On the regularity of stochastic currents, fractional brownian motion and applications to a turbulence model

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www.imstat.org/aihp 2009, Vol. 45, No. 2, 545–576

DOI: 10.1214/08-AIHP174

On the regularity of stochastic currents, fractional Brownian motion and applications to a turbulence model

Franco Flandoli

^a

, Massimiliano Gubinelli

^b

and Francesco Russo

^c,d

aDipartimento di Matematica Applicata, Università di Pisa, Via Bonanno 25B, I-56126 Pisa, Italia. E-mail: flandoli@dma.unipi.it bLaboratoire de Mathématiques, Université de Paris-Sud, Bâtiment 425, F-91405 Orsay Cedex, France.

E-mail: massimiliano.gubinelli@math.u-psud.fr

cInstitut Galilée, Mathématiques, Université Paris 13, 99, av. J.-B. Clément – F-93430 Villetaneuse, France. E-mail: russo@math.univ-paris13.fr dINRIA Paris-Rocquencort, Projet MATHFI, B.P. 105, F-78153 Le Chesnay Cedex, France.

Received 3 March 2007; revised 13 March 2008; accepted 25 March 2008

Abstract. We study the pathwise regularity of the map

ϕ→I (ϕ)= _T

0

ϕ(Xt),dX_t ,

whereϕis a vector function onR^dbelonging to some Banach spaceV,Xis a stochastic process and the integral is some version of a stochastic integral defined via regularization. A continuous version of this map, seen as a random element of the topological dual ofV will be calledstochastic current. We give sufficient conditions for the current to live in some Sobolev space of distributions and we provide elements to conjecture that those are also necessary. Next we verify the sufficient conditions when the processXis a d-dimensional fractional Brownian motion (fBm); we identify regularity in Sobolev spaces for fBm with Hurst indexH∈(1/4,1).

Next we provide some results about general Sobolev regularity of currents whenWis a standard Wiener process. Finally we discuss applications to a model of random vortex filaments in turbulent fluids.

Résumé. Nous étudions la régularité trajectorielle de l’opérateur

ϕ→I (ϕ)= _T

0

ϕ(Xt),dX_t ,

oùϕest une fonction vectorielle à valeurs dansR^dappartenant à un certain espace de BanachV,Xest un processus stochastique et l’intégrale est une certaine version d’une intégrale stochastique définie via régularisation. Une version continue d’un tel opérateur, interprétée comme une variable aléatoire à valeurs dans le dual topologique deV sera appeléecourant stochastique. Nous donnons des conditions suffisantes pour que le courant se situe dans un certain espace de Sobolev de distributions. De plus nous donnons des arguments qui permettent de conjecturer que ces conditions sont aussi nécessaires. Successivement nous vérifions la validité de ces conditions lorsque le processusXest un mouvement brownien fractionnaire (mbf)d-dimensionnel; en particulier, nous identifions la régularité de Sobolev pour un mbf d’indice de HurstH∈(1/4,1). Par suite, nous fournissons quelques résultats sur la régularité générale de Sobolev de courants relative à un mouvement brownien standard. Enfin nous discutons une application à un modèle de filaments de vorticité dans un fluide turbulent.

MSC:76M35; 60H05; 60H30; 60G18; 60G15; 60G60; 76F55

Keywords:Pathwise stochastic integrals; Currents; Forward and symmetric integrals; Fractional Brownian motion; Vortex filaments

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1. Introduction

We consider stochastic integrals, loosely speaking of the form I (ϕ)=

_T

0

ϕ(X_t),dXt

, (1)

where(X_t)is a Wiener process or a fractional Brownian motion with Hurst parameter H in a certain range. We are interested in the pathwise continuity properties with respect to ϕ: we would like to establish that the random generalized fieldϕ→I (ϕ)has a version that is a.s. continuous inϕin certain topologies. In the language of geometric measure theory, such a property means that the stochastic integral defines pathwise a current, with the regularity specified by the topologies that we have found.

This problem is motivated by the study of fluidodynamical models. In [4], the energy of a random vortex filament has an expression involving a stochastic double integral related to Wiener process in the form

[0,T]²f (X_s−X_t)dXsdXt, (2)

wheref (x)=K_α(x)and whereK_α(x)is the kernel of the pseudo-differential operator(1−)⁻^α(precise definitions will be given in Section 2.3).f is therefore a continuous singular function at zero.

