Regularity of Wiener functionals under a Hörmander type condition of order one

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Regularity of Wiener functionals under a Hörmander type condition of order one

Vlad Bally, Lucia Caramellino

To cite this version:

Vlad Bally, Lucia Caramellino. Regularity of Wiener functionals under a Hörmander type condition of order one. Annals of Probability, Institute of Mathematical Statistics, 2017, 45 (3), pp.1488-1511.

�hal-01413556�

arXiv:1307.3942v3 [math.PR] 11 Sep 2014

Regularity of Wiener functionals under a H¨ ormander type condition of order one

Vlad Bally^∗ Lucia Caramellino^†

Abstract. We study the local existence and regularity of the density of the law of a functional on the Wiener space which satisfies a criterion that generalizes the H¨ormander condition of order one (that is, involving the first order Lie brackets) for diffusion processes.

Keywords: Malliavin calculus; local integration by parts formulas; total variation distance;

variance of the Brownian path.

2010 MSC: 60H07, 60H30.

1 Introduction

H¨ormander’s theorem gives sufficient non degeneracy assumptions under which the law of a diffusion process is absolutely continuous with respect to the Lebesgue measure and has a smooth density. This condition involves the coefficients of the diffusion process as well as the Lie brackets up to an arbitrary order. The aim of this paper is to give a partial generalization of this result to general functionals on the Wiener space. We give in this framework a condition corresponding to the first order H¨ormander condition - we mean the condition which says that the coefficients and the first Lie brackets span the space. Roughly speaking our regularity criterion is as follows. Let F be a functional on the Wiener space associated to a Brownian motionW = (W¹, ..., W^d). We denote byDⁱ the Malliavin derivative with respect toWⁱ and, for someT >0, we define

λ(T) = inf

|ξ|=1

X^d

i=1

hDⁱ_TF, ξi²+ Xd

i,j=1

hDⁱ_TD^j_TF −D^j_TD_TⁱF, ξi²

(1.1) We fixx and we suppose that there existr, λ >0 such that

1_{{|F−x|≤r}}(λ(T)−λ)≥0 a.s. (1.2) Notice that, since s7→D_sF is defined as an element of L²([0, T]), the quantity D_TF in (1.1) makes no sense. So, we will replace it by ¹_δRT

T−δE_T,δ(D_sF)ds for small values of δ, where E_T,δ denotes a suitable conditional expectation (see (2.3) for details). Then, we actually replace (1.2) with an asymptotic variant (see next Remark 2.2).

∗Universit´e Paris-Est, LAMA (UMR CNRS, UPEMLV, UPEC), INRIA, F-77454 Marne-la-Vall´ee, France.

Email: bally@univ-mlv.fr.

†Dipartimento di Matematica, Universit`a di Roma - Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Roma, Italy. Email: caramell@mat.uniroma2.it

So, we assume that F is five times differentiable in Malliavin sense (actually in a slightly stronger sense) and that the above non degeneracy condition holds for some T >0. Then we prove that the restriction of the law ofF toB_r/2(x) is absolutely continuous and has a smooth density.

The analysis of the Malliavin covariance matrix under the non degeneracy hypothesis (1.2) is based on an estimate concerning the variance of the Brownian path. This is done by using its Laplace transform, which has been studied by Donati-Martin and Yor [5]. We employ also another important argument, which is the regularity criterion for the law of a random variable given in [2]: it allows one to use integration by parts formulas in an “asymptotic way”.

The main result is Theorem 2.1, and Section 2 is devoted to its proof, for which we use results on the variance of the Brownian path which are postponed to Appendix A. In Section 3 we illustrate the result with an example from diffusion processes with coefficients which may depend on the path of the process.

At our knowledge there are not many results concerning general vectors on the Wiener space - except of course the celebrated criterion given by Malliavin and the Bouleau Hirsh criterion for the absolute continuity. Another criterion proved by Kusuoka in [6] and further generalized by Nourdin and Poly [11] and Nualart, Nourdin and Poly [12] concerns vectors living in a finite number of chaoses. All these criterions suppose that the determinant of the Malliavin covariance matrix is non null in a more or less strong sense - but give no hint about the possible analysis of this condition. This remains to be checked using ad hoc methods in each particular example. So the main progress in our paper is to give a rather general condition under which the above mentioned determinant behaves well.

Acknowledgments. We are grateful to E. Pardoux who made a remark which allowed us to improve a previous version of our result.

2 Existence and smoothness of the local density

Let us recall some notations from Malliavin calculus (we refer to Nualart [10] or Ikeda and Watanabe [7]). We work on a probability space (Ω,F, P) with a d dimensional Brownian motionW = (W¹, ..., W^d) and we denote byFtthe standard filtration associated to W.We fix a time-horizonT₀ >0 and we denote byD^k,pthe space of the functionals on the Wiener space which arektimes differentiable inL^p in Malliavin sense on the time interval [0, T0] and we put D^k,∞=∩p≥1D^k,p.For a multi indexα= (α₁, ..., α_k)∈ {1, ..., d}^k and a functionalF ∈D^k,p we denote D^αF = (D_s^α_1,...,skF)_{s1,...,sk∈[0,T0]} with D^α_s1,...,skF = D^α_sk^k...D^α_s1¹F. Moreover, for |α| =k we define the norms

|D^αF|^p_Lp[0,T0]^k :=

Z

[0,T0]^k

D_s^α_1,...,s

kF^pds1, ..., ds_k and (2.1) kFk_k,p=kFk_p+

Xk

r=1

X

|α|=r

E(|D^αF|^p_L2[0,T0]^r)^1/p.

