Pierre-Louis Louis Lions

Benoît Perthame

Panagiotis E Souganidis

Stochastic averaging lemmas for kinetic equations

Keywords: Stochastic kinetic equations, stochastic conservation laws, averaging lemmas, fractional Sobolev spaces. AMS Class. Numbers. 35L65, 35R60, 60H15, 82C40

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Introduction

The kinetic formulation of the stochastic conservation laws, as developed in [START_REF] Lions | Scalar conservation laws with rough (stochastic) fluxes[END_REF], motivates to study the stochastic kinetic equation    ∂ ∂t f (x, ξ, t) + Ḃ(t) • ξ.∇ x f = g(x, ξ, t) in R 2d × (0, ∞), f (t = 0, x, ξ) = f 0 (x, ξ).

(

) 1
where both f 0 and g are deterministic. The notation for the flux means

Ḃ(t) • ξ.∇ x f = Ḃ(t) • d i=1 ξ i ∂f ∂x i .
That is we we use Stratonovich convention. Also we introduce a single brownian motion (for simplicity). At variance with the examples of Hamilton-Jacobi equations [START_REF] Lions | Équations aux dérivées partielles stochastiques nonlinéaires et solutions de viscosité, Seminaire: Équations aux Dérivées Partielles[END_REF][START_REF] Lions | Viscosity solutions of fully nonlinear stochastic partial differential equations, Viscosity solutions of differential equations and related topics (Japanese) Kyoto[END_REF] or x-dependent fluxes in scalar conservation laws [START_REF] Lions | Scalar conservation laws with rough (stochastic) fluxes[END_REF], this does not play an important role here.

Our results are mainly motivated by the case when the B is a brownian motion. But for reference we will use also the standard deterministic case B(t) = t. In this deterministic case, the averaging lemmas are now sophisticated [START_REF] Diperna | L p regularity of velocity averages[END_REF][START_REF] Gérard | Moyennisation et régularité deux-microlocale. (French) [Second-microlocal averaging and regularity[END_REF][START_REF] Perthame | A limiting case of velocity averaging lemmas[END_REF][START_REF] Bouchut | Introduction to the mathematical theory of kinetic equations[END_REF][START_REF] Perthame | Mathematical Tools for Kinetic Equations[END_REF] and have proven to be useful for treating nonlinear kinetic equations as the Vlasov-Maxwell [START_REF] Diperna | Global weak solutions of Vlasov-Maxwell systems[END_REF] or the Boltzmann equation [START_REF] Diperna | On the Cauchy problem for the Boltzmann equation: global existence and weak stability results[END_REF]22].

We understand this equation in the spirit of the works on Hamilton-Jacobi equations in the series of papers by Lions and Souganidis, see [START_REF] Lions | Viscosity solutions of fully nonlinear stochastic partial differential equations, Viscosity solutions of differential equations and related topics (Japanese) Kyoto[END_REF][START_REF] Lions | Équations aux dérivées partielles stochastiques nonlinéaires et solutions de viscosité, Seminaire: Équations aux Dérivées Partielles[END_REF] and the references therein. That is, we define solutions as the limit of standard distributional solutions when B(t) is regularized; in other words they correspond to a Stratonovich rather than an Itō-Doeblin integral. This is a difference compared to stochastic transport equation also treated by Flandoli [START_REF] Flandoli | The Interaction Between Noise and Transport Mechanisms in PDEs[END_REF][START_REF] Flandoli | Random Perturbation of PDEs and Fluid Dynamic Models[END_REF] or with conservation laws with stochastic semilinear terms treated by Debussche and Vovelle [4] or with random kinetic equations, with a stochastic semilinear term that are also treated by Debussche and Vovelle in [START_REF] Debussche | Diffusion limit for a stochastic kinetic problem[END_REF] where the diffusion limit is studied.

The question of the existence is not important here because of linearity and the method of characteristics gives the representation formula

d dt f x + B(t) • ξ, ξ, t = g x + B(t) • ξ, ξ, t .
This defines our solutions (see [START_REF] Lions | Équations aux dérivées partielles stochastiques nonlinéaires et solutions de viscosité, Seminaire: Équations aux Dérivées Partielles[END_REF][START_REF] Lions | Viscosity solutions of fully nonlinear stochastic partial differential equations, Viscosity solutions of differential equations and related topics (Japanese) Kyoto[END_REF] for a motivation and [START_REF] Lions | Scalar conservation laws with rough (stochastic) fluxes[END_REF] for a more interesting nonlinear case). This is possible because we consider the Stratonovich convention; as usual it can be transformed in an Itō integral with an additional diffusion term, [START_REF] Flandoli | The Interaction Between Noise and Transport Mechanisms in PDEs[END_REF][START_REF] Flandoli | Random Perturbation of PDEs and Fluid Dynamic Models[END_REF]. For a given test function ψ(ξ) with compact support, we define the random average

ρ ψ (x, t) = ψ(ξ)f (x, ξ, t)dξ.
Our interest lies in the velocity averaging lemmas which are regularity statements on ρ ψ ; how to state them? how do the fractional exponents reflect the scales of the brownian motion?

We answer these questions by a series of results covering a part of what is known for averaging lemmas as stated in long series of papers, from the initial remark on the phenomena [START_REF] Golse | Un résultat de compacité pour les équations du transport[END_REF][START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF] for p = 2, to the optimal cases of full space derivatives in the right hand side [START_REF] Gérard | Moyennisation et régularité deux-microlocale. (French) [Second-microlocal averaging and regularity[END_REF][START_REF] Perthame | A limiting case of velocity averaging lemmas[END_REF], including the case of L p spaces for p = 2 [START_REF] Diperna | L p regularity of velocity averages[END_REF].

We work in L 2 and we measure time decay thanks to an additional time damping. This is a way to better visualize the different scales between space and time that appear in the stochastic case compared to the deterministic case.

Space regularity, global in time

When considering space regularity only, we can obtain particularly simple results which are global in time and thus also express a time decay. They are useful to make a difference between deterministic and stochastic scales.

Theorem 2.1 (Stochastic averaging). Take B a brownian motion and λ > 0.

