Probab. Theory Relat. Fields https://doi.org/10.1007/s00440-018-0849-6

Weak universality for a class of 3d stochastic reaction–diffusion models M. Furlan1

· M. Gubinelli2

Received: 13 August 2017 / Revised: 14 April 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract We establish the large scale convergence of a class of stochastic weakly nonlinear reaction–diffusion models on a three dimensional periodic domain to the dynamic 43 model within the framework of paracontrolled distributions. Our work extends previous results of Hairer and Xu to nonlinearities with a finite amount of smoothness (in particular C 9 is enough). We use the Malliavin calculus to perform a partial chaos expansion of the stochastic terms and control their L p norms in terms of the graphs of the standard 43 stochastic terms. Keywords Weak universality · Paracontrolled distributions · Stochastic quantisation equation · Malliavin calculus · Partial chaos expansion Mathematics Subject Classification 60H15 · 60H07

1 Introduction Consider a family of stochastic reaction–diffusion equations in a weakly nonlinear regime: L u(t, x) = −εα Fε (u(t, x)) + η(t, x),

B

(t, x) ∈ [0, T /ε2 ] × (T/ε)3

(1)

M. Furlan [email protected] M. Gubinelli [email protected]

1

CEREMADE, Université Paris Dauphine, Paris, France

2

IAM & HCM, Universität Bonn, Bonn, Germany

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with ε ∈ (0, 1], T > 0, initial condition u¯ 0,ε : (T/ε)3 → R, Fε ∈ C 9 (R) with exponential growth at infinity, α > 0 and L := (∂t − ) the heat flow operator and T = R/(2π Z). Here η denotes a family of centered Gaussian noises on [0, T /ε2 ] × (T/ε)3 with stationary covariance E(η(t, x)η(s, y)) = C˜ ε (t − s, x, y) such that C˜ ε (t − s, x, y) = (t − s, x − y) if dist(x, y)  1 and 0 otherwise where : R × R3 → R+ is a smooth function compactly supported in [−1, 1] × BR3 (0, 1). We assume also that there exists a compactly supported function ψ such that ψ ∗ψ = (this is true e.g. when η is obtained by space-time convolution of the white noise with ψ). We look for a large scale description of the solution to Eq. (1) and we introduce the “mesoscopic” scale variable u ε (t, x) = ε−β u(t/ε2 , x/ε) where β > 0. Substituting u ε into (1) we get L u ε (t, x) = −εα−2−β Fε (εβ u ε (t, x)) + ε−2−β η



t x , ε2 ε

 .

(2)

In order for the term ε−2−β η(t/ε2 , x/ε) to converge to a non–trivial random limit we need that β = 1/2. Indeed the Gaussian field ηε (t, x) := ε−5/2 η(t/ε2 , x/ε) has covariance ε−5 C˜ ε (t/ε2 , x/ε) and converges in distribution to the space-time white noise on R × T3 . For large values of α the non–linearity will be negligible with respect to the additive noise term. Heuristically, we can attempt an expansion of the reaction term around the stationary solution Yε to the linear equation L Yε = −Yε + ηε ,

(3)

t |x|2 1 ˇ x) = ˇ e− 4t e−t 1t0 . x−y)ηε (s, y)dsdy with P(t, i.e. Yε (t, x) = −∞ P(t−s, (4π t)3/2 Let us denote with Cε the covariance of Yε . We approximate the reaction term as εα−5/2 Fε (ε1/2 u ε (t, x))  εα−5/2 Fε (ε1/2 Yε (t, x)). The Gaussian r.v. ε1/2 Yε (t, x) has variance σε2 = εE[(Yε (t, x))2 ] = εE[(Yε (0, 0))2 ] = εCε (0, 0) independent of (t, x). Although σε2 depends on ε, it can be bounded from above and below by two positive constants uniformly on ε ∈ (0, 1]. We can expand the random variable Fε (ε1/2 Yε (t, x)) according to the chaos decomposition relative to ε1/2 Yε (t, x) and obtain Fε (ε1/2 Yε (t, x)) =



f n,ε Hn (ε1/2 Yε (t, x), σε2 ),

(4)

n0

where Hn (x, σε2 ) are standard Hermite polynomials with variance σε2 and highestorder term normalized to 1. Note also that the coefficients ( f n,ε )n0 do not depend on

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Weak universality for a class of 3d stochastic…

(t, x) by stationarity of the law of ε1/2 Yε (t, x) since they are given by the formula f n,ε =

   1 1 1/2 1/2 2 ε = E F (ε Y (t, x))H Y (t, x), σ E [Fε (σε G) Hn (G)] ε ε n ε ε 2n n!σε n!σεn

where G is a standard Gaussian variable of unit variance. Let X be the stationary solution to the equation L X = −X + ξ, with ξ the space–time white noise on R × T3 and denote by X N  the generalized random fields given by the N -th Wick power of X , which are well defined as random elements of S  (R × T3 ) as long as N  4. The Gaussian analysis which we set up in this paper shows in particular that if ε(n−N )/2 f n,ε → gn as ε → 0 for every 0  n  N , N  4, and (Fε )ε ⊆ C N +1 (R) with exponential growth, then the family of random fields FεN : (t, x) → ε−N /2 Fε (ε1/2 Yε (t, x)),

(t, x) ∈ R × T3 ,

N converges in law in S  (R × T3 ) as ε → 0 to n=0 gn X n . Consider the smallest n such that f n,ε converges to a finite limit as ε → 0. Since Hn (ε1/2 Yε , σε2 ) = εn/2 Yεn , the n-th term in the expansion (4) is f n,ε εα+(n−5)/2 Yεn . Therefore, the equation yields a non-trivial limit only if α = (5 − n)/2. We are interested mainly in the case n = 3 ⇒ α = 1 and n = 1 ⇒ α = 2. The case α = 2 gives rise to a Gaussian limit and its analysis is not very difficult. In the following we will concentrate on the analysis of the α = 1 case where the limiting behaviour of the model is the most interesting and given by the 43 family of singular SPDEs. In this case we obtain the family of models 3

1

L u ε (t, x) = −ε− 2 Fε (ε 2 u ε (t, x)) + ηε (t, x)

(5)

1

with initial condition u 0,ε (·) := ε− 2 u¯ 0,ε (ε−1 ·) where u¯ 0,ε is the initial condition of the microscopic model (1). In order to state our main result, Theorem 1.2 below, let us introduce some notations and specify our assumptions. Let F˜ε be the centering (up to the third Wiener chaos relative to ε1/2 Yε (t, x)) of the function Fε , i.e. F˜ε (x) := Fε (x) − f 0,ε − f 1,ε x − f 2,ε H2 (x, σε2 ) =



f n,ε Hn (x, σε2 ).

(6)

n3

The decomposition of F˜ε is obviously the same as in (4) except for the fact that we (m) have discarded the orders 0,1,2. Let F˜ε be the m-th derivative of the function F˜ε for 0  m  9 and define the following ε–dependent constants: dε :=

ε−2 9

 s,x

 Ps (x)E F˜ε(1) (ε1/2 Yε (s, x)) F˜ε(1) (ε1/2 Yε (0, 0)) ,

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d¯ε := 2ε−1/2 f 3,ε f 2,ε dε := dε

ε−2 6



ε−5/2 := 3

s,x

 s,x

Ps (x)(Cε (s, x))2 ,

 Ps (x)E F˜ε(0) (ε1/2 Yε (s, x)) F˜ε(2) (ε1/2 Yε (0, 0)) ,



s,x

 Ps (x)E F˜ε(0) (ε1/2 Yε (s, x)) F˜ε(1) (ε1/2 Yε (0, 0)) ,

where Ps (x) is the heat kernel and

 s,x

(7)

denotes integration on R+ × T3 .

Assumption 1 All along the paper we enforce the following assumptions: a) (u 0,ε )ε converges in law to a limit u 0 in C −1/2−κ and is independent of η; b) (u¯ 0,ε )ε is uniformly bounded in L ∞ in probability, i.e. ∃C > 0 such that ∀ε ∈ (0, 1]

u¯ 0,ε L ∞ ((T/ε)3 )  C; c) (Fε )ε ⊆ C 9 (R) and there exists constants c, C > 0 such that sup ε,x

9 

|∂xk Fε (x)|  Cec|x| ,

(8)

k=0

d) the family of vectors λε = (λ0,ε , λ1,ε , λ2,ε , λ3,ε ) ∈ R4 given by λ3,ε := f 3,ε

λ1,ε := ε−1 f 1,ε − 9dε − 6dε

λ2,ε := ε−1/2 f 2,ε λ0,ε := ε−3/2 f 0,ε − ε−1/2 f 2,ε dε − 3dε − 3d¯ε

(9)

has a finite limit λ = (λ0 , λ1 , λ2 , λ3 ) ∈ R4 as ε → 0. Remark 1.1 Note that under Assumption 1 we do not allow initial data (u 0,ε )ε to be mollifications at size ε of a distribution u 0 ∈ C −1/2−κ . Indeed, that would verify 1.a) but not 1.b). The reason we need the rescaling u¯ 0,ε = ε1/2 u 0,ε (ε·) to be bounded in probability in L ∞ is to be able to perform the estimations of Sect. 4.1 on the remainder Rε (which accounts for the exponential divergence of Fε ). Simply assuming 1.a) would result on a non-integrable singularity in t = 0, as noted in [10] and [11]. Note also that our Assumption 1 does not allow to restart the solution in time, but since we are aiming for a local-in-time convergence of the solutions u ε , this is not a big problem. It is possible to show that 1.b) follows from more specific assumptions on Fε (for example by assuming Fε : R → R coercive and using a maximum principle to obtain 1.b)), but since the main point of this work is the estimation of stochastic terms we don’t develop this idea further. Theorem 1.2 (Convergence of the solution) Under Assumption 1 the family of random fields (u ε )ε given by the solution to Eq. (5) converges in law and locally in time to a limiting random field u(λ) in the space C T C −α (T3 ) for every 1/2 < α < 2/3. The law of u(λ) depends only on the value of λ and neither on the other details of the nonlinearity nor on the covariance of the noise term. We call this limit the dynamic 43 model with parameter vector λ ∈ R4 .

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Weak universality for a class of 3d stochastic…

Here C T C −α (T3 ) denotes the space of continuous functions from [0, T ] to the −α (T3 ) (see Appendix A.1 for the notation on Besov Besov space C −α (T3 ) = B∞,∞ spaces and paraproducts). Theorem 1.2 is actually just a corollary of the more precise result Theorem 2.2, in which we identify the paracontrolled equation satisfied by the limiting random field u(λ). Remark 1.3 We are interested only in local-in-time convergence of u ε , as a way to show the potential of our method for controlling stochastic terms with infinite chaos decomposition (developed in Sect. 3). Nevertheless, we expect it to be possible to obtain global-in-time convergence of the solution with more stringent assumptions on Fε , although we do not treat this problem here. Remark 1.4 As a special case we can take Fε (x) = λ3 H3 (x, σε2 ) + ε1/2 λ2 H2 (x, σε2 ) + ε(λ1 + γ1,ε )H1 (x, σε2 ) + ε3/2 (λ0 + γ0,ε ) so that f 3,ε = λ3 , ε−1/2 f 2,ε = λ2 , ε−1 f 1,ε = λ1 + γ1,ε , ε−3/2 f 0,ε = λ0 + γ0,ε , and dε = (λ3 )2 L ε , d¯ε = λ3 λ2 L ε , dε = dε = 0, where L ε := 2

 s,x

Ps (x)(CY,ε (s, x))2 . Choosing

γ1,ε := 9dε = 9(λ3 )2 L ε ,

γ0,ε := 3d¯ε = 3λ3 λ2 L ε ,

we obtain λε → (λ0 , λ1 , λ2 , λ3 ). This shows that all the possible limits λ ∈ R4 are attainable. In this case (5) takes the form  L u ε = −λ3 u 3ε − λ2 u 2ε − λ1 − 3λ3 ε−1 σε2 + 9(λ3 )2 L ε u ε − λ0 +λ2 σε2 − 3λ3 λ2 L ε + ηε .

(10)

The name dynamic 43 equation (or stochastic quantisation equation) derives from the fact that the simplest class of models which approximate the limiting random field u(λ) is precisely obtained by choosing a cubic polynomial like in (10) as non-linear term (which is the gradient of a fourth order polinomial playing the role of local potential). In two dimensions, this model has been subject of various studies for more than thirty years [1,5,12]. For the three dimensional case, the kind of convergence results described above are originally due to Hairer [8,9] and constitute one of the first groundbreaking applications of his theory of regularity structures. Similar results were later obtained by Catellier and Chouk [4] using the paracontrolled approach of Gubinelli, Imkeller and Perkowski [6]. Kupiainen [13] described a third approach using renormalization group ideas.

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Weak universality is the observation that the same limiting object describes the large scale behaviour of solutions of more general equations, in particular that of the many parameters present in a general model, only a finite number of their combinations survive in the limit to describe the limiting object. The adjective “weak” is related to the fact that, in order to control the large scale limit, the non-linearity has to be very small in the microscopic scale. This sets up a perturbative regime which is well suited to the analysis via regularity structures or paracontrolled distributions. The first result of weak universality for a singular stochastic PDE has been given by Hairer and Quastel [10] in the context the Kardar–Parisi–Zhang equation. Using the machinery developed there Hairer and Xu [11] proved a weak universality result for three dimensional reaction–diffusion equations in the case of Gaussian noise and a polynomial non–linearity, within the context of regularity structures. Weak universality for reaction–diffusion equations driven by non Gaussian noise is analysed in Shen and Xu [23]. Recently, important results concerning the stochastic quantisation equation were obtained by Mourrat and Weber. In particular they proved the convergence to the dynamic 42 model for a class of Markovian dynamics of discrete spin systems [15] and also the global wellposedness of 42 in space and time [16] and in time for 43 [17]. The recent preprint [22] analyzes an hyperbolic version of the stochastic quantisation equation in two dimensions, including the associated universality in the small noise regime. The present work is the first to consider in detail the weak universality problem in the context of paracontrolled distributions, showing that on the analytic side the a priori estimates can be obtained via standard arguments and that the major difficulty is related to showing the convergence of a finite number of random fields to universal limiting objects. The main novelty of our work is our use of the Malliavin calculus [20,21] to perform the analysis of these stochastic terms without requiring polynomial non– linearity as in the previous works cited above. In particular we were inspired by the computations in [19] and in general by the use of the Malliavin calculus to establish normal approximations [20]. The main technical results of our paper, Theorem 3.1 below, is not particularly linked to paracontrolled distributions. A similar analysis is conceivable for the stochastic models in regularity structures. Moreover the same tools can also allow to prove similar non-polynomial weak universality statements for the KPZ along the lines of the present analysis. This is the subject of ongoing work.

2 Analysis of the mesoscopic model The goal of this section is to obtain a paracontrolled structure for Eq. (5) analogous to that introduced in [4] for the cubic polynomial case and use it to set up the limiting procedure. Convergence of the stochastic terms and some a priori estimates will be the subject of the following sections. Definitions and a reminder of the basic results of paradifferential calculus needed here can be found in “Appendix A”. 2.1 Paracontrolled structure Let us start our analysis by centering the reaction term Fε (ε1/2 u ε ) in (5) using decomposition (6) to obtain:

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Weak universality for a class of 3d stochastic…

L u ε = − ε− 2 F˜ε (ε 2 u ε (t, x)) + ηε 3

1

 1  − ε−3/2 f 0,ε − ε−1 f 1,ε u ε − ε−3/2 f 2,ε H2 ε 2 u ε , σε2 .

We write u ε = Yε +vε with Yε as in (3), and perform a Taylor expansion of F˜ε (ε1/2 Yε + ε1/2 vε ) around ε1/2 Yε up to the third order to get 3 1 1 1 1 1 L u ε = ηε − ε− 2 F˜ε (ε 2 Yε ) − ε−1 F˜ε(1) (ε 2 Yε )vε − ε− 2 F˜ε(2) (ε 2 Yε )vε2 2 1 ˜ (3) 1 3 −3/2 −1 − Fε (ε 2 Yε )vε − ε f 0,ε − ε f 1,ε (Yε + vε ) 6 − ε−1/2 f 2,ε (Yε2  + 2vε Yε + vε2 ) − Rε (vε ).

(11)

where Rε (vε ) is the remainder of the Taylor series and we use the fact that H2 (ε1/2 Yε , σε2 ) = εYε2 . Notice that we stopped the Taylor expansion at the first term (3) 1 for which ε does not appear anymore with a negative exponent (that is F˜ε (ε 2 Yε )). One can then expect the remainder Rε (vε ) to converge to zero in some sense. On the 1 other hand, all the other terms except F˜ε(3) (ε 2 Yε ) and Rε (vε ) appear to diverge in the limit ε → 0, but in analogy with well-known renormalization methods for random fields, we try to find a combination of them that can be made to converge in some function space. Define the following random fields: L Yε := −Yε + ηε := ε−1/2 f 2,ε Yε2  Y¯ε 3 1 Yε := ε− 2 F˜ε (ε 2 Yε ) 1 Yε := 13 ε−1 F˜ε(1) (ε 2 Yε ) 1 1 Yε := 16 ε− 2 F˜ε(2) (ε 2 Yε ) Y¯ε := Y¯ε ◦ Yε − d¯ε Yε

:= Yε ◦ Yε − dε ,

L Y¯ε L Yε L Yε Yε∅ Yε Yε

:= := := :=

Y¯ε , Yε , Yε 1 ˜ (3) 21 6 Fε (ε Yε )

(12)

:= Yε ◦ Yε − dε ,

:= Yε ◦ Yε − dε Yε − dε ,

with Yε stationary solution, while Yε , Yε , Yε , Y¯ε have 0 initial condition in t = 0. The last four trees Y¯ε , Yε , Yε , Yε are obtained from the others via the resonant

Bony’s paraproduct ◦ recalled in “Appendix A”, and d¯ε , dε , dε , dε , dε are just ε-dependent constants whose exact value will matter only in Sect. 3. Indeed, in the scope of this section we only need the following relation to be verified: dε = 2dε + 3dε .

(13)

The notation Y¯ε denotes that this tree has finite chaos expansion and can be treated with the well-known techniques of [4] or [18] (we put a bar on Y¯ε just because it is the only tree obtained from Y¯ε ).