IfF = (F¹, ..., Fⁿ), we set

|D^αF|^p_L^p_[0,T0]^k = Xn

i=1

D^αF^ip

L^p[0,T0]^k and kFkk,p= Xn

i=1

Fⁱ

k,p.

Moreover we will use the following seminorms:

|||F|||k,p,q = Xk

r=3

X

|α|=r

E(|D^αF|^p_L^q_[0,T0]^r)^1/p

= Xk

r=3

X

|α|=r

E Z

[0,T0]^r

D^α_s1,...,s

kF^qds₁...ds_rp/q1/p

.

Notice thatk| · |kk,p,q does not take into accountkFkp and the norm of the first two derivatives.

Moreover, for q = 2 we find out the usual norms but if q > 2 the control given by |||F|||_k,q,p (on the derivatives of order larger or equal to three) is stronger than the one given by||F||k,p. We define the spaces

D^k,p={F :kFkk,p <∞}, D^k,p,q =D^k,p∩ {F :k|F|kk,p,q <∞}. Clearly D^k,p,q ⊂D^k,p forq >2 and for q= 2 we have equality. We also denote

D^k,∞=∩p≥1D^k,p, D^k,∞,q =∩p≥1D^k,p,q, D^k,∞,∞=∩p≥1∩q≥2D^k,p,q (2.2) Fors < t we denote

Fs^t=Fs∨σ(W_u−W_t, u≥t) =σ(W_v, v ≤s)∨σ(W_u−W_t, u≥t).

Now, for a fixed instantT ∈(0, T0], we denote byE_T,δthe conditional expectation with respect to FT^T−δ that is

E_T,δ(Θ) =E(Θ| FT^T−δ). (2.3)

We will use the following slight extension of the Clark-Ocone formula: for F ∈ D^1,2 and for 0≤δ < T one has

F =E_T,δ(F) + Xd

i=1

Z T

T−δ

E_T,T−s(Dⁱ_sF)dW_sⁱ. (2.4) (2.4) is immediate for simple functionals, and then can be straightforwardly generalized to functionals inD^1,2.

Forδ ∈(0, T), we consider a family of random vectors

a(T, δ) = (a_i(T, δ), a_k,j(T, δ))i,k,j=1,...,d

and we assume that a(T, δ) isFT^T−δ measurable. We denote [a]_i,j(T, δ) =a_i,j(T, δ)−a_j,i(T, δ),

a(T, δ) =X^d

i=1

|a_i(T, δ)|²+ Xd

i,j=1

|a_i,j(T, δ)|²1/2

,

λ(T, δ) = inf

|ξ|=1

X^d

i=1

ha_i(T, δ), ξi²+ Xd

i,j=1

h[a]_i,j(T, δ), ξi² .

(2.5)

Forp≥1, α >0,0< δ < T we define ε_α,p,T,δ(a, F) :=

Xd

i=1

E1

δ Z T

T−δ

E_T,δ(Dⁱ_sF)−a_i(T, δ) δ^12+α

2

dsp1/2p

+

+ Xd

i,j=1

1 δ²

Z T

T−δ

Z s1

T−δ

E

E_T,δ(D^js2D_sⁱ₁F)−a_i,j(T, δ) δ^α/2

2p

ds₁ds₂1/2p

. (2.6)

Our main result is the following.

Theorem 2.1 Let F = (F¹, .., Fⁿ) be FT0-measurable with Fⁱ ∈ D^2,∞, i = 1, ..., n. We fix y∈Rⁿ andr >0 and we suppose that there exists α, λ∗ >0, γ∈[0,¹₂),a time T ∈(0, T0] and a family a(T, δ) = (a_i(T, δ), a_i,j(T, δ))i,j=1,...,d,of F_T^T_−δ measurable vectors such that for every p≥1

i) lim sup

δ→0

ε_α,p,T,δ(a, F)<∞, ii) lim sup

δ→0

δ^pγE(a^p(T, δ))<∞, iii) lim sup

δ→0

δ^−pP({|F−y| ≤r} ∩ {λ(T, δ)< λ∗})<∞.

(2.7)

Then the following statements hold.

A. Suppose that Fⁱ ∈ ∪p>6D^5,∞,p, i= 1, ..., n. Then the law of F on B_r/2(y) :={x:|x−y|<

r/2}is absolutely continuous with respect to the Lebesgue measure. We denote byp_F the density of the law.

B. Suppose that for some k≥5 one has Fⁱ ∈D^k,∞,∞, i= 1, ..., n. Then p_F ∈ ∩p≥1W^k−5,p(B_r/2(y)).

Remark 2.2 Morally, Dⁱ_TF ∼ ¹_δRT

T−δE_T,δ(D_sⁱF)ds. Then, condition i) in (2.7) says that we may replaceD_TⁱF by a_i(T, δ), and we have a precise control of the error. The same forD_TⁱD^j_tF, which is replaced bya_i,j(T, δ). Then,iii)in (2.7) gives the asymptotic non-degeneracy condition in terms of λ(T, δ), which is associated to a(T, δ).