For g = 0 E e -λt ρ ψ 2 L 2 R + ; Ḣ1/2 (R d ) ≤ C(supp ψ) λ 1/2 ψf 0 2 L 2 (R d ×R d ) , E ρ ψ 2 L 2 R + ; Ḣ1/2 (R d ) ≤ C(supp ψ) ψf 0 L 2 (R d ×R d ) ψf L 2 (R + ×R d ×R d ) .
For

f 0 = 0 E e -λt ρ ψ 2 L 2 R + ; Ḣ1/2 (R d ) ≤ C(supp ψ) λ 3/2 e -λt ψg L 2 (R + ×R d ×R d ) , E ρ ψ 2 L 2 R + ; Ḣ1/2 (R d ) ≤ C(supp ψ) ψg 1/2 L 2 (R + ×R d ×R d ) ψf 3/2 L 2 (R + ×R d ×R d ) .
For f 0 = 0 and g = div h, we have

E ρ ψ 2 L 2 R + ; Ḣ1/3 (R d ) ≤ C(ψ) h 2 Ḣ-2/3 x ∩L 2 (R + ×R d ×R d ) + ψf 2 L 2 (R + ×R d ×R d ) .
Even though some exponents may seem similar, all the scales differ from the deterministic case B(t) = t. Indeed we recall the Theorem 2.2 (Deterministic averaging). Take B(t) = t and λ ≥ 0.

(i) For g = 0 e -λt ρ ψ 2 L 2 R + ; Ḣ1/2 (R d ) ≤ C(supp ψ) f 0 2 L 2 (R d ×R d ) .
(ii) For f 0 = 0

ρ ψ 2 L 2 R + ; Ḣ1/2 (R d ) ≤ C(supp ψ) ψg 1/2 L 2 (R + ×R d ×R d ) ψf 3/2 L 2 (R + ×R d ×R d ) .
(iii) For f 0 = 0 and g = divh, we have

ρ ψ 2 L 2 R + ; Ḣ1/4 (R d ) ≤ C(ψ) h 2 Ḣ-1/2 x ∩L 2 (R + ×R d ×R d ) + ψf 2 L 2 (R + ×R d ×R d ) .
The first two classes of formulas are not the most traditional and express in particular time decay, which is usually not the aim of averaging lemmas which aim at compactness. This is because other methods based on dispersion are more adapted to prove time decay as those in [START_REF] Bardos | Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data[END_REF][START_REF] Glassey | The Cauchy problem in kinetic theory[END_REF][START_REF] Perthame | Mathematical Tools for Kinetic Equations[END_REF].

One can find related L p formulas; to do so it is enough to interpolate them with the other two elementary pathwise inequalities in L 1 and

L ∞ R d |ρ ψ (x, t)|dx ≤ R 2d |f 0 (x, ξ)|dxdξ, ρ ψ (t) L ∞ (R d ) ≤ ψ L 1 (R d ) f 0 L ∞ (R 2d ) .
The usual proof of averaging lemmas, as initiated in [START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF], uses the Fourier transform in space and time. It is not very convenient because the transport term depends on time here. Therefore we prefer a method without time Fourier transform. Such a method has been developed in [START_REF] Bouchut | Desvillettes Averaging lemmas without time Fourier transform and application to discretized kinetic equations[END_REF] but it uses the Fourier transform in ξ. We prefer to still develop another variant that is particularly fit to the stochastic case.

Proof. We use the Fourier transform f (k, ξ, t) of f in the variable x. The equation on the Fourier transform of f becomes

∂ ∂t f (k, ξ, t) + i Ḃ(t)k.ξ f = g. (2)
As usual we add and subtract a damping term with λ > 0 (except for the first result where we do not need the extra-term in the right hand side because the damping term gives it naturally)

∂ ∂t f (k, ξ, t) + i Ḃ(t) • k.ξ f + λ f = g + λ f .
Because we use the Stratonovich rule, the solution is given by the representation formula

f (k, ξ, t) = f 0 (k, ξ)e -λt-iB(t)k.ξ + t 0 e -λs g + λ f ](k, ξ, t -s) e ik.ξ B(t-s)-B(t) ds
and thus

ρ ψ (k, t) = ψ f 0 (k, ξ)e -λt-iB(t)k.ξ dξ + t 0 e -λs ψ g + λψ f ]e ik.ξ B(t-s)-B(t) dξds. (3)
We obtain the general inequality that we will however particularize later

1 2 | ρ ψ (k, t)| 2 ≤ ψ f 0 (k, ξ)e -λt-iB(t)k.ξ dξ 2 + t 0 e -λs ψ g + λψ f ](k, ξ, t -s)e ik.ξ B(t-s)-B(t) dsdξ 2 .
First class of results; g = 0. For the first result, we use formula (3) with λ = 0 and write

E ∞ t=0 |e -λt ρ ψ (k, t)| 2 = E ∞ t=0 ψ f 0 (k, ξ)e -λt-iB(t)k.ξ dξ 2 dt = E ∞ t=0 ψ f 0 (k, ξ 1 )e -λt-iB(t)k.ξ 1 dξ 1 ψ f 0 (k, ξ 2 )e -λt-iB(t)k.ξ 2 dξ 2 dt = E ∞ t=0 ψ f 0 (k, ξ 1 ) ψ f 0 (k, ξ 2 )e -2λt-iB(t)k.(ξ 1 -ξ 2 ) dξ 2 dξ 1 dt = ∞ t=0 ψ f 0 (k, ξ 1 ) ψ f 0 (k, ξ 2 ) R e -2λt-iwk.(ξ 1 -ξ 2 ) e -w 2 2t dw √ 2πt dξ 2 dξ 1 dt = ∞ t=0 ψ f 0 (k, ξ 1 ) ψ f 0 (k, ξ 2 )e -2λt exp - t k.(ξ 1 -ξ 2 ) 2 2 dξ 2 dξ 1 dt = 2 ψ f 0 (k, ξ 1 ) ψ f 0 (k, ξ 2 ) 1 4λ + |k.(ξ 1 -ξ 2 )| 2 dξ 2 dξ 1 ≤ C √ λ|k| |ψ f 0 (k, ξ)| 2 dξ
The last line is not a direct Cauchy-Schwarz inequality. It uses the usual observation leading to averaging lemmas. It can be proved in choosing an orthonormal basis for the ξ space which first vector is k/|k|. In the orthogonal hyperplane the Cauchy-Schwarz inequality is enough and thus the constant C(supp ψ). In the first direction one uses the Young inequality u * v 2 ≤ C u 2 v 1 and reduces the inequality to compute (with

k ′ = k/λ) R R 1 4λ + |k| 2 |η| 2 dη = 1 λ R R 1 4 + |k ′ | 2 |η| 2 dη ≤ C √ λ( √ λ + |k|) ≤ C √ λ|k| .
The first result for g = 0 is proved.

For the second result with g = 0, we use (3) and estimate the lefthand side term λ f as in the proof of the second class of results below. We borrow from there the estimate for the second term which gives

E ∞ t=0 |k| | ρ ψ (k, t)| 2 ≤ C √ λ |ψ f 0 (k, ξ)| 2 dξ + Cλ √ λ |ψ f (k, ξ, t)| 2 dk dξ dt.
We choose λ = g 2 / f 2 and obtain the announced result.