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With the definitions (12), Eq. (11) takes the form L vε = Yε − Y¯ε − Yε − 3Yε vε − 3Yε vε2 − Yε∅ vε3 − ε−3/2 f 0,ε − ε−1 f 1,ε (Yε + vε ) − ε−1/2 f 2,ε (2Yε vε + vε2 ) − Rε (vε ). (14) At this point it is worth noting that the trivial case F˜ε (x) = H3 (x, σε2 ) yields Yε = Yε3 , Yε = Yε2 , Yε = Yε , Y ∅ = 1. By comparing these random fields to the ones defined in [4] we can guess that Yε , Yε , Yε , Y ∅ can be controlled respectively in C 1/2−κ , C −1−κ , C −1/2−κ , C −κ ∀κ > 0 for any Fε satisfying Assumption 1, and carry on the paracontrolled analysis of (14) as if it were the case. Clearly, the paracontrolled structure is robust and does not depend on how the terms Yε , Yε , Yε , Y ∅ are defined as long as they have the desired regularity. From these observations, we do not expect to be able to control the products Yε vε , Yε vε2 and Yε vε in Eq. (14) uniformly in ε > 0. In order to proceed with the analysis we make the Ansatz: u ε = Yε + vε ,

vε = −Yε − Y¯ε − 3vε ≺≺ Yε + vε

(15)

and proceed to decompose the ill-defined products using the paracontrolled techniques recalled in “Appendix A”. We start by writing vε Yε = vε ≺ Yε + vε  Yε + vε ◦ Yε . The resonant term, together with Ansatz (15), yields: vε ◦ Yε = −Yε ◦ Yε − Y¯ε ◦ Yε − 3vε (Yε ◦ Yε ) − 3com1 (vε , Yε , Yε ) + vε ◦ Yε ,

with the definition and bounds of com1 (·, ·, ·) given in Lemma A.7. Then we define ˆ ε := vε Yε − vε ≺ Yε + (3vε dε + dε Yε + dε + d¯ε ) Yε v = vε  Yε − Y¯ε − Yε − 3vε Yε + vε ◦ Yε − 3com1 (vε , Yε , Yε ). Moreover we have for vε Yε : vε Yε = ϕε Yε − Yε ≺ Yε − Yε  Yε − Yε ◦ Yε , where we introduced the shorthand ϕε := vε + Yε . So we let vε  Yε := vε Yε + dε = ϕε Yε − Yε ≺ Yε − Yε  Yε − Yε , Finally to analyse the product Yε vε2 we write Yε vε2 = Yε (Yε )2 − 2Yε Yε ϕε + Yε ϕε2 ,

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Weak universality for a class of 3d stochastic…

and consider the products involving only Y τ factors: first Yε Yε = Yε  Yε + Yε ≺ Yε + Yε + dε =: Yε  Yε + dε , and then we define the term Yε  (Yε )2 as follows: Yε  (Yε )2 := Yε (Yε )2 − 2dε Yε = Yε ≺ (Yε )2 + Yε  (Yε )2 + Yε ◦ (Yε ◦ Yε ) +2 com1 (Yε , Yε , Yε ) + 2Yε Yε , so that Yε  vε2 := Yε vε2 + 2dε vε = Yε  (Yε )2 − 2(Yε  Yε )ϕε + Yε ϕε2 . We note also that L vε = −L Yε − L Y¯ε + L vε − 3vε ≺ L Yε − 3 com3 (vε , Yε ) −3 com2 (vε , Yε ), with com2 (·, ·) and com3 (·, ·) specified in Lemma A.7 of the Appendix. Substituting these renormalized products into (14) and recalling the definition (9) for  λε = (λ0,ε , λ1,ε , λ2,ε , λ3,ε ), we obtain the following equation for vε : L vε = 3 com3 (vε , Yε ) + 3 com2 (vε , Yε ) ˆ ε − Yε∅ vε3 − 3Yε  vε2 − 3Yε v + Yε − λ2,ε (2vε  Yε + vε2 )  − λ1,ε (Yε + vε ) + 9dε + 6dε − 3dε vε − λ0,ε − Rε (vε ), where we can use the constraint (13) to remove the term proportional to vε . Summarizing, we obtain the following equation, together with Ansatz (15): vε = −Yε − Y¯ε − 3vε ≺≺ Yε + v  L vε = U (λε , Yε ; vε , vε ) − Rε (vε )

(16)

with initial condition vε,0 = u 0,ε − Yε (0) and U given by U (λε , Yε ; vε , vε ) := 3 com3 (vε , Yε ) + 3 com2 (vε , Yε ) − Yε∅ vε3 ˆ ε + Yε − λ2,ε (2vε  Yε + vε2 ) − 3Yε  vε2 − 3Yε v − λ1,ε (Yε + vε ) − λ0,ε − Rε (vε ).

(17)

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The enhanced noise vector Yε is defined by Yε := (Yε∅ , Yε , Yε , Y¯ε , Yε , Yε , Yε , Y¯ε , Yε ) 2  1 XT := C T C −κ × C T C − 2 −κ × C T C −1−κ × L

1/2−κ T

3 × C T C −κ

1

×C T C − 2 −κ

(18)

for every κ > 0, T > 0. We use the notation Yε XT = τ Yτε X τ for the associated norm where Yετ is a generic tree in Yε . The homogeneities |τ | ∈ R are given by Yετ = Yε∅ Yε Yε Y¯ε Yε Yε Yε Y¯ε Yε |τ | = 0 −1/2 −1 −1 1/2 0 0 0 −1/2 Note that for every ε > 0 Eq. (16) is equivalent to Eq. (5). Remark 2.1 The paracontrolled structure we developed in this section is the same as in the work of Catellier and Chouk [4], plus an additive source term (which is Rε (vε ) in Eq. (16)). Therefore, there exists T = T Yε XT , u ε,0 C −1/2−κ , |λε | such that we can define for α ∈ (1/2, 2/3), p ∈ [4, ∞), γ > 41 + 23 κ a solution map γ,p

 : C −1/2−κ × X τ × R4 × MT L p (T3 ) → C T C −α (T3 ) (u ε,0 , Yε , λε , R) → u ε so that u ε = (u ε,0 , Yε , λε , R) with u ε = Yε + vε and vε that solves (16) with the γ remainder Rε (vε ) replaced by R. The space MT L p (T3 ) is specified in Appendix A.1. Indeed, we can use Lemma A.1 and Lemma A.3 to control I R as

I R L

1/2,1+2κ T

 T δ R Mγ , p L p T

for δ > 0 small enough, and thanks to this bound it is easy to see that the the fixed point procedure of Section 3 of [4] still holds with a fixed additive source term R. In the same way, the continuity of the solution map  follows easily as in Theorem 1.2 of [4]. 2.2 Identification of the limit In order to identify interesting limits for Eq. (5), we introduce ∀λ = (λ0 , λ1 , λ2 , λ3 ) ∈ R4 the enhanced noise Y(λ) which is constructed from universal noises X τ as Y(λ) := (Y ∅ (λ), Y (λ), Y (λ), Y¯ (λ), Y (λ), Y (λ), Y (λ), Y (λ), Y (λ)) := (λ3 , λ3 X, λ3 X , λ2 X , λ3 X , (λ3 )2 X , (λ3 )2 X , λ3 λ2 X , (λ3 )2 X ) (19)

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Weak universality for a class of 3d stochastic…

where X is the stationary solution to to the linear equation L X = −X + ξ and ξ is the time-space white noise on R × T3 . We will sometimes use the shorter notation Y(λ) = (Y τ (λ))τ for (19). We define the universal fields X τ through their Littlewood–Paley decomposition ∀(t, x) ¯ ∈ R+ × T3 as: X

:=  X 3 ,

L X =X

with X (t = 0) = 0,

X :=  X , 2

 ¯ = (1 − J0 )( X 2 (ζ1 ) X 2 (ζ2 ))μq,ζ1 ,ζ2 , ¯ := q (1 − J0 )(X ◦ X )(t, x) q X (t, x) ζ1 ,ζ2  ¯ =  X 3 (ζ1 ) X (ζ2 )μq,ζ1 ,ζ2 , ¯ := q (X ◦ X )(t, x) q X (t, x) ζ1 ,ζ2  ¯ := (1 − J1 )( X 3 (ζ1 ) X 2 (ζ2 ))μq,ζ1 ,ζ2 q X (t, x) ζ1 ,ζ2    q X (t + s, x¯ − x) − q X (t, x) +6 ¯ Ps (x)[C(s, x)]2 , (20) s,x

where as before · stands for the Wick product, ζi = (xi , si ) ∈ R × T3 , C(·, ·) is the covariance of X and μq,ζ1 ,ζ2 is the measure 

μq,ζ1 ,ζ2 := δ(t − s2 )1[0,+∞) (s1 )dζ1 dζ2

x,y

K q,x¯ (x)



K i,x (y)K j,x (x2 )Pt−s1 (y − x1 )

i∼ j |x|2

with the usual heat kernel Pt (x) = (4π 1t)3/2 e− 4t 1t0 . We commit an abuse of notation by writing X (ζ ) since X is actually a distribution in space: the integrals in (20) should obviously be intended as functionals. Standard computations (see e.g. [4] or [18]) show that, ∀λ ∈ R4 and for any T > 0, 0 < κ < κ     1 1   2   2 − 1 −2κ  Y(λ) ∈ C Tκ C − 2 −2κ × C Tκ C −1−2κ × C Tκ C 2 −2κ × C Tκ C 0−2κ × C Tκ C T 2 , almost surely. Using the paracontrolled structure we developed in Sect. 2.1 we can identify the limiting solution u(λ) introduced in Theorem 1.2. Theorem 2.2 The family of random fields u ε given by the solutions of Eq. (5) converges in law and locally in time to a limiting random field u(λ) in the space C T C −α (T3 ) for every 1/2 < α < 2/3. The limiting random field u(λ) solves the paracontrolled equation u(λ) = X + v(λ) v(λ) = −λ3 X − λ2 X − 3λ3 v(λ) ≺≺ X + v  (λ) L v  (λ) = U (λ, Y(λ); v(λ), v  (λ)) v (λ)(t = 0) = v0 + λ3 X (t = 0) + λ2 X (t = 0) + 3λ3 vε,0 ≺ X (t = 0) (21) 

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with U defined in (17) and v0 = u 0 − X (t = 0). Proof Fix T > 0. Let u ε = Yε + vε be the solution of Eq. (5) for fixed ε > 0, which is seen to be unique in the (ε-dependent) time interval [0, Tε ] by a usual fixed-point argument on the original equation (without resorting to the paracontrolled decomposition). Let u ε = (u ε,0 , Yε , λε , Rε (vε )) on [0, Tε ] with  defined as in Remark 2.1 and of Rε (vε ) seen as an exogenous source term. We know from the a priori estimations

Sect. 4.2 that there exists a time T = T Yε XT , u ε,0 C −1/2−κ , |λε | and a family of events (Eε )ε>0 such that P(Eε ) → 1 for ε → 0 and we can control vε M1/4+3κ/2 L ∞ . T

Thus, we can control the L ∞ norm of vε (t) in [Tε /2, Tε ] and extend the solution vε on [0, T ] for every ε. Denote by u ε =  (u ε,0 , Yε , λε , Rε (vε )) the process u ε stopped at time T and  the corresponding stopped solution map. Note that u(λ) solves the same equation as u ε with Yε replaced by Y(λ), u ε,0 replaced by u 0 , λε replaced by λ and Rε (vε ) = 0. So u(λ) = u  =  (u 0 , Y(λ), λ, 0) up to time T . Let us introduce the random field u¯ ε =  (u ε,0 , Yε , λε , 0) with v¯ε = u¯ ε − Yε that solves the paracontrolled Eq. (16) but with remainder Rε (vε ) = 0. Consider the n-tuple of random variables (u ε,0 , Yε , u ε , u¯ ε ) and let με be its law on

2 that  is continuous Z = C −α × XT × C T C −α conditionally on Eε . Observe

as discussed in Remark 2.1, and this gives that ∀δ > 0, με u ε − u¯ ε C T C −α > δ → 0 γ,p as ε → 0. Indeed Rε (vε ) → 0 in probability in the space MT L p (T3 ) by Lemma 4.5. 

2  This shows that με concentrates on C −α × XT × (z, z) ∈ C T C −α . Let μ any  

2  accumulation point of (με )ε . Then μ C −α × XT × (z, z) ∈ C T C −α = 1. The a priori estimations of Sect. 4.2 yield the tightness of με and from the concentration of μ on the diagonal we know that there exists a subsequence such that for any test function ϕ,  ϕ(x, y, z, t)dμε (x, y, z, t) → ϕ(x, y, z, t)dμ(x, y, z, t) Z  Z = ϕ(x, y, t, t)dμ(x, y, z, t).



Z

(22)

Moreover, still along subsequences we have that for any bounded continuous function ϕ E(ϕ(u ε,0 , Yε , u¯ ε )) = E(ϕ(u ε,0 , Yε ,  (u ε,0 , Yε , λε , 0))) → E(ϕ(u 0 , Y(λ),  (u 0 , Y(λ), λ, 0))) since by Theorem 3.1 the vector Yε converges in law to Y(λ), u ε,0 to u 0 , and  is a continuous function as discussed in Remark 2.1. This shows that  ϕ(x, y, t, t)dμε (x, y, z, t) Z  → ϕ(x, y,  (x, y, λ, 0),  (x, y, λ, 0))dμ(x, y, z, t) (23) Z

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Weak universality for a class of 3d stochastic…

and then by comparing (22) and (23) we can conclude that there exists a subsequence such that  ϕ(x, y, z, t)dμε (x, y, z, t)  → ϕ(x, y,  (x, y, λ, 0),  (x, y, λ, 0))dμ(x, y, z, t).

Z

Z

We can identify the limit distribution μ by noting that since P(Eε ) → 1 we have   E ψ(u ε,0 , Yε )IEε E ψ(u ε,0 , Yε )|Eε = P(Eε ) → E[ψ(u 0 , Y(λ))] 



for any test function ψ. So the first two marginals of μ have the law of (u 0 , Y(λ)) and they are independent since (u ε,0 , Yε ) are independent for any ε. Calling ν the law of (u 0 , Y(λ)) we have that  Z

 ϕ(x, y, z, t)dμ(x, y, z, t) =

C −α ×XT

ϕ(x, y,  (x, y, λ, 0),  (x, y, λ, 0))dν(x, y)

that the whole family (με )ε converges to μ. We can conclude that u ε → u  in law  with u = u(λ) up to the time T Y(λ) XT , u 0 C −1/2−κ , |λ| since the function T  is lower semicontinuous (as obtained from the a priori estimates).  

3 Convergence of the enhanced noise This is the central section of the paper, in which we present a new method to estimate certain random fields that do not have a finite chaos decomposition, and we apply it to the treatment of the random fields Yε of (12). 3.1 An example of convergence We choose to give first a complete example (the convergence of the tree Yε to Y (λ)) in order to put in evidence the main idea in the proof of Theorem 3.1. Recall its definition (12): Yε =

ε−1 ˜ (1) 1 F (ε 2 Yε ), 3 ε

(1) with F˜ε being the first derivative of the centered function F˜ε defined in (6). Since d 2 2 dx Hn (x, σε ) = n Hn−1 (x, σε ) the Wiener chaos decomposition of Yε reads:

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M. Furlan, M. Gubinelli

ε−1 ˜ (1) 1 ε−1  n f n,ε Hn−1 (ε1/2 Yε , σε2 ) Fε (ε 2 Yε ) = 3 3 n3

= ε−1 f 3,ε H2 (ε1/2 Yε , σε2 ) +

ε−1  n f n,ε Hn−1 (ε1/2 Yε , σε2 ) 3 n4

= f 3,ε Yε2  +

ε−1 3



n f n,ε Hn−1 (ε1/2 Yε , σε2 ),

(24)

n4 n

where · is the Wiener product and we used the fact that ε− 2 Hn (ε1/2 Yε , σε2 ) = Yεn . Now one can use hypercontractivity (as done in [10,11]) to control the L p norm of each chaos order by its L 2 norms. However this strategy does not give useful bounds for the infinite series in the second term of (24). Instead, we just observe that 

n f n,ε Hn−1 (ε1/2 Yε , σε2 ) = (id −J0 − · · · − J2 ) F˜ε(1) (ε 2 Yε ), 1

n4

where Ji is the projection of on the i-th chaos, and look for a different way to write this remainder. One of the main insights of this paper is that we can write it as: 3 ˜ (1) 2 (id −J0 − · · · − J2 ) F˜ε(1) (ε 2 Yε ) = δ 3 G [3] [1] D Fε (ε Yε ) 1

1

where D, δ are the Malliavin derivative and divergence operators, and G [3] [1] = (1 − −1 −1 −1 L) (2 − L) (3 − L) with L the Ornstein–Uhlenbeck operator. This is proven in Lemma B.1. (1) 1 To compute the Malliavin derivative of F˜ε (ε 2 Yε ) we observe that for every ε > 0 (t, x) ∈ R × T3 there exists h (t,x) ∈ L 2 (R × T3 ) such that the Gaussian random variable Yε,(t,x) := Yε (t, x) can be written as law

Yε,(t,x) = ξ, h (t,x) .

(25)

Here ξ is the Gaussian white noise on R × T3 , which can be seen as a Gaussian Hilbert space ξ, hh∈H = {W (h)}h∈H indexed by the Hilbert space H := L 2 (R × T3 ). This is the framework in which we apply the Malliavin calculus results of “Appendix B”. Notice that by construction   h (t,x) , h (t  ,x  )  = Cε (t − t  , x − x  ) := E Yε,(t,x) Yε,(t  ,x  ) .

(26)

The function h (t,x) can actually be written as the space-time convolution h (t,x) = Pˇ ∗ ψε (t, x), 2

|x| 1 e− 4t e−t 1t0 and ψε such that ηε = ψε ∗ ξ , ψε (t, x) = (4π t)3/2 ε−5/2 ψ(ε−2 t, ε−1 x). We omit the dependence on ε of h (t,x) not to burden the notation.

ˇ x) = with P(t,

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Weak universality for a class of 3d stochastic… (1)

Going back to the calculations, we obtain from (25) that D F˜ε (ε 2 Yε ) = 1 ε F˜ε(2) (ε 2 Yε )h. Then (noting that F˜ε(4) = Fε(4) ): 1

1 2

(4) ⊗3 2 Yε = f 3,ε Yε2  + ε 3 δ 3 G [3] [1] Fε (ε Yε )h . = f 3,ε Yε2  + Yˆε , 1

1/2

(27)

It can be easily seen from (25) that Yε has the same law of a time-space mollification of X by convolution (with X defined in Sect. 2.2). Then the convergence in law of f 3,ε Yε2  to Y (λ) can be easily established by standard techniques (see [4] or [18]). We are only left to show that Yˆε in (27) converges to zero in C T C −1−κ . It is well known (see Sect. 3.2 for details) that in order to control the norm of Yˆε (t, ·) for t ∈ (0, T ] in the Besov space C −α−κ ∀κ > 0 and in probability, it is enough to have suitable estimates for     sup q Yˆε (t, x) ¯ 

x∈T3

L p ()

    = q Yˆε (t, x) ¯ 

L p ()

,

for any x¯ ∈ T3 since Yˆε is stationary in space. We then proceed to compute:     ¯  q Yˆε (t, x)

L p ()

ε1/2 = 3

    3 [3]  ⊗3 (4) 21 δ G K q,x¯ (x)Fε (ε Yε,(t,x) )h (t,x) dx  [1]  

L p ()

where K q,x¯ (x) is the kernel associated to the Littlewood–Paley block q . Observe that  p (4) (4)

Fε (ε1/2 Yε,(t,x) ) L p = R |Fε (z)| p γ (dz) where γ (dz) is the density of a centered Gaussian with variance σε2 . This norm is then finite by the bound (8) of Assumption 1. Another fundamental idea of this work is that we can “estimate out” the bounded (4) term Fε (ε1/2 Yε,(t,x) ) L p , which has an infinite chaos decomposition, and obtain a standard 43 diagram that can be treated with well understood techniques. Using Lemma B.2 and Corollary B.6, one has     3 [3]  ⊗3 (4) 21 δ G K q,x¯ (x)Fε (ε Yε,(t,x) )h (t,x) dx  [1]   

3  

 k [3] D G [1] 



L p ()

  1  dx K q,x¯ (x)Fε(4) (ε 2 Yε,(t,x) )h ⊗3 (t,x) 

k=0

    ⊗3 (4) 21  K  (x)F (ε Y )h dx q,x¯ ε,(t,x) (t,x)  ε 

L p (,H ⊗3 )

L p (,H ⊗3+k )

. 1/2

Then note that we can decompose the norm u L p (,H ⊗3 ) =

u 2H ⊗3 L p/2 for u ∈ L p (, H ⊗3 ) and since the norm of the Hilbert space H ⊗3 is given by the scalar product h ⊗3 2H ⊗3 = h, h3H we obtain

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M. Furlan, M. Gubinelli

    ¯  q Yˆε (t, x)

Lp

ε

 2   ⊗3 (4) 21  K (x)F (ε Y )h dx  q,x¯ ε,(t,x) (t,x)  ε 

1/2    ⊗3 H  p/2

1/2 

L        (4) 21 (4) 21  ε |K q,x¯ (x)K q,x¯ (x )|  Fε (ε Yε,(t,x) )Fε (ε Yε,(t,x  ) )

×|h (t,x) , h (t,x  ) |3 dxdx 

L p/2

1 2

.

