Convergence of the stochastic data

t^ν}L_µ^εpf1ăăf2q ´ f1ăăL_µ^εf₂qptq}_C^γ`β´2

p pG^ε,ρ1ptqρ2ptqq À }f1}_Lp,t^ν,γ_pG^ε_,ρ1q}f2ptq}C^βpG^ε,ρ2ptqq. where Lµ^ε“ Bt´ L^ε_µ is a discrete diffusion operator induced by some µ P µpωq as in Definition 3.3.3. The involved constants are independent of ε.

Proof. Again we can almost follow along the lines of the proof in [GP15b, Lemma 6.5]

with the only difference that in the derivation of the second estimate the application of the “product rule” ofLµ^ε does not yield a term ´2∇făă∇f2 but a more complex object, namely

ż

R^d

dµpyq

ε² D_y^εf₁ăăD^ε_yf₂, (4.7) where D^ε_yf₁pt, xq “ f1pt, x ` εyq ´ f1pt, xq and similar for f2. The bound on (4.7) follows from Lemma 4.1.5 once we can show

}D_y^εϕ}_C^γ´1

p pG^ε,ρ1q À }ϕ}C_p^γpG^ε,ρ1q|y|ε (4.8) for any γ P R. Note that due to Lemma 3.1.11 we can write

∆_jD_ε^yϕ “

´Ψ˜^ε,j_¨`εy´ ˜Ψ^ε,j

¯

˚G^ε ϕ , where ˜Ψ^ε,j “E^εΨ^Gε,j “ 2^jdϕ_xjyεp2^j¨q with ϕxjyε PS^ωpR^dq. With

Ψ˜^ε,j_x`εy´ ˜Ψ^ε,j_x “ 2^j ż1

0

2^jdϕ_xjyεp2^jpx ` ζεyqq dζ ¨ yε we get (4.8) by applying Lemma 3.1.7.

Morally, the reason why the diffusion operator Lµ^ε can be pulled on the second factor f₂ in the product f₁ăăf2 is that f₂ describes the small-scale behavior of this object which is the regime whereLµ^ε acts.

4.2 Convergence of the stochastic data

Let G^ε be, as in Definition 3.1.2, a sequence of refining lattices build from some Bravais latticeG in dimension d. Let further`ξ^εpzq˘

zPG^ε be a discrete approximation to white noise onG^ε as in Section 3.4.

Fix a symmetric χ P C_ω,c⁸ pR²q, independent of ε, which is 0 on ¹₄¨ pG and 1 outside

where l_µ^ε is as in (3.32) the multiplier of the diffusion operator Lµ^ε associated to µ. Note that Lµ^εY_µ^ε “ ´L^ε_µY_µ^ε “ χpDG^εqξ^ε :“ FG^´1ε pχ ¨ F^Gεξ^εq so that Y_µ^ε is a time independent solution to the heat (or Poisson) equation on G^ε induced by our operator Lµ^ε. Note that χ is not scaled by ε and only serves as a cut-off for the

“Fourier modes” around 0, where _l¹ε

µ diverges.

Our first task will be to measure the regularity of the sequences pξ^εq, pY_µ^εq in the discrete Besov spaces introduced in Subsection 3.1.2.

As an application of the discrete Wick calculus we introduced in Section 3.4 we can bound the moments of ξ^ε and Y_µ^ε in Besov spaces. We also want to control the resonant product Y_µ^ε˝ ξ^ε, for which we introduce the renormalization constant

c^ε_µ:“

We define a renormalized resonance product by

Y_µ^ε‚ ξ^ε :“ Y_µ^ε˝ ξ^ε´ c^ε_µ.

Remark 4.2.1. Since l^ε_µ «|¨|² (Lemma 3.3.5 together with the easy estimate l_µ^ε À

|¨|²) we have c^ε_µ« ´ log ε in dimension 2.

Using Lemma 3.4.1 we can derive the following bounds.

Lemma 4.2.2. Let ξ^ε, Y^ε and Y_µ^ε‚ ξ^ε be defined on G^ε as above with p_ξ ě 4 (where The involved constant is independent of ε.

Proof. Let us bound the regularity of Y_µ^ε. Recall that by Lemma 3.1.9 we have the continuous embedding (with norm uniformly bounded in ε) B^ζ`d{ppξ,pξ ^ξpG^ε, xxy^´κq Ď

96 4.2 Convergence of the stochastic data By assumption we have κp_ξ ą d and can bound ř

zPG^ε|G^ε|p1 ` |z|q^´κpξ À 1 uni-formly in ε (for example by Lemma 3.5.1). It thus suffices to derive a bound for Er|∆^G

ε

j Y_µ^εpxq|^pεs, uniform in ε and x. Note that by (3.6)

∆^G_j^εY_µ^εpxq “ ÿ

zPG^ε

|G^ε|Kj^εpx ´ zqξ^εpzq

withKj^ε “FG^´1εφ^G_j^εχ{l^ε_µ so that Lemma 3.4.1, Parseval’s identity (3.5) and l_µ^ε Á|¨|² on xG^ε (from Lemma 3.3.5) imply

Er|∆^G

ε

j Y_µ^εpxq|^pξs À }Kj^ε}^p_L^ξ2pG^εqÀ 2^jpξpd{2´2q,

which proves the bound for Y_µ^ε. The bound for ξ^ε follows from the same arguments or with Lemma 3.3.4.