The difficulty there comes from the appearance of anticipating integrands and from the singularity off at zero.

[4] gives sense to this integral in some Stratonovich sense. Moreover that paper explores the connection with self- intersection local time considered for instance by Le Gall in [10].

The work [17] considers a similar double integral in the case of fractional Brownian motion with Hurst index H >¹₂using Malliavin–Skorokhod anticipating calculus.

A natural approach is to interpret the previous double integral as a symmetric (or eventually) forward integral in the framework of stochastic calculus via regularization, see [21] for a survey. We recall that whenXis a semimartingale, the forward (respectively symmetric) integral_t

0ϕ(X)d⁻X(resp._t

0ϕ(X)d^◦X) coincides with the corresponding Itô (resp. Stratonovich) integral. The double stochastic integral considered by [4] coincides in fact with the symmetric integral introduced here. So (2) can be interpreted as

[0,T]²f (Xs−Xt)d^◦Xsd^◦Xt. (3)

In this paper, X will be a fractional Brownian motion with Hurst index H >1/4 but a complete study of the existence of integrals (3) will be not yet performed here because of heavy technicalities. We will only essentially consider their regularized versions. Now, those double integrals are naturally in correspondence with currents related toI defined in (1). The investigation of those currents is strictly related to “pathwise stochastic calculus” in the spirit ofrough pathstheory by Lyons and coauthors [15,16]. Here we aim at exploring the “pathwise character” of stochastic integrals via regularization. A first step in this direction was done in [13], where the authors showed that forward integrals of the type_T

0 ϕ(X)d⁻X, whenX is a one-dimensional semimartingale or a fractional Brownian motion with Hurst indexH > ¹₂, can be regarded as a.s. uniform approximations of their regularizationI_ε⁻(ϕ)(see Section 3), instead of the usual convergence in probability.

This analysis of currents related to stochastic integrals was started in [9] using an approach based on spectral analysis. The approach presented here is not based on the Fourier transform and contains new general ideas with respect to [9]. Informally speaking, it is based on the formula

_T

0

ϕ(X_t),dXt

=

R^d

(1−)^αϕ(x), _T

0

K_α(x−X_t)dXt

dx. (4)

This formula decouplesϕandXand replaces the problem of the pathwise dependence ofI (ϕ)on the infinite dimensional parameterϕ with the problem of the pathwise dependence of_T

0 K_α(x−X_t)dXt on the finite dimensional parameterx∈R^d. Another form of decoupling is also one of the ingredients of the Fourier approach of [9] but the

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novelty here is that we can take better advantages from the properties of the underlying process (like, for example, the existence of a density). Moreover formula (4) produces at least two new results.

First, we can treat in an essentially optimal way the case of fractional Brownian motion, making use of its Gaussian properties. ForH >1/2 results in this direction can be extracted from the estimates proved in [17] again by spectral analysis. However, with the present approach we may treat the caseH∈(1/4,1/2)as well.

Second, in the case of the Brownian motion, we may work with functionsϕ in the Sobolev spaces of Banach typeH_p^α, withp >1, instead of only the Hilbert topologiesH₂^α considered in [9], with the great advantage that it is sufficient to ask less differentiability onϕ(anyα >1 suffices), at the price of a largerp(depending onαand the space dimension). In this way we may cover, for instance, the classϕ∈C^1,εtreated in [16], see also [14]; the approach here is entirely different and does not rely on rough paths, see Remark 39.

Finally, we apply these ideas to random vortex filaments. In the case of the fractional Brownian motion we prove new results about the finiteness of the kinetic energy of the filaments (such a property is expected to be linked to the regularity of the pathwise current). In the case of the Brownian motion optimal conditions for a finite energy were already proved in [4,5], while a sufficient condition whenH >1/2 has been found in [17]. The results of the present work provide new regularity properties of the random filaments, especially for the parameter rangeH∈(1/4,1/2).

2. Generalities

2.1. Stochastic currents

Let(X_t)be a stochastic process such thatX₀=0 a.s. on a probability space(Ω,F, P )with values inR^d. LetT >0.

LetV be a Banach space of vector fieldsϕ:R^d→R^dandD⊂V be a dense subset. Assume that a stochastic integral I (ϕ)=

_T

0

ϕ(X_t),dXt

is well defined, in a suitable sense (Itô, etc.), for everyϕ∈D. Our first aim is to define it for everyϕ∈V. In addition, we would like to prove that it has a pathwise redefinition according to the following definition.

