2 Reduction of the problem

ELECTRONIC COMMUNICATIONS in PROBABILITY

WEAK APPROXIMATION OF FRACTIONAL SDES: THE DONSKER SETTING

XAVIER BARDINA¹

Departament de Matemàtiques, Facultat de Ciències, Edifici C, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

email: [email protected] CARLES ROVIRA²

Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain email: [email protected]

SAMY TINDEL

Institut Élie Cartan Nancy, B.P. 239, 54506 VandIJuvre-lès-Nancy Cedex, France email: [email protected]

SubmittedNovember 17, 2009, accepted in final formJune 6, 2010 AMS 2000 Subject classification: 60H10, 60H05

Keywords: Weak approximation, Kac-Stroock type approximation, fractional Brownian motion, rough paths

Abstract

In this note, we take up the study of weak convergence for stochastic differential equations driven by a (Liouville) fractional Brownian motionB with Hurst parameterH ∈(1/3, 1/2), initiated in [3]. In the current paper, we approximate thed-dimensional fBm by the convolution of a rescaled random walk with Liouville’s kernel. We then show that the corresponding differential equation converges in law to a fractional SDE driven byB.

1 Introduction

The current article can be seen as a companion paper to [3], to which we refer for a further introduction. Indeed, in the latter reference, the following equation on the interval [0, 1] was considered (the generalization to[0,T]being a matter of trivial considerations):

d y_t=σ y_t

d B_t+b y_t

d t, y₀=a∈Rⁿ, (1)

whereσ:Rⁿ→R^n×d,b:Rⁿ→Rⁿare two bounded and smooth enough functions, andBstands for ad-dimensional fBm with Hurst parameterH>1/3.

1RESEARCH SUPPORTED BY THE MINISTERIO DE CIENCIA E INNOVACIÓN GRANT MTM2009-08869

2RESEARCH SUPPORTED BY THE MINISTERIO DE CIENCIA E INNOVACIÓN GRANT MTM2009-07203

314

Let us be more specific about the driving process for equation (1): we consider in the sequel the so-calledd-dimensional Liouville fBmB, with Hurst parameterH∈(1/3, 1/2). Namely,Bcan be written asB= (B¹, . . . ,B^d), where theBⁱ’s aredindependent centered Gaussian processes of the form

Bⁱ_t= Zt

0

(t−r)^H−12dW_rⁱ, (2)

for a d-dimensional Wiener process W = (W¹, . . . ,W^d). This process is very close to the usual fBm, in the sense that they only differ by a finite variation process (as pointed out in[1]), and we shall see that its simple expression (2) simplifies some of the computations throughout the paper. In any case, B falls into the scope of application of the rough paths theory, which means that equation (1) can be solved thanks to the semi-pathwise techniques contained in[9, 10, 13]. The natural question raised in[3]was then the following: is it possible to approximate equations like (1) in law by ordinary differential equations, thanks to a Wong-Zakai type approximation (see [12, 16, 17]for further references on the topic)?

Some positive answer to this question had already been given in[8], where some Gaussian se- quences approximations were considered in a general context. In[3], we focused on a natural and easily implementable (non Gaussian) scheme for B, based on Kac-Stroock’s approximation to white noise (see [11, 15]). However, another very natural way to approximate B relies on Donsker’s type scheme (see[14]for the caseH>1/2 and[4]for the Brownian case), involving a rescaled random walk. We have thus decided to investigate weak approximations to (1) based on this process.

More precisely, as an approximating sequence ofB, we shall choose(X^")">0, whereX^",iis defined as follows fori=1, . . . ,d: consider a family of independent random variables{ηⁱk;k≥1, 1≤i≤ d}, satisfying the

Hypothesis 1.1. The random variables{ηⁱk;k≥1, 1≤i≤d}are independent and share the same law as another random variableη. Furthermore,ηis assumed to satisfy E η

=0,E€ 粊

=1and is almost surely bounded by a constant k_η.