Second class of result; f 0 = 0. The integral term for g is more technical due to an additional time integration, but the calculation follows the same ideas. For the result with λ, we consider only the term containing g in ( 3) with λ = 0,

E ∞ t=0 |e -λt ρ ψ (k, t)| 2 = E ∞ t=0 t 0 e -λt ψ g(k, ξ, t -s)e ik.ξ B(t-s)-B(t) dξds 2 dt = E ∞ t=0 t s 1 ,s 2 =0 e -2λt ψ g(k, ξ 1 , t -s 1 )e ik.ξ 1 B(t-s 1 )-B(t) ψ g(k, ξ 2 , t -s 2 )e ik.ξ 2 B(t-s 2 )-B(t) ds 1 ds 2 dξ 2 dξ 1 dt.
To continue we may restrict ourselves to 0 < s 1 < s 2 < t and compute

Ee ik.ξ 1 B(t-s 1 )-B(t) e -ik.ξ 2 B(t-s 2 )-B(t) = Ee ik.ξ 1 B(t-s 1 )-B(t) e -ik.ξ 2 B(t-s 2 )-B(t-s 1 ) e -ik.ξ 2 B(t-s 1 )-B(t) = e ik.(ξ 2 -ξ 1 )w e -w 2 2s 1 dw √ 2πs 1 e ik.ξ 2 w e -w 2 2(s 2 -s 1 ) dw 2π(s 2 -s 1 ) = exp - s 1 |k.(ξ 1 -ξ 2 )| 2 2 exp - (s 2 -s 1 )|k.ξ 2 | 2 2 .
At this stage it is easier to change variables and set τ i = t -s i , t > τ 1 > τ 2 > 0. Then, combining the two ingredients above, we obtain the expression for the term of interest

1 2 E ∞ t=0 |e -λt ρ ψ (k, t)| 2 = = ∞ t=0 t τ 1 >τ 2 =0 e -2λt ψ g(k, ξ 1 , τ 1 )ψ g(k, ξ 2 , τ 2 )e -(t-τ 1 )|k.(ξ 1 -ξ 2 )| 2 2 e -(τ 1 -τ 2 )|k.ξ 2 | 2 2 dτ 1 dτ 2 dξ 2 dξ 1 dt = ∞ τ 1 >τ 2 =0 ∞ t=τ 1 e -2λt ψ g(k, ξ 1 , τ 1 )ψ g(k, ξ 2 , τ 2 )e -(t-τ 1 )|k.(ξ 1 -ξ 2 )| 2 2 e -(τ 1 -τ 2 )|k.ξ 2 | 2 2 dτ 1 dτ 2 dξ 2 dξ 1 dt = ∞ τ 1 >τ 2 =0 e -2λτ 1 ψ g(k, ξ 1 , τ 1 )| ψ g(k, ξ 2 , τ 2 ) 2 2λ + |k.(ξ 1 -ξ 2 )| 2 e -(τ 1 -τ 2 )|k.ξ 2 | 2 2 dτ 1 dτ 2 dξ 2 dξ 1 = ∞ τ 1 >τ 2 =0 e -λτ 1 e -λτ 2 ψ g(k, ξ 1 , τ 1 )| ψ g(k, ξ 2 , τ 2 ) 2 2λ + |k.(ξ 1 -ξ 2 )| 2 e -(τ 1 -τ 2 )(2λ+|k.ξ 2 | 2 ) 2 dτ 1 dτ 2 dξ 2 dξ 1 .
Next, we treat the time convolution using the Young inequality

u 1 (τ 1 ) u 2 * K(τ 1 )dτ 1 ≤ u 1 2 u 2 2 K 1
with the convolution K given by the truncated exponential

1 2 E ∞ t=0 |e -λt ρ ψ (k, t)| 2 ≤ |e -λτ 1 ψ g(k, ξ 1 , τ 1 )| 2 dτ 1 |e -λτ 2 ψ g(k, ξ 2 , τ 2 )| 2 dτ 2 1/2 2 2λ + |k.(ξ 1 -ξ 2 )| 2 2 λ + |k.ξ 2 | 2 dξ 2 dξ 1 ≤ 1 λ |e -λτ 1 ψ g(k, ξ 1 , τ 1 )| 2 dτ 1 1/2 |e -λτ 2 ψ g(k, ξ 2 , τ 2 )| 2 dτ 2 1/2 4 2λ + |k.(ξ 1 -ξ 2 )| 2 dξ 2 dξ 1 ≤ C(supp ψ) λ 3/2 |k| |e -λτ ψ g(k, ξ, τ )| 2 dτ dξ.
As before, the last inequality follows from the Young inequality used in η, the one dimensional component of ξ 1 -ξ 2 parallel to k, with

dη 2λ + |k| 2 η 2 = C √ λ|k| .
This proves the first inequality with f 0 ≡ 0.

For the second inequality, we use again (3) and begin with the term containing g in the right hand side.

I := E ∞ t=0 t 0 e -λs ψ g(k, ξ, t -s)e ik.ξ B(t-s)-B(t) dξds 2 dt = = E ∞ t=0 t s 1 ,s 2 =0 e -λ(s 1 +s 2 ) ψ g(k, ξ 1 , t -s 1 )e ik.ξ 1 B(t-s 1 )-B(t) ψ g(k, ξ 2 , t -s 2 )e ik.ξ 2 B(t-s 2 )-B(t) ds 1 ds 2 dξ 2 dξ 1 dt.
To continue we may again restrict ourselves to 0 < s 1 < s 2 < t and use the calculation above to compute the expectations. With the variables τ i = t -s i , t > τ 1 > τ 2 > 0 we obtain

I/2 = = ∞ t=0 t τ 1 >τ 2 =0 e -λ(2t-τ 1 -τ 2 ) ψ g(k, ξ 1 , τ 1 )ψ g(k, ξ 2 , τ 2 )e -(t-τ 1 )|k.(ξ 1 -ξ 2 )| 2 2 e -(τ 1 -τ 2 )|k.ξ 2 | 2 2 dτ 1 dτ 2 dξ 2 dξ 1 dt = ∞ τ 1 >τ 2 =0 ∞ t=τ 1 e -λ(τ 1 -τ 2 ) ψ g(k, ξ 1 , τ 1 )ψ g(k, ξ 2 , τ 2 )e -(t-τ 1 )[4λ+|k.(ξ 1 -ξ 2 )| 2 ] 2 e -(τ 1 -τ 2 )|k.ξ 2 | 2 2 dτ 1 dτ 2 dξ 2 dξ 1 dt = ∞ τ 1 >τ 2 =0 ψ g(k, ξ 1 , τ 1 ) ψ g(k, ξ 2 , τ 2 ) 2 4λ + |k.(ξ 1 -ξ 2 )| 2 e -(τ 1 -τ 2 )(λ+|k.ξ 2 | 2 ) 2 dτ 1 dτ 2 dξ 2 dξ 1 ≤ |ψ g(k, ξ 1 , τ 1 )| 2 dτ 1 1/2 |ψ g(k, ξ 2 , τ 2 )| 2 dτ 2 1/2 2 4λ + |k.(ξ 1 -ξ 2 )| 2 2 2λ + |k.ξ 2 | 2 dξ 2 dξ 1 .
Again this follows from the Young inequality u 1 (τ 1 ) u 2 * K(τ 1 )dτ 1 ≤ u 1 2 u 2 2 K 1 with the convolution K given by the truncated gaussian. Then, we conclude because

