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,γpG^ε,ρ1q}f₂ptq}_C^β_pG^ε_,ρ2ptqq. where Lµ^ε“ Bt´L^ε_µ is a discrete diffusion operator induced by some µPµpωq 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}_Cp^γ´1_pGε,ρ1q À }ϕ}C_p^γpG^ε,ρ1q|y|ε (4.8) for any γ PR. Note that due to Lemma 3.1.11 we can write

∆_jD_ε^yϕ “

´Ψ˜^ε,j_¨`εy´Ψ˜^ε,j

¯

˚G^ε ϕ , whereΨ˜^ε,j “E^εΨ^Gε,j “2^jdϕ_xjyεp2^j¨qwith ϕ_xjyε PS^ωpR^dq. With

Ψ˜^ε,j_x`εy´Ψ˜^ε,j_x “2^j ż1

0

2^jdϕ_xjyεp2^jpx`ζεyqqdζ¨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 latticeGin dimensiond. Let further`

ξ^εpzq˘

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

Fix a symmetricχPC_ω,c⁸ pR²q, independent ofε, which is0on ¹₄¨Gpand 1outside

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” around0, where _l¹ε

µ diverges.

Our first task will be to measure the regularity of the sequencespξ^εq, pY_µ^εqin 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 productY_µ^ε˝ξ^ε, for which we introduce the renormalization constant

c^ε_µ:“

We define arenormalized 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« ´logε 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 ě4 (where The involved constant is independent of ε.

Proof. Let us bound the regularity ofY_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_j^εY_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) andl_µ^ε Á|¨|² onxG^ε (from Lemma 3.3.5) imply

Er|∆^G_j^εY_µ^εpxq|^pξs À }Kj^ε}^p_L^ξ2pG^εqÀ2^jpξpd{2´2q,

which proves the bound forY_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 ErpY_µ^ε˝ξ^εqpxqs “ErpY_µ^ε¨ξ^εqpxqs “c^ε_µ, xP G^ε, and, by a similar argument as above, it now suffices to boundY_µ^ε‚ξ^ε in

Bp^βξ{2,pξ{2pR^d,xxy^´2κqfor β ă2´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∆^G_i^εY_µ^ε∆^G_j^εξ^εs. But

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

À2^{ipd{2´2qpξ{2}2^jd{2¨pξ{2 »2^{jpd´2q¨pξ{2}, where we used Parseval’s identity, l^ε_µÁ|¨|² onxG^ε, and that |i´j| ď1.

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 ´L_µY_µ“χpD_R^dqξ :“F_R^´1d`

χFR^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 ´ pY_µăξ`ξăY_µqpφq (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 Y_µ‚ 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 dă4

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¨qwhere the smear functionψ is symmetric and satisfies in particularψ P C_ω,c⁸ pR^dqandψ “1 on some ball around 0.

Since from Lemma 4.2.2 we already know that the sequencepE^εξ^ε,E^εY_µ^ε,E^εpY_µ^ε‚ ξ^εqqis tight inC^ζ´2pR^d,xxy^´κqˆC^ζpR^d,xxy^´κqˆC^2ζ´2pR^d,xxy^´2κq, it suffices to prove the convergence after testing againstφPS^ωpR^dq:

pE^εξ^εpφq,E^εY_µ^εpφq,E^εpY_µ^ε‚ξ^εqpφqqÑ p^d ξpφq, Y_µpφq, Y_µ‚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 Gx^ε and F_R^´1dpψ^εFR^dφq “ φ for ε small enough, which we will assume from now on. Note that suppFR^dφĎxG^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 “φby our choice of φandε. Using Lemma 3.4.2 and relation (4.14) the convergence of E^εξ^εpφq toξpφq 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 onGx^ε and converges to

FR^dφ χ{pp2πq²∥¨∥²µq˘ by the explicit formula forl^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^εÑYµ we obtain via the (E)-Property of the paraproduct

limεÑ0E^εpY_µ^εăξ^εq “ lim

εÑ0E^εY_µ^εăE^εξ^ε“Y_µăξ

and similarly one gets E^εpξ^εăY_µ^εq Ñ ξăY_µ. We can therefore show instead E^ε`

Y_µ^εξ^ε´ErY_µ^εξ^εs˘

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

E^ε`

Y_µ^εξ^ε´ErY_µ^εξ^εs˘

pφq “ ÿ

z1,z2PG^ε

|G^ε|²φpz₁qKµ^εpz₁ ´z₂qξ^εpz₁q ˛ξ^εpz₂q, pY_µ‚ξ`ξăY<sub>µ</sub>`Y_µăξqpφq “

ż

R²

ż

R²

φpz₁qKµ⁰pz₁´z₂qξpdz₁q ˛ξpdz₂q

withKµ^εas above andKµ⁰as in (4.11). ThepG^εq²-Fourier transform ofφpz₁qKµ^εpz₁´ z₂q is φˆ_extpx₁ ´ x₂qχpx₂q{l_µ^εpx₂q for x₁, x₂ P xG^ε, where φˆ_ext denotes the periodic

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

We can therefore apply Lemma 3.4.2 since for dă4the 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.

Im Dokument Refinements of the Solution Theory for Singular SPDEs (Seite 106-111)