We then defineX^",iin the following way:

X^i,"(t) = Z t

0

(t+"²−r)^H−12θ^i,"(r)d r, (3)

where

θ^i,"(r):=1

"

X+∞

k=1

ηⁱkI_[k−1,k)

 r

"²

‹

. (4)

Notice thatX^" is really a process given by the convolution of the rescaled random walkθ^" with Liouville’s kernel.

Let us then consider the process y^"solution to equation (1) driven byX^", namely:

d y^"_t =σ€

y_t^"Š

d X_t^"+b€

y^"_tŠ

d t, y₀^"=a∈Rⁿ, t∈[0,T]. (5)

Our main result is as follows:

Theorem 1.2. Let(y^")_">0be the family of processes defined by (5), and let1/3< γ <H, where H is the Hurst parameter of B. Then, as"→0, y^" converges in law to the process y obtained as the solution to (1), where the convergence takes place in the Hölder space C^γ([0, 1];Rⁿ).

Let us make a few comments about this theorem:

(i)We have presented our results for the Liouville fBmBbecause of some simplifications, apparent in[3], in the manipulations of some stochastic integrals. However, as mentioned in[1], the usual fractional Brownian motion Bˆ can be decomposed as Bˆ = B+V, where V is a finite variation process andBis the Liouville fBm. By writing the Lévy area ofBˆaccording to this decomposition, it is certainly possible to extend our results to fBm, which is in a sense a more natural process for its invariance properties. We did not explore this possibility for sake of conciseness.

(ii)We also have chosen the range(1/3, 1/2)for the coefficientH. Indeed, the caseH>1/2 can be deduced easily from[5]since in this case, the solution yto (1) is a continuous function ofB.

The Brownian caseH =1/2 is proved easily following the methods in[6]. We could also have dealt with a coefficient H lying in(1/4, 1/3] according to[8], but this case is much harder for two reasons: (1) The approximation of third order integrals is also required in this case. (2) The assumption H >1/3 is needed for the convergence of certain terms in [3, Proof of Proposition 5.1](see our termB_2,1,1there).

(iii)Notice that ift∈[N"²,(N+1)"²)one has

X^i,"(t) = 1 (H+1/2)"

– N

X

k=1

ηⁱk

”(t−(k−2)"²)^H+1/2−(t−(k−1)"²)^H+1/2—

+ηⁱN+1

”(t−(N−1)"²)^H+1/2−"^2H+1—

™ . Another possibility in order to approximate Xⁱ would have been to define a process Xˆ^i," at any point of the formN"²by

ˆ

X^i,"(N"²) = 1 (H+1/2)"^−2H

N

X

k=1

ηⁱ_k”

(N−(k−2))^H+1/2−(N−(k−1))^H+1/2— ,

and then computeXˆ^i," at any other point by linear interpolation. This new approximation might be closer to the spirit of Donsker type results, since the process Xˆ^i," is piecewise linear. In any case, it should be easy (though technical) to show that Xˆ^i,"−X^i," vanishes when" →0, as L²- random variables taking values in an appropriate rough path space (including the Lévy area of both processes). Thus Theorem 1.2 certainly holds true whenX^i,"is replaced byXˆ^i,". Here again, we did not explore this possibility for sake of conciseness.

Here is how our paper is structured: as the reader shall see, many of the techniques introduced in[3]are also useful in our context. In the end, as explained at Section 2, most of the technical differences between the two articles arise in the way to evaluate the moments of quantities like R1

0 f(r)θ^i,"(r)d rfor a given Hölder function f, and to compare them with the moments of Gaus- sian random variable. This is thus where we shall concentrate our efforts, and this essential point will be handled at Section 3.

2 Reduction of the problem

We shall recall here briefly some preliminary steps contained in [3], which allow to reduce our problem to the evaluation of the moments of a specific type of Wiener integrals.