I/2 ≤ 2 λ |ψ g(k, ξ 1 , τ 1 )| 2 dτ 1 1/2 |ψ g(k, ξ 2 , τ 2 )| 2 dτ 2 1/2 1 4λ + |k.(ξ 1 -ξ 2 )| 2 dξ 2 dξ 1 ≤ C λ √ λ|k| |ψ g(k, ξ, t)| 2 dξ dt.
As before, the last inequality follows from the Young inequality used in the one dimensional component of ξ 1 -ξ 2 parallel to k. These are the usual estimates for the averaging lemma but we win an extra-factor 1 √ λ compared to the deterministic case. It does not play a role for regularity in the second result of Theorem 2.1 because there we choose λ = g 2 / f 2 in the upper bound

E ∞ 0 |k|| ρ ψ (k, t)| 2 dtdk ≤ C λ √ λ | g(k, ξ, t)| 2 dk dξ dt + Cλ √ λ | f (k, ξ, t)| 2 dk dξ dt (4)
that is obtained after combining the two terms in g and λf in (3).

Third result; g = div h. We go back to the formula for ρ ψ (k, t) and write

ψdiv ξ h = -h.∇ ξ ψ+div[ψ h].
The first term in this Leibniz formula can be treated as before and we examine the second one. It gives a contribution

ρ ψ (k, t) = first term + t 0 e -λs ∂ ∂ξ [ψ h]e ik.ξ B(t-s)-B(t) dsdξ = first term -i t 0 e -λs ψ h.k B(t -s) -B(t) e ik.ξ B(t-s)-B(t) dsdξ
Therefore, its contribution to

E ∞ 0 | ρ ψ (k, t)| 2 dt is E ∞ t=0 t s 1 ,s 2 =0 e -λ(s 1 +s 2 ) ψ h(k, ξ 1 , t -s 1 ).k B(t -s 1 ) -B(t) e ik.ξ 1 B(t-s 1 )-B(t) ψ h(k, ξ 2 , t -s 2 ).k B(t -s 2 ) -B(t) e -ik.ξ 2 B(t-s 2 )-B(t) dξ 1 dξ 2 ds 1 ds 2 dt
which we evaluate as before. The contribution for 0 < s 1 < s 2 < t is the s i , ξ i integral of the expectation that we evaluate as

w z we -ik.(ξ 1 -ξ 2 )w (z + w)e ik.ξ 2 z e -w 2 2s 1 e - z 2 2(s 2 -s 1 ) dw √ 2πs 1 dz 2π(s 2 -s 1 ) = w z w 2 e -ik.(ξ 1 -ξ 2 )w e ik.ξ 2 z e -w 2 2s 1 e - z 2 2(s 2 -s 1 ) dw √ 2πs 1 dz 2π(s 2 -s 1 ) = s 1 (1 -s 1 |k.(ξ 1 -ξ 2 )| 2 ) exp - s 1 |k.(ξ 1 -ξ 2 )| 2 2 exp - (s 2 -s 1 )|k.ξ 2 | 2 2 .
After changing the variable s i in τ i = t -s i , we have to evaluate

∞ t=0 τ 2 ≤τ 1 ≤t e -λ(2t-τ 1 -τ 2 ) ψ h(k, ξ 1 , τ 1 ).k ψ h(k, ξ 2 , τ 2 ).k (t -τ 1 ) 1 -(t -τ 1 )|k.(ξ 1 -ξ 2 )| 2 exp - (t -τ 1 )|k.(ξ 1 -ξ 2 )| 2 2 exp - (τ 1 -τ 2 )|k.ξ 2 | 2 2
But we notice that

∞ t=τ 1 (t -τ 1 ) 1 -(t -τ 1 )|k.(ξ 1 -ξ 2 )| 2 exp - (t -τ 1 )[4λ + |k.(ξ 1 -ξ 2 )| 2 ] 2 dt ≤ C 4λ + |k.(ξ 1 -ξ 2 )| 2 2 .
Finally the formula for

E t 0 | ρ ψ (k, t)| 2 dt is controlled by first term + 0<τ 2 <τ 1 ψ h(k, ξ 1 , τ 1 ).k ψ h(k, ξ 2 , τ 2 ).k C 4λ + |k.(ξ 1 -ξ 2 )| 2 2 e -λ(τ 1 -τ 2 ) ≤ first term + C |k| λ 2 √ λ ψ h(k, ξ, t -s) 2 L 2 t,ξ because dη 4λ + |k| 2 η 2 2 = C |k|λ √ λ , ∞ 0 e -τ λ dτ = 1 λ .
When balancing this contribution with that of λ f we end up with

C |k| λ 2 √ λ + √ λ |k| ψ h(k, ξ, t) 2 L 2 t,ξ + ψ f (k, ξ, t) 2 L 2 t,ξ
which is optimal for λ = |k| 2/3 and gives

C 1 |k| 2/3 ψ h(k, ξ, t) 2 L 2 t,ξ + ψ f (k, ξ, t) 2 L 2 t,ξ .
This means a regularizing term in Ḣ1/3 regularity as announced.

The contribution of the First Term, analogous to the second class of results (4), gives a contribution controlled by

C λ √ λ |k| ∇ψ h(k, ξ, t) 2 L 2 t,ξ
that gives with our previous choice of λ = |k| 2/3

C |k| 2/3 1 |k| 4/3 ∇ψ h(k, ξ, t) 2 L 2 t,ξ .
These controls together give the third result.