(4) The norm containing F˜ε can then be easily estimated using Hölder’s inequality as

  1  (4) 21   Fε (ε Yε,(t,x) )Fε(4) (ε 2 Yε,(t,x  ) )

L p/2

  1     Fε(4) (ε 2 Yε,(t,x) )

L

   (4) 21   F (ε Y )  ε,(t,x )  ε p

Lp

 1. This yields     ¯  q Yˆε (t, x)

L p ()

  1 2  ε |K q,x¯ (x)K q,x¯ (x  )||h (t,x) , h (t,x  ) |3 dxdx 

which is a standard 43 diagram that can be analysed with the techniques of [8] (recalled in Appendix A.3). We just remark that h (t,x) , h (t,x  )  = Cε (0, x − x  ) and the bound ε|Cε (t, x)|  1 of Lemma A.13 yields ∀δ ∈ (0, 1):     ¯  q Yˆε (t, x)

δ

L p ()

 ε2



|K q,x¯ (x)K q,x¯ (x  )||Cε (0, x − x  )|2+δ dxdx 

1 2

δ

 ε 2 2(1+δ/2)q and then Yˆε (t, ·) converges to zero in probability in the space C −1−δ/2 ∀δ ∈ (0, 1), as ε → 0. The time regularity of Yˆε needed to obtain the convergence in C T C −1−δ/2 does not need new ideas, and it is done in Sect. 3.3. The method shown in this section is valid verbatim for the trees Yε∅ , Yε , Yε , while for the composite trees in (12) (namely Yε , Yε , Y¯ε , Yε ) that are obtained via paraproducts of simple trees, one has to be able to write the remainder Yˆετ as an iterated Skorohod integral δ n (. . .) in order to exploit the boundedness of this operator. Moreover, these trees require a second renormalization (on top of the Wick ordering) which is not easy to control for infinite chaos decompositions. We deal with both these difficulties introducing the product formula (94), which allows to write products of iterated Skorohod integrals as combinations of iterated Skorohod integrals. The details and calculations for composite trees can be found in Sect. 3.4.

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Weak universality for a class of 3d stochastic…

3.2 Main theorem and overview of the proof Theorem 3.1 Under Assumption 1 there exists C > 0 such that for any p ∈ [2, ∞) we have Yε XT < C in L p (). Moreover, Yε → Y(λ) ∈ XT and Yε → X ∈ C T C −1/2−κ in law. The rest of Sect. 3 is dedicated to the proof of Theorem 3.1. From the definition Y(λ) = (Y τ (λ))τ of (19) it is clear that we need to prove that Yετ → Y τ (λ) for every tree τ . Note that we can write each tree Y τ (λ) as Y τ (λ) = f τ (λ)K τ (X ) for a the measurable function K τ of the Gaussian process X ∈ C T C −1/2−κ defined (20), and a suitable deterministic function f τ (λ) of λ. For example, we can write Y (λ) = λ3 X 2  with K (X ) = X = X 2  and f (λ) = λ3 . We will show [Eqs. (36) and (44)] that every random field Yετ defined in (12) can be decomposed with the same functions f τ (·) and K τ as Yετ = f τ (λε )K τ (Yε ) + Yˆετ

(28)

where Yˆετ are suitable remainder terms. For all p  2 it is well-known (see [4,8]) that the term f τ (λε )K τ (Yε ) is uniformly bounded in L p (; X τ ) (with X τ given by (18)). Thus, we will prove that Yˆετ converges to zero in L p (; X τ ). This can be done by showing that, by Besov embedding, for p ∈ [2, ∞) and ∀α < |τ | we have p p E( Yˆετ (t) C α−3/ p )  E( Yˆετ (t) B α )  p, p

 q

2

αpq

 T3

p

q Yˆετ (t, x) L p () dx → 0

(29) thanks to the stationarity of the process Y (t, x). In order to prove the bound (29) it suffices to show that ∀t ∈ [0, T ]  q

2αpq sup q Yˆετ (t, x) L p () → 0 as ε → 0, p

(30)

x

which is one of the key estimation of this paper and will be performed in Sects. 3.3 and 3.4.2. In order to obtain uniform convergence for t ∈ [0, T ] it suffices to show that ∀σ ∈ [0, 1/2], q  −1: p sup q Yˆετ (t, x) − q Yˆετ (s, x) L p ()  Cε |t − s|σ p 2−(α−2σ ) pq with Cε → 0. x

(31) Indeed, by the Garsia-Rodemich-Rumsey inequality we obtain for δ > 0 small enough and p large enough

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M. Furlan, M. Gubinelli

sup E( Yˆετ

p σ −2/ p α−2σ −δ ) CT B p, p

ε

 T2



2(α−2σ −δ) pq

q

q Yˆετ (t, x) − q Yˆετ (s, x) L p () p

×

sup

sup

s
 Cε T 2



|t − s|σ p

2−δpq

q

  which by Besov embedding yields an estimation on E Yετ C σ −κ/2 C α−2σ −κ for κ > 0 T small enough. This gives us the necessary tightness to claim that Yε has weak limits along subsequences. The only thing left after proving (28), (30) and (31) is that for each τ we have K τ (Yε ) → K τ (X ) in law. However this is clear and already well-known, since by hypothesis we can introduce a space-time convolution regularisation of X (let’s call it X ε ) which has the same law of Yε for any ε > 0. This yields immediately the convergence Yε → X in law. At this point an approximation argument gives that K τ (Yε ) has the same law of K τ (X ε ). Transposing the regularisation to the kernels of the chaos expansion we can write K τ (X ε ) = K ετ (X ) and now it is easy to check that K ετ (X ) → K τ (X ) in probability (as done systematically in [4,18]). We can then conclude that K τ (Yε ) → K τ (X ) and therefore Yετ → Y τ (λ) in law for every τ , since from Assumption 1 we have immediately f τ (λε ) → f τ (λ). Let us give some more details on how to prove the decomposition (28) and the bounds (30) and (31). As seen in Sect. 3.1 we have Yε,ζ = ξ, h ζ  in law for ζ = (t, x) ∈ R × T3 and this gives Dn F˜ε(m) (ε1/2 Yε,ζ ) = F˜ε(m+n) (ε1/2 Yε,ζ )h ⊗n ζ . We define for m ∈ N, ζ ∈ R × T3 : [m] ζ := ε

m−3 2

F˜ε(m) (ε1/2 Yε,ζ )

(32)

Note that the term [m] above is not the m-th derivative of some function  (we use the square parenthesis notation to emphasize this fact). It easy to see from (32) [m+k] ⊗k h ζ . Therefore, the partial chaos expansion (91) takes a more that Dk [m] ζ = ζ

explicit form when applied to [m] ζ : [m] ζ = =

  [m+k] n−1 E ζ k=0

n−1 k=0

k! ε(m+k−3)/2

[m+n] ⊗n k Yε,ζ  + δ n (G [n] hζ ) [1] ζ

(m + k)! ˜ [m+n] ⊗n k  + δ n (G [n] h ζ ) (33) f m+k,ε Yε,ζ [1] ζ k!

n n ⊗n with G [n] [1] defined in (89). Here we used the fact that δ (h ζ ) = Yε,ζ  (see 3 Remark B.12) and that by the definition of [m] ζ we obtain ∀ζ ∈ R × T :

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Weak universality for a class of 3d stochastic…

  = ε(m+k−3)/2 (m + k)! f˜m+k,ε E [m+k] ζ 1 with f˜n,ε the coefficients in the decomposition F˜ε (ε 2 Yε ) := Choosing n = 4 − m in Eq. (33) we obtain

[m] ζ =



1 2 ˜ 2 n0 f n,ε Hn (ε Yε , σε ).

3! 3−m ˆ [m] + f 3,ε Yε,ζ ζ (3 − m)!

(34)

and a remainder with an infinite chaos decomposition strictly greater that 3 − m: ˆ [m] = δ 4−m (G [4−m] [4] h ⊗4−m ).  ζ ζ ζ [1]

(35)

This is a key step in the proof of Theorem 3.1. Indeed, it suffices to substitute (34) into definition (12) to identify the remainder Yˆετ in decomposition (28) that has to ˆ [m] . Moreover, the strucconverge to zero, and see that it always contains the term  ζ ˆ [m] makes it possible to bound its L p norm and obtain (30), (31) in the ture (35) of  ζ same way as done in Sect. 3.1 for Yε . We will consider separately simple trees (namely Yε∅ , Yε , Yε , Y¯ε , Yε ) which are linear functions of [m] in Sect. 3.3, and composite trees (namely Yε , Yε , Y¯ε , Yε ) which are quadratic in simple trees and need to be further renormalized in order to converge to some limit as ε → 0. We will show the decomposition (28) for composite trees in Sect. 3.4.1 and the bounds (30), (31) in Sect. 3.4.2. Remark 3.2 We can easily estimate terms of the form ε−(m−3)/2 [m] ζ for 3  m  9 and every p ∈ [2, ∞). We have (as already observed for Fε(1) ):

   m−3  − 2 [m]  p p (m) 1/2 ζ  p = Fε (ε Yε,ζ ) L p = |Fε(m) (x)| p γ (dx) ε L

R

where γ (dx) is the density of a centered Gaussian with variance σε2 . The integral is finite by Assumption 1: in particular we only need to assume that the first m derivatives of Fε have exponential growth (actually, it is easy to see that one can require even weaker growth conditions).

3.3 Analysis of simple trees First of all note that the term Y¯ε has no remainder, and then it can be shown to converge in law to λ(2) Y by usual techniques (see [4]). In this section we show the convergence of the trees Yε∅ , Yε , Yε , Yε . We obtain easily from (34):

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M. Furlan, M. Gubinelli

 (3 − m)! [m] ζ μq,ζ 3! ζ   (3 − m)! (3−m) ˆ [m] μq,ζ  = f 3,ε Yε,ζ μζ + ζ 3! ζ ζ = f τ (λε )q K τ (Yε )(t, x) + q Yˆετ (t, x),

q Yετ (t, x) :=

(36)

with ζ = (s, y) and either μq,ζ = δ(t − s)K q,x (y)dsdy, for q Yε∅ , q Yε , q Yε ,  (37) μq,ζ = 1[0,+∞) (s)dsdy K q,x (z)Pt−s (z − y)dz, for q Yε , where K q,x (y) is the kernel associated to the Littlewood–Paley block q and Pt (x) is the heat kernel.  (3−m) As said before, f 3,ε ζ Yε,ζ μq,ζ converges in law in L p for every 2  p <  +∞ to λ3 ζ Yζ(3−m) μq,ζ since f 3,ε → λ3 by Assumption 1. We can bound the  [m] ˆ μq,ζ in L p () using Lemma B.2 and the definition of the remainder term ζ  ζ norm · D4−m, p (H ⊗4−m ) to obtain:       ˆ [m] μq,ζ  ζ   ζ

L p ()

     4−m [4−m] [4] ⊗4−m  = δ G [1] ζ h ζ μq,ζ   ζ L p ()      [4−m] ⊗4−m  [4] μq,ζ  ζ hζ  G [1] 

 4−m 

ζ

 k [4−m] D G [1] 

k=0

 ζ

D4−m, p (H ⊗4−m )

  ⊗4−m  [4] h μ q,ζ  ζ ζ

L p (,H ⊗4−m+k )

.

From Corollary B.6 we know that ( j − L)−1 and D( j − L)−1 are bounded in L p for every p ∈ [2, ∞) and every j  1. Applying repeatedly these estimations we obtain:     k [4−m]  [4] ⊗4−m D G   h μ q,ζ  ζ ζ [1]  ζ

L p (,H ⊗4−m+k )

    [4] ⊗4−m    h μ q,ζ  ζ ζ  ζ

L p (,H ⊗4−m )

.

Now we can proceed toimplement  the idea we already described in Sect. 3.1, i.e.  − 12 [4]  estimating out the term ε ζ  p (which is bounded by Remark 3.2 but with L ()

infinite chaos decomposition) and considering the finite-chaos term that is left. We do this by decomposing the L p (, H ⊗4−m ) norm as norms on H ⊗4−m and L p/2 () as follows:

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Weak universality for a class of 3d stochastic…

     [4] h ⊗4−m μq,ζ  ζ ζ   ζ L p (,H ⊗4−m )  1/2 2      [4] ⊗4−m   ζ h ζ μq,ζ     ζ  H ⊗4−m L p/2 ()  1/2   [4] [4] ⊗4−m ⊗4−m    ζ ζ  h ζ , hζ   H ⊗4−m μq,ζ μq,ζ    ζ

L p/2 () 1/2

 

[4] 4−m

[4] |μq,ζ μq,ζ  | ζ ζ  L p/2 () |h ζ , h ζ  |

ζ,ζ 

.

Finally, putting the estimations together and using Hölder’s inequality, together with the bound ε|h ζ , h ζ  | = ε|Cε (ζ − ζ  )|  1 of Lemma A.13, we obtain for every δ ∈ (0, 1]:       ˆ [m] μq,ζ  ζ   ζ

   ε   ε

ζ,ζ 

δ

L p ()

 1   − 2 [4]  ε ζ 



ζ,ζ 

L p ()

 1   − 2 [4]  ε ζ 

 1   − 2 [4]  ε ζ  

L p ()

1 L p ()

 1   − 2 [4]  ε ζ  

|h ζ , h ζ  |4−m |μq,ζ μq,ζ  |

L p ()

2

1 |h ζ , h ζ  |

3−m+δ

|μq,ζ μq,ζ  |

2

.

1

Now using Remark 3.2 (note that to bound ε− 2 [4] we only need to control the first 4 derivatives of Fε ) and the fact that h ζ , h ζ   H = Cε (ζ − ζ  ) we obtain as a final estimation       ˆ (m) μq,ζ  ζ   ζ

δ

 ε2 L p ()



|Cε (ζ − ζ  )|3−m+δ |μq,ζ μq,ζ  |

1/2 .

(38)

From the definition (37) of the measure μq,ζ , the l.h.s of (38) can be estimated in a standard way using Lemma A.18 to obtain for every x ∈ T3 , q > 0:   δ 1−δ    ε 2 2− 2 q q Yˆε (t, x) p L ()   δ 2+δ    ε22 2 q q Yˆε (t, x) p L ()

  δ 1+δ    ε 2 2 2 q, q Yˆε (t, x) p L ()   δ δ    ε 2 22q. q Yˆε∅ (t, x) p L ()

Time regularity of trees We want to show (31). In order to do that, we compute in the same way as before:

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M. Furlan, M. Gubinelli

    [m]  (  ˆ [m] ˆ −  )μ q,ζ  t,x s,x  ζ L p ()       4−m  [4−m] [4] ⊗4−m [4] ⊗4−m  t,x h t,x μq,ζ   δ G [1] − s,x h s,x  ζ

 2      [4] [4] ⊗4−m t,x − s,x h s,x μq,ζ      ζ

   +  [4] (h ⊗4−m  ζ s,x t,x

1/2     H ⊗4−m L p/2 () 1/2 2    ⊗4−m  − h s,x )μq,ζ  .   p/2 ⊗4−m H L

L p ()

()

We focus on the first term above to obtain that it is bounded by   1/2     [4] [4] [4] ⊗4−m [4] ⊗4−m      h −  −  , h  μ μ ⊗4−m    q,ζ q,ζ t,x H s,x s,x   t,x s,x s,x ζ

L p/2 ()





ζ,ζ 

   ε 

 εδ

1/2    [4] [4] [4] 4−m  

[4] −  −  |h , h | |μ μ | 

p/2   s,x q,ζ s,x q,ζ t,x L () s,x t,x s,x

ζ,ζ 

1    2 [4] [4] [4] 4−m   

ε−1 [4] −  −  |h , h | |μ μ | p/2 s,x q,ζ q,ζ s,x t,x L () s,x t,x  s,x 



ζ,ζ 

1    2 [4] [4] [4] 3−m+δ   

ε−1 [4] −  −  |h , h | |μ μ | . s,x q,ζ q,ζ s,x t,x L p/2 () s,x t,x  s,x 

Now note that   1 [4] (4) 21 (4) 21 ε− 2 [4] −  t,x s,x = F (ε Yε (t, x)) − F (ε Yε (s, x))  1 1 1 1 2 F (5) [ε 2 Yε (s, x) + τ ε 2 (Yε (t, x) =ε 0

−Yε (s, x))](Yε (t, x) − Yε (s, x)),  1   [4]  and we can estimate ε− 2 [4] t,x − s,x  Lemma A.14 as

p ε

  

1/2 

0

1

F

(5)



L p ()

  ε Yε (s, x) + τ ε (Yε (t, x) − Yε (s, x))   1 2

1 2

Yε (t, x) − Yε (s, x) L 2 ()  ε1/2 [Cε (0, 0) − Cε (t − s, 0)]1/2  ε−2σ |t − s|σ

123

by hypercontractivity and using

L 2 p ()

Weak universality for a class of 3d stochastic…

for any σ ∈ [0, 1/2]. The other term can be estimated more easily by  ε

δ

1

 ζ,ζ 

|h s,x , h s,x  |

2−m+δ

   ε−2κ |t − s|σ εδ

|h t,x − h s,x , h t,x  − h s,x  ||μq,ζ μq,ζ  |

2

1 ζ,ζ 

|h s,x , h s,x  |3−m+δ+2σ |μq,ζ μq,ζ  |

2

,

and finally obtain     [m]  (  ˆ [m] ˆ t,x − s,x )μq,ζ   ζ

 εδ/2−2κ |t − s|σ



L p ()

ζ,ζ 

1

|h s,x , h s,x  |3−m+δ+2σ |μq,ζ μq,ζ  |

2

.

Which yields estimation (31) by applying Lemma A.18 as before. This concludes the treatment of simple trees. Notice that in this section we only needed Fε ∈ C 5 (R) with the first 5 derivatives having exponential growth: indeed we need to take 4 derivatives to bound ε1/2 [4] ζ as of Remark 3.2, plus one more derivative for the time regularity of ε1/2 [4] ζ .

3.4 Analysis of composite trees In this section we show the decomposition (28) and the bound (30) for the trees Yε , Yε , Y¯ε , Yε . The time regularity (31) of Yˆετ can be obtained with the same technique as in the previous section, assuming that we can control one more derivative of Fε than what is needed to prove the boundedness of Yετ (thus we will need Fε ∈ C 9 (R) with exponential growth, as discussed in Remark 3.7). Looking at the definitions in (12) it is clear that we can write the q-th Littlewood–Paley blocks of Yε , Yε , Y¯ε and Yε ∀ε > 0 as:  1 [0] [2] μq,ζ1 ,ζ2 6 ζ1 ,ζ2 ζ1 ζ2  1 [1] [1] μq,ζ1 ,ζ2 q Yε (ζ¯ ) = 9 ζ1 ,ζ2 ζ1 ζ2  1 ¯ [1] [1] μq,ζ1 ,ζ2  q Y¯ε (ζ¯ ) = 3 ζ1 ,ζ2 ζ1 ζ2  1 [0] [1] μq,ζ1 ,ζ2 q Yε (ζ¯ ) = 3 ζ1 ,ζ2 ζ1 ζ2

q Yε (ζ¯ ) =

− dε q (1)(ζ¯ ), − dε q (1)(ζ¯ ), − d¯ε q (1)(ζ¯ ), − dε q Yε (ζ¯ ) − dε q (1)(ζ¯ ),

(39)

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M. Furlan, M. Gubinelli

for any time-space point ζ¯ = (t, x) ¯ that we keep fixed throughout this section. In order to keep the notation shorter we defined ¯ [1] := ε−1/2 f 2,ε Yε2 (ζ1 ),  ζ1 which can be thought of as a finite-chaos equivalent of [1] ε (modulo a constant f 2,ε / f 3,ε ) in the same way as Y¯ε is a finite-chaos equivalent of Yε . The measure μq,ζ1 ,ζ2 on (R × T3 )2 is given by μq,ζ1 ,ζ2

⎡  := ⎣ x,y

K q,x¯ (x)



⎤ K i,x (y)K j,x (x2 )Pt−s1 (y − x1 )⎦ 1[0,+∞) (s1 )δ(t − s2 )dζ1 dζ2 ,

i∼ j

with ζi:= (si , xi ) for i = 1, 2, K being the kernel associated to the Littlewood–Paley decomposition and P being the heat kernel. The first step for decomposing (39) is to expand them using the partial chaos expansion (91) to obtain    [2] [0] [2] [0] [2] [0] ζ1 ζ2 = E ζ1 ζ2 + δG 1 D ζ1 ζ2 ,    [1] [1] [1] [1] [1] [1] + δG  = E   D   1 ζ1 ζ2 ζ1 ζ2 ζ1 ζ2 ,       [1] [0] [1] [0] [1] [1] 2 [0] + δ 2 G [2] [0] ζ1 ζ2 = E ζ1 ζ2 + δ J0 D ζ1 ζ2 ζ 1 ζ 2 [1] D    [1] [1] [1] [0] [2] = E [0] + Y + Y  (ζ )E   (ζ )E   ε 1 ε 2 ζ1 ζ2 ζ1 ζ2 ζ1 ζ2   [0] [1] 2 [2] 2 +δ G [1] D ζ1 ζ2 . (40) Like the trees appearing in the 43 model, we expect composite trees to require a further renormalisation, on top of the Wick ordering. We developed (40) to the smallest order that allows us to see the effect of renormalization. 3.4.1 Renormalisation of composite trees In this section we show how to renormalize (39) by estimating terms of the type [n] E[[m] ζ1 ζ2 ] in expansion (40). This poses an additional difficulty, as in principle we

[n] would need to compute an infinite number of contractions between [m] ζ1 and ζ2 . However, we can again decompose [m] as in (34), and then the product formula (94) ensures that we only need to control a finite number of contractions. This is another important step in the proof and will be carried out in Lemma 3.5. First we need some preparatory results:

Lemma 3.3 We have    [1] μ Yε (ζ1 )E [1]  = q,ζ1 ,ζ2 ζ1 ζ2 ζ1 ,ζ2

123

s,x

q Yε (s, x¯ − x)G(t − s, x).