First of all, we need to recall the definition of some Hölder spaces, in which our convergences take place. We call for instance Cj([0, 1];R^d) the space of continuous functions from[0, 1]^j toR^d, which will mainly be considered for j=1 or 2 variables. The Hölder norms on those spaces are then defined in the following way: forf ∈ C2([0, 1];R^d)let

kfk_µ= sup

s,t∈[0,T]

|f_st|

|t−s|^µ, and C₂^µ([0, 1];R^d) =¦

f ∈ C2([0, 1];R^d);kfk_µ<∞© . The usual Hölder spacesC₁^µ([0, 1];R^d)are then determined by settingkgk_µ=kδgk_µfor a contin- uous functiong∈ C1([0, 1];R^d), whereδg∈ C2([0, 1];R^d)is defined byδg_st=g_t−g_s. We then say thatg∈ C₁^µ([0, 1];R^d)iffkgk_µis finite. Note thatk · k_µis only a semi-norm onC1([0, 1];R^d), but we will work in general on spaces of the type

C_1,a^µ ([0, 1];R^d) =¦

g:[0,T]→V;g₀=a,kgkµ<∞©

, (6)

for a givena∈V, on whichkgk_µis a norm.

The second crucial point one has to recall is the natural definition of a Lévy area for Liouville’s fBm. To this purpose, considerE the set of step-functions on[0,T]with values inR^d. LetH be the Hilbert spaceH defined as the closure ofE with respect to the scalar product induced by

¬(1_[0,t1], . . . ,1_[0,td]),(1_[0,s1], . . . ,1_[0,sd])¶

H =

d

X

i=1

R(t_i,s_i), s_i,t_i∈[0,T], i=1, . . . ,d, whereR(t,s):=E[Bⁱ_tB_sⁱ]. Then a natural representation of the inner product inH is given via the operatorK, defined fromE toL²([0,T]), by:

Kϕ(t) = (T−t)^H−12ϕ(t)− 1

2−H Z ^T

t

[ϕ(r)−ϕ(t)](r−t)^H−32d r,

and it can be checked thatK can be extended as an isometry betweenH and the Hilbert space L²([0,T];R^d). Thus the inner product inH can be defined as:

ϕ,ψ

H ¬

Kϕ,Kψ

L²([0,T];R^d). The mapping(1_[0,t1], . . . ,1_[0,td])7→Pd

i=1Bⁱ_t

i can also be extended into an isometry between H and the first Gaussian chaosH₁(B)associated withB= (B¹, . . . ,B^d). We denote this isometry by ϕ 7→ B(ϕ), and B(ϕ)is called the Wiener-Itô integral of ϕ. It is shown in [7, page 284] that C₁^γ(R^d)⊂ H wheneverγ >1/2−H, which allows to defineB(ϕ)for such kind of functions.

Proposition 2.1. Let B be a d-dimensional Liouville fBm, and suppose that its Hurst parameter satisfies H∈(1/3, 1/2). Then

(1)B is almost surely aγ-Hölder path for any1/3< γ <H.

(2)A Lévy area based on B can be defined by setting

B²_st= Z t

s

d B_u⊗ Zu

s

d B_v, i. e. B²_st(i,j) = Z t

s

d B_uⁱ Zu

s

d B_v^j, i,j∈ {1, . . . ,d},

for0≤s<t≤T . Here, the stochastic integrals are defined as Wiener-Itô integrals when i6= j, while, when i= j, they are simply given by

Z t

s

d B_uⁱ Zu

s

d Bⁱ_v=1 2

€Bⁱ_t−B_sⁱŠ2

.

(3)The processB²is almost surely an element ofC₂^2γ([0, 1];R^d×d), and satisfies the algebraic rela- tion

B²_st−B²_su−B²_ut= B_u−B_s

⊗ B_t−B_u , for all0≤s≤u≤t≤1.

These algebraic and analytic properties of the fBm path allow to invoke the rough path machinery (see[9, 10, 13]) in order to solve equation (1):

Theorem 2.2. Let B be a Liouville fBm with Hurst parameter1/3<H<1/2, andσ:Rⁿ→R^n×d be a C²function, which is bounded together with its derivatives. Then

(1)Equation (1) admits a unique solution y ∈ C₁^γ(Rⁿ)for any1/3< γ <H, with the additional structure of weakly controlled process introduced in[10].

(2)The mapping(a,B,B²)7→y is continuous fromRⁿ× C₁^γ(R^d)× C₂^2γ(R^d×d)toC₁^γ(Rⁿ).