Space regularity, general velocity field a(ξ)

Our method can be extended to the case of the kinetic equation with a more general transport field, still motivated by the case of scalar conservation laws,

   ∂ ∂t f (x, ξ, t) + Ḃ(t) • a(ξ).∇ x f = g(x, ξ, t) in R d × R × (0, ∞), f (t = 0) = f 0 on R d × R. (5)
We assume that the transport field a(ξ) ∈ C 1 (R; R d ) satisfies a natural non-degeneracy condition (in view of the deterministic averaging lemmas): there is a α ≥ 1 and a constant A(α) > 0 such that inf

e∈R d , |e|=1 |e.a(ξ 1 ) -e.a(ξ 2 )| ≥ A |ξ 1 -ξ 2 | α , 1 ≤ α < ∞. ( 6
)
Because it uses pointwise control, this assumption is stronger than the usual one based on measures of the sets {|e.a(ξ) -τ | ≤ ε} but it is in the same spirit of imposing nonlinearity of the curves ξ → a(ξ) (that is this d-dimensional curve is not locally contained in a hyperplane) as in the deterministic case [START_REF] Bouchut | Desvillettes Averaging lemmas without time Fourier transform and application to discretized kinetic equations[END_REF][START_REF] Diperna | L p regularity of velocity averages[END_REF][START_REF] Gérard | Moyennisation et régularité deux-microlocale. (French) [Second-microlocal averaging and regularity[END_REF][START_REF] Golse | Un résultat de compacité pour les équations du transport[END_REF][START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF][START_REF] Perthame | Mathematical Tools for Kinetic Equations[END_REF][START_REF] Perthame | A limiting case of velocity averaging lemmas[END_REF]. Theorem 3.1 (Stochastic averaging for general velocity field). Take B a brownian motion and assume (6) in the stochastic kinetic equation [START_REF] Debussche | Diffusion limit for a stochastic kinetic problem[END_REF]. Take λ > 0 (i) For g = 0, we have

E e -λt ρ ψ 2 L 2 R + ; Ḣ1/(2α) (R d ) ≤ C λ 1-1/2α ψf 0 2 L 2 (R d ×R) , E ρ ψ 2 L 2 R + ; Ḣ1/(2α) (R d ) ≤ C ψf 0 1+1/2α L 2 (R d ×R) ψf 1-1/2α L 2 (R d ×R×R + ) .
(ii) For f 0 = 0, we have

E e -λt ρ ψ 2 L 2 R + ; Ḣ1/(2α) (R d ) ≤ C λ 1-1/2α e -λt ψg 2 L 2 (R d ×R×R + ) , E ρ ψ 2 L 2 R + ; Ḣ1/(2α) (R d ) ≤ C ψg 1+1/2α L 2 (R d ×R×R + ) ψf 1-1/2α L 2 (R d ×R×R + ) .
Proof. First class of results. The representation formula ( 3) is simply changed to

e -λt ρ ψ (k, t) = ψ f 0 (k, ξ)e -λt-iB(t)k.a(ξ) dξ + t 0
e -λs ψ g + λψ f ]e ik.a(ξ) B(t-s)-B(t) dsdξ, therefore, we can carry out the same analysis until we evaluate the convolution in ξ. For the term in f 0 for instance, we find

E ∞ 0 e -2λt | ρ ψ (k, t)| 2 = 2 ψ f 0 (k, ξ 1 ) ψ f 0 (k, ξ 2 ) 1 4λ + |k.(a(ξ 1 ) -a(ξ 2 )| 2 dξ 2 dξ 1 ≤ C ψ f 0 (k, ξ 1 ) ψ f 0 (k, ξ 2 ) 1 4λ + |k| 2 |ξ 1 -ξ 2 | 2α dξ 2 dξ 1 ≤ C ψ f 0 (k, ξ) 2 L 2 ξ 1 4λ + |k| 2 |ξ| 2α L 1 ξ .
We obtain the estimate

1 4λ + |k| 2 |ξ| 2α L 1 ξ = C λ √ λ |k| 1/α
, and the first result follows.

For the second result, we use the inequality obtained in section 2 on the contribution to

E ∞ 0 | ρ ψ (k, t)| 2
from the quantity (with g = λf )

I := E ∞ t=0 t 0 e -λs ψ g(k, ξ, t -s)e ik.ξ B(t-s)-B(t) dξds 2 dt, I ≤ 2 λ |ψ g(k, ξ 1 , τ 1 )| 2 dτ 1 1/2 |ψ g(k, ξ 2 , τ 2 )| 2 dτ 2 1/2 1 4λ + |k.(ξ 1 -ξ 2 )| 2 dξ 2 dξ 1 ≤ 2 λ |ψ g(k, ξ 1 , τ 1 )| 2 dτ 1 1/2 |ψ g(k, ξ 2 , τ 2 )| 2 dτ 2 1/2 1 4λ + |k| 2 |ξ 1 -ξ 2 | 2α dξ 2 dξ 1 ≤ C ψ g(k, ξ, t) 2 L 2 ξ,t 1 4λ + |k| 2 |ξ| 2α L 1 ξ ≤ C λ √ λ |k| 1/α ψ g(k, ξ, t) 2 L 2 ξ,t .
This gives

|k| 1/α E ∞ 0 | ρ ψ (k, t)| 2 ≤ C λ λ 1/2α ψ f 0 (k, ξ) 2 L 2 ξ + Cλ λ 1/2α ψ f (k, ξ, t) 2 L 2 ξ,t
.

And, after integrating in k and then optimizing the value of λ

R d |k| 1/α E ∞ 0 | ρ ψ (k, t)| 2 dk ≤ C ψ f 0 1+1/2α L 2 ψ f 1-1/2α L 2
. This is the second result.

Second class of results. For the result with λ, we use again our previous inequality in section 2 which we modify to change ξ in a(ξ). We obtain

1 2 E ∞ t=0 |e -λt ρ ψ (k, t)| 2 ≤ 1 λ |e -λτ 1 ψ g(k, ξ 1 , τ 1 )| 2 dτ 1 |e -λτ 2 ψ g(k, ξ 2 , τ 2 )| 2 dτ 2 1/2 4 2λ + |k.(a(ξ 1 ) -a(ξ 2 ))| 2 dξ 2 dξ 1 ≤ 1 λ |e -λτ 1 ψ g(k, ξ 1 , τ 1 )| 2 dτ 1 1/2 |e -λτ 2 ψ g(k, ξ 2 , τ 2 )| 2 dτ 2 1/2 4 2λ + |k| 2 |ξ 1 -ξ 2 | 2α dξ 2 dξ 1 ≤ C λ √ λ |k| 1/α |e -λτ ψ g(k, ξ, τ )| 2 dτ dξ.
This gives the first result. We do not prove the second result which follows from the same combination as in section 2 with these ingredients.

Time regularity

We can also obtain time regularity with the method developed before. The gain of regularity is however lower than in the deterministic case (see Appendix B) which is a more visible symptom of the difference of scale between brownian motion and classical time. For deterministic kinetic transport equations, space and time play a similar role but this no longer true in the stochastic case (see the scale |k| ∼ √ λ below rather than |k| ∼ λ in the deterministic case).

