Furthermore, ErXptqs " 0. Formula (18) allows us to write where Vpωq :" ω 1 p0q, for ω " pω 1 , ω 2 q. Using the so-called Yaglom-reversibility (see Section 1.4 of [20]), we get that, for every 0 ď s ă t, the random variables Bptq´Bpsq and ş t s Vpη r q dr are uncorrelated, so that This in turn implies that we can rewrite (11) as D T pλq " D B pλq`D V pλq, where for all λ ą 0, and therefore, we may focus on D V pλq, which requires a good understanding of the process pη t q tě0 . Since the drift field is stationary (see Proposition 8) and divergence-free, the law of ω is invariant also for pη t q tě0 (see e.g. Chapter 11 in [11]). This ensures that, by Lemma 5.1 in [6], we can write where L s denotes the infinitesimal generator of pη t q tě0 , defined in (32) below, and with a slight abuse of notation we use E to denote the expectation with respect to the law of ω.

Operators on Fock space
In order to analyse expression (21), we describe the infinitesimal generator of the infinite dimensional Markov process t Þ Ñ ω t . With a small abuse of notation, let P denote the law of ω and consider F P L 2 pPq of the form F pωq " f pω i1 px 1 q, . . . , ω in px n qq for arbitrary points x 1 , . . . , x n P R 2 and for an f P C 2 p pR n , Rq, the C 2 functions with polynomially growing partial derivatives of order less or equal than 2. In this section, to emphasise its dependence in s P r0, 8q, let us denote by L s 0 the infinitesimal generator of pω t q tě0 . For every s P r0, 8q, an application of Itô's formula gives where p¨,¨q V is given by (6), B k f denotes the function y " py 1 , . . . , y n q Þ Ñ B y k f pyq and for every x P R 2 , the expression with δ x is well defined by Remark 1.
Let us introduce the Wiener chaos with the respect to P, following the same convention and notation as [7]. Let x 1:n :" px 1 , . . . , x n q, i :" pi 1 , . . . , i n q and :¨¨¨: denotes the Wick product with respect to P. Define H 0 as the set of constant random variables and for n ě 1 let H n be the set # where the functions f i are symmetric and such that satisfies ψ n pp 1:n q| 2 dp 1:n ă 8 .
Here, pp K k,1 , p K k,2 q :" pp k,2 ,´p k,1 q for p k " pp k,1 , p k,2 q andf i denotes the Fourier transform of f i , given bŷ f i px 1:n qe´ι x1:n¨p1:n dx 1:n , where x 1:n¨p1:n denotes the canonical inner product in R 2n and ι " ?´1 .
Remark 6. Note that since we have the mollification in the noise, the objects f i can be distributions of any negative regularity, such as the delta Dirac distribution. The random variable which we are most interested in here, namely Vpωq " ω 1 p0q, defined in the previous section, can be seen as Furthermore,Vppq " p 2 for p " pp 1 , p 2 q.
It is well known, see e.g. Nualart [17] or Janson [9], that and for F i P L 2 pPq, i " 1, 2 given by F i " ř 8 n"0 ψ i n , for ψ i n P H n , the expectation ErF 1 F 2 s can be written as x ψ 1 n pp 1:n q x ψ 2 n pp 1:n q dp 1:n .
Remark 7. Henceforth we will implicitly identify a random variable F P H n Ă L 2 pPq of the form (23) with its kernel x ψ n in Fourier space. In the same philosophy, we will denote linear operators acting on L 2 pPq with the correspondent operators acting on Fock space À n L 2 sym pR 2n q, and we will denote them by the same symbol.
: ω i1 px 1 q¨¨¨p´p´∆q s qω i k px k q¨¨¨ω in px n q : on Wick monomials, and in Fourier variables by { p´L s 0 qψ n pp 1:n q " Furthermore, the law of ω is invariant under the dynamics governed by L s 0 , i.e., the infinite dimensional Markov process pω t q tě0 is stationary and it is distributed according to the law of ω for every t ě 0.