Weak universality for a class of 3d stochastic…

and   [2] μ Yε (ζ2 )E [0]  = q Yε (t, x¯ − x)H (t, x), q,ζ1 ,ζ2 ζ1 ζ2

 ζ1 ,ζ2

x

where we introduced the kernels:  G(t − s, x) :=  H (t, x) :=

 x1 ,x2 i∼ j

 [1] K i,x (x1 )K j,x (x2 )Pt−s (x1 )E [1] 0 (t−s,x2 ) ,



s,x1 ,x1 i∼ j

 [2] K i,x (x1 )K j,x (0)Pt−s (x1 − x1 )E [0]  0 (t−s,−x1 ) .

Remark 3.4 Some caveat on the notation: although we use the same letter for the kernel G(·, ·) and the Green operator G [n] [m] , those two are not related in any possible way. It is always clear which one the notation refers to. Proof We have 

 [1] Yε (ζ1 )E [1]  ζ1 ζ2 μq,ζ1 ,ζ2 ζ1 ,ζ2   = K q,x¯ (x) K i,x (x1 )K j,x (x2 )Pt−s1 (x1 s1 ,x1 ,x2 ,x,x1

−x1 )Yε (s1 , x1 )E



i∼ j [1] [1] 0 (t−s1 ,x2 −x1 )



and by change of variables, exploiting the translation invariance of the problem we obtain:  = K q,x¯ (x + x1 )Y (s1 , x1 ) s ,x ,x  1 1  [1] K i,x (x1 )K j,x (x2 )Pt−s1 (x1 )E [1]  0 (t−s1 ,x2 ) . x1 ,x2 i∼ j

Using the definition of K q we have  =



s1 ,x

q Yε (s1 , x¯ − x)

 x1 ,x2 i∼ j

 [1] K i,x (x1 )K j,x (x2 )Pt−s1 (x1 )E [1]  0 (t−s1 ,x2 ) .

Finally we can write  ζ1 ,ζ2

  [1] μ Yε (ζ1 )E [1]  = q,ζ1 ,ζ2 ζ1 ζ2

s1 ,x

q Yε (s1 , x¯ − x)G(t − s1 , x).

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M. Furlan, M. Gubinelli

Similar computations holds for the other term, indeed 

 [2] Yε (ζ2 )E [0]  ζ1 ζ2 μq,ζ1 ,ζ2 ζ1 ,ζ2   = K q,x¯ (x) K i,x (x1 )K j,x (x2 )Pt−s1 (x1 s1 ,x1 ,x2 ,x,x1

i∼ j

 [2] −x1 )Yε (t, x2 )E [0] 0 (t−s1 ,x2 −x1 )    = K q,x¯ (x + x2 )Yε (t, x2 ) K i,x (x1 )K j,x (0)Pt−s1 (x1 x2



[2] −x1 )E [0] 0 (t−s1 ,−x1 )   = q Yε (t, x¯ − x)



s1, x1 ,x,x1 i∼ j



s1, x1 ,x1 i∼ j

x

K i,x (x1 )K j,x (0)Pt−s1 (x1

 [2] −x1 )E [0]  0 (t−s1 ,−x1 )  = q Yε (t, x¯ − x)H (t, x) x

  Substituting the lemma above and (40) in the expressions (39), we can write them as:

q Y¯ε

q Yε

q Yε

123



  [1] δG 1 D [1]  ζ1 ζ2 μq,ζ1 ,ζ2 ζ1 ,ζ2    1 ¯ +q (1)(ζ ) G(t − s, x) − dε 9 s,x  1 ¯ [1] [1] )μq,ζ1 ,ζ2 (ζ¯ ) = δG 1 D( ζ1 ζ2 3 ζ1 ,ζ2    1 ¯ − s, x) − d¯ε +q (1)(ζ¯ ) G(t 3 s,x    1 [2] (ζ¯ ) = δG 1 D [0]  ζ1 ζ2 μq,ζ1 ,ζ2 6 ζ1 ,ζ2    1 +q (1)(ζ¯ ) H (t, x) − dε 6 x    1 [0] [1] 2  (ζ¯ ) = δ 2 G [2] D  ζ1 ζ2 μq,ζ1 ,ζ2 [1] 3 ζ1 ,ζ2     1 [1] μ +q (1)(ζ¯ ) E [0]  − d q,ζ1 ,ζ2 ε ζ1 ζ2 3 ζ1 ,ζ2

q Yε (ζ¯ ) =

1 9

Weak universality for a class of 3d stochastic…

    1 1 G(t − s, x) + H (t, x) − dε 3 s,x 3 x 1 1 + q Rε (ζ¯ ) + q Rε (ζ¯ ) 3 3

+q Yε (ζ¯ )

with the additional definitions ¯ − s, x) := G(t q Rε (ζ¯ ) := q Rε (ζ¯ ) :=

 

 x1 ,x1 i∼ j

s,x



x

 ¯ [1] [1] K i,x (x1 )K j,x (0)Pt−s1 (x1 − x1 )E  0 (t−s,−x1 ) ,

  ¯ G(t − s, x), q Yε (s, x¯ − x) − q Yε (t, x)

  q Yε (t, x¯ − x) − q Yε (t, x) ¯ H (t, x).

[n] Now we can characterise the local behaviour of E[[m] ζ1 ζ2 ] appearing in the integrals [n] above. Decomposing separately [m] ζ1 and ζ2 as in (34) we obtain:

 [n]  E [m] ζ1 ζ2 =

 3!2 3−m 3−n ( f 3,ε )2 E Yε,ζ Y  ε,ζ2 1 (3 − m)!(3 − n)!  3! 3−m ˆ [n] f 3,ε E Yε,ζ + ζ2 1 (3 − m)!   3! 3−n ˆ [m] ˆ [n] , ˆ [m]  + f 3,ε E Yε,ζ + E    ζ1 ζ1 ζ2 2 (3 − n)!

3−m 3−n where E[Yε,ζ Yε,ζ ] = (3 − m)!δ(3 − m, 3 − n)Cε (ζ1 − ζ2 )3−n and to bound all 1 2 other terms we introduce the following result.

Lemma 3.5 Under Assumption 1 (in particular if F ∈ C 8 (R) with exponentially growing derivatives) we have, for every 0  m, n  3 and m  n: 4−n   n−m m+n ˆ [n] |  ˆ [m]  |E  ε 1+ 2 +i |h ζ1 , h ζ2 |4−m+i  ε δ |h ζ1 , h ζ2 |3− 2 +δ , ζ1 ζ2

∀δ ∈ [0, 1].

i=0

Moreover for every 0  m, n  3,  m+n−3 m ˆ [n] |  ε 2 |h ζ1 , h ζ2 |m if m  4 − n, |E Yε,ζ   ζ2 1 m ˆ [n] ] = 0 if m < 4 − n. E[Yε,ζ  ζ2 1

123

M. Furlan, M. Gubinelli

Proof Using formula (92) we decompose  ˆ [n] ˆ [m]  E  ζ1 ζ2  [4−m] [4] ⊗4−m 4−n [4−n] [4] ⊗4−n = E δ 4−m (G [1] ζ1 h ζ1 )δ (G [1] ζ2 h ζ2 ) =

  4−n   4−m 4−n [8−m−n−i] [8−n−i] [8−m−n−i] [8−m−i] i!E(G [5−n−i] ζ 1 G [5−m−i] ζ2 )h ζ1 , h ζ2 8−m−n−i . i i i=0

  4−n   4−m 4−n [8−m−n−i] [8−n−i] [8−m−n−i] [8−m−i] i!E(G [5−n−i] ζ 1 G [5−m−i] ζ2 )h ζ1 , h ζ2 8−m−n−i = i i i=0

We can bound the term ε

[8−m−n−i] [8−n−i] [8−m−n−i] [8−m−i] E(G [5−n−i] ζ G [5−m−i] ζ2 )   1 m+i−5   n+i−5   2   ζ[8−m−i]  ε 2 ζ[8−n−i]  2 ε  2 1 2

m+n 2 +i−5

L

L

knowing 8 − n − i ∨ 8 − m − i  8 derivatives of Fε (see Remark 3.2) and using the bound ε|Cε (ζ1 − ζ2 )|  1 of Lemma A.13 with |h ζ1 , h ζ2 | = |Cε (ζ1 − ζ2 )| we have: 4−n   n−m m+n ˆ [m]  ˆ [n] |  |E  ε1+ 2 +i |h ζ1 , h ζ2 |4−m+i  εδ |h ζ1 , h ζ2 |3− 2 +δ . ζ1 ζ2 i=0 m  = δ m (h ⊗m ) (Remark B.12) and compute For the second bound we recall that Yε,ζ ζ1 1

 m ˆ [n] E Yε,ζ  ζ2 ζ2 1  [4−n] [4] ⊗4−n 4−n = E δ m (h ⊗m (G [1] ζ2 h ζ2 ) ζ1 )δ =

m∧4−n   i=0

m i

  4−n [m+4−n−i] [4+m−i] ⊗m+4−n−i i!E(D4−n−i (h ⊗m h ζ2  H ⊗m+4−n−i ). ζ1 ), G [m+1−i] ζ2 i

m ˆ [n] Since Dh ⊗m ζ1 = 0 we obtain E[Yε,ζ1 ζ2 ] = 0 if m < 4 − n and

 3−m−n  m+n−3 [m+n] m ˆ [n] |  ε 2 E ε− 2 G [m] |h ζ1 , h ζ2 |m    |E Yε,ζ ζ ζ [m+n−3] 1 2 2 if m  4 − n, with  3−m−n [m+n]  1.  E ε− 2 G [m] [m+n−3] ζ2   Using Lemma 3.5 we obtain   [1] 2 2 ˆ [1] )( f 3,ε Yε,ζ ˆ [1] ) = 18( f 3,ε )2 [Cε (ζ1 − ζ2 )]2 = 9E ( f 3,ε Yε,ζ E [1] + + ζ1 ζ2 ε,ζ1 ζ2 1 2  ˆ [1] ˆ [1]  +E  ζ1

123

ζ2

Weak universality for a class of 3d stochastic…



and thus G(t − s, x) = 18( f 3,ε )2 x  ,x2 i∼ j K i,x (x1 )K j,x (x2 )Pt−s (x1 )[Cε (ζ1 − 1 ˆ − s, x) with the remainder term defined by ζ2 )]2 + G(t ˆ − s, x) := G(t



 x1 ,x2 i∼ j

 ˆ [1] ˆ [1]  K i,x (x1 )K j,x (x2 )Pt−s (x1 )E  0 (t−s,x2 ) .

We have the estimation  ˆ [1] |  εδ Cε (ζ1 − ζ2 )2+δ . ˆ [1]  |E  ζ1 ζ2

(41)

Similarly   2 2 ˆ [1] ) ¯ [1] [1] = 3ε−1/2 f 2,ε E Yε,ζ ( f Y  +  E  3,ε ε,ζ ζ1 ζ2 ζ2 1 2 = 6ε−1/2 f 2,ε f 3,ε [Cε (ζ1 − ζ2 )]2 , and   [2] [0] [2] 3 ˆ ˆ | = |E ( f  Y  +  )(6 f Y +  ) | |E [0] 3,ε 3,ε ε,ζ 2 ε,ζ1 ζ1 ζ2 ζ1 ζ2   3 ˆ [2] ˆ [0]  ˆ [2] | + |E    | f 3,ε ||E Yε,ζ ζ2 ζ1 ζ2 | 1  εδ (| f 3,ε | + 1)Cε (ζ1 − ζ2 )2+δ ,

(42)

and     [1] [0] 2 3 ˆ [0] ˆ [1] ˆ [1] ˆ [1] |E [0] ζ1 ζ2 | = |E  (3 f 3,ε Yε,ζ2  +  ) |  | f 3,ε |E Yε,ζ1 ζ2 + E ζ1 ζ2  ε1/2 (| f 3,ε | + 1)Cε (ζ1 − ζ2 )3 .

(43)

ˆ − s, x)|  εδ (|t − s|1/2 We have by Lemma A.17 that for all δ ∈ (0, 1) |G(t −5−δ +|x|) . Using estimate (87) together with Lemma A.10, we have for all δ ∈ (0, 1),  δ  ∈ (0, δ) that |H (t, x)|  εδ |x|−δ . Furthermore, we have    1 2 q Rε = 6( f 3,ε ) q Yε (t + s, x¯ − x) − q Yε (t, x) ¯ Ps (x) [Cε (s, x)]2 3 s,x 1 + q Rˆ ε 3 with the remainder term q Rˆ ε given by q Rˆ ε =

 s,x

  ˆ − s, x). ¯ G(t q Yε (t, x¯ − x) − q Yε (t, x)

123

M. Furlan, M. Gubinelli

The term  6( f 3,ε )

2 s,x

  ¯ Ps (x) [Cε (s, x)]2 q Yε (t + s, x¯ − x) − q Yε (t, x)

can be shown to converge in law to    2 q Y (t + s, x¯ − x) − q Y (t, x) 6(λ3 ) ¯ Ps (x)E(Y (0, 0)Y (s, x))2 s,x

in C Tκ C −1/2−2κ with the standard techniques used in the analysis of the 43 model. On the other hand, for all δ > 0 sufficiently small we have the bounds     q Rε 

L∞

    + q Rˆ ε 

L∞

 εδ Yε C κ C −1/2−2κ 2q(1/2+2κ+2δ)



T

(|x| + |t − s|1/2 )δ−5 s,x

 εδ Yε C κ C −1/2−2κ 2q(1/2+2κ+2δ) , T

which shows that these remainders go to zero in C −1/2−2κ as ε → 0, since

Yε C κ C −1/2−2κ is bounded in L p (). Moreover, it is easy to see that q Rε − T

q Rˆ ε L p () ∼ O L ∞ (2q(1/2+2κ+2δ) ). Note that 

   [1] 2 = 18( f G(t − s, x) = Ps (x)E [1]  ) Ps (x) [Cε (s, x)]2 3,ε 0 (s,x) s,x s,x s,x   ˆ [1] , ˆ [1]  + Ps (x)E  0 (s,x) s,x



 H (t, x) = x

s,x

  [2] Ps (x)E [0]  0 (s,x) =

s,x

 ˆ [2] Ps (x)E [0] 0 (s,x) .

Here we used the fact that     K i,x (x1 )K j,x (0) = K i,x (x1 )K j,x (0) = δ(x1 ), x i∼ j

x i, j

 since x K i,x (x1 )K j,x (0) = 0, where |i − j| > 1. This is readily seen in Fourier space taking into account the support properties of the Littlewood–Paley blocks. Now,     [0] ˆ [2] ˆ [1] , ˆ [1]   Ps (x)E  P (x)E  s 0 0 (s,x) (s,x) , s,x

s,x

converge to finite constants due to the bounds (41) and (42) and by Lemma A.15  2 s,x Ps (x)C ε (s, x)  | log ε|.

123

Weak universality for a class of 3d stochastic…

Finally, from (43) we have  ζ1 ,ζ2

  [1] μ E [0]  = q,ζ1 ,ζ2 ζ1 ζ2

s,x

 [1] −1/2 Ps (x)E [0]  ). 0 (s,x) = O(ε

 Indeed Lemma A.15 again yields ε s,x Ps (x)Cε (s, x)3  1.  [1] Thus ζ1 ,ζ2 E[ζ[0] ζ2 ]μq,ζ1 ,ζ2 gives a diverging constant which depends on all the 1 ( f n,ε )n . Making the choice to define the renormalisation constants d τ as in Eq. (7) we cancel exactly these contributions which are either (Fε )ε dependent and/or diverging. In particular we verify that we can satisfy the constraint (13). 2 Finally, noting that δG 1 D = (1 − J0 ) and δ 2 G [2] [1] D = (1 − J0 − J1 ) (as seen in (91)), we can write the trees of (39) as  2 2 (1 − J0 )(Yε,ζ Yε,ζ )μq,ζ1 ,ζ2 q Yε (ζ¯ ) = ( f 3,ε )2 1 2 ζ1 ,ζ2  f 3,ε 2 2 ˆ [1] Yε,ζ ˆ [1] )μq,ζ1 ,ζ2 + δG 1 D(  + Yε,ζ  ζ1 ζ2 2 1 3 ζ1 ,ζ2  1 ˆ [1] ˆ [1]  + δG 1 D( ζ1 ζ2 )μq,ζ1 ,ζ2 , 9 ζ1 ,ζ2  1 2 2 (1 − J0 )(Yε,ζ Yε,ζ )μq,ζ1 ,ζ2 q Y¯ε (ζ¯ ) = ε− 2 f 2,ε f 3,ε 1 2 ζ1 ,ζ2  1 ¯ [1]  ˆ [1] + δG 1 D( ζ1 ζ2 )μq,ζ1 ,ζ2 , 3 ζ1 ,ζ2  3 Yε,ζ Yζ2 μq,ζ1 ,ζ2 q Yε (ζ¯ ) = ( f 3,ε )2 1 ζ1 ,ζ2  f 3,ε 3 ˆ [0] Yε,ζ2 + Yε,ζ ˆ [2] )μq,ζ1 ,ζ2 + δG 1 D(6  ζ1 ζ2 1 6 ζ1 ,ζ2  1 ˆ [2] ˆ [0]  + δG 1 D( ζ1 ζ2 )μq,ζ1 ,ζ2 , 6 ζ1 ,ζ2  1 3 2 (1 − J1 )(Yε,ζ Yε,ζ )μq,ζ1 ,ζ2 + q Rε (ζ¯ ) q Yε (ζ¯ ) = ( f 3,ε )2 1 2 3 ζ1 ,ζ2  [q Yε (t + s, x¯ − x) − q Yε (t, x)]P ¯ s (x)[Cε (s, x)]2 +6( f 3,ε )2 s,x

1 + q Rˆ ε 3 1 2 3 ˆ [0] 2 ˆ [1] + δ 2 G [2] [1] D (3ζ1 Yε,ζ2  + Yε,ζ1 ζ2 )μq,ζ1 ,ζ2 3 ζ1 ,ζ2  1 2 ˆ [0] ˆ [1] + δ 2 G [2] [1] D (ζ1 ζ2 )μq,ζ1 ,ζ2 . 3 ζ1 ,ζ2

(44)

123

M. Furlan, M. Gubinelli

Let us summarize our results so far. We have shown that q Rˆ ε (ζ¯ ) + q Rε (ζ¯ ) ∼ O L ∞ (εδ 2q(1/2+2κ+2δ) ) in L p () and then these terms converge to 0 in the right topology as ε → 0. As already mentioned, the convergence in law of 