Theorem 4.1 (Stochastic time averaging for general velocity field). Take B a brownian motion, λ > 0 and g = 0 in equation [START_REF] Bardos | Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data[END_REF]. Then for high Fourier frequencies in x, e -λt ρ ψ ∈ H 1/4 R + ; L 2 (R d ) , more precisely

E |k| √ λ √ λ + |k| e -λt ρ ψ (k, t) 2
H 1/4 R + ;L 2 (R d ) ≤ C f 0 2 L 2 (R d ×R d ) .
For a(ξ) satisfying (6) in the stochastic kinetic equation (5), we have for all β = 1/4α

E |k| 1/α λ 1/2α λ 1/2α + |k| 1/α e -λt ρ ψ (k, t) 2 H 1/4 R + ;L 2 (R d ) ≤ C f 0 2 L 2 (R d ×R) .
Our strategy of proof uses the integral characterization of the fractional Sobolev space ([21], p139) and thus we define for 0 < β < 1,

u 2 Ḣβ (R + ) = [0,∞]×[0,∞] |u(t) -u(s)| 2 [t -s] 1+2β ds dt.
The proof can be used also in the deterministic case and gives the correct regularity H 1/2 . See Appendix B.

Proof. We compute for t > s and with a 1 = a(ξ 1 ), a 2 = a(ξ 2 ),

E|e -λt ρ ψ (k, t) -e -λs ρ ψ (k, s)| 2 = E ψ f 0 (k, ξ 1 )ψ f 0 (k, ξ 2 ) e -λt-iB(t)k.a 1 -e -λs-iB(s)k.a 1 e -λt+iB(t)k.a 2 -e -λs+iB(s)k.a 2 dξ 2 dξ 1 = E ψ f 0 (k, ξ 1 ) ψ f 0 (k, ξ 2 ) e -λ(t-s)-i(B(t)-B(s))k.a 1 -1 e -λ(t-s)+i(B(t)-B(s))k.a 2 -1 e -2λs+iB(s)k.(a 2 -a 1 ) dξ 2 dξ 1 = 1 2π ψ f 0 (k, ξ 1 ) ψ f 0 (k, ξ 2 ) e -λ(t-s)-iw.k.a 1 -1 e -λ(t-s)iw.k.a 2 -1 e -2λs+izk.(a 2 -a 1 ) e - |w| 2
2(t-s) e -

du ds u 1+2β = 2π ∞ s=0 ∞ u=0 e -u[2λ+|k.(a 2 -a 1 )| 2 /2] -e -u[λ+|k.a 1 | 2 /2] -e -u[λ+|k.a 2 | 2 /2] + 1 e -s[2λ+|k.(a 2 -a 1 )| 2 /2] du ds u 1+2β = 2π 2λ + |k.(a 2 -a 1 )| 2 ∞ u=0 e -u[2λ+|k.(a 2 -a 1 )| 2 /2] -e -u[λ+|k.a 1 | 2 /2] -e -u[λ+|k.a 2 | 2 /2] + 1 du u 1+2β ≤ C 2λ + |k.(a 2 -a 1 )| 2 2λ + |k.(a 2 -a 1 )| 2 /2 2β + λ + |k.a 1 | 2 /2] 2β + λ + |k.a 2 | 2 /2 2β ≤ C(β)λ 2β-1 2 + | k.(a 2 -a 1 )| 2 |4 + | k.(a 2 -a 1 )| 2 | 2β + |2 + | k.a 1 | 2 ]| 2β + |2 + | k.a 2 | 2 | 2β
The result follows.

We go back to our estimate on E| ρ ψ (k, t) -ρ ψ (k, s)| 2 and we have to estimate the quantity

I(k) = E ∞ s=0 ∞ t=s | ρ ψ (k, t) -ρ ψ (k, s)| 2 |t -s| 1+2β dt ds.
With the above lemma we conclude that, in the case a(ξ) = ξ,

I(k) ≤ C |ψ f 0 (k, ξ 1 )| |ψ f 0 (k, ξ 2 )| C(β)λ 2β-1 (1 + | k| 4β ) 2 + | k.(a 2 -a 1 )| 2 dξ 2 dξ 1 ≤ C(β, suppψ) ψ f 0 2 L 2 ξ λ 2β-1 (1 + | k| 4β ) 2 + | k.ξ| 2 L 1 ξ ≤ C(β, suppψ) ψ f 0 2 L 2 ξ 1 + | k| 4β |k| λ 2β-1/2
which leads us to choose β = 1/4.

In the general case we conclude with (still with a appropriate choice of coordinates in ξ)

I(k) ≤ |ψ f 0 (k, ξ 1 )| |ψ f 0 (k, ξ 2 )| C(β)λ 2β-1 (1 + | k| 4β ) 2 + | k| 2 |(ξ 2 -ξ 1 )| 2α dξ 2 dξ 1 ≤ C(β, suppψ) ψ f 0 2 L 2 ξ λ 2β-1 (1 + | k| 4β ) 2 + | k| 2 |ξ| 2α L 1 ξ ≤ C(β, suppψ) ψ f 0 2 L 2 ξ 1 + | k| 4β |k| 1/α λ 2β-1/(2α)
which leads to the choice β = 1/4α.

A Deterministic averaging lemma, space regularity

In the first statement of Theorem 2.1, a coefficient √ λ appears; it expresses that the time decay in the stochastic case is slow. In the deterministic result of Theorem 2.2 (i), this parameter λ is not needed and thus, the averaging lemma in space also expresses faster time decay. Being a time scale parameter, this slower decay is not surprising when a brownian motion is involved. To better understand how it appears, we give the corresponding proof of the deterministic averaging lemma.

We consider the regularizing effect with respect to the initial data (first statement of Theorem 2.2, i.e., the solution to

   ∂ ∂t f (x, ξ, t) + ξ.∇ x f + λf = 0 in R 2d × (0, ∞), f (t = 0) = f 0 on R 2d . ( 7
)
The solution is given by

f (k, ξ, t) = f 0 (k, ξ)e (-λ+iξ.k)t , ρ ψ (k, t) = ψ f 0 (k, ξ)e (-λ+iξ.k)t dξ, ∞ 0 e -2λt | ρ ψ (k, t)| 2 dt = ∞ t=0 ψ f 0 (k, ξ 1 )ψ f 0 (k, ξ 2 )e -2λ-i(ξ 1 -ξ 2 ).k t dtdξ 1 dξ 2 = ψ f 0 (k, ξ 1 ) ψ f 0 (k, ξ 2 ) 1 2λ -i(ξ 1 -ξ 2 ).k dξ 1 dξ 2 ≤ C |k| ψ f 0 (k, ξ) 2 L 2 (R ξ )
Before we explain this last line, we point out that the difference with the stochastic case appears in the Fourier multiplier, here 1 2λ-i(ξ 1 -ξ 2 ).k and in the stochastic case 1 2λ+|(ξ 1 -ξ 2 ).k| 2 . The homogeneity in k compared to ξ is the same (hence the same gain of regularity) but the decay in λ is different: hyperbolic in the deterministic case, parabolic in the stochastic case.