Proof. By the definition of Wick monomials, we have that where a ✁ bc :" ac for a, b, c P R. Now, the above applied to (22) with F " : ω i1 px 1 q¨¨¨ω in px n q : gives Note that on Wick monomials, multiplication by ω i k px k q, as in (30), produces both a term in one higher homogeneous chaos and a term in one lower homogeneous chaos. Precisely, for each 1 ď k ď n in (30) we have : Summing over k, the first term after the equal sign gives us (28) and the second term after the equality cancels out with (31). (29) is a direct consequence of (28) and (24). Now we move to the invariance of the law of ω. It is known that a necessary and sufficient condition for this is that pL s 0 q˚1 " 0, where pL s 0 q˚denotes the adjoint of the operator L s 0 in L 2 pPq and 1 denotes the constant function equal to 1, see e.g. [16,Theorem 3.37]. Also, by (26), it is enough to consider F " : ω i1 px 1 q¨¨¨ω in px n q : , so that Er: ω i1 px 1 q¨¨¨p´p´∆q s qω i k px k q¨¨¨ω in px n q :s " 0 completes the proof.
So far we gathered all the ingredients necessary to characterise the full generator L ": L s of pη t q tě0 . Putting together the generator L s 0 of the environmental process pω t q tě0 with Proposition 8, the arguments in Section 2.1 of Tóth and Valkó [21] and the main result of Komorowski [10], we get that the generator L s is given by where V∇ :" V 1 D 1`V2 D 2 , with V i pωq " ω i p0q and D i is the infinitesimal generator of the spatial shifts in the canonical directions of R 2 , for i " 1, 2, see [10]. Also, V∇ " A`´A˚can be decomposed into a creation and annihilation parts, one being minus the adjoint of the other, and it comes from the drift part of (1), i.e., the environment, while ∆ " ∇ 2 comes from the Brownian part in (1), see [21]. We have that L s 0 , ∆ : H n Ñ H n , A`: H n Ñ H n`1 and A˚: H n Ñ H n´1 .
As noted in Toninelli et al. [7], adopting the conventions on Fock space discussed earlier, one has { p´∆qψ n pp 1:n q "ˇˇˇˇn ÿ where p 1:n`1zl :" pp 1 , . . . , ✚ ✚ p l , . . . , p n`1 q and for p, q P R 2 , pˆq denotes the scalar given by the third coordinate of the cross product of p with q, when thought as vectors in R 3 , precisely, pˆq " p 1 q 2´p2 q 1 " |p||q| sin θ, where θ is the angle between p and q.
Here we can see that if s " 1 in (29), the difference between the operators ∆ and L 1 0 is simply the cross terms in (33). The most important observation here is that if s ě 1 and |p| ď 1, in view of (21) and Remark 6, for any function ψ 1 P H 1 , we have that This is a good evidence to suggest (12), as can be further seen in Remark 11. Also, a good heuristics for the drastic change in behaviour in s contained in Theorem 3 is that in Fourier variables, the operator L s 0 acts much more severely in large scales when s ă 1 than when s ě 1, since |p| 2s ăă |p| 2s 1 for |p| ăă 1, if s 1 ă s. Now we proceed to the analysis of the resolvent equation in (21).

Iterative analysis of the resolvent equation
We can write ErVpλ´L s q´1Vs as ErVVs, where V is the solution to the resolvent equation pλ´L s qV " V. Note however that V P H 1 is in the first Wiener chaos and that the operator L s maps H n to H n´1 'H n 'H n`1 , one should expect that the solution V to the resolvent equation has non-trivial componentes in all Wiener chaoses. Following the idea introduced by Landim et al. [15] we truncate the generator L s by using L s n :" P ďn L s P ďn , where P ďn denotes the orthogonal projection onto the inhomogeneous chaos of order n, i.e., P ďn : L 2 pPq Ñ À n k"0 H k . Denote by V n P À n k"0 H k the solution to the resolvent equation truncated at level n, i.e., pλ´L s n qV n " V and V n " Now, writing one equation for each of the components of V above we get that the equation above is equivalent to the system of equations Note that as it was observed in [21, Section 2], A˚F " 0 for every F P H 1 , so that V n 0 " 0 and we do not write an equation for it. Note that since V P H 1 to evaluate (21) at the level of the truncation, only the component in the first Wiener chaos is necessary, i.e., V n 1 . For that, the system above can be solved and shows that V n 1 " pλ´∆´L s 0`Hn q´1V , where # H 1 :" 0 , H k`1 " A˚pλ´∆´L s 0`Hk q´1A`, k ě 1 .