One of the nice aspects of rough paths theory is precisely the second point in Theorem 2.2, which allows to reduce immediately our weak convergence result for equation (1), namely Theorem 1.2, to the following result on the approximation of(B,B²):

Theorem 2.3. Recall that the random variablesηⁱ_ksatisfy Hypothesis 1.1, and let X^"be defined by (3). For any" >0, let X^2," = (X^2,"_st (i,j))s,t≥0;i,j=1,...,d be the natural Lévy’s area associated to X^", given by

X^2,"_st (i,j) = Z t

s

(X_u^j,"−X_s^j,")d X_u^i,", (7) where the integral is understood in the usual Lebesgue-Stieltjes sense. Then, as"→0,

(X^",X^2,") −→^Law (B,B²), (8)

whereB²denotes the Lévy area defined in Proposition 2.1, and where the convergence in law holds in the spacesC₁^µ(R^d)× C₂^2µ(R^d×d), for anyµ <H.

The remainder of our work is thus devoted to the proof of Theorem 2.3.

As usual in the context of weak convergence of stochastic processes, we divide the proof into weak convergence for finite-dimensional distributions and a tightness type result. Furthermore, the tightness result in our case is easily deduced from the analogous result in[3]:

Proposition 2.4. The sequence(X^",X^2,")">0defined at Theorem 2.3 is tight inC₁^µ(R^d)×C₂^2µ(R^d×d). Proof. The proof follows exactly the steps of[3, Proposition 4.3], the only difference being that our Lemma 3.1 has to be applied here in order to get the equivalent of inequality (28) in[3]. Details are left to the reader.

With these preliminaries in hand, we can now turn to the finite dimensional distribution (f.d.d. in the sequel) convergence, which can be stated as:

Proposition 2.5. Under the assumption 1.1, let(X^",X^2,")be the approximation process defined by (3) and (7). Then

f.d.d.−lim

"→0(X^",X^2,") = (B,B²), (9)

wheref.d.d.−limstands for the convergence in law of the finite dimensional distributions. Otherwise stated, for any k≥1and any family{s_i,t_i;i≤k, 0≤s_i<t_i≤T}, we have

L −lim

"→0(X_t^"

1,X^2,"_s

1t₁, . . . ,X_t^"

k,X^2,"_s

kt_k) = (B_t1,B²_s

1t₁, . . . ,B_tk,B²_s

kt_k). (10)

Proof. The structure of the proof follows again closely the steps of[3, Proposition 5.1], except that other kind of estimates will be needed in order to handle the Donsker case.

To be more specific, it should be observed that the first series of simplifications in the proof of[3, Proposition 5.1]can be repeated here. They allow to pass from a convergence of double iterated integrals to the convergence of some Wiener type integrals with respect toX^". Namely, fori=1, 2 and 0≤u<t≤1, set

Yⁱ(u,t) = Z t

u

(Bⁱ_v−B_uⁱ)(v−u)^H−32d v,

and for 0≤u<t≤1 and(u₁, . . . ,u₆)in a neighborhood of 0 inR⁶, set also Z_u = u₁+u₂B_u²+u₃Y²(u,t) +u₄

Z t

u

(v−u)^H−12dW_v² +u₅

Zt

u

d w Zw

u

(w−v)^H−32 (w−u)^H−12−(v−u)¹² dW_v²

+u₆ Zt

u

d w Zu

0

(w−v)^H−32(w−u)^H−12dW_v².

Consider the analogous processesY^i,",Z^"defined by the same formulae, except that they are based on the approximationsθ^i," of white noise. We still need to recall a little more notation from[3]: for f ∈L²([0, 1])andt∈[0, 1], we set

Φ_"(f) =E

eⁱ

Rt

0f(u)θ^",1(u)du

, φ^"_f =

Z1

0

Z1

0

f²(x)f²(y)I_{|x−y|<"²_}d x d y, (11) and

Φ(f) =E eⁱ

Rt

0f(u)dW_u¹

=e¹²

Rt 0f²(u)du

.