Following the argument for the proof of Theorem 2.1, in the case at hand, after reduction to the direction of k for ξ, the convolution kernel

1 2λ-iξ|k| is not in L 1
ξ . However, it is still bounded by C |k| as an operator on L 2 ξ , e.g., because its Fourier transform is 1 |k| H(η)e -λη/|k| and it is bounded by 1/|k| independently of λ; that this operator is bounded in L 2 ξ relies however on the general structure of a Calderón-Zygmund operator (see [START_REF] Stein | Singular Integrals and Differentiability Properties of Functions[END_REF] for instance) and not on its specific form that allows to compute exactly its Fourier transform.

B Deterministic averaging lemma, time regularity

For the stochastic averaging lemma, we have presented a proof of time regularity that leads to a gain of 1 4 th derivative. We present here the same proof in the deterministic case so as to show it recovers the optimal result with a gain of 1 2 derivatives.

Theorem B.1 (Deterministic time averaging). Take B(t) = t, λ ≥ 0 and g = 0 in equation [START_REF] Bardos | Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data[END_REF], then e -λt ρ ψ ∈ Ḣ1/2 R + ; L 2 (R d ) , more precisely e -λt ρ ψ (k, t) 2

H 1/4 R + ;L 2 (R d ) ≤ C f 0 2 L 2 (R d ×R d ) .
Proof. We compute for t > s |e -λt ρ ψ (k, t) -e -λs ρ ψ (k, s)| 2 = ψ f 0 (k, ξ 1 )ψ f 0 (k, ξ 2 ) e -λt-itk.ξ 1 -e -λs-isk.ξ 1 e -λt+itk.ξ 2 -e -λs+isk.ξ 2 dξ 2 dξ 1 = ψ f 0 (k, ξ 1 ) ψ f 0 (k, ξ 2 ) e -λ(t-s)-i(t-s)k.ξ 1 -1 e -λ(t-s)+i(t-s)k.ξ 2 -1 e -2λs+isk.(ξ 2 -ξ 1 ) dξ 2 dξ 1 .

And we compute

∞ t=s e -λ(t-s)-i(t-s)k.ξ 1 -1 e -λ(t-s)+i(t-s)k.ξ 2 -1 e -2λs+isk.(ξ 2 -ξ 1 ) ds dt (t -s) 1+2β ∞ u=0 e -2λu-iuk.(ξ 1 -ξ 2 ) -e -λu+iuk.ξ 2 -e -λu-iuk.ξ 1 + 1 du u 1+2β

Compared to the stochastic case, there is one more cancellation in the singularity at u = 0 which makes that the integral converges for β = 1/2. Indeed we can write

1 ( 1 -

 11 |z| 2 2s dw (t -s) d/2 dz s d/2 dξ 2 dξ 1 For the fractional Sobolev exponent 0 < β < 1, we need theLemma4.2. ∞ s=0 ∞ t=s w,z e -λ(t-s)-iw.k.a 1 -1 e -λ(t-s)iw.k.a 2 -1 e -2λs+izk.(a 2 -a 1 ) e -)λ 2β-1 (1 + | k| 4β ) 2 + | k.(a 2 -a 1 )| 2 with k = k/ √ λ and C(β) = O( 2β) 2β ). Proof of Lemma 4.2. With the change of variable t → u = t -s, we estimate the expression of interest as ∞ s=0 ∞ u=0 w,z e -λu-iw k.a 1 -1 e -λu+iw k.a 2 -1 e -2λsiz k.(a 2 -a 1 ) e -|w| 2 2u e -|z| 2 2s dw u d/2 dz s d/2

e

  -λu-iuk.ξ 1 -1 e -λu+iuk.ξ 2 -1 e -2λs+isk.(ξ 2 -ξ 1 ) ds du u 1+2β = 1 2λ -ik.(ξ 2 -ξ 1 )

Fdξ 2 dξ 1 ≤ 2 L 2 ξ(λ 2 + 2 L 2 ξ

 122222 (a, b) = ∞ u=0 e -(a+b)u -e -au -e -bu + 1 du u 2β = R u=0 e -(a+b)u -e -au -e -bu + 1 du u 2β + ∞ R .... |F (a, b)| ≤ C(|a| 2 + |b| 2 )R 2-2β + CR -2β ≤ C(|a| 2 + |b| 2 ) β .Therefore, we can act the Fourier multiplier as in Appendix A and find∞ s=0 ∞ t=s | ρ ψ (k, t) -ρ ψ (k, s)| 2 |t -s| 1+2β dt ds = ψ f 0 (k, ξ 1 ) ψ f 0 (k, ξ 2 ) F 2λ + ik.ξ 1 , 2λ -ik.ξ 2 ) λ + ik.(ξ 2 -ξ 1 C(suppψ) ψ f 0 |k| 2 ) β λ + |k| ≤ C(suppψ) ψ f 0after choosing β = 1/2, and we arrive to the result.


Lei Yu
email: [lei.yu@univ-rennes1.fr

Mohamed Laaraiedh
email: mohamed.laaraiedh@univ-rennes1.fr

Stéphane Avrillon
email: stephane.avrillon@univ-rennes1.fr

Bernard Uguen
email: bernard.uguen]@univ-rennes1.fr

Fingerprinting Localization based on Neural Networks and Ultra-wideband signals

Keywords: Fingerprinting, Localization, Neural networks, UWB

Fingerprinting techniques have been proved as an effective techniques for determining the position of a mobile user in an indoor environment and in challenging environments such as mines, canyons, and tunnels where common localization techniques based on time of arrival (TOA) or received signal strength (RSS) are subject to big positioning errors. In this paper, a fingerprinting based localization technique using neural networks and ultra-wideband signals (UWB) is presented as an alternative. The fingerprinting database is built with signatures extracted from channel impulse responses (CIR) obtained by processing an IR-UWB indoor propagation measurement campaign. The construction of the neural networks and the adopted approach are described. Positioning performances are evaluated with different selected signatures and different sizes of the fingerprinting database.

I. INTRODUCTION

Indoor localization has received a lot of attention over the past few decades [START_REF] Liu | Survey of wireless indoor positioning techniques and systems[END_REF]. A good indoor localization system must have a high accuracy, a short training phase, a low cost, and a good robustness in the face of previously unobserved conditions. Fingerprinting based localization techniques have been proposed for indoor and challenging environments such as mines, tunnels, and hospitals where the propagation is highly affected by noise and multipath components. Fingerprinting techniques refer to identifying a specified location by relying on the data which can represent this location.