2 2 (1 − J0 )(Yε,ζ Yε,ζ )μq,ζ1 ,ζ2 → (λ3 )2 q Y (ζ¯ ) 1 2 ζ1 ,ζ2  − 12 2 2 (1 − J0 )(Yε,ζ Yε,ζ )μq,ζ1 ,ζ2 → λ3 λ2 q Y (ζ¯ ) ε f 2,ε f 3,ε 1 2 ζ1 ,ζ2  3 Yε,ζ Yζ2 μq,ζ1 ,ζ2 → (λ3 )2 q Y (ζ¯ ) ( f 3,ε )2 1 ζ ,ζ 1 2 2 3 2 (1 − J1 )(Yε,ζ Yε,ζ )μq,ζ1 ,ζ2 ( f 3,ε ) 1 2 ζ1 ,ζ2    q Yε (t + s, x¯ − x) − q Yε (t, x) +6( f 3,ε )2 ¯ Ps (x)[Cε (s, x)]2

( f 3,ε )

2

s,x

→ (λ3 ) q Y (ζ¯ ) 2

is easy to establish with standard techniques (as done in [4,18]) assuming the convergence of λε as in (9) to λ. Then, comparing (44) with the canonical trees in (20) we can identify the remainder terms q Yˆετ that still need to be bounded, that are precisely ˆ [n] appears. Estimating these terms is the content of next section. those in which  ζ 3.4.2 Estimation of renormalised composite trees In this section we prove the bound (30) for composite trees. The difficulty we encounter here is that the remainder Yˆετ cannot be written as an iterated Skorohod integral as in Sect. 3.1, but instead as a product of iterated Skorohod integrals. We will then use the product formula (94) to write the remainder in the desired form. We can write (44) in a much shorter way as:    1 [1] δG 1 D [1] ζ1 ζ2 μq,ζ1 ,ζ2 , 9 ζ ,ζ 1 2 1 ¯ [1] [1] )μq,ζ1 ,ζ2 , D( q Y¯ε (ζ¯ ) = δG 1 ζ1 ζ2 3 ζ ,ζ 1 2   1 [2] D [0] q Yε (ζ¯ ) = δG 1 ζ1 ζ2 μq,ζ1 ,ζ2 , 6 ζ1 ,ζ2    1 2 [2] 1 1 [1] ˆ ¯ q Yε (ζ¯ ) = δ G [1] D2 [0] ζ1 ζ2 μq,ζ1 ,ζ2 + q Rε (ζ ) + q Rε 3 3 3 ζ1 ,ζ2    q Yε (t + s, x¯ − x) − q Yε (t, x) +6( f 3,ε )2 ¯ Ps (x)[Cε (s, x)]2 ,

q Yε (ζ¯ ) =

s,x

123

Weak universality for a class of 3d stochastic… 2 just substituting again δG 1 D = (1 − J0 ) and δ 2 G [2] [1] D = (1 − J0 − J1 ). In order to treat all trees at the same time, we can write the first terms in the r.h.s. above (modulo a constant that we discard) as:    ] [i] [ j] r  D  δr G [r ζ1 ζ2 μq,ζ1 ,ζ2 with r = 1, i + j = 2 or r = 2, i + j = 1. [1] ζ1 ,ζ2

First notice that by Lemma B.2 we have    r [r ] δ G [1]  

r

ζ1 ,ζ2

D

r   θ=0

 θ [r ] D G [1] 



[ j] [i] ζ 1 ζ 2

 ζ1 ,ζ2



  μq,ζ1 ,ζ2  

L p ()

    [ j]  μ Dr [i]  q,ζ1 ,ζ2  ζ1 ζ2

L p (H ⊗r +θ )

] and from the boundedness of the operator Dθ G [r [1] given by Corollary B.6 we obtain:

      r [r ]  [i] [ j] r δ G   μ D  q,ζ1 ,ζ2  ζ1 ζ2 [1]  ζ1 ,ζ2 L p ()       [ j]   Dr [i] ζ1 ζ2 μq,ζ1 ,ζ2   ζ1 ,ζ2

L p (,H ⊗r )

.

Computing the r -th derivative of the integrand we obtain ⊗ [4−n] h ⊗k ζ[4−m] ζ2 ζ1 ⊗ h ζ2 1    3! f 3,ε 3! f 3,ε [4−m] [4−n] m−1 n−1 ⊗ ˆ  h ⊗k Y Y =  +  ζ1  + ζ 2 ζ1 ⊗ h ζ2 (45) (3 − m)! ζ1 (3 − n)! ζ2

for m + n = 5 and 0  k +   2. The constraints on m, n, k,  are related to the number of branches in the graphical notation of the trees: each tree has m + k − 1 leaves with height 2 and n +  − 1 leaves with height 1, as follows Yε ↔ m + k = 3, n +  = 3 Y¯ε ↔ m = 3, k = 0, n = 2,  = 1 Yε ↔ m + k = 4, n +  = 2 Yε ↔ m + k = 4, n +  = 3.

(46)

ˆ [n] will generate finite contributions in In (45), the terms which do not contain  ζ the limit, as seen in Sect. 3.4.1 by writing the decomposition (44). We just consider ˆ [4−n] , because all the other similar terms featuring ˆ [4−m]  the terms proportional to  ζ1 ζ2 [m]  at least one remainder  ζ can be estimated with exactly the same technique, and are easily shown to be vanishing in the appropriate topology.

123

M. Furlan, M. Gubinelli

We can use one of the key observations of this paper, the product formula (94), to rewrite products of Skorohod integrals in the form δ m (u)δ n (v) as a sum of iterated Skorohod integrals δ  (w), which are bounded in L p by Lemma B.2. We obtain ˆ [4−m]  ˆ [4−n]  ζ1 ζ2 [4] ⊗m n [n] [4] ⊗n = δ m (G [m] [1] ζ1 h ζ1 )δ (G [1] ζ2 h ζ2 )  [4] ⊗m q−i [n] [4] ⊗n = Cq,r,i δ m+n−q−r (Dr −i G [m] G [1] ζ2 h ζ2  H ⊗q+r −i ) [1] ζ1 h ζ1 , D (q,r,i)∈I



=

Cq,r,i ε1+

r +q 2 −i

(q,r,i)∈I [n+q−i]

⊗n+q−i

[1+q−i] (ζ2 )h ζ2

[m+r −i] ⊗m+r −i δ m+n−q−r ([1+r , −i] (ζ1 )h ζ1

 H ⊗q+r −i )

with I = {(q, r, i) ∈ N3 : 0  q  m, 0  r  n, 0  i  q ∧ r } and the notation shortcut: [ j]

[ j]

. [i] (ζ ) := ε− 2 G [i] [3+i] ζ i

By Remark B.3, for every n, m  1 and  ∈ Dom δ n we can write δ n ()h ⊗m = δ n ( ⊗ h ⊗m ), and therefore 

ˆ [4−n] h ⊗k ⊗ h ⊗ μq,ζ1 ,ζ2 ˆ [4−m]   ζ1 ζ2 ζ1 ζ2 = 



Cq,r,i ε

2+q+r −2i 2

δ m+n−q−r

I [n+q−i]

⊗m−q

[m+r −i] [1+r −i] (ζ1 )[1+q−i] (ζ2 )h ζ1

⊗ q+r −i ⊗ h ζ⊗n−r ⊗ h ⊗k μq,ζ1 ,ζ2 ζ1 ⊗ h ζ2 |h ζ1 , h ζ2 | 2

[ j]

[ j]

proceed as in Sect. 3.1 to estimate separately the terms [i] (ζ ) = ε− 2 G [i] [3+i] in ζ L p (), which are bounded as discussed in Remark 3.2. i

Lemma 3.6 Under Assumption 1 (in particular if Fε ∈ C 8 (R) and the first 8 derivatives have exponential growth) we have the bound:  m+n−q−r δ  [n+q−i] ⊗m−q [m+r −i] ⊗ h ζ⊗n−r ⊗ h ⊗k [1+r ζ1 −i] (ζ1 )[1+q−i] (ζ2 )h ζ1 2 2  q+r −i ⊗h ⊗ μq,ζ1 ,ζ2  p ⊗k+ ζ2 |h ζ1 , h ζ2 | ) L (H   |h ζ1 , h ζ1 |m+k−q |h ζ2 , h ζ2 |n+−r |h ζ1 , h ζ2 |q+r −i |h ζ1 , h ζ2 |q+r −i |μq,ζ1 ,ζ2 ||μq,ζ1 ,ζ2 |.

123

Weak universality for a class of 3d stochastic…

Proof Thanks to Lemma B.2 the integral can be estimated with m+n−q−r 

   Dh [m+r −i] (ζ1 )D j−h [n+q−i] (ζ2 )h ⊗m−q ζ1 [1+r −i] [1+q−i] 

j=0,h j

2  ⊗k ⊗ q+r −i ⊗h ζ⊗n−r ⊗ h ⊗ h |h , h | μ  p ζ ζ q,ζ ,ζ 1 2 1 2 ζ ζ 2 1 2

L (V )

,

with V = H ⊗m+k+n+−q−r + j . We have that · 2L p (H ⊗k+ ) =

· 2H ⊗k+ L p/2 and therefore we can bound each term in the sum above as  [m+r −i] j−h [n+q−i] [1+q−i] (ζ2 ), 

Dh [1+r −i] (ζ1 )D 1/2

[n+q−i]

[m+r −i]  j−h Dh [1+r [1+q−i] (ζ2 ) H ⊗ j L p/2 −i] (ζ1 )D

×|h ζ1 , h ζ1 |m+k−q |h ζ2 , h ζ2 |n+−r |h ζ1 , h ζ2 |q+r −i 1/2 |h ζ1 , h ζ2 |q+r −i |μq,ζ1 ,ζ2 ||μq,ζ1 ,ζ2 | Using Hölder’s inequality we get the estimate [n+q−i]

[n+q−i]

[m+r −i] [m+r −i]  j−h j−h

Dh [1+r [1+q−i] (ζ2 ), Dh [1+r [1+q−i] (ζ2 ) H ⊗ j L p/2 −i] (ζ1 )D −i] (ζ1 )D [n+q−i]

[m+r −i] h [m+r −i]  j−h p  Dh [1+r [1+q−i] (ζ2 ), −i] (ζ1 ), D [1+r −i] (ζ1 ) H ⊗h L D [n+q−i]

D j−h [1+q−i] (ζ2 ) H ⊗ j−h L p h [m+a]  p Now to bound Dh [m+a] [1+a] (ζ ), D [1+a] (ζ ) H ⊗h L (with h  j  m+n−q −r

[m+a] and a = r − i) we can use the boundedness of the operator Dh G [1+a] for h  2m given by Corollary B.6. Consider the two regions h  2m and h > 2m. In the first region we just use Corollary B.6 to obtain: [m+a] [m+a]  (ζ ), Dh [1+a] (ζ ) H ⊗h L p

Dh [1+a] 2      h [m+a] − 1+a  [m+a] − 1+a [4+a] 2 2   D  Dh G [1+a] ε 2 [4+a] G ε    4 p ⊗h ζ1 [1+a] ζ1 L 4 p (H ⊗h ) L (H )  1+a 2  1+a 2  − 2 [4+a]   − 2 [4+a]   ε ζ1  4 p ε ζ   4 p . L

1

L

[m+a] If h > 2m we first use the bound D2m G [1+a] and then take the remaining h − 2m

derivatives on ε−

1+a 2

[4+a] : ζ

   h [m+a] − 1+a [4+a] 2 D G [1+a] ε 2 ζ1  4 p L

(H ⊗h )

2  1+a    Dh−2m ε− 2 [4+a]  4 p ⊗h−2m ζ1 L (H )   1+a+h−2m  − [4+a+h−2m]  2  ε ζ 1  4 p ⊗h−2m L

(H

)

123

M. Furlan, M. Gubinelli

From Remark 3.2 we see that this last term is bounded by a constant if F ∈ C 4+a+h−2m (R) with the first 4 + a + h − 2m derivatives having an exponential growth, with a = r − i and h  m + n − q − r . [n+q−i] [n+q−i] Applying the same reasoning to D j−h [1+q−i] (ζ2 ), D j−h [1+q−i] (ζ2 ) H ⊗ j−h L p we conclude that we need to control 4 +n ∨ 4 +m derivatives of Fε in order to perform the estimates of this Lemma. From the constraints (46) we see that 4 + n ∨ 4 + m  8.   From Lemma 3.6 we obtain ∀δ ∈ [0, 1/2): ε

2+q+r −2i 2

 2+q+r −2i  ε 2 δ m+n−q−r  [n+q−i] ⊗m−q [m+r −i] ⊗ h ζ⊗n−r ⊗ h ⊗k [1+r ζ1 −i] (ζ1 )[1+q−i] (ζ2 )h ζ1 2   q+r −i ⊗h ⊗ μq,ζ1 ,ζ2  p ⊗k+ ζ2 |h ζ1 , h ζ2 | L (H )    εδ ε2+q+r −2i−δ |h ζ1 , h ζ1 |m+k−q |h ζ2 , h ζ2 |n+−r |h ζ1 , h ζ2 |q+r −i q+r −i

|h ζ1 , h ζ2 | δ 2

    21    μq,ζ1 ,ζ2  μq,ζ  ,ζ   1 2

1 2

:= ε (I) . Our aim now is to estimate the quantity I. The idea is to use the bound ε|h ζ , h ζ  | = εCε (ζ − ζ  )  1 of Lemma A.13 to cancel strategically some of the covariances |h ζ , h ζ  |. We will consider three regions: If q + r  2 we use the bounds εq+r −2i |h ζ1 , h ζ2 |q+r −i |h ζ1 , h ζ2 |q+r −i  ε2 |h ζ1 , h ζ2 |q |h ζ1 , h ζ2 |r and then (we suppose r < 2) ε2−r −δ |h ζ2 , h ζ2 |n+−r  |h ζ2 , h ζ2 |n+−2+δ to obtain Iε

r −δ

 |h ζ1 , h ζ1 |m+k−q |h ζ2 , h ζ2 |n+−2 |h ζ1 , h ζ2 |q

|h ζ1 , h ζ2 |r |μq,ζ1 ,ζ2 ||μq,ζ1 ,ζ2 |   |h ζ1 , h ζ1 |m+k−q |h ζ2 , h ζ2 |n+−2+δ |h ζ1 , h ζ2 |q |μq,ζ1 ,ζ2 ||μq,ζ1 ,ζ2 |. (47) (If vice-versa q < 2 it suffices to put δ on the term |h ζ1 , h ζ2 |q+δ .) Notice that in this case m + k − q > 0.

123

Weak universality for a class of 3d stochastic…

In the case q + r = 3 if m + k − q  2 we estimate like before to obtain  I  ε2−δ

|h ζ1 , h ζ1 |m+k−q |h ζ2 , h ζ2 |n+−r |h ζ1 , h ζ2 |

q+r 2

q+r

|h ζ  , h ζ  | 2 |μq,ζ1 ,ζ2 ||μq,ζ1 ,ζ2 |  1 2  |h ζ1 , h ζ1 |m+k−q |h ζ2 , h ζ2 |n+−r |h ζ1 , h ζ2 |1+δ |μq,ζ1 ,ζ2 ||μq,ζ1 ,ζ2 |.

(48)

Note that m + k − q + δ − 1 > 0 and m + k − q + 2δ − 3 > −1 here. If m + k − q = 1 we bound  3+δ I  |h ζ1 , h ζ1 |m+k−q |h ζ2 , h ζ2 |n+−r −2 |h ζ1 , h ζ2 | 2 |h ζ1 , h ζ2 |

3+δ 2

|μq,ζ1 ,ζ2 ||μq,ζ1 ,ζ2 |

(49)

and note that m + k − q − 1/2 + δ/2 > 0, m + k − q − 1 + δ > 0, n +  − r − 2  0. Finally if m + k − q = 0 we can only have m + k = 3, q = 3, r = 0, i = 0 and thus  I  ε3−2δ |h ζ2 , h ζ2 |n+ |h ζ1 , h ζ2 |2−δ |h ζ1 , h ζ2 |2−δ |μq,ζ1 ,ζ2 ||μq,ζ1 ,ζ2 |   |h ζ2 , h ζ2 |n++m+k−6 |h ζ1 , h ζ2 |2−δ |h ζ1 , h ζ2 |2−δ |μq,ζ1 ,ζ2 ||μq,ζ1 ,ζ2 | (50) If q + r  4 we bound first δ

δ

ε2q+2r −2i+δ−4 |h ζ1 , h ζ2 |q+r −i |h ζ1 , h ζ2 |q+r −i  |h ζ1 , h ζ2 |2− 2 |h ζ1 , h ζ2 |2− 2 (note that 2q + 2r − 2i + δ − 4  δ) to obtain: Iε

6−q−r −δ



δ

δ

|h ζ1 , h ζ1 |m+k−q |h ζ2 , h ζ2 |n+−r |h ζ1 , h ζ2 |2− 2 |h ζ1 , h ζ2 |2− 2

|μq,ζ1 ,ζ2 ||μq,ζ1 ,ζ2 | Now in the cases m + k = 3, n +  = 3 and m + k = 4, n +  = 2 we can just write ε6−q−r −δ = εm+k−q ε6−m−k−r −δ and cancel the corresponding number of covariances to obtain  δ δ I  |h ζ2 , h ζ2 |δ |h ζ1 , h ζ2 |2− 2 |h ζ1 , h ζ2 |2− 2 |μq,ζ1 ,ζ2 ||μq,ζ1 ,ζ2 | (51) while for the case m + k = 4, n +  = 3 we have either   1 or k  1 and therefore with one of the following bounds εm+k−1−q εn+−r −δ |h ζ1 , h ζ1 |m+k−q |h ζ2 , h ζ2 |n+−r  |h ζ1 , h ζ1 ||h ζ2 , h ζ2 |δ εm+k−q εn+−1−r −δ |h ζ1 , h ζ1 |m+k−q |h ζ2 , h ζ2 |n+−r  |h ζ2 , h ζ2 |1+δ

123

M. Furlan, M. Gubinelli

we obtain the estimates  δ δ (52) I  |h ζ2 , h ζ2 |1+δ |h ζ1 , h ζ2 |2− 2 |h ζ1 , h ζ2 |2− 2 |μq,ζ1 ,ζ2 ||μq,ζ1 ,ζ2 |  δ δ I  |h ζ1 , h ζ1 ||h ζ2 , h ζ2 |δ |h ζ1 , h ζ2 |2− 2 |h ζ1 , h ζ2 |2− 2 |μq,ζ1 ,ζ2 ||μq,ζ1 ,ζ2 |. (53) We can use directly Lemma A.19 to obtain a final estimate of (47), (48), (49), (52). For (50), (51) and (53) notice that the integral over ζ1 , ζ1 is finite and thus the whole quantity is proportional to |h ζ2 , h ζ2 |n . Globally, we have I  2(m+k+n+−6)q as needed to prove (30). Remark 3.7 Finally, by controlling one more derivative of Fε as done in Sect. 3.3, we κ/2 can show (31) for Y τ = Yε , Yε , Y¯ε , Yε , thus proving that Yˆ τ → 0 in C T C α−κ in probability ∀α < |τ |. From the proof of Lemma 3.6 together with this observation, we conclude that we need to control the derivatives of Fε up to order 9 to be able to show the convergence for composite trees.

4 Convergence of the remainder and a priori bounds In this section we prove the convergence of the remainder (Lemma 4.5), as well as some technical results on the norm of the solution, needed in the proof of Theorem 2.2. In order to prove Lemma 4.5 we need first to prove Lemmas 4.1, 4.2 and 4.3 in this order. 4.1 Boundedness of the remainder We show that the remainder Rε (vε ) that appears in Eq. (16) can be controlled by a stochastic term Mε,δ that converges to zero in probability, times a function of the solution vε . Let. Mε,δ (Yε , u 0,ε ) := εδ/2 ecε

1/2 |Y

ε |+cε

1/2 |P (u −Y (0))| · 0,ε ε

L p [0,T ]L p (T3 ) .

(54)

for p ∈ [1, ∞), δ ∈ [0, 1] and define vε := vε − vε with vε : t → Pt (u 0,ε − Yε (0)).