Now let us turn to Y_µ^ε‚ ξ^ε. A short computation shows that ErpYµ^ε˝ ξ^εqpxqs “ ErpY_µ^ε¨ ξ^εqpxqs “ c^ε_µ, x P G^ε, and, by a similar argument as above, it now suffices to bound Y_µ^ε‚ ξ^ε in

Bp^βξ{2,pξ{2pR^d, xxy^´2κq for β ă 2 ´ d. We are therefore left with the task of bounding the pξ{2-th moment of ř

|i´j|ď1∆^G_i^εY_µ^ε∆^G_j^εξ^ε´ Er∆^Gi^εY_µ^ε∆^G_j^εξ^εs. But

∆^G_i^εY_µ^εpxq∆^G_j^εξ^εpxq ´ Er∆^Gi^εY_µ^εpxq∆^G_j^εξ^εpxqs

“ ÿ

z1,z2

|G^ε|²Ki^εpx ´ z1qΨ^jpx ´ z2q pξ^εpz1qξ^εpz2q ´ Erξ^εpz1qξ^εpz2qsq

“ ÿ

z1,z2

|G^ε|²K_i^εpx ´ z1qΨ^jpx ´ z2qξ^εpz1q ˛ ξ^εpz2q ,

so that Lemma 3.4.1 yields E

”ˇ

ˇ∆^G_i^εY_µ^ε∆^G_j^εξ^ε´ Er∆^Gi^εY_µ^ε∆^G_j^εξ^εsˇ ˇ

pξ{2ı

À }Ki^ε}^p_L^ξ2^{2pG^εq}Ψ^j}^p_L^ξ2^{2pG^εq

À 2^{ipd{2´2qpξ{2}2^jd{2¨pξ{2 » 2^{jpd´2q¨pξ{2}, where we used Parseval’s identity, l^ε_µÁ|¨|² on xG^ε, and that |i ´ j| ď 1.

By the compact embedding result in Lemma 2.1.26 we see that the sequences pE^εξ^εq, pE^εY_µ^εq, and pE^εpY_µ^ε ‚ ξ^εqq have convergent subsequences in distribution.

We will see in Lemma 4.2.3 below that E^εξ^ε converges to the white noise ξ on R². Consequently, the solution Y_µ^ε to ´L^ε_µY_µ^ε “ χpDG^εqξ^ε should, if all goes well, approach the solution of ´LµY_µ“ χpD_R^dqξ :“F_R^´1d`χ FR^dξ˘

, i.e.

Y_µ“ 1

p2πq²}D_R²}²_µχpD_R²qξ “K_µ⁰˚ ξ, K_µ⁰ :“F_R^´1d χ p2πq²∥¨∥²µ

. (4.11)

where } ¨ }µ is defined as in Definition 3.3.1. The limit of E^εpY_µ^ε‚ ξ^εq will turn out to be the distribution

Y_µ‚ ξpφq :“

ż

R^d

ż

R²

Kµ⁰pz1´ z2qφpz1qξpdz1q ˛ ξpdz2q ´ pYµăξ ` ξăYµqpφq (4.12)

for φ P S^ωpR^dq, where the first term on the right hand side denotes as in Section 3.4 the second order Wiener-Itô (or Skorohod) integral with respect to the Gaussian stochastic measure ξpdzq induced by the white noise ξ. Note that Yµ‚ ξ is not a continuous functional of ξ, so the last convergence is not a trivial consequence of the convergence for E^εξ^ε. To identify the limit of E^εpY_µ^ε ‚ ξ^εq we could use a diagonal sequence argument that first approximates the bilinear functional by a continuous bilinear functional as in [MW17b, HS15, CGP17]. However, having already established the machinery in Section 3.4 we can apply Lemma 3.4.2 instead.

Lemma 4.2.3. In the setup of Lemma 4.2.2 with ξ, Y_µ and Y_µ‚ ξ defined as above and with ζ, κ as in Lemma 4.2.2 we have for d ă 4

pE^εξ^ε,E^εY_µ^ε,E^εpY_µ^ε‚ ξ^εqqÝÑ pξ, Y^εÑ0 µ, Y_µ‚ ξq

in distribution inC^ζ´2pR^d, xxy^´κq ˆC^ζpR^d, xxy^´κq ˆC^2ζ´2pR^d, xxy^´2κq.