Definition 1. The family of r.v.{I (ϕ)}ϕ∈D has a pathwise redefinitiononV if there exists a measurable mapping ξ:Ω→Vsuch that for everyϕ∈D

I (ϕ)(ω)=

ξ(ω) (ϕ) forP-a.e.ω∈Ω. (5)

Then, if we succeed in our objective:

• for everyϕ∈V we consider the r.v.ω→(ξ(ω))(ϕ)as the definition of the stochastic integralI (ϕ)(now extended to the classϕ∈V ), and

• for P-a.e.ω∈Ω, we consider the linear continuous mapping ϕ→(ξ(ω))(ϕ) as a pathwise redefinition of the stochastic integral onV.

Formally, the candidate forξis the expression ξ(x)=

_T

0

δ(x−Xt)dXt,

whereδis here thed-dimensional Dirac measure. Indeed, always formally, ξ(ϕ)=

R^d

ξ(x), ϕ(x) dx=

_T

0

R^dδ(x−X_t)ϕ(x)dx

,dXt

= _T

0

ϕ(X_t),dXt

=I (ϕ).

We remark that this viewpoint is inspired by the theory of currents; with other methods (spectral ones) it was developed in [9].

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2.2. Decoupling by duality

As we said in the introduction, our approach is based on a proper rigorous version of formula (4). One way to interpret it is by the following duality argument, which we describe only at a formal level.

LetW be another Banach space andΛ:V →W be an isomorphism. Proceeding formally as above we have _T

0

ϕ(X_t),dXt

R^d = ϕ, ξ _{V ,V} =

Λ⁻¹Λϕ, ξ

V ,V= Λϕ,

Λ⁻¹ ^∗ξ

W,W

(notice thatΛ⁻¹:W→V,(Λ⁻¹)^∗:V→W). Our aim essentially amounts to proving that(Λ⁻¹)^∗ξ:Ω→Wis a well-defined random variable.

This reformulation becomes useful if the spacesW,Ware easier to handle thanVandV, and the operator(Λ⁻¹)^∗ has a kernelK(x, y)as an operator in function spaces:

Λ⁻¹ ^∗f (x)=

K(x, y)f (y)dy.

In such a case, formally Λ⁻¹ ^∗ξ (x)=

K(x, y) T

0

δ(y−Xt)dXt

dy

= _T

0

K(x, y)δ(y−Xt)dy

dXt

= _T

0

K(x, X_t)dXt.

Below we make a rigorous version of this representation by choosingV =H_p^α(R^d),W=L^p(R^d),Λ=(1−)^α/2, K(x, y)=Kα/2(x−y)(notations are given in the next section).

2.3. Rigorous setting

Denote byS(R^d)the space of rapidly decreasing infinitely differentiablevector fieldsϕ:R^d→R^d, byS(R^d)its dual (the space of tempered distributional fields) and byFthe Fourier transform

(Fϕ)()=(2π)⁻^d/2

R^de⁻ⁱ^x,ϕ(x)dx, ∈R^d

which is an isomorphism in bothS(R^d)andS(R^d). LetF⁻¹denote the inverse Fourier transform. For everys∈R, letΛs:S(R^d)→S(R^d)be the pseudo-differential operator defined as

Λsϕ=F⁻¹

1+ ||² ^s/2Fϕ.

We shall also denote it by(1−)^s/2. In this paperDof Section 2.1 will always indicateS(R^d).

LetH_p^s(R^d), withp >1 ands∈R, be the Sobolev space of vector fieldsϕ∈S(R^d)such that ϕ^p_Hs

p:=

R^d

(1−)^s/2ϕ(x)^pdx <∞;

see [22], Section 2.3.3, where the definition chosen here for brevity is given as a characterization. From the very definitions ofΛs andH_p^s(R^d), the operatorΛsis an isomorphism fromH_p^s(R^d)ontoL^p(R^d)(the Lebesgue space of p-integrablevector fields).

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Another fact often used in the paper is that the dual space(H_p^s(R^d))isH_p⁻^s(R^d):

H_p^s R^d

=H_p⁻^s

R^d , 1 p+ 1

p =1;

see [22], Section 2.6.1. Moreover, beingΛs an isomorphism fromH_p^s(R^d)ontoL^p(R^d), its dual operatorΛ_s is an isomorphism fromL^p(R^d)ontoH_p⁻^s(R^d).