(55)

Lemma 4.1 (Boundedness of remainder) For every γ ∈ (0, 1), δ ∈ [0, 1] we have

Rε (vε , vε , vε )(t, x) Mγ /(3+δ), p L p (T3 )  Mε (Yε , u 0,ε ) vε 3+δγ /(3+δ) T

123

MT

L ∞ (T3 )

ecε

1/2 v  ε CT L ∞

Weak universality for a class of 3d stochastic… γ /(3+δ), p

in (54), vε as in (55) and MT

γ /(3+δ) ∞ 3 L (T )

L p (T3 ), MT

defined in (60).

Proof We can write the remainder in two ways: 

 1  (1 − τ )2 1 1 dτ Fε(3) (ε 2 Yε + τ ε 2 vε ) − Fε(3) ε 2 Yε 2! 0  1 1 1 1 (1 − τ )3 = ε 2 vε4 dτ. Fε(4) (ε 2 Yε + τ ε 2 vε ) 3! 0

Rε (vε ) = vε3

1

From Assumption (8) on Fε we obtain by interpolation of these two expressions, ∀δ ∈ [0, 1], ∀t  0, x ∈ T3 , |Rε (vε )(t, x)|  εδ/2 |vε (t, x)|3+δ ecε

1 1  1 2 |Yε (t,x)|+cε 2 |vε (t,x)|+c|ε 2 vε (t,x)|

,

and we estimate, ∀γ ∈ [0, 1), γ

cε 3+δ

t → t γ Rε (vε )(t, x) L p ((0,T ],L p (T3 ))  t 3+δ vε (t) C ∞e TL    δ cε 21 |Y (t,x)|+cε 21 |v  (t,x)|  ε ε ε 2 e  .   p p 3

1/2 v  ε CT L ∞

L [0,T ]L (T )

  We can also verify that Mε,δ → 0 in probability for every δ > 0: Lemma 4.2 (Convergence of the stochastic term) Under Assumption 1 the random variable Mε,δ (Yε , u 0,ε ) defined in (54) converges to zero in probability ∀δ ∈ (0, 1]. Proof We can use Young’s inequality to estimate Mε,δ (Yε , u 0,ε ) for some c > 0 as  1/2 |Y

Mε,δ (Yε , u 0,ε )  εδ/2 ec ε

+εδ/2 T 1/ p e

ε|

 1/2 |P Y (0)| . ε

L p [0,T ]L p (T3 ) + εδ/2 ec ε

c ε1/2 u

0,ε L ∞

L p [0,T ]L p (T3 )

.

Under Assumptions 1 the term ε1/2 u 0,ε L ∞ (T3 ) is uniformly bounded, so the third term above converges to zero in probability. Note that ε1/2 Yε (t, x) and Pt ε1/2 Yε (t = 0)  1/2 p are centered Gaussian random variables, and then both E ec ε |Yε | L p [0,T ]L p (T3 ) and  1/2

E ec ε |P. Yε (0)| L p [0,T ]L p (T3 ) are uniformly bounded in ε > 0 for every p ∈ [1, ∞).   This yields the convergence in probability of Mε,δ (Yε , u 0,ε ). p

In order to show that Rε (vε )

γ , p

MT

Lp

→ 0 in probability for γ  >

1 3 4 + 2κ 3+δ

needed in the proof of Theorem 2.2, we still need to control the norms vε and vε C T L ∞ that appear in Lemma 4.1. This is done in next section.

as

γ /(3+δ) ∞ L

MT

123

M. Furlan, M. Gubinelli

4.2 A priori bounds on the solution Lemma 4.3 Fix T > 0. There exists κ > 0, (A priori bound on the solution)

T = T Yε XT , u ε,0 C −1/2−κ , |λε | ∈ (0, T ] a lower semicontinuous function depending only on Yε XT , u ε,0 C −1/2−κ , |λε | and a collection of events (Eε )ε>0 such that P(Eε ) → 1

as ε → 0

and conditionally on Eε there exists a universal constant C > 0 such that:

vε

1

C T C − 2 −κ

+ vε

1 + 3κ MT4 2

L∞



3  C(1 + |λε |)(1 + Yε XT )3 1 + u ε,0 C −1/2−κ

for any vε that solves Eq. (16). Moreover, still conditionally on Eε we have

3

vε C T L ∞  C(1 + |λε |)(1 + Yε XT )3 1 + u ε,0 C −1/2−κ with vε as in (55). Proof We know from Lemma 4.4 that the bounds above on vε

vε κ/2

T

1 + 3κ 2

MT4

L∞

and vε C T L ∞ hold whenever Mε,δ 

1

C T C − 2 −κ κ/2 T . The event Eε = {Mε,δ

} has P(Eε ) → 1 by Lemma 4.2 and this proves the result.

+   

The only thing left to prove is Lemma 4.4, which just a standard application of some well-known bounds on paraproducts, that are recalled in Appendices A.1, and A.2.  First observe that for ε > 0 a pair (vε , vε ) solves the paracontrolled Eq. (16) if and     only if vε = vε + vε and (vε , vε ) solves: vε = −Yε − Y¯ε − 3vε ≺≺ Yε + vε + vε L vε = U (λε , Yε ; vε , vε + vε ) − Rε (vε ) vε (0) = Yε (0) + Y¯ε (0) + 3vε,0 ≺ Yε (0)

(56)

Here U is the same as in (16). The initial condition of (56) is given by vε,0 := u 0,ε − Yε (0). The a priori bounds of Lemma 4.4 come from being able to find closed estimates for (56). Let us specify now all the notations we are going to use in the rest of this section. We consider the spaces 

VT := L

2κ T

∩L

1/4,1/2+2κ T

∩L

1/2,1+2κ , T

VT := L

1/2,1/2−κ T

∩L

1/4+3κ/2,2κ , T

with the corresponding norms

vε V  := vε L T

vε VT := vε L

123

2κ T

+ vε L

1/2,1/2−κ

1/4,1/2+2κ T

+ vε L

+ vε L

1/4+3κ/2,2κ

.

1/2,1+2κ T

,

(57) (58)

Weak universality for a class of 3d stochastic…

We refer to Appendix A.1 for the definition of the parabolic spaces L

γ ,α T .

We let

  vε := vε − v  = −Yε − Y¯ε − 3(vε + vε ) ≺≺ Yε + vε ,   vε := vε + Yε = −Y¯ε − 3(vε + vε ) ≺≺ Yε + vε ,

and vε (t = 0) = vε (t = 0) = 0. We define also the norm

vε V  := v  L T

2κ T

+ v  M1/4 C 1/2+2κ , T

1/4

with MT C 1/2+2κ given in Appendix A.1. In order not to get lost in these definitions the reader can keep in mind the following: – vε is the solution without the linear term;  – vε is the contribution of the initial condition, which give origin to some explosive norm (near the initial time);  – vε is the regular part of the solution; – vε , vε enter in the estimation of the remainder, they are just convenient shortcuts for certain contributions appearing in vε .

Lemma 4.4 There exists T = T Yε XT , u ε,0 C −1/2−κ , |λε | ∈ (0, T ] a lower semicontinuous function depending only on Yε XT , u ε,0 C −1/2−κ and |λε |, a constant Mε,δ = Mε,δ (Yε , u 0,ε ) > 0 defined by (54), and a universal constant C > 0 κ/2 such that, whenever Mε,δ  T we have

3

vε V   C(1 + |λε |)(1 + Yε XT )3 1 + u ε,0 C −1/2−κ , T   

vε VT  C Yε XT + u ε,0 C −1/2−κ + vε V  . T

Proof Using the well-known Schauder estimates of Lemma A.2 (and the fact that

f L κ,α  T κ f L αT ) we obtain for κ, θ > 0 small enough T

I f L −θ+2κ,2κ + I f L 1/4−θ+2κ,1/2+2κ + I f L T T  

f M1−θ C −κ + f M1/2+2κ C −1/2−2κ .

κ

1/2−θ+2κ,1+2κ T

T2 (59)

T

We choose θ > 2κ small enough so that L

−θ+3κ/2,2κ T

∩L

1/4−θ+3κ/2,1/2+2κ T

∩L

1/2−θ+3κ/2,1+2κ T



⊆ VT .

123

M. Furlan, M. Gubinelli

Now

vε V   Yε + Y¯ε V  + vε C T L ∞ ( Yε C T C 1−κ + Yε C T C −1−κ ) T T    + vε C T C −1/2−κ + vε M1/4 C −κ ( Yε C T C 1−κ + Yε C T C −1−κ ) T

+ vε V  T  Yε XT + T κ vε V  + vε,0 C −1/2−κ + vε V  T

T

where we used that v  (0) = 0 and as a consequence that vε C T L ∞  T κ vε C κ L ∞  T κ vε  to gain a small power of T . So provided T is small T

VT

enough (depending only on Yε ) this yields the following a priori estimation on vε :

vε C T L ∞  vε V   Yε XT + vε,0 C −1/2−κ + vε V  . T

T

Therefore we have an estimation on vε :

vε VT  vε VT + v  VT  vε,0 C −1/2−κ + vε V   Yε XT T

+ vε,0 C −1/2−κ + vε V  . T





In order to estimate terms in U (λε , Yε ; vε , vε + vε ) we decompose the renormalised products as ˆ ε = vε  Yε − Y¯ε − Yε − 3vε Yε + vε ◦ Yε + vε ◦ Yε − 3com1 (vε , Yε , Yε ) Yε v ≺ Yε )Yε + Yε  (vε + vε ) + Yε  (vε + vε ) vε  Yε = −Y¯ε Yε − 3(vε ≺ Yε  vε2

−Yε ≺ Yε − Yε  Yε − Yε = Yε  (Yε )2 + 2(Yε  Yε )(Y¯ε + 3vε ≺ ≺ Yε ) − 2(Yε  Yε )  (vε + vε ) +2(Yε  Yε )  (vε + vε ) + Yε  (vε + v  )2 + Yε  (vε + v  )2 . 



We decompose U (λε , Yε ; vε , vε + vε ) as U (λε , Yε ; vε , vε + vε ) = Q −1/2 (λε , Yε , v0,ε , vε , vε ) + Q 0 (λε , Yε , v0,ε , vε , vε ) + Q λε ,Yε  Q −1/2 := −3 vε  Yε − 3com1 (vε , Yε , Yε ) + Yε  (vε + v  )2  −6 (Yε  Yε )(3vε ≺ ≺ Yε ) + (Yε  Yε )  (vε + vε ) +2λ2,ε (3(vε ≺ ≺ Yε )Yε − Yε  (vε + vε )) +3 com3 (vε , Yε ) + 3 com2 (vε , Yε )  Q 0 := 3 3vε Yε − vε ◦ Yε − vε ◦ Yε + 2(Yε  Yε )  (vε + vε ) − Yε  (vε + v  )2

123

Weak universality for a class of 3d stochastic…

 −Yε∅ vε3 − λ2,ε vε2 + 2Yε  (vε + vε )  := (1 − λ1,ε )Yε − λ0,ε + 3 Y¯ε + Yε − Yε  (Yε )2 −2(Yε  Yε )Y¯ε

Q λε ,Yε

+2λ2,ε (Y¯ε Yε + Yε ≺ Yε + Yε  Yε + Yε ).

Here Q λε ,Yε does not depend from the solution but only on λε , Yε (as the notation suggests) and we have grouped the other terms which we expect to have regularity C −1/2−2κ in Q −1/2 , (and the same for Q 0 and regularity C −k ). With the same technique we used above for vε , we obtain the following estimate on vε

vε L

1/2+3κ/2,1/2+2κ T

+ vε L

1/4+κ,κ T

 Yε XT + vε,0 C −1/2−κ + vε V 

T

and this yields

(vε )2 L

3/4+5κ/2,1/2+2κ T

+ (vε )2 L

1/2+2κ,κ T

 2  Yε XT + vε,0 C −1/2−κ + vε V  . T

Then we are ready to bound Q −1/2 , Q 0 , Q λε ,Yε using the standard paraproducts estimations recalled in Appendix A.2:

Q −1/2 M1/2+2κ C −1/2−2κ + Q 0 M1−θ C −κ T T  3  (1 + |λε |)(1 + Yε XT )3 1 + vε,0 C −1/2−κ + vε V  T

Q λε ,Yε C T C −1/2−κ  (1 + |λε |)(1 + Yε XT )3 . 

In order to conclude the estimation of vε V  we have to control I Rε (vε ) V  . T T This is achieved easily by the using the results of Sect. 4.1. Thanks to Lemma 4.1 1−θ 1 3κ ∀δ ∈ (0, 1), ∀θ > 0 such that 3+δ > 4 + 2 (note that it is possible to choose θ > 2κ that satisfies this property as long as k and δ are small enough) we have:

Rε (vε ) M1−θ, p L p (T3 )  Mε,δ (Yε , u 0,ε ) vε 3+δ VT e

cε1/2 vε

VT

T

.

By Lemma A.3 together with (61) we obtain then

I Rε (vε ) V   Mε,δ (Yε , u 0,ε ) vε 3+δ VT e T

cε1/2 vε

VT

.

Using that



P· vε (0) V   vε (0) C T C 1/2−2κ  1 + vε,0 C −1/2−κ Yε XT T

123

M. Furlan, M. Gubinelli

we obtain that ∃C  > 0 such that

3

vεn V   C  (1 + |λεn |)(1 + Yεn XT )3 1 + vε,0 C −1/2−κ T

+C  T κ/2 (1 + |λε |)(1 + Yε XT )3 vε 3 

VT

+C  Mε,δ (Yε , u 0,ε )ecε

1/2

Yε XT + vε,0 C 

 D + C Mε,δ (Yε , u 0,ε )e +C Mε,δ (Yε , u 0,ε )e

cε1/2 vε

 cε1/2 vε  VT

 VT

−1/2−κ

cε1/2 vε

e

 VT

vε 3+δ VT

+ C T κ/2 vε 3 

VT

vε 3+δ  VT

with  C := C  (1 + |λε |)(1 + Yε XT )3



3+δ  1/2 , +ecε Yε XT + vε,0 C −1/2−κ 1 + Yε XT + vε,0 C −1/2−κ and

3 D := C  (1 + |λεn |)(1 + Yεn XT )3 1 + vε,0 C −1/2−κ . Let T ∈ (0, T ] such that: κ/2

C T

 1 1/2 1/2 κ/2 , and C T ecε (5C)  D. (5C)2 + ecε (5C) (5C)2+δ  2 κ/2

Assume that Mε,δ  T



. Define a closed interval [0, S] = {t ∈ [0, T ] : vεn V   

t

4D} ⊆ [0, T ]. This interval is well defined and non–empty since t → vεn V  is t



continuous and nondecreasing and vεn V   4D. Let us assume that S < T , then 0



we can take > 0 small enough such that S+ < T and by continuity vε V   5C, S+ then

vεn V 

S+



 D + C Mε,δ (Yε , u 0,ε )e +C Mε,δ (Yε , u 0,ε )e

cε1/2 vε

 V S+

 cε1/2 vε  V S+

 D + C Mε,δ (Yε , u 0,ε )e

cε1/2 (5C)

V S+ κ/2

+ C T S+

123

V S+

vε 3+δ 

1/2 κ/2 +C T ecε (5C) (5C)2+δ vε V 

1  2D + vε V  S+ 2

+ C(S + )κ/2 vε 3 

(5C)2 vε V 

S+

Weak universality for a class of 3d stochastic… 

which gives vεn V 

 4D. This implies S = T (by contradiction). From the

construction of T it is easy to see that T Yε XT , u ε,0 C −1/2−κ , |λε | is lower semicontinuous.   S+

4.3 Convergence of the remainder It suffices to put together the results obtained in Sects. 4.1 and 4.2 to obtain the convergence of Rε (vε ): Lemma 4.5 The remainder Rε (vε ) that appears in Eq. (16) converges in probability γ,p to 0 as ε → 0 in the space MT L p (T3 ). Proof From the estimation on Rε (vε ) of Lemma 4.1, together with the fact that Mε,δ → and vε C T L ∞ of 0 in probability (Lemma 4.2) and the bounds on vε 1 + 3κ MT4

Lemma 4.3 we see immediately that

2

L∞

Rε (vε ) Mγ , p L p → 0 T

in probability.

 

Acknowledgements The authors would like to thank the anonymous referee for the detailed and constructive critique which contributed to improve the overall exposition of the results. Support via SFB CRC 1060 is also gratefully acknowledged.

Appendix A: Paracontrolled analysis and kernel estimations In this section we first recall the the basic results of paracontrolled calculus first introduced in [6], without proofs. For more details on Besov spaces, Littlewood–Paley theory, and Bony’s paraproduct the reader can refer to the monograph [2]. We then proceed to give some results on the convolution of functions with known singularity and the estimation of finite-chaos Gaussian trees. We refer to Section 10 of [8] and to the nice pedagogic exposition [18] for further details. A.1 Notation Throughout the paper, we use the notation a  b if there exists a constant c > 0, independent of the variables under consideration, such that a  c · b. If we want to emphasize the dependence of c on the variable x, then we write a(x) x b(x). If f is a map from A ⊂ R to the linear space  Y , then we write f s,t = f (t) − f (s). For p f ∈ L p (Td ) we write f (x) L p (T3 ) := T3 | f (x)| p dx. x Given a Banach space X with norm · X and T > 0, we note C T X = C([0, T ], X ) for the space of continuous maps from [0, T ] to X , equipped with the supremum norm

· C T X , and we set C X = C(R+ , X ). For α ∈ (0, 1) we also define C Tα X as the space of α-Hölder continuous functions from [0, T ] to X , endowed with the seminorm

123

M. Furlan, M. Gubinelli α X for the space

f C Tα X = sup0s 0, p ∈ [1, ∞), we define γ,p

MT X = {v : L p ((0, T ], X ) : v Mγ , p X = t → t γ v(t) L p ((0,T ],X ) < ∞}, T

γ

MT X = {v : C((0, T ], X ) : v Mγ X = t → t γ v(t) C T X < ∞}. T

(60)

The space of distributions on the torus is denoted by D  (T3 ) or D  . The Fourier transform is defined with the normalization  F u(k) = u(k) ˆ =

Td

e−ιk,x u(x)dx,

k ∈ Z3 ,

so that the inverse Fourier transform is given by F −1 v(x) = (2π )−1 k eιk,x v(k). Let (χ , ρ) denote a dyadic partition of unity such that supp(ρ(2−i ·))∩supp(ρ(2− j ·)) = ∅ for |i − j| > 1. The family of operators ( j ) j≥−1 will denote the Littlewood–Paley projections associated to this partition of unity,

that is −1 u = F −1 (χ F u) and −1 − j ρ(2 ·)F u for j ≥ 0. Let S j = i< j i , and K q be the kernel of j = F q so that  q f (x) ¯ =

T3

K x,q ¯ (x) f (x)dx.

For the precise definition and properties of the Littlewood–Paley decomposition f =

 3 α 3 q−1 q f in D (T ), see Chapter 2 of [2]. The Hölder-Besov space B p,q (T , R) for α ∈ R, p, q ∈ [1, ∞] with B αp,q (T3 , R) =: C a is and equipped with the norm α

f α = f B∞,∞ = sup (2iα i f L ∞ (T3 ) ),

i−1

f B αp,q = 2iα i f L p (T3 ) q . If f is in C α−ε for all ε > 0, then we write f ∈ C α− . For α ∈ (0, 2), we define the α/2 space L αT = C T L ∞ ∩ C T C α , equipped with the norm

f L

α T

  = max f C α/2 L ∞ , f C T C α . T

The notation is chosen to be reminiscent of L := ∂t −, by which we will always denote the heat operator with periodic boundary conditions α/2 on Td . We also write L α = Cloc L ∞ ∩ CC α . When working with irregular initial

123

Weak universality for a class of 3d stochastic…

conditions, we will need to consider explosive spaces of parabolic type. For γ  0, α ∈ (0, 1), and T > 0 we define the norm

f L

γ ,α T

  = max t → t γ f (t) C α/2 L ∞ , f Mγ C α , T

T

 < ∞ . In particular, we

0,α We introduce the linear operator I : C R+ , D  (T) → have L T  = C R+ , D (T) given by γ ,α T L αT .

and the space L

=



f : [0, T ] → R : f L

 I f (t) =

t

γ ,α T

Pt−s f (s)ds,

0 |x|2

where (Pt )t0 is the heat semigroup with kernel Pt (x) = (4π 1t)3/2 e− 4t It0 . Paraproducts are bilinear operations introduced by Bony [3] in order to linearize a class of non-linear PDE problems. They appear naturally in the analysis of the product of two Besov distributions. In terms of Littlewood–Paley blocks, the product f g of two distributions f and g can be formally decomposed as f g = f ≺ g + f  g + f ◦ g, where f ≺ g = g  f :=

j−2  

i f  j g and f ◦ g :=



i f  j g.

|i− j|1

j−1 i=−1

This decomposition behaves nicely with respect to Littlewood–Paley theory. We call f ≺ g and f  g paraproducts, and f ◦ g the resonant term. We use the notation f  g = f ≺ g + f ◦ g. The basic result about these bilinear operations is given by the following estimates, essentially due to Bony [3] and Meyer [14]. When dealing with paraproducts in the context of parabolic equations it would be natural to introduce parabolic Besov spaces and related paraproducts. But to keep a simpler setting, we choose to work with space–time distributions belonging to the scale of spaces (C T C α )α∈R for some T > 0. To do so efficiently, we will use a modified paraproduct which introduces some smoothing in the time variable that is tuned to the parabolic scaling. Let therefore ϕ ∈ C ∞ (R, R+ ) be nonnegative with compact support contained in R+ and with total mass 1, and define for all i  −1 the operator Q i : CC β → CC β ,

 Q i f (t) =

∞

2−2i ϕ(22i (t − s)) f (s)ds.