Proof. Recall that the extension operator E^ε is constructed from ψ^ε “ ψpε¨q where the smear function ψ is symmetric and satisfies in particular ψ P C_ω,c⁸ pR^dq and ψ “ 1 on some ball around 0.

Since from Lemma 4.2.2 we already know that the sequence pE^εξ^ε,E^εY_µ^ε,E^εpY_µ^ε‚ ξ^εqq is tight inC^ζ´2pR^d, xxy^´κqˆC^ζpR^d, xxy^´κqˆC^2ζ´2pR^d, xxy^´2κq, it suffices to prove the convergence after testing against φ PS^ωpR^dq:

pE^εξ^εpφq,E^εY_µ^εpφq,E^εpY_µ^ε‚ ξ^εqpφqqÑ pξpφq, Y^d µpφq, Yµ‚ ξpφqq . (4.13) We can even restrict ourselves to those φ P S^ωpR^dq with FR^dφ P C_ω,c⁸ pR^dq, which implies suppFR^dφ Ď xG^ε and F_R^´1dpψ^εFR^dφq “ φ for ε small enough, which we will assume from now on. Note that suppFR^dφ Ď xG^ε implies

F^Gεφ “FR^dφ|_Gxε (4.14) since by definition ofFG^´1ε

FG^´1εFR^dφ “ pF_R^´1dFR^dφq|G^ε “ φ|G^ε.

Let us first show the convergence of (4.13) in every component.

98 4.2 Convergence of the stochastic data To show the convergence of E^εξ^ε to white noise note that we have from (3.21) the following formula

E^εξ^εpφq “ ÿ

zPG^ε

|G^ε| pF_R^´1dψ^ε˚ φqpzqξ^εpzq “ ÿ

zPG^ε

|G^ε|F_R^´1dpψ^εFR^dφqpzqξ^εpzq

“ ÿ

zPG^ε

|G^ε|φpzqξ^εpzq

where we used in the first step that ψ^ε is symmetric and in the last step that F_R^´1dpψ^εFR^dφq “ φ by our choice of φ and ε. Using Lemma 3.4.2 and relation (4.14) the convergence of E^εξ^εpφq to ξpφq follows.

For the limit ofE^εY^ε we use the following formula, which is derived by the same argument as above:

E^εY_µ^εpφq “ ÿ

z1, z2PG^ε

|G^ε|²φpz1qK_µ^εpz2´ z1qξ^εpz2q

with Kµ^ε“FG^´1ε χ

l^ε_µ. In view of Lemma 3.4.2 it then suffices to note that fˆ^ε :“F^Gεpφ ˚G^εKµ^εq “F^Gεφ ¨_l^χε

µ

(4.14)

“ FR^dφ ¨_l^χε

µ is due to Lemma 3.3.5 dominated by χ{|¨|² on xG^ε and converges to

FR^dφ χ{pp2πq²∥¨∥²µq˘ by the explicit formula for l^ε_µ in (3.32).

We are left with the convergence of the third component. Since E^εξ^ε Ñ ξ and E^εY_µ^εÑ Yµ we obtain via the (E )-Property of the paraproduct

limεÑ0E^εpY_µ^εăξ^εq “ lim

εÑ0E^εY_µ^εăE^εξ^ε“ Yµăξ

and similarly one gets E^εpξ^εăY_µ^εq Ñ ξăYµ. We can therefore show instead E^ε`Y_µ^εξ^ε´ ErYµ^εξ^εs˘

pφq Ñ pYµ‚ ξ ` ξăYµ` Yµăξqpφq . (4.15) Note that we have the representations

E^ε`Y_µ^εξ^ε´ ErYµ^εξ^εs˘

pφq “ ÿ

z1,z2PG^ε

|G^ε|²φpz1qKµ^εpz1 ´ z2q ξ^εpz1q ˛ ξ^εpz2q ,

pYµ‚ ξ ` ξăYµ` Yµăξqpφq “ ż

R²

ż

R²

φpz1qKµ⁰pz1´ z2q ξpdz1q ˛ ξpdz2q

withKµ^εas above andKµ⁰as in (4.11). The pG^εq²-Fourier transform of φpz1qKµ^εpz1´ z₂q is ˆφ_extpx1 ´ x2qχpx2q{l_µ^εpx2q for x1, x₂ P xG^ε, where ˆφ_ext denotes the periodic

extension from (3.10) for FR^dφ|_Gxε P C_ω,c⁸ pxG^εq (recall again that suppFR^dφ Ď xG^ε).

We can therefore apply Lemma 3.4.2 since for d ă 4 the function pχpx2q{l^εpx2qq² À 1_|x|Á1{|x|⁴ is integrable on xG^ε and thus we obtain (4.15).

We have shown the convergence in distribution of all the components in (4.13).

By Lemma 3.4.2 we can take any linear combination of these components and still get the convergence from the same estimates, so that (4.13) follows from the Cramér-Wold Theorem.

In document Refinements of the Solution Theory for Singular SPDEs (Page 106-111)