It is known that negative fractional powers of a positive self-adjoint operatorAin a Hilbert spaceH, such that−A generates the semigroupT (t ), have the representation

A⁻^α= 1 (α)

_∞

0

t^α⁻¹T (t )dt,

where is the standard Gamma function; see [18], formula (6.9). TakingA= (1−)in the Hilbert space H= L²(R^d), we have

T (t )ϕ (x)=(4πt )⁻^d/2

R^de^−|^x⁻^y^|²^{/(4t )}⁻^tϕ(y)dy and thus, forα >0,

(1−)⁻^αϕ (x)=(4πt )⁻^d/2 (α)

_∞

0

t^α⁻¹

R^de^−|^x⁻^y^|²^{/(4t )}⁻^tϕ(y)dydt

=

R^d

1 (α)(4π)^d/2

_∞

0

t^α⁻¹⁻^d/2e^−|^x⁻^y^|²^{/(4t )}⁻^tdt

ϕ(y)dy.

In fact this formula can be proved more elementarily from the definition of(1−)⁻^αϕand the formula λ⁻^α= 1

(α) _∞

0

t^α⁻¹e⁻^λtdt,

then taking λ=1+ ||² and the Fourier transform of the Gaussian density. This fact implies that the operator (1−)⁻^α, which originally is an isomorphism between H₂⁻^2α(R^d) and L²(R^d), considered by restriction as a bounded linear operator inL²(R^d), has a kernelK_α(·),

(1−)⁻^αϕ (x)=

R^dK_α(x−y)ϕ(y)dy (6)

given by

Kα(x)= 1 (α)(4π)^d/2

_∞

0

t^α⁻^d/2e^−|^x^|²^{/(4t )}⁻^tdt t .

The following estimates are not optimized as far as the exponential decay is concerned; we just state a version sufficient for our purposes. The proofs of the following two lemmas are in the Appendix.

Lemma 2. There exist positive constantscα,d,Cα,dsuch that:

(1) For0< α <^d₂,we have

Kα(x)= |x|^2α⁻^dρ(x), (7)

wherecα,de⁻²^|^x^|²≤ρ(x)≤Cα,de^−|^x^|^/8;

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(2) Forα >^d₂,we have

cα,de^−|^x^|²^/4≤Kα(x)≤Cα,de^−|^x^|^/8 (8)

for two positive constantsc_α,d,C_α,d .Moreover

K_α(x)=K_α(0)−ρ(x)|x|⁻^d⁺^2α≥0 (9)

with0< K_α(0) <∞and wherec_α,de⁻²^|^x^|²≤ρ(x)≤C_α,de^−|^x^|^/8; (3) Finally,whenα=d/2we have

Kα(x)≤Cα,dlog|x|e⁻^a^α^|^x^|, (10)

wherea_α is another positive constant.

Remark 3. In particularρandρare bounded.

In the applications we will need also some control onK_α(x)which is provided by the next lemma.

Lemma 4. It holds that−K_α(x)=K_α(x)−K_α₋₁(x).Then,whenα < d/2+1,

|−Kα(x)| ≤ρ(x)|x|^2α⁻^d⁻², (11) whereρis positive,bounded above,locally bounded away from zero below and depends onα.

2.4. Regularity of stochastic currents

With these notations and preliminaries in mind, we may state a first rigorous variant of formula (4). Given a continuous stochastic process(X_t)_t_≥₀on(Ω,F, P )with values inR^d, givenε >0, let(D_εX_t)_t_≥₀be any one of the following discrete derivatives:

X_t₊_ε−X_t

ε , X_t−X_t₋_ε

ε , X_t₊_ε−X_t₋_ε

2ε ,

where we understand thatX_t₋_ε=0 fort < ε. The following integral Iε(ϕ)=

_T

0

ϕ(Xt), DεXt

dt

is well definedP-a.s. as a classical Riemann integral, at least for every continuous vector fieldϕ.