0

We will often apply Q i and other operators on CC β to functions f ∈ C T C β which we then simply extend from [0, T ] to R+ by considering f (· ∧ T ). With the help of

123

M. Furlan, M. Gubinelli

Q i , we define a modified paraproduct f ≺≺ g :=

 (Q i Si−1 f )i g i



for f, g ∈ C R+ , D  (T) . We define the commutators com1 , com1 , com2 , com3 in |x|2

1 ˇ x) = e− 4t e−t 1t0 for a modified heat kernel Lemma A.7. We write P(t, (4π t)3/2 which has the same bounds of the usual heat kernel P(t, x). Let Yε as in (3) and recall that Cε is the covariance of Yε , i.e. Cε (t, x) = E(Yε (t, x)Yε (0, 0)). We will sometimes write Yε,ζ := Yε (t, x) for ζ = (t, x) ∈ R×T3 . Let σε2 = εE[(Yε (0, 0))2 ] = εCε (0, 0).

A.2 Basic paracontrolled calculus results First, let us recall some interpolation results on the parabolic time-weighted spaces γ ,α LT : Lemma A.1 For all α ∈ (0, 2), γ ∈ [0, 1), ε ∈ [0, α ∧ 2γ ), T > 0 and f ∈ L with f (0) = 0 we have

f L γ −ε/2,α−ε  f L γ ,α . T

T

γ ,α T

(61)

Let α ∈ (0, 2), γ ∈ (0, 1), T > 0, and let f ∈ L αT . Then for all δ ∈ (0, α] we have

f L δ  f (0) C δ + T (α−δ)/2 f L αT , T

f L γ ,δ  T (α−δ)/2 f L γ ,α . T

T

(62)

Schauder estimates Lemma A.2 Let α ∈ (0, 2) and γ ∈ [0, 1). Then

I f L γt ,α  f Mγt C α−2 ,

(63)

for all t > 0. If further β  −α, then

s → Ps u 0 L (β+α)/2,α  u 0 C −β .

(64)

t

For all α ∈ R, γ ∈ [0, 1), and t > 0 we have

I f Mγt C α  f Mγt C α−2 .

(65)

Proofs can be found e.g. in [7]. We need also some well known bounds for the solutions of the heat equation with sources in space–time Lebesgue spaces.

123

Weak universality for a class of 3d stochastic… β

p

Lemma A.3 Let β ∈ R and f ∈ L T B p,∞ , then for every κ ∈ [0, 1] we have I f ∈ κ/q C T C β+2(1−κ)−(2−2κ+d)/ p with

I f C κ/q C β+2(1−κ)−(2−2κ+d)/ p T f L p B β , T

T

p,∞

with q1 + 1p = 1. Moreover, for every γ < γ  < 1 − 1/ p and every 0 < α < (2 − 5/ p + β) ∧ 2 we have

I f

L

γ  ,α T

T f Mγ , p B β . p,∞

T

Proof We only show the second inequality as the first one is easier and obtained with similar techniques. Let u = I f , we have t γ i u(t) L ∞   t 1/q 2di/ p

1

0

γ ,q 2id/ p 2−2i/q

s −γ q e−cq2

2i t (1−s)

1/q  ds 0



t 0

s γ p i f (s) L p ds p

t

s γ p i f (s) L p ds p

1/ p

1/ p



which allows us to bound I f Mγ C α . In order to estimate t → t γ I f C α/2 L ∞ we T T write 



t γ i u(t) − s γ i u(s) L ∞  t   v γ −1 i u(v) L ∞ dv + |t − s|2i(d+2)/ p i f Mγ , p L p (T3 ) T s t    γ  +  v i f (v)dv  L∞

s

We can estimate the first term as  t  t  γ  −1 i(d+2)/ p γ , p ∞ v

i u(v) L dv  2

i f M L p (T3 ) v γ −γ −1 dv. T

s

s

For the third term we have  t     v γ i f (v)dv    s

 L∞

1/q 

t



t

dv s

s

v γ p i f (s) L ∞ dv p

1/ p

 2id/ p |t − s|1/q i f Mγ , p L p (T3 ) T We obtain then if 22i |t − s|  1 



t γ i u(t) − s γ i u(s) L ∞  2id/ p |t − s|1/q i f Mγ , p L p (T3 ) T

123

M. Furlan, M. Gubinelli

and if 22i |t − s| > 1 we just use the trivial estimate 



t γ i u(t) − s γ i u(s) L ∞  2id/ p 2−2i/q i f Mγ , p L p (T3 ) T

 2id/ p |t − s|1/q i f Mγ , p L p (T3 ) . T Therefore, for every κ ∈ [0, 1]:

t γ  i u(t) − s γ  i u(s) L ∞  2

( d+2 p −2)i 2κi/q

2

|t − s|κ/q i f Mγ , p L p (T3 ) . T

Choosing κ/q = α/2 we obtain the desired estimate.

 

Estimates on Bony’s paraproducts and commutators Lemma A.4 For any β ∈ R we have

f ≺ g C β β f L ∞ g C β ,

(66)

and for α < 0 furthermore

f ≺ g C α+β α,β f C α g C β .

(67)

f ◦ g C α+β α,β f C α g C β .

(68)

For α + β > 0 we have

A natural corollary is that the product f g of two elements f ∈ C α and g ∈ C β is well defined as soon as α +β > 0, and that it belongs to C γ , where γ = min{α, β, α + β}. We will also need a commutator estimation: Lemma A.5 Let α > 0, β ∈ R, and let f, g ∈ C α , and h ∈ C β . Then

f ≺ (g ≺ h) − ( f g) ≺ h C α+β  f C α g C α h C β . We collect in the following lemma various estimates for the modified paraproduct f ≺≺ g, proofs are again in [7]. Lemma A.6 a) For any β ∈ R and γ ∈ [0, 1) we have t γ f ≺≺ g(t) C β  f Mγt L ∞ g(t) C β ,

(69)

for all t > 0, and for α < 0 furthermore t γ f ≺≺ g(t) C α+β  f Mγt C α g(t) C β .

123

(70)

Weak universality for a class of 3d stochastic…

b) Let α, δ ∈ (0, 2), γ ∈ [0, 1), T > 0, and let f ∈ L L g ∈ C T C α−2 . Then

γ ,δ T ,

g ∈ C T C α , and



f ≺≺ g L γ ,α  f L γ ,δ g C T C α + L g C T C α−2 . T

(71)

T

We introduce various commutators which allow to control non-linear functions of paraproducts and also the interaction of the paraproducts with the heat kernel. Lemma A.7 a) For α, β, γ ∈ R such that α + β + γ > 0 and α ∈ (0, 1) there exists bounded trilinear maps com1 , com1 : C α × C β × C γ → C α+β+γ , such that for smooth f, g, h they satisfy com1 ( f, g, h) = ( f ≺ g) ◦ h − f (g ◦ h).

(72)

com1 ( f, g, h) = ( f ≺≺ g) ◦ h − f (g ◦ h).

(73)

b) Let α ∈ (0, 2), β ∈ R, and γ ∈ [0, 1). Then the bilinear maps com2 ( f, g) := f ≺ g − f ≺≺ g.

(74)

com3 ( f, g) := [L , f ≺≺] g := L ( f ≺≺ g) − f ≺≺ L g.

(75)

have the bounds t γ com2 ( f, g)(t) α+β  f L γt ,α g(t) C β ,

t > 0.

(76)

as well as t γ com3 ( f, g)(t) α+β−2  f L γt ,α g(t) C β ,

t > 0.

(77)

Proofs can be found in [7]. A.3 Estimation of finite-chaos diagrams In this section we give some estimations of Feynman-like diagrams that appear in both the 43 model and in Sect. 3. Such diagrams were already estimated in [8, Chap.10] and [18]. Let K q,x (y) = 23q K (2q (x − y)) be the kernel associated to the q-th Littlewood– 3 3 Paley block q on R . For a function f defined on the torus T we still write q f (x) = f (y)dy where with an abuse of notation K q,x stands for the kernel on T3 , T3 K q,x (y)

which is j∈Z3 K q,x (y + 2π j).Let P be the heat kernel on R × R3 , i.e. P(t, x) =

2 1 − |x| 4t 1t0 e (4π t)3/2

(78)

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M. Furlan, M. Gubinelli

and call P(t, x) also its periodized version

(with another abuse of notation) which for (t, x) ∈ R × T3 can be obtained as j∈Z3 P(t, x + 2π j). Let Yε be the stationary Gaussian field which solves L Yε = −Yε + ηε

(79)

where ηε is a centered Gaussian field on R × T3 with stationary covariance Cε (t − s, x − y) := E(ηε (t, x)ηε (s, y)) such that ∀ε ∈ (0, 1] Cε (t, x) = ε−5 (ε−2 t, ε−1 x)

(80)

where is a smooth even function compactly supported in [−1, 1] × BR3 (0, 1). We introduce the modified heat kernel 

ˇ x) = P(t,

k∈Z3

k|2 1 − |x+2π 4t e e−t 1t0 (4π t)3/2

(81)

and take  Yε (t, x) =

t



−∞ T3

ˇ − s, x − y)ηε (s, y)dsdy. P(t

(82)

We call Cε the covariance of Yε , i.e. Cε (t − s, x − y) := E[Yε (t, x)Yε (s, y)]. One can see easily that ˇ Cε = P(−·) ∗ Pˇ ∗ Cε

(83)

ˇ with time-space convolutions in R × T3 and the kernel P(−·) that has reversed time. In the following we use the notations |k|s = 2k1 + 4j=2 k j and ζ s = |t|1/2 +

3 4 3 j=1 |x j | for k ∈ N , ζ = (t, x) ∈ R × T . Lemma A.8 Let P(t, x) be defined in (78). Then ∀(t, x) ∈ R × R3 |P(t, x)|  (|t|1/2 + |x|)−3 . Moreover, for every multi-index k we have: |∂ k P(t, x)|  (|t|1/2 + |x|)−3−|k|s

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Weak universality for a class of 3d stochastic…

Proof  |P(t, x)|(|t|

1/2



+ |x|)  1 + 3

|x| |t|1/2

3 

|x|2

e− 4|t|

|α|

and ∃C > 0 such that (1 + |α|3 )e− 4  C for every α ∈ R. Calling kt = k1 and k x = (k2 , . . . , k4 ) one can see directly by taking derivatives of P(t, x) that |∂ kt ∂ k x P(t, x)| 

|kt |  |x|2 j j=0

|t| j

|t|−

3+|k x |+2|kt | 2

e−

|x|2 4t

and then |∂ kt ∂ k x P(t, x)|(|t|1/2 + |x|)3+|k x |+2|kt |  C.

 

Remark A.9 It is immediate to note that the estimation of Lemma A.8 holds as well ˇ x). for the kernel P(t, x) on the torus R × T3 and the stationary kernel P(t, We recall [8, Lemma 10.14] in a restricted formulation that is enough for our purposes. Lemma A.10 Let K 1 , K 2 : R4 \{0} → R smooth and such that ∃α, β ∈ (−5, 0), ∃m ∈ N such that ∀|k|s  m, ∀ζ ∈ R4 \{0} β−|k|s

α−|k|s

|∂ k K 1 (ζ )|  ζ s

and |∂ k K 2 (ζ )|  ζ s

.

(84)

Let γ = α + β + 5. If γ < 0 then γ −|k|s

|∂ k (K 1 ∗ K 2 )(ζ )|  ζ s

∀|k|s  m, ζ ∈ R4 \{0}.

Remark A.11 It is clear that the argument of Lemma A.10 works as well for spaceperiodic kernels K 1 , K 2 : R × T3 \{0} → R. Lemma A.12 ([8], Lemma 10.17) Let K : R4 \{0} → R smooth and such that ∃m ∈ N, ∃α ∈ (−5, 0) such that α−|k|s

|∂ k K (ζ )|  ζ s

∀|k|s  m, ζ ∈ R4 \{0}.

Let ψ ∈ Cc∞ (R4 ) with unit mass. Let ψε (t, x) = ε−5 ψ(ε−2 t, ε−1 x) and define K ε = K ∗ ψε . Then |∂ k K ε (ζ )|  ( ζ s + ε)α−|k|s

∀|k|s  m.ζ ∈ R4 .

Proof Let w.l.o.g. ψε be supported on Bs (0, ε) = {ζ ∈ R4 | ζ s < ε}. For ζ s > 2ε we bound ∂ k K ε as       ∂ k K (ζ − ζ  )ψε (ζ  )dζ    sup |∂ k K (ζ − ζ  )| |ψε (ζ  )|dζ   

ζ  <ε

α−|k|s

 ζ s

 εα−|k|s

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M. Furlan, M. Gubinelli

since for ζ  < ε we have ζ − ζ   ζ − ε  ζ /2. For ζ s  2ε we bound       K (ζ − ζ  )∂ k ψε (ζ  )dζ    ε−5−|k|s |K (ζ − ζ  )|dζ    ε

α−|k|s

ζ −ζ  3ε α−|k|  ζ s s .

  Lemma A.13 The covariance Cε on R × T3 has the bound, for every multi-index k ∈ N4 : |∂ k Cε (t, x)|  (|t|1/2 + |x| + ε)−1−|k|s

∀(t, x) ∈ R × T3 , ε ∈ (0, 1].

Proof Note that from Lemma A.8 it follows immediately that Pˇ satisfies the assumptions of Lemma A.10. We obtain then the estimation  ˇ ∗ Pˇ (ζ )|  ζ −1−|k|s |∂ k P(−·) for every multi-inded k ∈ N4 , and from Lemma A.12 we obtain the result.

 

Lemma A.14 We have for every σ ∈ [0, 1] sup |Cε (t, x) − Cε (0, x)|  ε−1−2σ |t|σ

x∈T3

Proof Since for every ε ∈ (0, 1] Cε is smooth, the result is immediately obtained by Taylor expansion and interpolation from the bound of Lemma A.13.   Lemma A.15 For every t > 0 we have  A2 := Ps (x) [Cε (s, x)]2 1 [0,t] (s)dxds  | log ε|, R×T3

and for every n  3  An := εn−2

R×T3

Ps (x)|Cε (s, x)|n 1 [0,t] (s)dxds  1.

Proof From the estimations of Lemmas A.8 and A.13 we have  1 1 A2  1 (s)dxds 1/2 + |x|)3 (|s|1/2 + |x| + ε)2 [0,t] 3 (|s| R×T 1 1 1 −2 (s)dxds  1/2 + |x|)3 (|s|1/2 + |x| + 1)2 [0,ε t] −1 3 (|s| R×(ε T)  1 1  1 −2 (s)1 B(0,ε−1 ) (x)dxds 1/2 + |x|)3 (|s|1/2 + |x| + 1)2 [0,ε t] 4 (|s| R  | log(ε)|.

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Weak universality for a class of 3d stochastic…

In the same way for n  3  1 1 An  εn−2  [0,t] (s)dxds 1/2 3 1/2 + |x|) (|s| + |x| + ε)n R×T3 (|s|  1 1 dxds  1/2 3 1/2 + |x|) (|s| + |x| + 1)n R4 (|s|  1.   Lemma A.16 We have for every α ∈ (0, 3) 

|K i,x (y)| dy  (|x| + |t|1/2 + 2−i )−α , (|y| + |t|1/2 )α  |K i,x (y)| dy  (|x| + 2−i )−α . |y|α

(85) (86)

with the integral over T3 or R3 . Moreover for P : R4 → R as in (78) we have ∀δ ∈ [0, 1]    

R3

  2−δi K i,x (y)P(t, y)dy   . (|x| + |t|1/2 + 2−i )3+δ

(87)

Proof We start with inequality (85), which can be obtained in a similar way as Lemma A.12. When |t|1/2  2−i ∨ |x| we have  T3

|K i,x (y)| 1 dy  α/2 (|y| + |t|1/2 )α t

 T3

|K i,x (y)|dy 

1  (|x| + t 1/2 + 2−i )−α . t α/2

When 2−i  |t|1/2 ∨ |x| we estimate for α ∈ (0, 3)  T3

|K i (x − y)| dy  2αi (|y| + |t|1/2 )α



|K (y)| dy i α T3 |2 x − y|  |K (y)| dy  2αi sup α z∈(2i T)3 (2i T)3 |z − y|  2αi  (|x| + |t|1/2 + 2−i )−α .

Finally, when |x|  2−i ∨|t|1/2 we split the domains |x|  2−i+1 |y| or |x| < 2−i+1 |y|. In the first region |x − 2−i y|  |x| so  T3

|K i (x − y)| dy  (|y| + |t|1/2 )α

 (2i T)3

|K (y)| dy  |x|−α  (|x| + |t|1/2 + 2−i )−α , |x − 2−i y|α

while in the second region |y|  2i |x|/2, then |K (y)|  |K (y)|1/2 f (2i |x|/2) where f is another rapidly decreasing function which is defined on the torus as f (·) =

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M. Furlan, M. Gubinelli

j∈Z3

f (· + 2π j) by an abuse of notation. Then for α ∈ (0, 3)  T3

|K i (x − y)| dy  f (2i |x|/2) (|y| + |t|1/2 )α 2

αi

 |x|

 (2i T)3

|K (y)|1/2 dy |2−i y|α

f (2 |x|/2) i

−α

 (|x| + |t|1/2 + 2−i )−α ,

concluding the argument. Taking the integral over R3 in (85) does not change the estimations, and the second inequality (86) is obtained  in the same way. Let us show (87). Note that since ∀i  0, ∀x ∈ R3 K i,x (y)dy = 0 (obvious from its Fourier transform) we have  I = =



R3  1

K i,x (y)P(t, y)dy =  dτ



0

|I | 

R3



1

dτ 0

 2−i 2

−i



R3 1

0

|(y − x)K i (x − y)||P  (t, x + τ (y − x))|dy  

1

dτ 0

K i,x (y) [P(t, y) − P(t, x)] dy

  K i,x (y) P  (t, x + τ (y − x))(y − x) dy

dτ 

R3

|y K (y)||P  (t, x + τ 2−i y)|dy

R3

|y K (y)|

R3

1 dy (|t|1/2 + |x + τ 2−i y|)4

where P  denotes the derivative of P with respect to the space variable and can be estimated with Lemma A.8. As before we can bound (|t|1/2 + |x + τ 2−i y|)−4 by considering three separate regions: when |t|1/2  2−i ∨ |x| we have |I |  2−i |t|−2  2−iδ |t|

−3−δ 2

 2−iδ (|x| + |t|1/2 + 2−i )−3−δ .