Remark 2.3 Notice that we may ask the non-degeneracy conditioniii)in (2.7) to hold in any intermediary time T ∈(0, T₀]and not only for T =T₀ (we thank to E. Pardoux for a remark in this sense).

The proof is postponed to Section 2.4. We first need to state some preliminary results.

2.1 A short discussion on the proof of Theorem 2.1

Let us give the main ideas and the strategy we are going to use to prove Theorem 2.1.

We will look to the law of F under P_U where U is a localization random variable for the set {|F −y| ≤ r}. We want to prove that this law is absolutely continuous with respect to the Lebesgue measure - this implies that the law of F restricted to {|F−y| ≤ r} is absolutely continuous (and this is our aim). In order to do it we proceed as follows: for each δ > 0

we construct some localization random variables U_δ in such a way that on the set {U_δ 6= 0} the random variable F has nice properties - this means that we may control the Malliavin derivatives and the Malliavin covariance matrix of F on the set {U_δ 6= 0}. This allows us to build integration by parts formulas forF underP_Uδ.TheL^p norms of the weights which appear in these integration by parts formulas blow up asδ →0 but we have a sufficiently precise control of the rate of the blow up. On the other hand we will estimate the total variation distance between the law of F under P_U and under P_Uδ. We prove that this distance goes to zero as δ → 0 and we obtain sufficiently precise estimates of the rate of convergence. Then we use Theorem 2.13 in [2], that we recall here in next Theorem 2.10, which guarantees that if one may achieve a good equilibrium between the rate of the blow up and the rate of convergence to zero, then one obtains a density for the limit law.

It worth to stress that the strategy employed here is slightly different from the usual one. In fact, in next (2.8) we decomposeF asF =E_T,δ(F) +Z_δ(a) +R_δand one would expect that we approximate F by E_T,δ(F) +Z_δ(a). But we do not proceed in this way. We keep all the time the same random variableF (which includesR_δ) but we change the probability measure under which we work in order to have a good localization: we replaceP_U by P_Uδ.The decomposition F =E_T,δ(F) +Z_δ(a) +R_δ is not used in order to produce the approximation E_T,δ(F) +Z_δ(a) but just to analyze the properties for F itself under different localizations given in P_Uδ. As we will see soon, such a decomposition appears as a Taylor expansion of order one in which Z_δ(a) represents the principal term and R_δ is a reminder in the sense that it is small on the set{U_δ6= 0}.

2.2 Preliminary results

LetF ∈D^4,2. Using twice Clark Ocone formula (2.4), we obtain

F−E_T,δ(F) =Z_δ(a) +R_δ(F) (2.8) with

Z_δ(a) = Xd

i=1

a_i(T, δ)(W_Tⁱ −W_Tⁱ_−δ) + Xd

i,j=1

a_i,j(T, δ) Z T

T−δ

(W_sⁱ−W_Tⁱ_−δ)dW_s^j (2.9) and R_δ(F) =R^′_δ(F) +R_δ^′′(F) with

R^′_δ(F) = Xd

i=1

Z T

T−δ

(E_T,δ(D_sⁱF)−a_i(T, δ))dW_sⁱ

+ Xd

i,j=1

Z T

T−δ

Z s1

T−δ

(E_T,δ(D_s^j₂Dⁱ_s1F)−a_i,j(T, δ))dW_s^j₂dW_sⁱ₁

R_δ^′′(F) = Xd

i,j,k=1

Z T

T−δ

Z s1

T−δ

Z s2

T−δ

E_T,T−s3(D^k_s3D_s^j₂Dⁱ_s1F)dW_s^k₃dW_s^j₂dW_sⁱ₁

(2.10)

Since T and δ are fixed we will use in the following shorter notation ai=ai(T, δ), ai,j =ai,j(T, δ), a=a(T, δ).

We will use the Malliavin calculus restricted toWs, s∈[T−δ, T].Straightforward computations give

D^j_sZ_δ(a) =a_j+X

i6=j

[a]_i,j(W_sⁱ−W_Tⁱ_−δ) +r_j, withr_j = Xd

i=1

a_i,j(W_Tⁱ −W_Tⁱ_−δ). (2.11) We denote

q₁(W) =|W_T −W_T−δ|, q₂(W) = 1 δ

Z T

T−δ|W_s−W_T−δ|²ds, G_δ=

Z T

T−δ|D_sR_δ|²ds

(2.12)

and we define Λ_T,δ=n

q₁(W)≤ 1 8a

√λ∗

d

o∩n

q₂(W)≤1o

∩n

G_δ≤ λ∗

34δ²o

∩n

λ(T, δ) ≥λ∗

o (2.13)

We set σ_F,T,δ as the Malliavin covariance matrix of F associated to the Malliavin derivatives restricted to W_s, s∈[T−δ, T] that is

σ_F^i,j =σ_F,T,δ^i,j = Z T

T−δ

D_sFⁱ, D_sF^j

ds, i, j= 1, . . . , n. (2.14) The main step of the proof is the following estimate. It is based on an analysis of the variance of the Brownian path, which is done in Appendix A.