Lemma 5. Givenα, ε >0,with probability one the functiont→Kα/2(x−Xt)DεXt is integrable for a.e.x∈R^d, the function

η_ε(x):=

_T

0

Kα/2(x−X_t)D_εX_tdt is inL¹(R^d)and for anyϕ∈S(R^d)we have

_T

0

ϕ(Xt), DεXt

dt=

R^d

(1−)^α/2ϕ(x), _T

0

Kα/2(x−Xt)DεXtdt

dx. (12)

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Proof. Notice that(1−)^α/2ϕ∈S(R^d)and, by (6), ϕ=(1−)⁻^α/2(1−)^α/2ϕ=

R^dK_α/2(· −x)

(1−)^α/2ϕ (x)dx.

Thus _T

0

ϕ(Xt), DεXt

dt= _T

0

R^dKα/2(Xt−x)

(1−)^α/2ϕ

(x)dx, DεXt

dt.

Denote byKα/2(x)the function equal toKα/2(x)forx=0, infinite forx=0. Suppose for a moment that P

_T

0

R^d

K_α/2(X_t−x)dxdt <∞

=1. (13)

Then the integrability properties stated in the lemma will hold. Since sup

(t,x)∈[0,T]×R^d

(1−)^α/2ϕ

(x), D_εX_t<∞ a.s.,

using the Fubini theorem we will get (12) [notice thatKα/2(X_t−x)=Kα/2(x−X_t)].

Thus we have only to prove (13). Since Kα/2 is positive, we may apply again the Fubini theorem and analyze _T

0 (

R^dK_α/2(X_t−x)dx)dt. But we have, for everyy∈R^d

R^d

K_α/2(y−x)dx=

R^d

K_α/2(x)dx.

This quantity is finite for everyα >0, from the estimates of Lemma 2. The proof is now complete.

Below we need a criterion to decide whenηε(defined in the previous lemma) belongs toL²(R^d). It is thus useful to introduce the following condition which ensures the existence of the representation given in Lemma 8.

Condition 6. _T

0

_T

0 K_α(X_t−X_s)dtds <∞P-a.s.

Remark 7. _T

0

_T

0 K_α(X_t−X_s)dtds <∞,implies that the function(t, s)→X_t−X_s is different from zero except possibly on a zero measure set of [0, T]²,and that the well-defined function (t, s)→K_α(X_t −X_s)is Lebesgue integrable on[0, T]².From now on the notationK_αwill simply be replaced byK_α.

Lemma 8. Under Condition6 we have the following double integral representation for the norm ofη_ε defined in Lemma5:

η_ε²_L2(R^d)= _T

0

_T

0

K_α(X_t−X_s)D_εX_t, D_εX_s dtds.

Proof. We have η_ε²_L2(R^d)=

R^d

η_ε(x), η_ε(x) dx

=

R^d

_T

0

K_α/2(x−X_t)D_εX_tdt, _T

0

K_α/2(x−X_s)D_εX_sds

dx

= _T

0

_T

0

R^dKα/2(x−Xt)Kα/2(x−Xs)DεXt, DεXs dxdtds

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if we can apply the Fubini theorem. Then it is sufficient to use the property

R^dK_α/2(x−y)K_α/2(x−z)dx=K_α(y−z).

Since the processX is continuous,(t, s)→D_εX_sD_εX_t is a continuous two-parameter process which on[0, T]²is a.s. bounded. Then a sufficient condition to apply the Fubini theorem is

_T

0

_T

0

R^dK_α/2(x−X_t)K_α/2(x−X_s)dxdtds <∞. Since the integrand is positive, it is equal to

_T

0

_T

0

R^dKα/2(x−Xt)Kα/2(x−Xs)dx

dtds= _T

0

_T

0

Kα(Xt−Xs)dtds.

Invoking Condition 6 we can conclude the proof.

The double integral representation of the norm ofη_ε will play a major role in the following, so we introduce the notation

Z_α,ε:=

_T

0

_T

0

K_α(X_t−X_s)D_εX_t, D_εX_s dtds.

Lemma 9. Assume Condition6holds for anyα≥α.Then the functionα→Zα,ε≥0is decreasing forα≥α.

Proof. Denoteη_α,ε(x)=_T

0 K_α/2(x−X_t)D_εX_tdt making explicit the dependence onα. It is not difficult to prove that, ifα≤α≤β, we haveη_β,ε=(1−)^(α⁻^β)/2η_α,ε. ThenZ_β,ε= η_β,ε²_L2(R^d)= (1−)^(α⁻^β)/2η_α,ε²_L2(R^d)≤ ηα,ε²_L2(R^d)=Zα,ε, since beingα−β≤0 , the operator(1−)^(α−β)/2has a norm bounded by one.