Then it is shown in[3, Proposition 5.1]that one is reduced to prove that lim_"→0v_"^a=0, wherev_"^a is given by

v_"^a=E

Φ_"(Z^")e^iw

Rt

0θ^",2(u)du

−E

Φ(Z^")e^iw

Rt

0θ^",2(u)du

, for an arbitrary real parameterwin a neighborhood of 0. Furthermore, boundinge^iw

Rt

0θ^",2(u)du

triv- ially by 1 and conditioning, it is easily shown thatv_"^ais controlled by the differenceE[|Φ_"(Z^"))−

Φ(Z^")|], for which Lemma 3.3 provides the bound

|Φ"(Z^"))−Φ(Z^")|

≤E



4 r1

5w³(φ^"_Z^")¹²kZ^"kL²k³_ηexp(4w²k²_ηkZ^"k²_L2) +w²"^2αkZ^"k_αkZ^"kL²exp(w²kZ^"k²_L2)

+ 8

p5w⁴(φ^"_Z")¹²kZ^"k²_L2k_η²exp(4w²k²_ηkZ^"k²_L2) +1

2w⁴φ^"_Z"exp(w²kZ^"k²_L2)

,

for anyα∈(0, 1). In order to reach our aim, it is thus sufficient to check the following inequalities:

sup" E€

kZ^"k²_αŠ

≤M, lim

"→0E€

(φZ^"")²Š

=0 and forw<w₀, wherew₀is a small enough constant,

sup" E

e^w2kZ"k2L2

≤M.

However, these relations can be deduced, as (39), (40) and (41) in[3], from Lemma 3.1 (it should be noticed however that a one-parameter version of[2, Lemma 5.1]is needed for the adaptation of the latter result to our Donsker setting). The proof is thus finished once the lemmas below are proven.

3 Moments estimates in the Donsker setting

In order to deal with our technical estimates, let us first introduce a new notation: set ρ1 = (1−5^1/2)/2 andρ2= (1+5^1/2)/2. Then the moments of any integral of a deterministic kernel f with respect toθ^i," can be bounded as follows:

Lemma 3.1. Let m∈N, f ∈L²([0, 1]), i∈ {1, 2}and" >0. Recall that the random variableηis assumed to be almost surely bounded by a constant k_η. Then we have

E





 Z1

0

f(r)θ^i,"(r)d r

!2m





(12)

≤ (2m)!

2^mm!kfk^2m_L2 + (2m)!

5^1/2(m−2)!k_η^2m€

ρ^2m−1₂ −ρ₁^2m−1Š

(φ^"_f)¹²kfk^2m−2_L2 ,

and E





 Z1

0

f(r)θ^i,"(r)d r

!2m+1





≤ (2m+1)!

5^1/2(m−1)!k^2m+1_η €

ρ₂^2m−ρ₁^2mŠ

(φ^"_f)¹²kfk^2m−1_L2 , (13)

whereφ^"f is the quantity defined at (11).

Proof. We focus first on inequality (12) and divide this proof into several steps.

Step 1: Identification of some key iterated integrals.Notice that

E





 Z1

0

f(r)θ^i,"(r)d r

!2m





≤ Z

[0,1]^2m

|f(r₁)| · · · |f(r_2m)||E(θ^i,"(r₁)· · ·θ^i,"(r_2m))|d r₁· · ·d r_2m. Transforming the symmetric integral on[0, 1]^2minto an integral on the simplex, and using expres- sion (4) forθ^", we can write the latter expression as:

(2m)!

"^2m

n(")

X

k1, . . . ,k2m=1 k1≥ · · · ≥k2m

Zk₁"²

(k₁−1)"²

· · ·

Z k_2m"²

(k_2m−1)"²

|f(r₁)| · · · |f(r_2m)|

× |E(ηⁱ_k1· · ·ηⁱ_k2m)|I_{{r1≥r2≥···≥r2m}}d r₁· · ·d r_2m, (14) wheren(") = [_"¹2] +1 and where we understand that f(x) =0 wheneverx>1.