Lemma 2.4 Let F = (F¹, ..., Fⁿ) with Fⁱ ∈D^4,2. Let 0≤δ < T be fixed and E_T,δ be defined in (2.3). Then for everyp≥1

E_T,δ(1_ΛT,δ(detσ_F,T,δ)^−p)≤ C_n,p

λ^pn∗ δ^2pn (2.15)

with

Cn,p= 2Γ(p) Z

Rⁿ|ξ|^n(2p−1)e^−341|ξ|2dξ.

Proof. By using Lemma 7-29, pg 92 in [4], for every n×n dimensional and non negative defined matrixσ one has

(detσ)^−p ≤Γ(p) Z

|ξ|^n(2p−1)e^−hσξ,ξidξ, so that

E_T,δ((detσ_F)^−p1_ΛT,δ)≤Γ(p) Z

|ξ|^n(2p−1)E_T,δ(1_ΛT,δe^{−hσFξ,ξi})dξ.

Since Λ_T,δ⊂ {G_δ ≤ ^λ₃₄^∗δ²}we have hσFξ, ξi ≥ 1

2

σ_Zδ(a)ξ, ξ

− hG_δξ, ξi ≥ 1 2

σ_Zδ(a)ξ, ξ

−G_δ|ξ|²

≥ 1 2

σ_Zδ(a)ξ, ξ

−λ_∗ 34δ²|ξ|²

so that

E_T,δ((detσ_F)^−p1_ΛT,δ)≤Γ(p) Z

|ξ|^n(2p−1)e^{λ∗34δ2|ξ|2}E_T,δ(1_ΛT,δe⁻h^σZδ(a)ξ,ξi)dξ.

We fixξ ∈Rⁿ and we choose j=j(ξ) such that ha_j, ξi²+X

i6=j

h[a]_i,j, ξi² ≥ λ∗

d |ξ|².

This is possible because we are on the set Λ_T,δ⊂ {λ(T, δ) ≥λ∗}.Then by (2.11) σ_Zδ(a)ξ, ξ

= Z T

T−δ

D_s^jZ_δ(a), ξ2

ds

= Z T

T−δ

ha_j, ξi+hr_j, ξi+X

i6=j

h[a]_i,j, ξi(W_sⁱ−W_Tⁱ_−δ)2

ds.

We define

β_j²(ξ) =X

i6=j

h[a]_i,j, ξi² and for β_j²(ξ)>0,

bs(j, ξ) = 1 β_j(ξ)

X

i6=j

h[a]i,j, ξi(W_Tⁱ_−δ+s−W_Tⁱ_−δ).

Notice that b(j, ξ) is a Brownian motion under P_T,δ. We also set b_s(j, ξ) = 0 in the case β_j²(ξ) = 0. Then the previous equality reads

σ_Zδ(a)ξ, ξ

= Z δ

0

(ha_j, ξi+hr_j, ξi+β_j(ξ)b_s(j, ξ))²ds.

We use now Lemma A.1 in Appendix A withα=ha_j, ξi, β =β_j(ξ), r=hr_j, ξiandb_s=b_s(j, ξ).

We have to check that the assumptions there are verified. Using Cauchy-Schwarz inequality we obtain

1 δ

Z δ

0

b_s(j, ξ)ds ≤1 δ

Z δ

0 |b_s(ξ)|²ds1/2

≤1 δ

Z δ

0 |W_T−δ+s−W_T−δ|²ds1/2

=p

q₂(W)≤1.

Moreover, sinceα²+β² ≥ ^λ_d^∗|ξ|² we have

|r|²≤ |r_j|²|ξ|² ≤da²q²₁(W)|ξ|²≤ 1 64

λ_∗

d |ξ|²≤ 1

64(α²+β²).

So the hypothesis are verified: by using (A.3) we obtain E_T,δ(1_ΛT,δe⁻h^σZδ(a)ξ,ξi)≤2e^−δ

2

17(|α|²+|β|²)≤2e^−δ

2λ∗ 17d |ξ|².

We come back and we obtain

E_T,δ((detσ_F)^−p1_ΛT,δ)≤2Γ(p) Z

|ξ|^n(2p−1)e^{34dλ∗δ2|ξ|2}e^−δ

2λ∗ 17d |ξ|²dξ

= 2Γ(p) Z

|ξ|^n(2p−1)e^−δ

2λ∗

34 |ξ|²dξ

= C_n,p λ^pn∗ δ^2pn

where the last equality easily follows by a change of variable . We also need the following estimate.

Lemma 2.5 Suppose that (2.7)i) holds and let G_δ be defined as in (2.12).

A. If Fⁱ ∈ ∪p>6D^4,∞,p, i= 1, ..., n, there existsε >0 such that lim sup

δ→0

δ^−εP(G_δ≥δ²)<∞. (2.16) B. If Fⁱ∈D^4,∞,∞, i= 1, ..., n then (2.16) holds for every ε >0.