When 2−i  |t|1/2 ∨ |x| we estimate simply  |I | 

R3

|K i (x − y)||P(t, y)|dy  23i  2−iδ (|x| + |t|1/2 + 2−i )−3−δ

since P(2−2i t, 2i y) = 2−3i P(t, y). When |x|  2−i ∨ |t|1/2 we have instead that either |x|  2τ 2−i |y| or |x| < 2τ 2−i |y|. In the first region |x + τ 2−i y|  |x| so |I |  2−i





1

dτ 0

−i

 2 |x|

123

−4

|y K (y)|

R3 −iδ

2

1 dy |x|4

(|x| + t 1/2 + 2−i )−3−δ .

Weak universality for a class of 3d stochastic…

In the region |y| > 2i |x|/(2τ ) we have |y K (y)|  (2i |x|/(2τ ))K (2i |x|/(2τ )) and then the integral can be estimated as follows: 



1

1 dy |x + τ 2−i y|4 0   2−i 1 τ 2−i i i dτ (2 |x|/(2τ ))K (2 |x|/(2τ )) (1 +  |y|)−4 dy 3 |x|4 0 |x| R   2−i 1 1 dτ (2i |x|/(2τ ))4 K (2i |x|/(2τ )) dy  3 |x|4 0 (1 + |y|)4 R

|I |  2−i

dτ

R3

|y K (y)|

 2−i |x|−4  2−iδ (|x| + t 1/2 + 2−i )−3−δ .

 

This concludes the proof for (87). Lemma A.17 We have for every α ∈ (0, 3)     |K j,x (y)| 1  K i,x (y)P(t, y)dy    (|y| + |t|1/2 )α dy  (|x| + |t|1/2 )3+α . i∼ j

Proof From (85) and (87) we deduce that       |K j,x (y)| 2−i  K i,x (y)P(t, y)dy  dy  .   (|y| + |t|1/2 )α (|x| + |t|1/2 + 2−i )4+α i∼ j

i

Bounding the sum over i with an integral, we conclude that 

1 0

 1/(|x|+|t|1/2 ) λ dλ dλ 1 = 1/2 4+α 1/2 3+α λ (|x| + |t| + λ) (|x| + |t| ) (1 + λ)4+α 0 1  . (|x| + |t|1/2 )3+α  

¯ ∈ R × T3 and ∀q ∈ Z, q  −1 define the Lemma A.18 For a fixed ζ¯ = (t, x) measures K q,x¯ (y)δ(t − s)dζ, with ζ = (s, y), μq,ζ :=   μ˜ q,ζ := R3 K q,x¯ (x)P(t − s, x − y)dx 1[0,+∞) (s)dζ, with ζ = (s, y). Let Cε (ζ ) for ζ ∈ R × R3 be (T3 -periodic) covariance of Yε . Then for every α < 3  ¯ := Iα (t, x)

(R×T3 )2

|Cε (ζ − ζ  )|α |μq,ζ ||μq,ζ  |  2αq .

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M. Furlan, M. Gubinelli

For every β ∈ (3, 5)\{4} ¯ := I¯β (t, x)

 (R×T3 )2

|Cε (ζ − ζ  )|β |μ˜ q,ζ ||μ˜ q,ζ  |  2(β−4)q

Proof We have  ¯ = Iα (t, x)

|K q,x¯ (y)K q,x¯ (y  )||y − y  |−α dydy 

by Lemma A.13. By (86) we have 

|K q,x¯ (y)||y − y  |−α dy  (|x¯ − y  | + 2−q )−α  |x¯ − y  |−α

and using again (86) we obtain the result. For the second estimation we obtain from (87) that |μ˜ q,ζ |  2−q (|t − s|1/2 + |x¯ − y| + 2−q )4 dsdy, and from Lemma A.8  |μ˜ q,ζ  | 

R3

|K q,x¯ (x  )|(|t − s  |1/2 + |x¯ − y  |)3 dx  ds  dy  .

By the estimation on convolutions of Lemma A.10 we obtain then ¯  2−q I¯β (t, x)



|K q,x¯ (x  )||x¯ − x  |3−β dx 

 2(β−4)q  

by (86), and this concludes the proof.

Lemma A.19 For a fixed ζ¯ = (t, x) ¯ ∈ R × T3 and ∀q ∈ Z, q  −1 define the measure μq,ζ1 ,ζ2 ⎡



:= ⎣

R6

K q,x¯ (x)



⎤ K i,x (y)K j,x (x2 )P(t − s1 , y − x1 )dxdy ⎦

i∼ j

1[0,+∞) (s1 )δ(t − s2 )dζ1 dζ2 .

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Weak universality for a class of 3d stochastic…

with the notation ζi = (si , xi ) ∈ R × Td for i = 1, 2. For k,  ∈ [0, 2) and m ∈ (0, 5), n ∈ (0, 3) define Ik,,m,n  :=

|Cε (ζ1 − ζ2 )|k |Cε (ζ1 − ζ2 )| |Cε (ζ1 − ζ1 )|m

(R×Td )2

|Cε (ζ2 − ζ2 )|n |μq,ζ1 ,ζ2 ||μq,ζ1 ,ζ2 |

(88)

If k + m − 1 ∈ (0, 5) or  + m − 1 ∈ (0, 5), k +  + m − 4 ∈ (−2, 5) we have the bound Ik,,m,n  2(k++m+n−4)q . Proof From the Fourier support properties of the kernel K we have μq,ζ1 ,ζ2 ⎡  ⎣ =

R6

K q,x¯ (x)



⎤ K i,x (y)K j,x (x2 )P(t − s1 , y − x1 )dxdy ⎦

i∼ j∼q

×1[0,+∞) (s1 )δ(t − s2 )dζ1 dζ2 ⎡ ⎤   +⎣ K q,x¯ (x) K i,x (y)P(t − s1 , y − x1 )δ(x − x2 )dxdy ⎦ R6

iq

×1[0,+∞) (s1 )δ(t − s2 )dζ1 dζ2 = μ¯ q,ζ1 ,ζ2 + μ˜ q,ζ1 ,ζ2 . By Cauchy–Schwarz inequality it suffices then to bound the terms I¯k,,m,n and I˜k,,m,n where μq,ζ1 ,ζ2 in (88) is replaced respectively by μ¯ q,ζ1 ,ζ2 and μ˜ q,ζ1 ,ζ2 . The first term can be estimated by repeated change of variables, using the fact that P(2−2i s, 2−i y) = 2−3i P(s, y) and |Cε (2−2i s, 2−i y)|  2i (|s|1/2 + |y|)−1 . To bound the term μ˜ q,ζ1 ,ζ2 we use the estimation (87) choosing δ ∈ [0, 1] such that the hypotheses of Lemma A.10 are satisfied. A repeated application of these convolution estimations yields the result.  

Appendix B: Some Malliavin calculus We recall here some tools from Malliavin calculus that are widely used in the rest of the paper. An introduction to Malliavin calculus and the proofs of some results of this Appendix can be found in [20,21,25]. Lemma B.7 was inspired by the calculations of [19].

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M. Furlan, M. Gubinelli

B.1 Notation Let {W (h)}h∈H be an isonormal Gaussian process indexed by a real separable Hilbert space H . Let (, F, P) a probability space  with F generated by the isonormal Gaussian process W , we note L 2 () = n0 Hn the well-known Wiener chaos decomposition of L 2 (). For any real separable Hilbert space V and k ∈ N, let Dk : L 2 (; V ) → L 2 (; H #k ⊗ V ) be the Malliavin derivative and δ k : L 2 (; H ⊗k ⊗ V ) → L 2 (; V ) the divergence operator (also called Skorohod integral) defined as the adjoint of Dk . For p  1 we will write Dk, p (V ) ⊂ L p (; V ) for the closure of smooth random variables with respect to the norm ⎤1/ p k    p p = ⎣E(  V ) + E D j  H ⊗ j ⊗V ⎦ ⎡

 Dk, p (V )

j=1

with the notation Dk, p := Dk, p (R). Let {Pt }t∈R+ the Ornstein–Uhlenbeck semigroup and L : L 2 () → L 2 () its generator (i.e. et L = Pt ). Following [25] we introduce the Green operator G j = ( j − L)−1 with the notation G [m] [ j] :=

m 

G k for 1  j  m

(89)

k= j [ j]

so that G [ j] = G j . To avoid confusion, it is worth stressing that G [m] [ j] is not the m-th m power of the operator G j but just a shortcut for k= j G k . B.2 Partial chaos expansion

Let  ∈ L 2 () which has the Wiener chaos decomposition  = n0 Jn . Then by Proposition 1.2.2 of [21] DJn  = Jn−1 D, and knowing that L Jn  = −n Jn  we obtain the commutation property D( j − L)−α  = D

 n0

=



n1

1 Jn  ( j + n)α

1 Jn−1 D = ( j + 1 − L)−α D ( j + n)α

(90)

for every α > 0, j > 0. The above formula holds also for j = 0 if E() = 0. The results we have recalled so far let us write an nth-order Wiener chaos expansion for a random variable in Dn,2 :

123

Weak universality for a class of 3d stochastic… [n] n Lemma B.1 Let  ∈ Dn,2 and G [m] [ j] as in (89). Then for every n ∈ N\{0} G [1] D  ∈ Dom δ n , J0 Dk  ∈ Dom δ k ∀0  k < n and

n δ n G [n] [1] D  = (id −J0 − · · · − Jn−1 ) =  −

n−1  1 k δ J0 Dk . k!

(91)

k=0

Proof We have for any  ∈ L 2 (), since L = −δD ([21], Proposition 1.4.3):  − E() = L L −1 ( − J0 ) = −δDL −1 ( − J0 ) = δ(1 − L)−1 D where we used (90), and the fact that (1−L)−1 D ∈ Dom δ is obvious by construction. This yields the first order expansion  = E() + δ(1 − L)−1 D. Iterating the expansion up to order n we obtain (91). It is clear that J0 Dk  ∈ Dom δ k since J0 Dk  is constant with values in H ⊗k . The second equality comes from the fact that δ k J0 Dk  ∈ Hk ∀k ∈ N, indeed ∀  ∈ Dk,2 : E(δ k (J0 Dk )Jk   ) = J0 Dk , J0 Dk    L 2 (;H ⊗k ) = E(δ k (J0 Dk )  )   n In order to obtain L p estimations of the remainder term δ n G [n] [1] D  generated by expansion (91), we used the following lemmas:

Lemma B.2 ([21], Prop. 1.5.7) Let V be a real separable Hilbert space. For every p > 1 and every q ∈ N, k  q and every  ∈ Dk, p (H q ⊗ V ) we have

δ q () Dk−q, p (V ) k, p  Dk, p (H q ⊗V ) Remark B.3 Let V be a real separable Hilbert space. For every v ∈ V and every  ∈ Dq,2 (H ⊗q ) with q ∈ N we have  ⊗ v ∈ Dom δ q and δ q ()v = δ q ( ⊗ v). Indeed, notice that for every smooth   ∈ Dq,2 (V ) and every smooth  ∈ Dq,2 (H ⊗q ) we have E(δ q ( ⊗ v),   V ) = E( ⊗ v, Dq    H ⊗q ⊗V ) = E(δ q ()v,   V ). Now since Dq ( ⊗v) = Dq  ⊗v and  ∈ Dq,2 (H ⊗q ), we have  ⊗v ∈ Dq,2 (H ⊗q ⊗ V ). Lemma B.2 yields the bound δ q ( ⊗ v) L 2 (V )   ⊗ v Dq,2 (H ⊗q ⊗V ) which allows to pass to the limit for   and δ q ( ⊗ v) in L 2 (V ). Lemma B.4 ([25], Prop. 4.3) For every j > 0 the operator ( j − L)−1/2 is bounded in L p for every 1  p < ∞.

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M. Furlan, M. Gubinelli

Lemma B.5 Let j ∈ N\{0} and V a real separable Hilbert space. There exists a finite constant c p such that for every  ∈ L p (, V ):

D( j − L)−1/2  L p (,H ⊗V )  c p  L p (,V ) (where the operator D( j − L)−1/2 is defined on every  which is polynomial in W (h 1 ), . . . , W (h n ) and can be extended by density on L p ). Proof First notice that we can suppose w.l.o.g. E() = 0 thanks to (90). Therefore 1 we can write D( j − L)− 2 as D( j − L)−1/2 = D(−C)−1 (−C)( j − L)−1/2 √ with C = − −L. We decompose the second part as −C( j − L)−1/2  =

∞

∞  n 1/2 Jn  = Tφ , with Tφ  := n=1 j+n n=0 φ(n)Jn . We apply Theorem 1.4.2 of [21] to show that Tφ is bounded in L p , indeed φ(n) = h(1/n) and h(x) = ( j x + 1)−1/2 which is analytic in a neighbourhood of 0. Finally, we can apply Proposition 1.5.2 of [21] to show that DC −1 is bounded in L p , thus concluding the proof.   The two lemmas above give the following immediate corollary: n −1 is Corollary B.6 For every 1  m  n the operator G [n] j=m ( j − L) [m] := p bounded in L for every 1  p < ∞. Moreover, Let j ∈ N\{0} and V a real separable Hilbert space. Then for every  ∈ L p (, V ) we have:

D( j − L)−1  L p (,H ⊗V )   L p (,V ) . Moreover, for every 0  k  2m, i  0 we have [i+m]  L p (,H ⊗k ⊗V )   L p (,V ) .

Dk G [i+1]

The next lemma is one of the most useful tools of this paper. It allows us to write products of decompositions of the type (91) as sums of iterated Skorohod integrals. From now on we will note ·, · H ⊗r the r -th contraction, which to avoid inconsistency has to be taken between symmetric tensors. We also note h #n v1 ,...,vn := h v1 # · · · # h vn for h v1 , . . . , h vn ∈ H . m+n (R) and Lemma B.7 Let u = f (W (h u ))h ⊗m and v = Fh #n u v1 ,...,vn with f ∈ C m+n,2 . Then F ∈D  Cm,n,q,r,i δ m+n−q−r δ m (u)δ n (v) = (q,r,i)∈Im,n

 f

123

(r −i)

#n (W (h u ))h u⊗m−i , Dq−i F H ⊗q−i h ⊗r u , h v1 ,...,vn  H ⊗r

(92)

Weak universality for a class of 3d stochastic…

with Cm,n,q,r,i := mq nr qi ri i! and Im,n := {(q, r, i) ∈ N3 : 0  q  m, 0  r  n, 0  i  q ∧ r }. A trivial change of variables gives also: 

δ m (u)δ n (v) =

Cm,n,q+i,r +i,i δ m+n−q−r −2i

 (i,q,r )∈Im,n



× f (r ) (W (h u ))h u⊗m−i , Dq F H ⊗q h u⊗r +i , h #n v1 ,...,vn  H ⊗r +i

(93)

 := {(i, q, r ) ∈ N3 : 0  i  m ∧ n, 0  q  m − i, 0  r  n − i}. with Im,n

Remark B.8 In the special case v = g(W (h v ))h ⊗n v Eq. (92) takes the form δ m (u)δ n (v) =



Cm,n,q,r,i δ m+n−q−r (Dr −i u, Dq−i v H ⊗q+r −i )

(94)

(q,r,i)∈Im,n

which is just a generalization to Skorohod integrals of the multiplication formula for multiple Wiener integrals ([24,26]). We can write the above formula more explicitly as   ⊗m−q Cm,n,q,r,i δ m+n−q−r f (r −i) (W (h u ))g (q−i) (W (h v ))h u δ m (u)δ n (v) = (q,r,i)∈Im,n

⊗h v⊗n−r h u , h v q+r −i .

Remark B.9 Note that one can assume w.l.o.g. the argument of δ m+n−q−r in (92) to be symmetric, and this would allow to iterate Lemma B.7. Remark B.10 We can give the following intuition for the second formula in Lemma B.7. The random variables u and v have an infinite chaos decomposition, and following the tree-like notation of [6] or [8] they could be thought of as having an infinite number of leaves which need to be contracted with each other. It is apparent that the index i in the second equation denotes contractions between the already existing leaves of the trees u, v. The indexes r and q count new leaves in each vertex that are created by the Malliavin derivatives, which are then contracted with other leaves from the other tree. There are then m + n − r − q − 2i overall unmatched leaves which are arguments to the iterated Skorokhod integral. The more intuitive interpretation of the second equation in Lemma B.7 is the reason why we gave two distinct expression for the same quantity. Nevertheless, the formula (92) is more practical in the calculations and is more widely used throughout the paper. Proof [Lemma B.7] Using Cauchy–Schwarz inequality and Lemma B.2 we can show that Dr δ n (v), δ j (u) H ⊗r ∈ L 2 (, H ⊗m− j−r ) for every 0  r + j  m. Then we apply Lemma B.11 to get: δ m (u)δ n (v) =

n    n n−r δ (Dr δ m (u), v H ⊗r ). r r =0

123

M. Furlan, M. Gubinelli

Using the commutation formula (95) we rewrite the r.h.s. as n   r ∧m     n r m δ (u)δ (v) = i!δ n−r (δ m−i (Dr −i u), v H ⊗r ). r i i m

n

r =0

i=0

We obtain #n δ m−i (Dr −i u), v H ⊗r = δ m−i ( f (r −i) (W (h u ))h u⊗m−i )Fh ⊗r u , h v1 ,...,vn  H ⊗r

and using again Lemma B.11 we obtain δ

m−i

r −i

(D

u), v H ⊗r =

m−i  =0

 m − i m−i− (r −i) (f (W (h u ))h u⊗m−i , D F H ⊗ δ 

h ⊗r u , h v1 # · · · # h vn  H ⊗r ) where we used δ k ()h ⊗n−r = δ k ( ⊗ h ⊗n−r ) for  ∈ Dom δ k , as seen in Remark B.3. Substituting this expression into δ m (u)δ n (v) we get δ m (u)δ n (v) =

 Am,n,r,i, δ m+n−r −i− f (r −i) (W (h u ))h u⊗m−i , D F H ⊗

 (r,i,)∈Jm,n

h ⊗r u , h v1 # · · · # h vn  H ⊗r



r 3 where we set Am,n,r,i, := mi nr m−i  i i! and Jm,n := {(r, i, ) ∈ N : 0  r  n, 0  i  r ∧ n, 0    m − i}. In order to complete the proof we just have basic changes of

to perform q msome

= we have Am,n,r,i, = indexes. Taking q =  + i and noting that mi m−i q−i i q m q n r q i r i i! and this yields (92). Finally, we perform the change of variables q − i → q, r − i → r to get the second formula.   We give below the results we used to prove Lemma B.7. Lemma B.11 ([19], Lemma 2.1) Let q ∈ N\{0},  ∈ Dq,2 , u ∈ Dom δ q and symmetric. Assume also that ∀0  r + j  q Dr , δ j (u) H ⊗r ∈ L 2 (, H ⊗q−r − j ). Then ∀0  r < q Dr , ur ∈ Dom δ q−r and q    q q−r δ (Dr , u H ⊗r ). δ (u) = r q

r =0

Remark B.12 Note that δ n (h ⊗n ) = W n (h)

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Weak universality for a class of 3d stochastic…

where · stands for the Wick product. Indeed ∀ ∈ D1,2 we know that E[δ(h ⊗n )] = E[W (h)h ⊗n−1 ] using the definition of δ, and then δ n (h ⊗n ) = δ n−1 (W (h)h ⊗n−1 ). From Lemma B.11 we have, since DW (h) = h: δ n−1 (W (h)h ⊗n−1 ) = δ n−1 (h ⊗n−1 )W (h) − (n − 1)h, hδ n−2 (h ⊗n−2 ) which gives by induction δ n (h ⊗n ) = W n (h). Lemma B.13 Let j, k ∈ N, u ∈ D j+k,2 (H ⊗ j ) symmetric and such that all its derivatives are symmetric. We have Dk δ j (u) =

k∧ j     k j i!δ j−i (D k−i u) i i

(95)

i=0

Proof If j = 0, k = 1 or k = 0, j = 1 Eq. (95) is trivial. Let j = k = 1 and u ∈ D2,2 (H ) ⊂ D1,2 (H ). We can apply Proposition 1.3.2 of [21] to obtain Dδ(u), h = u, h + δ(Du, h) ∀h ∈ H . Since by hypothesis Du is symmetric we have δ(Du, h) = δDu, h, and then Dδ(u) = u + δDu. The proof by induction is   easy noticing that Dδ j = δ j D + jδ j−1 .

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