We will assume below the following condition on the convergence of the regularized integrals:

Condition 10. For everyϕ∈S(R^d),Iε(ϕ)converges in probability to some r.v.,denoted byI (ϕ).

Under Condition 10, the mappingϕ→I (ϕ)isa prioridefined only onS(R^d)with values in the setL⁰(Ω)of random variables. Its extension toϕ∈H₂^α(R^d)is a result of the next theorem.

Theorem 11. Assume Conditions6and10,and the a priori bound sup

ε∈(0,1)

E _T

0

_T

0

K_α(X_t−X_s)D_εX_t, D_εX_s dtds

<∞. (14)

Then:

(i) The mappingsϕ∈S(R^d)→I_ε(ϕ), I (ϕ)∈L⁰(Ω)take values inL²(Ω)and extend(uniquely)to linear contin- uous mappings from H₂^α(R^d)toL²(Ω).Moreover,for everyϕ∈H₂^α(R^d),I_ε(ϕ)→I (ϕ)in probability and in L²⁻^δ(Ω)for everyδ >0.

(ii) In addition,there exist random elements ξ_ε, ξ:Ω →H₂⁻^α(R^d) [in fact belonging to L²(Ω;H₂⁻^α(R^d))] that constitute pathwise redefinitions ofI_ε and I onV =H₂^α(R^d),in the sense of Definition 1.Moreover,ξ_ε→ξ weakly inL²(Ω;H₂⁻^α(R^d)):this means that for allΨ ∈L²(Ω, H₂^α(R^d)).E(ξ_ε, Ψ )→E(ξ, Ψ ).

Remark 12. If the convergence in probability in Condition10is replaced by a convergence inL²(Ω),then we can takeδ=0in statement(i)of Theorem11.

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Proof of Theorem 11. Step 1 (mean square results). By the assumptions and the previous lemma we have sup_ε_∈_(0,1)E[η_ε²_L2(R^d)]<∞. From (12), forϕ∈S(R^d)we have

I_ε(ϕ)=

(1−)^α/2ϕ, η_ε

L²(R^d). Therefore, always forϕ∈S(R^d),

Iε(ϕ)≤ ηεL²(R^d)(1−)^α/2ϕ

L²(R^d)≤CαηεL²(R^d)ϕH₂^α(R^d), EI_ε(ϕ)²

≤C_αϕ²_Hα 2(R^d)E

η_ε²_L2(R^d)

≤C_αϕ²_Hα 2(R^d).

Immediately we have I_ε(ϕ)∈L²(Ω)for every ϕ∈S(R^d), and the mappingϕ →I_ε(ϕ) extends (uniquely by density) to a linear continuous mapping fromH₂^α(R^d)toL²(Ω).

Given ϕ ∈ S(R^d), since Iε(ϕ) → I (ϕ) in probability, uniform integrability arguments and E[|Iε(ϕ)|²] ≤ C_αϕ²_Hα

2(R^d), yieldIε(ϕ)→I (ϕ)inL²⁻^δ(Ω)for everyδ >0; moreover, it is not difficult to deduceI (ϕ)∈L²(Ω) andE[|I (ϕ)|²] ≤C_αϕ²_Hα

2(R^d). As before, this implies that the mappingϕ→I (ϕ)extends uniquely to a linear continuous mapping fromH₂^α(R^d)toL²(Ω). Now, with these extensions, it is not difficult to show thatI_ε(ϕ)→I (ϕ)in L²^−δ(Ω)for everyδ >0 also for everyϕ∈H₂^α(R^d).

Step 2 (pathwise results). We still have to constructξ_ε andξ. Recalling thatη_ε∈L²(Ω;L²(R^d)),ξ_ε is simply defined as[(1−)^α/2]η_ε, element ofL²(Ω;H₂⁻^α(R^d)), whereAdenotes the dual of an operatorA.

To this end, recall that(1−)^α/2 is an isomorphism betweenH₂^α(R^d)andL²(R^d), and thus the dual operator [(1−)^α/2]is an isomorphism between the dual spacesL²(R^d)andH₂⁻^α(R^d)(we identifyL²(R^d)with its dual).