Proof. A. Let F ∈(D^4,∞,p)ⁿ for somep > 6. We recall thatR^′_δ and R^′′_δ are defined in (2.10).

We writeR^′_δ(F) =Pd

i=1r_δⁱ +Pd

i,j=1r^i,j_δ and R_δ^′′=Pd

i,j,k=1r^i,j,k_δ , with r_δⁱ =

Z T

T−δ

(E_T,δ(Dⁱ_sF)−a_i(T, δ))dW_sⁱ, r_δ^i,j =

Z T

T−δ

Z s1

T−δ

(E_T,δ(D_s^j₂Dⁱ_s1F)−a_i,j(T, δ))dW_s^j₂dW_sⁱ₁, r_δ^i,j,k =

Z T

T−δ

Z s1

T−δ

Z s2

T−δ

E_T,T−s3(D_s^k₃D^j_s2D_sⁱ₁F)dW_s^k₃dW_s^j₂dW_sⁱ₁. Step 1. We estimate Gⁱ_δ = RT

T−δ

Dⁱ_srⁱ_δ²ds. For s∈ [T −δ, T] we have D_sⁱr_δⁱ =E_T,δ(Dⁱ_sF)− a_i(T, δ) so

Gⁱ_δ = Z T

T−δ

E_T,δ(Dⁱ_sF)−a_i(T, δ)²ds.

It follows that 1

δ^εP(Gⁱ_δ≥δ²)≤ 1

δ^εδ^−2pGⁱ_δ^p_p= 1

δ^εEδ⁻¹ Z T

T−δ

E_T,δ(Dⁱ_sF)−a_i(T, δ) δ^1/2

2

ds^p

≤ 1

δ^ε ×δ^2αpε^2p_α,p,T,δ(a, F)

and consequently, by our hypothesis (2.7)i),this term satisfies (2.16) for everyε >0 (it suffices to takep sufficiently large).

Step 2. We estimate G^i,j_δ =Pd ℓ=1

RT T−δ

D_s^ℓr_δ^i,j2ds. We have D^ℓ_sr^i,j_δ =1_i=ℓ

Z s

T−δ

(E_T,δ(D^j_s2D_sⁱF)−a_i,j(T, δ))dW_s^j₂+ + 1_j=ℓ

Z T

s

(E_T,δ(D^p_sDⁱ_s1F)−ai,p(T, δ))dW_sⁱ₁ =: 1_i=ℓu^i,j_s + 1_j=ℓv^i,j_s .

We have E

Z T

T−δ

u^i,j_s ²ds^p

≤δ^p−1 Z T

T−δ

E u^i,j_s ^2p ds

≤Cδ^p−1 Z T

T−δ

E Z s

T−δ

(E_T,δ(D^j_s2D_sⁱF)−a_i,j(T, δ))²ds₂^p ds

≤Cδ^2p−2 Z T

T−δ

Z s

T−δ

E E_T,δ(D^j_s2D_sⁱF)−a_i,j(T, δ)^2p ds₂ds

=Cδ^2p+αp 1 δ²

Z T

T−δ

Z T

T−δ

E

E_T,δ(D_s^j₂D_sⁱ₁F)−a_i,j(T, δ) δ^α/2

2p ds₁ds₂

≤Cδ^2p+αpε^2p_α,p,T,δ(a, F).

Using Chebyshev inequality we obtain P Z ^T

T−δ

u^i,j_s ²ds≥δ²

≤Cδ^−2pδ^2p+αpε^2p_α,p,T,δ(a, F) =Cδ^αpε^2p_α,p,T,δ(a, F).

which by (2.7)i),satisfies (2.16) for every ε >0. Forvs^i,j the argument is the same.

Step 3. We estimate G^i,j,k_δ =Pd ℓ=1

RT

T−δ|D_s^ℓr_δ^i,j,k|²ds. We have D_s^ℓr^i,j,k_δ =1_i=ℓ

Z s

T−δ

Z s2

T−δ

E_T,T−s3(D_s^k₃D^j_s2D_sⁱF)dW_s^k₃dW_s^j₂+ + 1_j=ℓ

Z T

T−δ

Z s2

T−δ

1_s<s1E_T,T−s3(D^k_s3D_s^jDⁱ_s1F)dW_s^k₃dW_sⁱ₁+ + 1_k=ℓ

Z T

T−δ

Z s1

T−δ

1_s<s2E_T,T−s(D^k_sD_s^j₂Dⁱ_s1F)dW_s^j₂dW_sⁱ₁+ +

Z T

T−δ

Z s1

T−δ

Z s2

T−δ

1_s<s3D^ℓ_sE_T,T−s3(D_s^k₃D_s^j₂Dⁱ_s1F)dW_s^k₃dW_s^j₂dW_sⁱ₁

=:1_i=ℓu^i,j,k_s + 1_j=ℓv^i,j,k_s + 1_k=ℓw_s^i,j,k+z^i,j,k,ℓ_s .