Let us study now the quantities E(ηⁱ_k1· · ·ηⁱ_kl). If there exists l such that k_l 6= k_j for all j 6= l then E(ηⁱ_k1· · ·ηⁱ_kl) = 0. On the other hand, when k_2l−1 = k_2l > k_2l+1 for any l, we clearly have E(ηⁱ_k1· · ·ηⁱ_kl) = 1. Finally, in the general case, for all l ∈ N, |E(ηⁱ_k1· · ·ηⁱ_kl)| ≤ k_η^l. Sepa- rating the cases in this way for |E(ηⁱk₁· · ·ηⁱk_2m)|, we end up with a decomposition of the form E[(R1

0 f(r)θ^i,"(r)d r)^2m] =T_m¹+T_m², where T_m¹=(2m)!

"^2m

n(")

X

k1, . . . ,km=1 k1>· · ·>k_m

Zk₁"²

(k₁−1)"²

Zk₁"²

(k₁−1)"²

· · ·

Zk_m"²

(k_m−1)"²

Zk_m"²

(k_m−1)"²

|f(r₁)| · · · |f(r_2m)| ×I_{{r1≥r2≥···≥r2m}}d r₁· · ·d r_2m, and where the termT_m²is defined by:

T_m²=(2m)!k_η^2m

"^2m

X

n1, . . . ,ns≥2;s∈ {1, ...,m−1} n1+· · ·+ns=2m

U_n1,...,n

s, (15)

with

U_n1,...,n

s=

n(")

X

k1, . . . ,ks=1 k1>· · ·>ks

Z

D_k1···ks

|f(r₁)| · · · |f(r_2m)|I_{{r1≥r2≥···≥r2m}}d r₁· · ·d r_2m, (16)

and where we have setD_k

1···ks=Qs

j=1[(k_j−1)"²,k_j"²]^nj.

Let us observe at this point that we have split our sum into T_m¹ and T_m² because T_m¹ represents the dominant contribution to our moment estimate. This is simply due to the fact that T_m¹ is obtained by assuming some pairwise equalities among the random variablesηⁱ_k, whileT_m²is based

on a higher number of constraints. In any case, both expressions will be analyzed through the introduction of some iterated integrals of the form

K_ν(k;v,w) = 1

"^ν Z

[(k−1)"²,k"²]^ν

Yν

j=1

|f(r_j)|I_{w≥r1>···>r_ν≥v}d r₁· · ·d r_ν,

defined forν,k≥1 and 0≤v<w≤1.

Step 2: Analysis of the integrals K_ν. Those iterated integrals are treated in a slightly different way according to the parity ofν. Indeed, forν=2n, thanks to the elementary inequality 2a b≤a²+b², we obtain a bound of the form:

n(")

X

k=1

K_2n(k;v,w) (17)

≤

n(")

X

k=1

1

"²ⁿ Z

[(k−1)"²,k"²]²ⁿ n

Y

i=1

‚f²(x_2i−1) +f²(x_2i) 2

Œ I_{w≥x

1≥···≥x_2n≥v}d x₁· · ·d x_2n

≤

n(")

X

k=1

1

"²ⁿ Z

[(k−1)"²,k"²]²ⁿ

f²(x₁)· · ·f²(x_n)I_{{w≥x1≥···≥xn≥v}}d x₁· · ·d x_2n

=

n(")

X

k=1

Z

[(k−1)"²,k"²]ⁿ

f²(x₁)· · ·f²(x_n)I_{w≥x

1≥···≥xn≥v}d x₁· · ·d x_n

≤ Z

[0,1]ⁿ

f²(x₁)· · ·f²(x_n)I_{x1−xn<"²_}I_{{w≥x1≥···≥xn≥v}}d x₁· · ·d x_n.

The caseν=2n+1 can be treated along the same lines, except for the fact that one has to cope with some expressions of the form

n(")

X

k=1

K₃(k;v,w) ≤

n(")

X

k=1

1

"

Z

[(k−1)"²,k"²]²

|f(x₁)|f²(x₂)I_{w≥x

1≥x2≥v}d x₁d x₂

≤ 1

"

Z

[0,1]²

|f(x₁)|f²(x₂)I_{x1−x2<"²_}I_{{w≥x1≥x2≥v}}d x₁d x₂. (18)