The family{ηε}is bounded inL²(Ω;L²(R^d)), hence there exist a sequenceηε_n weakly convergent to someηin L²(Ω;L²(R^d)):Eηε_n, Y _L2(R^d)→Eη, Y _L2(R^d)for everyY ∈L²(Ω;L²(R^d)). We set

ξ:=

(1−)^α/2

η, ξε:=

(1−)^α/2

ηε (15)

random element of H₂⁻^α(R^d). We shall see that this definition does not depend on the sequence ε_n. We have to prove that for everyϕ∈S(R^d),I (ϕ)(ω)=(ξ(ω))(ϕ)forP-a.s.ω∈Ω. Equivalently we have to prove that for every ϕ∈S(R^d)

I (ϕ)(ω)=

η(ω), (1−)^α/2ϕ

L²(R^d) forP-a.s.ω∈Ω. (16)

We already know thatI_ε(ϕ)= (1−)^α/2ϕ, η_ε _L2(R^d) for everyε >0. ChooseY above of the formY =F (1− )^α/2ϕwith genericF∈L²(Ω). Givenϕ∈S(R^d), we know that

E F

ηε_n, (1−)^α/2ϕ

L²(R^d)

→E F

η, (1−)^α/2ϕ

L²(R^d)

for everyF ∈L²(Ω). Hence E

F Iε(ϕ)

→E F

η, (1−)^α/2ϕ

L²(R^d)

but we also know thatIε(ϕ)→I (ϕ)inL²⁻^δ(Ω)for everyδ >0. We get E

F I (ϕ)

=E F

η, (1−)^α/2ϕ

L²(R^d)

at least for every bounded random variableF, hence (16) holds true. This also implies that the definition ofξ does not depend on the sequenceε_n.

Step 3(weak convergence). A consequence of (15) is that ξε²_H−α

2 (R^d)= ηε²_L2(R^d).

(10)

Using Lemma 8 and Assumption 14, it follows thatξ_ε²_L2(Ω,H₂^−α(R^d))is bounded inε >0. We remark that the linear spaceC of simple random elements of the type_n

i=1ψ_iZ_i, where Z_i ∈L^∞(Ω)andψ_i ∈H₂^α(R^d)is dense in the Hilbert spaceH:=L²(Ω, H₂⁻^α(R^d)). We consider the linear formsT^ε, T:H→Rdefined byT^ε(Ψ )=E(ξ^ε, Ψ ), T (Ψ )=E(ξ, Ψ ). Sinceξεare pathwise redefinitions ofI, then point (i) implies that, for everyΨ ∈C,T^ε(Ψ )→ T (Ψ )whenε→0. Finally by the Banach–Steinhaus theorem we obtainT^ε(Ψ )→T (Ψ )for anyΨ ∈Hand soξ_ε

weakly converges toξ.

To state a possible converse of Theorem 11, it is useful to introduce a weaker version of Definition 1.

Definition 13. LetA∈Fbe such thatP (A) >0.We say that{I (ϕ)}ϕ∈Dhas apathwise redefinition onV restricted toAif there exists a measurable mappingξ :A→V such that for everyϕ∈S(R^d)Eq. (5)holds true forP-a.e.

ω∈A.

Theorem 14. Assume Conditions6and10,and the a priori bound E

T 0

K_α(X_t−X_s)D_εX_t, D_εX_s dtds

<∞

for everyε >0.If there existsA∈FwithP (A) >0such that{I (ϕ)}ϕ∈H^α(R^d)has a pathwise redefinition onH₂^α(R^d) restricted toA,then

lim sup

ε→0

_T

0

_T

0

K_α(X_t−X_s)D_εX_t, D_εX_s dtds <∞ P-a.s.ω∈A.

Proof. The first part of Step 1 of the previous proof is still valid (except for the uniformity inεof the constants). Thus in particularE[η_ε²_L2(R^d)]<∞, I_ε(ϕ)= (1−)^α/2ϕ, η_ε ,|I_ε(ϕ)| ≤C_α,εϕ²_Hα

2(R^d), I_ε(ϕ)∈L²(Ω)for every ϕ∈S(R^d)and the mappingϕ→I_ε(ϕ)extends to a linear continuous mapping fromH₂^α(R^d)toL²(Ω).

We know, by Condition 10 and the existence ofξ, that for everyϕ∈S(R^d) (1−)^α/2ϕ, η_ε(ω)

L²(Ω) →

ε→0

ξ(ω) (ϕ) forP-a.e.ω∈A.