By using H¨older and Burkholder inequality as in step 1, one obtains E

Z T

T−δ

u^i,j,k_s ²ds^p

≤δ^3p−3 Z T

T−δ

Z T

T−δ

Z T

T−δ

E |D^k_s3D_s^j₂Dⁱ_sF|^2p

ds₃ds₂ds

≤δ^3p−3|||F|||^2p_3,2p,2p. An identical bound holds for E(|RT

T−δ|v^i,j,k_s |²ds|^p) and E(|RT

T−δ|w^i,j,k_s |²ds|^p). As for z^i,j,k,ℓ, one more further integral appears, so we get E(|RT

T−δ|z^i,j,k,ℓ_s |²ds|^p) ≤ δ^4p−4|||F|||^2p4,2p,2p. By summarizing, we get

E Z T

T−δ

D_sR_δ^′′2ds^p

≤δ^3p−3|||F|||^2p4,2p,2p

so that for every p >1 P Z ^T

T−δ

D_s^ℓR^′′_δ²ds≥δ²

≤Cδ^−2pδ^3p−3|||F|||^2p_4,2p,2p =Cδ^p−3|||F|||^2p_4,2p,2p.

Suppose first that Fⁱ ∈ ∪p>6D^4,∞,p. Then we may find p >3 such that |||F|||4,2p,2p <∞ and consequently the above quantity is upper bounded byCδ^p−3.This means that (2.16) holds for ε < p−3.IfFⁱ ∈D^4,∞,∞then we may take parbitrary large and so we obtain (2.16) for every ε >0.

We will also need the following property forG_δ. Lemma 2.6 If F ∈D^k+1,2p then

kG_δkk,p ≤C kFk²k+1,2p+δka(T, δ)k²4p

, where C denotes a constant depending on k, p, d only.

Proof. ForG∈(D^k,p)ⁿ, we set|D^(k)G|=Pk ℓ=0

P

|γ|=ℓ|D^γG|², where, for |γ|=ℓ,

|D^γG|² = Z

[0,T]^ℓ|D_s^γ_1...sℓG|²ds₁· · ·ds_ℓ,

that is|D^γG|is the one given in (2.2) withp= 2. Here, the case|γ|= 0, that isγ =∅, reduces to the original random variable: D^∅G=Gand |D⁽⁰⁾G|=|G|.

In the following, we letCdenote a positive constant, independent ofδand the random variables we are going to write. And we let C vary from line to line.

We takeG_δ =RT

t−δ|D_sR_δ|²ds and we first prove the following (deterministic) estimate: there exists a constant C depending onk and dsuch that

|D^(k)G_δ| ≤C|D^(k+1)R_δ|². (2.17) For k = 0, this is trivial. Consider k = 1. One has Dⁱ_uG_δ = Pd

ℓ=1

RT

t−δ2D^ℓ_sR_δD_uⁱD^ℓ_sR_δds, so that, by using the CauchySchwarz inequality, we get

|DG_δ|² ≤4 Xd

i,ℓ=1

Z _T

T−δ

Z _T

T−δ

2D_s^ℓR_δDⁱ_uD_s^ℓR_δds²du

≤4 Xd

i,ℓ=1

Z T

T−δ

du Z T

T−δ

2|D_s^ℓR_δ|²ds Z T

T−δ

2|D_uⁱD_s^ℓR_δ|²ds

≤C|D⁽¹⁾R_δ|²|D⁽²⁾R_δ|²≤C|D⁽²⁾R_δ|⁴

and (2.17) holds for k = 1. For k ≥ 2, we use the following straightforward formula: if α denotes a multi-index of lengthk, then

D^αG_δ= Xd

ℓ=1

Z T

T−δ

2D_s^ℓR_δD^αR_δ+ X

β∈Pα

D^βD_s^ℓR_δD^α\βD_s^ℓR_δ ds,

wherePα is the set of the non empty multi indecesβ which are a subset of αand α\β stands for the multi index of length|α| − |β| given by eliminating from α the entries of β. By using the above formula and the CauchySchwarz inequality, one easily gets

Z

[T−δ,T]^k|D_s^α_1,...,skG_δ|²ds1· · ·ds_k≤C

|D⁽¹⁾R_δ|²|D^(k)R_δ|²+ Xk

r=1

|D^(r+1)R_δ|²|D^(k−r+1)R_δ|²

≤C|D^(k+1)R_δ|⁴

and (2.17) follows. Passing to expectation in (2.17), it follows that kG_δkk,p ≤CkR_δk²k+1,2p

and by recalling thatR_δ=F −E_T,δ(F)−Z_δ(a), we obtain kG_δkk,p≤C kFk²k+1,2p+kZ_δ(a)k²k+1,2p

. From (2.9), by using H¨older’s inequality we get

kZ_δ(a)kk+1,2p≤ Xd

i=1

ka_i(T, δ)k4pkW_Tⁱ −W_Tⁱ_−δkk+1,4p

+ Xd

i,j=1

ka_i,j(T, δ)k4p

Z T

T−δ

(W_sⁱ−W_Tⁱ_−δ)dW_s^j_k+1,4p

≤Cka(T, δ)k4pδ^1/2+Cka(T, δ)k4pδ

≤Cδ^1/2ka(T, δ)k4p, and the statement follows.