It is important to note that H k : H n Ñ H n for every k, n P N. Recall that by (27) we can write ErVpλĹ s n q´1Vs " xV, V n 1 y. As it was first noticed in [15, eq. (2.4)], the following monotonicity formula follows from the fact that λ´∆´L s 0 is a positive operator. Lemma 10. Let S :" λ´∆´L s 0 , then, for every n ě 1, we get the bounds xV, pS`H 2n q´1Vy " xV, V 2n 1 y ď xV, pλ´L s q´1Vy ď xV, V 2n´1 1 y " xV, pS`H 2n´1 q´1Vy .
Remark 11. Let us look to the first upper bound when taking n " 1 in Lemma 10 above. Recall that V P H 1 and thatVppq " p 2 for p " pp 1 , p 2 q. Thus by considering the solution V 1 to the truncation at the first level, we arrive at xV, pλ´L s q´1Vy ď xV, pλ´∆´L s 0 q´1Vy " for a constant C ą 0. Note now that for the case s ă 1, the inequalities in (11) imply the diffusive bounds (13) in Theorem 4, see (63) in Section 6 and the following discussion. On the other hand, for the case of s ě 1, the estimates in (11) together with the first lower bound obtained with n " 1 in Lemma 10, by the same argument for the lower bound in Section 7 for the case γ " 1, gives for constants A, B ą 0. These are precisely the estimates obtained in [21] for the static case. In particular this already implies that the dynamics of SHE is not enough to remove the super diffusivity caused by the random environment.
The estimates in (37) can be iterated for higher levels and be improved at each step. Indeed, to get (12), it is necessary to use Lemma 10 in full by taking the level k to diverge with λ Ó 0. Moreover, an understanding of the estimates for every level is necessary, and for that it suffices to analyse the operators H k . For this, we make use of the following three lemmas, taken from Toninelli et al. [7]. In what follows, S is an operator which acts diagonally in Fock space with Fourier multiplier denoted by σ, such that y Sψ n pp 1:n q " σ n pp 1:n q x ψ n pp 1:n q for any ψ n P H n , which will later be taken to be S " S`H n , for n ě 1.
Lemma 12. For any ψ n P H n , it holds that xψ n , A˚SA`ψ n y " xψ n , A˚SA`ψ n y Diag`x ψ n , A˚SA`ψ n y Off , where xψ n , A˚SA`ψ n y Diag :" ψ n pp 1:n q| 2 σ n`1 pp 1:n`1 q˜p n`1ˆn ÿ k"1 p k¸2 dp 1:n`1 and xψ n , A˚SA`ψ n y Off x ψ n pp 1:n q x ψ n pp 1:n`1zn qσ n`1 pp 1:n`1 q˜p n`1ˆn`1 ÿ k"1 p k¸˜pnˆn`1 ÿ k"1 p k¸d p 1:n`1 Lemma 13. If for every n P N and any p 1:n P R 2n with ř n k"1 p k ‰ 0 ż R 2 p V pqqpsin θq 2 σ n`1 pp 1:n , qq dq ďσ n pp 1:n q with θ the angle between q and ř n k"1 p k , then for every ψ n xψ n , A˚SA`ψ n y Diag ď xψ n , p´∆qSψ n y (39) whereS is the diagonal operador whose Fourier multiplier isσ. If the inequality in (38) is ě, then (39) holds with ě as well.
Lemma 14. If for every n P N and any p 1:n P R 2ň V pqq psin θq 2 σ n`1 pp 1:n , qqˇˇq`ř n´1 k"1 p kˇd q ďσ n pp 1:n q with θ the angle between q and ř n k"1 p k , then for every ψ n |xψ n , A˚SA`ψ n y Off | ď nxψ n , p´∆qSψ n y whereS is the diagonal operador whose Fourier multiplier isσ.
Here are some preliminary definitions, needed to state and prove the next theorem. Expressions (40) and (41) arise naturally when iterating the estimates for different levels k in Lemma 10. For k P N, x ą 0 and z ě 0, let L, LB k and U B k be given by LB k px, zq :" and, for k ě 1, define σ k as We have that σ k " 1. Also, for n P N, let z k pnq " K 1 pn`kq 2`2ε and f k pnq " K 2 a z k pnq , where K 1 , K 2 are constants to be chosen sufficiently large later and ε is the small positive constant appearing in the main Theorem 3. Now, for k ě 1, let δ k be an operator such that its Fourier multiplier is σ k , meaning where N denotes the so-called Number Operator, the infinitesimal generator of B t u "´u`?2p´∆q´1 2 ξ, which acts diagonally on the n-th Wiener chaos by multiplying by n: N ψ n " nψ n for every ψ n P H n .