One can find a countable setD⊂S(R^d)with the following two properties: (i)(1−)^α/2Dis dense inL²(R^d)and (ii) forP-a.e.ω∈A

(1−)^α/2ϕ, η_ε(ω)

L²(Ω) →

ε→0

ξ(ω) (ϕ) for everyϕ∈D. (17)

Let us prove the claim by contradiction. Assume there isA∈F,A⊂A,P (A) >0, such that lim sup

ε→0

_T

0

_T

0

K_α(X_t−X_s)D_εX_t, D_εX_s dtds= ∞ for everyω∈A. By Lemma 8, lim sup_ε_→₀η_ε(ω)²_L2(R^d)= ∞for everyω∈A.

Consequently, there is a subsetA ofAwithP (A) >0 such that (17) holds true for everyω∈A. Thus, given ω∈A, there is an infinitesimal sequence{ε_n(ω)}such that limn→∞η_ε_n_(ω)(ω)²_L2(R^d)= ∞but at the same time

nlim→∞

(1−)^α/2ϕ, ηε_n(ω)(ω)

L²(Ω)=

ξ(ω)(ϕ)

for everyϕ∈D. This is impossible because of the density of(1−)^α/2DinL²(R^d). The proof is complete.

Definition 15. We say that a.s.a stochastic currentI does not belong toVif there is noA∈F,P (A) >0such that {I (ϕ)}ϕ∈V has a pathwise redefinition onV restricted toA.

(11)

Corollary 16. Under the conditions of Theorem14,if lim sup

ε→0

_T

0

_T

0

Kα(Xt−Xs)DεXt, DεXs dtds= +∞ P-a.s.

the currentIdoes not belong toH₂^α(R^d).

3. Application to the fractional Brownian motion

We recall here that a fractional Brownian motion (fBm)B=(B_t)with Hurst indexH∈(0,1), is a Gaussian mean- zero real process whose covariance function is given by

Cov(Bs, B_t)=1 2

|s|^2H+ |t|^2H− |s−t|^2H , (s, t )∈R²₊.

This process has been widely studied: for some recent developments, we point to [3] as a relevant monograph. For instance we recall that when H= ¹₂, B is a classical Wiener process. Its trajectories are Hölder continuous with respect to any parameterγ < H. Recall that

E

|Bt−Bs|² = |t−s|^2H.

For our stochastic integral redefinition, we have chosen the framework of stochastic calculus via regularization; for a survey about the topic and recent developments, see [21].

LetX=(X¹, . . . , X^d)be ad-dimensional fractional Brownian motion with Hurst indexH∈(0,1), i.e., anR^d- valued process whose componentsX¹, . . . , X^dare real independent fractional Brownian motions. In this section we study the regularity of the current generated byXusing the results of the previous section and in particular Theorem 11 which gives sufficient conditions for regularity in the Hilbert spacesH₂^α(R^d).

We will consider thesymmetricandforwardintegrals, respectively defined as the limit in probability, asε→0, of I_ε^◦(ϕ):=

_T

0

ϕ(Xt),X_t₊_ε−X_t₋_ε 2ε

dt and

I_ε⁻(ϕ):=

T 0

ϕ(X_t),X_t₊_ε−X_t ε

dt.

Whenever they exist we will write I^◦(ϕ)=

_T

0

ϕ(X_t), d⁰X_t

and I⁻(ϕ)= _T

0

ϕ(X_t),d⁻X_t .

In [1] the authors show that the symmetric integral exists for fBm with anyH >1/4 (the proof is about thed=1 case but it extends without problem to higher dimensions). ForH >1/6 necessary and sufficient conditions for the existence of the symmetric integral are given in [12]; see also [11] for a complete characterization whenH >¹₄.

As far as the forward integral is concerned, it is known that, in dimension 1, it does not exist at least for some (very smooth) functionsϕ. Existence of the forward integral in dimension one is guaranteed if and only ifH≥1/2 [20,21].

Observe that, whenH <¹₂ the forward integral_T

0 Bd⁻B does not exist sinceB is not of finite quadratic variation process.

WhenH≥1/2 and ford >1 the forward integral (equal to the Young integral in the caseH >1/2), is equal to the symmetric integral minus the covariation[ϕ(X), X]/2. This exists ifXhas all itsmutual covariationsand theith component of that bracket gives

f (X), Xⁱ

t= _t

0

d j=1

∂jf (Xs)d Xⁱ, X^j

s,