Remark 2.7 If hypothesis (2.7)ii) holds then lim sup_δ→0δka(T, δ)k²4p = 0 because in this case one takes γ <1/2, so that forF ∈(D^k+1,2p)ⁿ one has

sup

δ>0kG_δkk,p<∞. 2.3 Localization

We will use a localization argument from [2] that we recall here.We consider a random variable U taking values in [0,1] and we denote

dP_U =U dP. (2.18)

This is a non negative measure (but generally not a probability measure - one must divide with E(U) to get a probability measure). We denote

kFkU,p :=E_U(|F|^p)^1/p=E(|F|^pU)^1/p and (2.19) kFk_U,k,p:=kFk_U,p+

Xk

r=1

X

|α|=r

E_U(|D^αF|^p_L2[0,T0]^r)^1/p. Clearly kFkU,k,p≤ kFkk,p.For a random variableF ∈(D^1,2)ⁿ we denote

σ_U,F(p) =E_U((detσ_F)^−p)^1/p. (2.20) We assume thatU ∈D^1,∞ and that for everyp≥1

m_p(U) :=E_U(|DlnU|^p)<∞. (2.21) In Lemma 2.1 in [1] we have proved the following:

Lemma 2.8 Assume that (2.21) holds. Let F ∈ (D^2,∞)ⁿ be such that detσF 6= 0 on the set {U 6= 0}. We denote σb_F the inverse of σ_F and we assume that σ_U,F(p)<∞ for every p∈N.

Then for every V ∈D^1,∞ and every f ∈C_b^∞(Rⁿ) one has

E_U(∂_if(F)V) =E_U(f(F)H_i,U(F, V)) (2.22) with

H_i,U(F, V) = Xn

j=1

Vbσ_F^j,iLF^j−D

D(Vbσ^j,i_F ), DF^jE

−Vσb^j,i_F

D(lnU), DF^j . (2.23) Suppose that lnU ∈ D^k,∞. Iterating (2.22) one obtains for a multi index α = (α₁, ..., α_k) ∈ {1, ..., n}^k

E_U(∂αf(F)V) =E_U(f(F)Hα,U(F, V)), with Hα,U(F, V) =Hαk,U(F, Hα,U(F, V)), (2.24) where α= (α₁, ..., α_k−1).

We will use this result with a localization random variableU constructed in the following way.

Fora∈(0,1) we define ψa :R₊→R₊ by

ψ_a(x) = 1_[0,a)(x) + 1_[a,2a)(x) exp

1− a²

a²−(x−a)²

. (2.25)

Then for every multi indexα and every p∈N there exists a universal constantC_α,p such that sup

x∈R+

ψ_a(x)|∂_αlnψ_a(x)|^p ≤ C_α,p

a^p|α|. (2.26)

Leta_i >0 and Q_i ∈D^1,p, i= 1, ..., l and U =Ql

i=1ψ_ai(Q_i).As an easy consequence of (2.26) we obtain the following estimates

m_p(U)≤C Xl

i=1

1

a^p_i kQ_ik^p_U,1,p≤C Xl

i=1

1

a^p_i kQ_ik^p1,p (2.27) where C is a universal constant. And moreover, for every k, p ∈ N there exists a universal constant C such that

klnUk_U,k,p≤C Xl

i=1

1

a^k_i kQ_ik_k,p. (2.28)

The function ψ_a is suited for localization around zero. In order to localize far from zero we have to use the following alternative version:

φ_a(x) = 1_[a,∞)(x) + 1_[a/2,a)(x) exp

1− a² (2x−a)²

. (2.29)

The property (2.26) holds forφ_aas well. And if one employs bothψ_ai andφ_ai in the construc- tion ofU, that is if one sets

U = Yl

i=1

ψ_ai(Q_i)×

l^′

Y

j=1

φ_al+j(Q_l+j), (2.30)

both properties (2.27) and (2.28) hold again. Then we have the following estimate.

Lemma 2.9 Letk, l, l^′ ∈N, Qi∈D^k+1,∞, i= 1, ..., l+l^′ and setU as in (2.30). Consider also some F ∈ (D^k+1,∞)ⁿ. Then for every p ≥ 1 there exist some universal constants C > 0 and p^′ > p(depending on k, n, p only) such that for every multi indexα with|α| ≤k one has

kHα,U(F,1)k_U,p≤C 1 +σU,F p^′k+1 1 +

l+l^′

X

i=1

1

a^k_i kQik_k,p^′

1 +kFk^2nk_k+1,p^′ . Proof. For G ∈ (D^r,p)ⁿ, let |D^(r)G| = Pr

ℓ=0

P

|γ|=ℓ|D^γG|² as in the proof of Lemma 2.6.

Then the following deterministic estimate for the Malliavin weights holds:

|H_α,U(F, V)| ≤ CX^k

r=0

|D^(r)V|

× 1 +

Xk

r=1

|D^(r)lnU|

×

× 1 +|detσ_F|^−(k+1)

×

× 1 +

Xk+1

r=1

|D^(r)F|+ Xk−1

r=0

|D^(r)LF|2nk

.

(2.31)

The proof of (2.31) is straightforward, although non trivial, and can be found in the preprint version of the present paper, see [3]. The statement now easily follows by applying to the r.h.s.

of (2.31) the H¨older inequality and the Meyer inequalitykLFkU,r,p ≤ kLFkr,p ≤CkFkr+2,p. We finally recall the result in Theorem 2.13 from [2], on which the proof of Theorem 2.1 is based.