Remark 15. Note that the functions L, LB k and U B k are the same as in Toninelli et al. [7], while the operators δ k carry the the generator L s 0 , which is the difference between the dynamic and the static settings. Gathering these we put them into the next theorem.
Theorem 16. If s ě 1 in (2), then for every ε ą 0, we may choose K 1 and K 2 in (42) to be large enough so that, for 0 ă λ ď 1 and k ě 1, the following operator estimates hold true.
where c 1 " 1 and c 2k " π c 2k´1ˆ1`1 k 1`ε˙, c 2k`1 " π c 2kˆ1´1 pk`1q 1`ε˙. (45) Remark 17. We shall emphasise here that the sequences c 2k and c 2k`1 in (45) do converge to finite, strictly positive constants, as k Ñ 8, provided that ε ą 0. Furthermore, the limits are strictly greater than 2π and strictly smaller than 1, respectively. This can be seen, e.g. for the even sequence, c 2k`2 c 2k " p1`1 k 1`ε qp1´1 pk`1q 1`ε q´1 ą 1 and c 2 " 2π. Also, by iterating the definition for c 2k , it can be shown that convergence of the sequence is equivalent to the convergence of ř 8 l"1 l´p 1`εq , which only holds when ε ą 0. Now we will prove Theorem 16 by induction on k. Note that the induction alternates between lower (43) and upper (44) bounds, being one the consequence of the other, and so forth.
Proof of the lower bound (43). Recall that s ě 1. For k " 1 we note that, by definition, H 1 " 0 and δ 1 is non-positive if we choose the constant K 2 in (42) to be large enough.
where in the last inequality we have again used Lemma 18 and chosen the constant K 2 in (42) large enough so that for all k, n P N, it holds that So by Lemma 13 we get that the diagonal part of (47) is lower bounded by xψ, p´∆qSψy, wherẽ Here we have twice lower bounded z 2k`1 " z 2k`1 pnq ě z 2k`1 p1q and f 2k`1 " f 2k`1 pnq ě f 2k`1 p1q.
For the off-diagonal part of (47) we use Lemma 14. For that, denote p " ř n i"1 p i and p 1 " pqqpsin θq 2 dq rλ`|p 1:n | 2s`| p`q| 2 c 2k f 2k`1 U B k´1 pλ`|p 1:n | 2s`| p`q| 2`| q| 2 , z 2k`1 qs|p 1`q | , where in the last inequality we have used the monotonicity of U B k´1 and that sinceV is supported on |q| ď 1, we have that |q| 2s ď |q| 2 if s ě 1.Thanks to Lemma 20 the functions f px, zq " c 2k f 2k`1 U B k´1 px, zq and gpx, zq " 1 c 2k f 2k`1 LB k´1 px, zq satisfy the assumptions of Lemma 23 and we obtain the upper bound where we have used that LB k´1 ď LB k , the definition of z 2k`1 " z 2k`1 pnq in (42) and the fact that Altogether, Lemmas 13 and 14 combined with expressions (51) and (53), we obtain that the operator A˚pλ´∆p1`c 2k δ 2k q´L s 0 q´1A`is lower bounded by which by (46) is also a lower bound for H 2k`1 . Again, making the constants K 1 and K 2 in (42) as large as necessary, we obtain that A ě 1´1 pk`1q 1`ε and B ď 1´1 pk`1q 1`ε , which combined with the definition of c 2k`1 in (45) concludes the proof of the lower bound in (43).

Proof of (12) in Theorem 3
In this section we finish proving Theorem 3 by using the full power of the iterative estimates provided by Lemma 10. This is done by choosing the level of the truncation depending on λ, i.e., as λ Ñ 0, n Ñ 8 in Lemma 10 Again, C denotes a constant, which may change from line to line, but is independent of p, z, λ and k.