Combining (18) and (17) we can state the following general formula: let ν ≥ 1, and define a couple(ν^∗,ν)ˆ as: (i)ν^∗=ν/2,νˆ=0 ifνis even, (ii)ν^∗= (ν+1)/2,νˆ=1 ifνis odd. With this notation in hand, we have:

n(")

X

k=1

K_ν(k;v,w)≤ 1

"^νˆ Z

[0,1]^ν∗

|f(x₁)|^2−ˆνf²(x₂)· · ·f²(x_ν∗)I_{x

1−x_ν∗<"²}

×I_{w≥x

1≥···≥x_ν∗≥v}d x₁· · ·d x_ν∗. (19) Step 3: Bound on T_m¹. It is readily checked that T_m¹ can be decomposed into blocks of the form

K₂(k;w,v), for which one can apply (19). This yields T_m¹≤(2m)!

2^m

n(")

X

k₁,...,k_m=1

Z k₁"²

(k1−1)"²

· · ·

Z k_m"²

(km−1)"²

f²(r₁)· · ·f²(r_m)I_{r

1≥r2≥···≥rm}d r₁· · ·d r_m

≤(2m)! 2^mm!kfk^2m_L2. Step 4: Bound on U_n

1,...,n_s. Recall that U_n

1,...,n_s is defined by (16). We introduce now a recursion procedure in order to control this term. Namely, integrating with respect to the lastn_svariables, one obtains that

1

"^2mU_n

1,...,n_s= 1

"^2m−ns

n(")

X

k1, . . . ,ks−1=1 k1>· · ·>k_s−1

Z

D_{k1···ks−1} 2m−ns

Y

l=1

|f(r_l)|I_{r

1≥r2≥···≥r_2m−ns}

×K_ns(k_s; 0;r_2m−ns)d r₁· · ·d r_2m−ns Plugging our bound (19) onK_n

sinto this expression, we get 1

"^2mU_n

1,...,n_s≤ 1

"^2m−ns+ˆns

n(")

X

k₁, . . . ,k_s−1=1 k₁>· · ·>k_s−1

Z

D_k

1···ks−1

Z

[0,1]^n∗s 2m−n_s

Y

l=1

|f(r_l)| |f(y₁)|^2−ˆns

×

n^∗_s

Y

j=2

f²(y_j)I_{y1−y

n∗s<"²}I_{{r1≥r2≥···≥r}

2m−ns≥y₁≥···≥y_n∗s}d r₁· · ·d r_2m−nsd y₁· · ·d y_n∗ s. We can now proceed, and integrate with respect to the variablesr_lfor 2m−n_s−n_s−1<l≤2m−n_s. In the end, sinceP

n_s=2m, the remaining singularity in"is of the formQ

"^−ˆns. However, each of the singularity"^−ˆns comes with an integral that compensates the singularity"^−ˆns (recall that ˆ

n_s≤1). Hence, iterating the integrations with respect to the variablesr, we end up with a bound of the form

1

"^2mU_n1,...,ns≤ 1

(m−2)!kfk²⁽_L2^m−2)φ^"_f ≤ 1

(m−2)!kfk^2m−2_L2 (φ^"_f)¹². (20)

Notice that when one of the termsn₁,n₂, . . . ,n_sof the decomposition of 2mis even the last bound is easily obtained. On the other hand, we will illustrate with an example how the bound can be obtained when all the terms are odd. Let us consider the casem=n₁=n₂=3. Following our procedure we obtain that

1

"⁶U_3,3 ≤ 1

"² Z

[0,1]⁴

f²(y₂)· |f(y₁)| ·f²(y₄)· |f(y₃)|

×I_{y1−y2<"²_}I_{y3−y4<"²_}I_{{y3≥y4≥y1≥y2}}d y₁· · ·d y₄

≤ 1

"² Z

[0,1]⁴

f²(y₂)f²(y₄)

‚f²(y₁) +f²(y₃) 2

Œ

×I_{y

1−y2<"²}I_{y

3−y4<"²}I_{y

3≥y4≥y1≥y2}d y₁· · ·d y₄

≤ φ^"_f 1

"² Z

[0,1]²

f²(y₄)I_{{0≤y3−y4<"}²_}d y₃d y₄

= kfk²_L2φ^"_f.