Consider a random variableF, a probability measureQand a family of probabilitiesQ_δ,δ >0.

We denote by µ the law of F under Q and by µ_δ the law of F under Q_δ. In the following, we will take Q= P_U and Q_δ = P_Uδ as given in (2.18), where U and U_δ are both of the form (2.30). Actually, P_U and P_Uδ are not probability measures but they are both finite with total mass less or equal to 1, and this is enough.

We let E_Q and E_Qδ denote expectation under Qand Q_δ respectively.

Fixδ > 0. Form ∈N_∗ and p≥1, we say that F ∈ Rm,p(Q_δ) if for every multi indexα with

|α| ≤ m there exists a random variable H_α,δ such that the following abstract integration by parts formula holds:

E_Qδ(∂_αf(F)) =E_Qδ(f(F)H_α,δ) ∀f ∈C_c^∞, with E_Qδ(|H_α,δ|^p)<∞. (2.32) By using Theorem 2.13A in [2] withm= 1 andk= 0, we have

Theorem 2.10 Let q∈Nandp >1 be fixed and letr_n= 2(n+ 1). Let F ∈ ∩δ>0Rq+3,rn(Q_δ).

Suppose that there exist θ > 0, C ≥ 1 and η > ^q+n/p₂ ^∗, with p_∗ the conjugate of p, such that one has

lim sup

δ→0

E_Qδ(|F|^rn)^1/rn+ X

|α|≤q+3

δ^θ|α|E_Qδ(|H_α,δ|^rn)^1/rn

<∞, (2.33)

d₀(µ, µ_δ)≤Cδ^ηθn2(q+3), (2.34)

whered₀ denotes the total variation distance, that is d₀(µ, ν) = sup{|R

f dµ−R

f dν| : kfk∞≤ 1}. Thenµ is absolutely continuous and has a density pF ∈W^q,p(Rⁿ).

Proof. Let us first notice that Theorem 2.13 in [2] concerns a family of r.v.’s F_δ,δ >0, and it is assumed that all these random variables F_δ are defined on the same probability space (Ω,F,P). But this is just for simplicity of notations. In fact the statement concerns just the law of (F_δ, H_α(F_δ,1),|α| ≤2m+q+ 1), whereH_α(F_δ,1) are the weights in the integration by parts formulas for F_δ. So we may assume that each F_δ is defined on a different probability space (Ω_δ,Fδ,Q_δ). In our case, we take F_δ =F for each δ, we work on the space (Ω,F,Q_δ) and we have H_α(F_δ,1) = H_α,δ. We then apply Theorem 2.13 in [2] with m = 1 and k = 0.

(2.33) immediately gives that sup_δE_Qδ(|F|ⁿ⁺³) <∞ because 2(n+ 1) ≥n+ 3. Moreover, in view of (2.39) in [2], the quantityT_q+3,2(n+1)(F_δ) in the statement of Theorem 2.13 therein can be upper bounded by

S_q+3,2(n+1)(δ) :=E_Qδ(|F|^rn)^1/rn+ X

|α|≤q+3

E_Qδ(|H_α,δ|^rn)^1/rn.

As an immediate consequence of (2.33) and (2.34), all the requirements in Theorem 2.13 in [2]

hold, and the statement follows.

2.4 Proof of Theorem 2.1

We are now ready to prove our main result.

Proof of Theorem 2.1. Step 1: construction of the localization r.v.’s U and U_δ. We consider the functions ψ =ψ_1/2 and φ =φ₂ defined in (2.25) and (2.29) with a= ¹₂ and a= 2 respectively. We recall that in hypothesis (2.7)ii) some γ < ¹₂ is considered. We denote λ= ¹₃(¹₂ −γ).Recall that q_i(W), i= 1,2 are defined in (2.12). Then we define

Q0=r⁻¹|F−y|, Q1 = 68d³

λ_∗δ²G_δ, Q2 =δ^−(γ+2λ)q1(W), Q3=q2(W), Q4 =δ^γ+λa, Q5= λ(T, δ)

λ_∗ and we set

U =ψ(Q₀), U_δ= Y4 i=0

ψ(Q_i)×φ(Q₅).

Step 2: construction and estimate of the weights H_α,δ (defined in(2.32)) under P_Uδ. We fixk∈N_∗ and we assume that F ∈(D^k+3,∞,∞)ⁿ.

Notice that for δ^λ ≤ _8d¹ √

λ_∗,on the set{U_δ6= 0}we have a(T, δ)q₁(W) = δ^γ+λa(T, δ)

(δ^−(γ+2λ)q₁(W))×δ^λ ≤δ^λ ≤ 1 8

√λ∗

d .

The other restriction required in Λ_T,δ (see (2.13) for the definition) are easy to check. So, we obtain

{U_δ 6= 0} ⊂ {|F−y| ≤r} ∩Λ_T,δ. Then, by using Lemma 2.4 we have

E_T,δ 1_{Uδ6=0}(detσ_F,T,δ)^−p

≤ C_n,p

λ^np∗ δ^2np (2.35)