Proof of Theorem 3 for s ě 1. Recall that for p " pp 1 , p 2 q P R 2 ,Vppq " p 2 and that V P H 1 implies that the multiplier of´∆´L s 0 is |p| 2`| p| 2s . Let us start with upper bound. By Lemma 10 and (21) we get that y " xV, pλ´∆´L s 0`H 2k`1 q´1Vy , which by (43) in Theorem 16 is upper bounded by xV, pλ´∆p1`c 2k`1 δ 2k`1 q´L s 0 q´1Vy ppq dp λ`|p| 2 LB k pλ`|p| 2 , z 2k`1 q where we have used (57). Note that since V P H 1 , the arguments in f 2k`1 and z 2k`1 are both 1 and therefore they are constants which only depend on k. Now we conclude exactly as [7], since the expression above is equal to expression (5.1) in their paper. We include the missing steps in Appendix C for completeness. Now we proceed to the lower bound. Again, by Lemma 10 and (21), we get that λ 2 2 D V pλq ě xV, V 2k 1 y " xV, pλ´∆´L s 0`H 2k q´1Vy , which in turn, by Theorem 16, is lower bounded by xV, pλ´∆p1`c 2k δ 2k q´L s 0 q´1Vy " where we have substituted c 2k by its limit as k Ñ 8 and used the monotonicity of U B k´1 . Now, note that since all the functions in (59) but p Þ Ñ p 2 2 are rotationally invariant, the integral has the exact same value as if we replace p Þ Ñ p 2 2 with p Þ Ñ p 2 1 . Summing the integrals with p Þ Ñ p 2 2 and p Þ Ñ p 2 1 and diving it by two, we get that expression (59) is equal to (the 1{2 is merged into C) ppq dp pλ`|p| 2 qU B k´1 pλ`|p| 2 , z 2k q`|p| 2s .
Thus, an application of (83) gives the lower bound where the second inequality is a consequence of (76) in Lemma 18. Once again, expression (60) above reduces to the exact same as the third line in display (5.7) in [7], and thus we include the end of the proof in Appendix C for completeness.

Proof of Theorem 4
In this section, we show that our model for s ă 1 is a particular case of the theory developed in Komorowski and Olla [13] of homogenisation for diffusions in divergence free, Gaussian and Markovian random environments. See also Chapters 11 and 12 of the monograph [11]. Let us consider here the function Vpωq :" ωp0q " pω 1 p0q, ω 2 p0qq " pV 1 pωq, V 2 pωqq. In view of Remark 6, we see that V i P H 1 , i " 1, 2. Now, we may write and focus on the additive functionals of pη t q tě0 given by ş V i pη s q ds, for i " 1, 2, since εBpt{ε 2 q d " Bptq for every ε ą 0 and t ě 0. Let C :" Ť n ' kďn H k be a core for L s and pL s q˚. Let S :" pL s`p L s q˚q{2 " L s 0`∆ be the symmetric part of the generator L s . For every ψ P C , let }ψ} 2 1 :" xψ,´L s ψy " xψ,´Sψy be a norm and }ψ} 2 1 :" lim λÑ0 xψ, pλ´Sq´1ψy be another norm. By [19,Theorem 2.2], for every t ě 0, it holds that where the last inequality is a consequence of ppq dp |p| 2s ď C since s ă 1, as discussed previously in (36) in Remark 11 for i " 1. Note that (63) proves the upper bound (13) in Theorem 4. The lower bound follows from the Yaglom-reversibility (19). So now we show that our model, for s ă 1, is a particular case of the general framework of divergence-free, Gaussian and Markovian environments treated in [13,Section 6].
Proof of Theorem 4. In Section 6 of [13], the same SDE as in (1) is considered, with a dynamic random environment pω t q tě0 which is divergence-free, Gaussian and Markovian. Moreover, they assume that, in d " 2, the space-time correlations of the drift field ω satisfy expression (1.2) in page 181, which reads as where a : r0, 8q Ñ r0, 8q is a compactly supported and bounded cut-off function, β ě 0 and α ă 1. Also, the notation p b p represents the canonical tensor product in R 2 and I the identity 2ˆ2 matrix. Since here we consider the dyamics in (2), we identify β in (65) with s. Also, sinceV is rotationally invariant and has compact support, we may identify ap|p|q in (65) withV ppq. Now, note that, for p " pp 1 , p 2 q P R 2 , In view of (27) and (24), we get that, for every ψ j P H 1 , j " 1, 2, given by ψ j pωq " ş R 2 f j 1 pxqω 1 pxq dxş R 2 f j 2 pxqω 2 pxq dx, (in what follows we suppress p fromf i j ppq) With this observation, we see that α ă 1 in (65) translates to α " 0. With the same argument, we conclude that the law of ω satisfies assumption (E) in Section 6 of [13] with α " 0. Therefore, since s " β ă 1, by [13,Theorem 6.3], we get Theorem 4 with the covariance matrix D given by where the objects ψ i , for i " 1, 2 satisfy lim λÓ0 }ψ i λ´ψ i } 1 " 0 for ψ i λ solution to the resolvent equations λψ i λ´L s ψ i λ "´V i , λ ą 0 . The inner product x¨,¨, y 1 is defined through polarisation by

Proof of Theorem 5
In this section we prove Theorem 5 by making use of the first upper and lower bounds provided by Lemma 10, i.e., the estimates obtained for n " 1. When ω t " ω γ t is the solution to (14), the dominant terms in the estimates are once again the ones coming from the dynamics of the environment, as in the case of s ă 1 in Theorem 3. This is the reason why we can find matching upper and lower bounds just going to the first two estimates.
Since p´∆q is a self-adjoint, positive operador, we can make sense of the operator plogpe`p´∆q´1qq γ for every γ ą 0 through its Fourier multiplier, in the spirit of Proposition 8, given by σ γ pp 1:n q " Expression (67) is associated with the generator L γ 0 of the process pω γ t q tě0 solution to (14). Therefore, since we have the correction plogpe`p´∆q´1qq γ 2 in front of the noise in (14), Proposition 8 holds true with´p´∆q s replaced by plogpe`p´∆q´1qq γ ∆ and thus the dynamics in (14) preserves the law of ω as invariant measure for every γ ą 0. Note also that plogpe`x´1qq γ ą 1 for every x ě 0.
The third inequality is a result of the same argument as in (59), the fourth inequality is true because γ P r 1 2 , 1s ñ 1´γ ď γ and thus we may absorb the lower order terms into |p| 2 plogpe`|p|´2qq γ by changing the constant C. The fifth inequality is due to an application of the Mean Value Theorem together with the inequality in (68), in the same spirit of (57). Once again, by adapting (81) in Lemma 19, we get that (73) is lower bounded by Therefore, if γ P r 1 2 , 1q, by (69) and (21), we get that and if γ " 1, by (76) with k " 0, we get the lower bound in (16), which concludes the proof of Theorem 5.

A Technical lemmas I
In this section, for completeness, we list some important technical lemmas used throughout the estimates in the proofs of Theorem 16 and Theorem 3, all of them due to Toninelli et al. [7].
Lemma 18. For k P N let L, LB k and U B k be the functions defined in (40) and (41). Then, the three are decreasing in the first variable and increasing in the second. For every x ą 0 and z ě 1, the following holds true Furthermore, for every 0 ă a ă b, one has Finally, it also holds that Lemma 19. Let V be as in (7). Let z ą 1 and f p¨, zq : r0, 8q Þ Ñ r1, 8q be a strictly decreasing and differentiable function, such that´f pxq x ď f 1 pxq ă 0 for all x P R (79) and the function gp¨, zq : r0, 8q Þ Ñ r1, 8q a strictly decreasing function such that gpx, zqf px, zq ě z. Then, there exists a constant C Diag ą 0 such that, for all z ą 1, one gets the bounďˇˇˇˇż pqqpsin θq 2 dq λ`|p`q| 2 f pλ`|p`q| 2 , zq´π 2 where p " ř n i"1 p i for some n P N and p 1 , . . . , p n P R 2 and θ is the angle between p and q. The second integral is zero if λ`|p| 2 ě 1. Moreover, for λ ď 1,ˇˇˇˇ1 2 ż R 2V pqq dq λ`|q| 2 f pλ`|q| 2 , zq´π 2 Lemma 20. The functions U B k p¨, zq and LB k p¨, zq satisfy the conditions of the previous lemmas.

B Technical lemmas II
The two lemmas in this section are modifications of Lemmas 19 and Lemma A.3 in Toninelli et al. [7]. Throughout this section we use a generic constant C which may change from line to line, but is always independent of p, q, z, λ, k and n.
This concludes the estimate of the first term.