Fingerprinting localization systems are based on two phases: the off-line phase (training) and the on-line phase (localization). In this paper, during the training phase, a fingerprinting database consisting of the signatures extracted from the channel impulse response (CIR) of the Ultra-wideband (UWB) measurement campaign is built. This database benefits from the high time resolution and multipath resolvability of the UWB signals [START_REF] Gezici | Position estimation via ultra-wideband signals[END_REF], [START_REF] Hayar | Performance analysis of some fingerprinting based localization techniques using uwb signaling[END_REF], [START_REF] Taok | Fingerprinting localization using ultra-wideband and neural networks[END_REF]. In the on-line phase, this database is used to locate targeted nodes. Different techniques can be applied to train the database such as Nearest Neighbour, Support Vector Regression, and Neural Networks. Assuming that the fingerprinting database cannot be analysed mathematically due to their complexities, the neural networks are proposed in our work.

In this direction, this paper presents a contribution to fingerprinting based localization techniques using neural networks and ultra-wideband CIR. We start in section II by introducing the fingerprinting technique and the construction of the fingerprinting database. Then, we describe the neural networks technique in section III by presenting its architecture and the algorithm applied during the learning process. In section IV, the UWB measurement campaign is presented, this measurement campaign is used to evaluate and compare the different fingerprints and sizes of the databases. Finally, the results are discussed in section V and our concluding remarks and work perspectives are given in section VI.

II. FINGERPRINTING BASED LOCALIZATION TECHNIQUE

A. Training phase

The fingerprinting based localization technique could be considered as a regression scheme that maps an input vector to an output vector by using a training set. During the training phase, the fingerprinting database is constructed with the location-dependent parameters (LDPs) extracted from the CIR. Two main factors should be considered for selecting the fingerprints. Firstly, the fingerprints must have considerable spacial variability. Secondly, the fingerprints must be consistent in time. In this paper, we focus on the following LDPs:

Etot

The total energy integrated over the CIR Emax

The energy integrated over the strongest received path τrms Delay spread τpercent Delay associated with the percentile of the received energy TOA Time of arrival of the first path A fingerprinting database is expressed by:

T = (f 1 , p 1 ), (f 2 , p 2 )...(f N , p N ) (1)
where f i represents the input fingerprint pattern for the i th location and p i is the position of the i th point, given by p i = [x i , y i , z i ]. N is the total number of elements in the fingerprinting database (i.e. the size of the fingerprinting database).

The fingerprint pattern f i can be made of one or several LDPs extracted from the CIR between the transmitter and a number of reference nodes.

f i = [m i1 , m i2 , m i3 , ..., m ij ] (2)
where m i1 , m i2 , m i3 , ..., m ij represent the different fingerprints of the i th point. E.g. if E tot is selected as the fingerprint and four different receivers are the reference nodes, the definition (2) becomes:

f i = [E toti1 , E toti2 , E toti3 , E toti4 ] (3)
Certainly, we can also combine some signal features for building a combined fingerprint pattern. Assuming that we select E max and TOA as the fingerprint, the definition (2) becomes:

f i = [E maxi1 , E maxi2 , E maxi3 , E maxi4 , T OA i1 , T OA i2 , T OA i3 , T OA i4 ] (4)
We distinguish two different types of LDPs depending on whether a time reference is available (TOA) or not (E tot , E max , τ rms , τ percent ). The consumption of a known time reference has an impact on the receiver complexity.

B. Location phase

The location phase consists in the modelling of the input vector and the output vector of the fingerprinting database and then using the learned model to estimate the targeted locations. Various techniques could be implemented in the location phase. In this paper, neural networks technique is proposed to be used during the location phase.

C. Advantages vs. drawbacks

The main advantage of the fingerprinting based localization technique is that it can provide accurate location estimation in the challenging multipath environments and thus outperforms some common localization techniques based on trilateration and triangulation. Moreover, in the on-line phase, the targeted location is calculated based on the carpet values, which avoids the need for communication with a central server. No more additional access to the network is required, and hence it provides for a standalone positioning solution independent of the environment. With the fingerprinting based localization technique, we just need to focus on the choice of fingerprints and the association between the chosen fingerprints and their positions.

However, the fingerprinting database has to be large enough in order to achieve a good positioning accuracy. In addition, the fingerprinting database has to be updated when the environments change significantly. For example, if some metallic furnitures are placed in an indoor environment, we have to rebuild the fingerprinting database to take into account their effects on the signal propagation. This drawback can be very expensive especially in outdoor environments which are usually very dynamic. That is why the fingerprinting based localization technique is more suitable for indoor localization systems.

III. NEURAL NETWORKS

Neural network is a software simulation of a biological brain (sometimes called Artificial Neural Network or ANN) [START_REF] Gezici | Position estimation via ultra-wideband signals[END_REF], [START_REF] Heidari | Neural network assisted identification of the absence of direct path in indoor localization[END_REF]. It is a mathematical model which aims to recognize patterns in a database. Once the neural network has been trained on samples of the known database, it is able to predict by detecting similar patterns for future input data even if they have never been seen previously. Generally, neural network is an adaptive system which can change its structure based on the external or internal information that flows through the network during the learning phase. Modern neural networks are nonlinear statistical data modelling tools. They are usually used to model the complex relationship between the input vector and the output vector, which makes it suitable for the fingerprinting based localization technique.

In this paper, we use the feed-forward network which allows the signals to travel from input to output without feedback. Feed-forward network tends to be straight forward network that associates inputs with outputs. They are extensively applied in pattern recognition applications.

The behaviour of the neural network depends on both the input-output transfer function and the appropriate weights of the neurons which can reduce the error between desired outputs and real outputs. During the learning process, the sigmoid activation function is used to translate the input signal to outputs signal. The value of logistic sigmoid transfer function ranges from 0 to 1 (Fig. 1). In order to make neural networks applicable to a specific task, we must choose how the units are connected to one another, and we must set weights on the connections appropriately in such a way that the error between the desired output and the real output can be reduced. This learning process requires that the neural networks calculate the error derivative of weights. In other words, it must compute how the error changes as each weight is increased or decreased slightly. The back-propagation algorithm [START_REF] Heung | Fast learning method for back-propagation neural network by evolutionary adaptation of learning rates[END_REF] is the most widely used method for determining the error derivative of weights.

Fig. 2 illustrates a back-propagation multilayer network with M layers. N j represents the number of neurons in j th layer. Here, the network presents the p th pattern of training sample set with N 0 dimensional input X p1 , X p2 , , X pN 0 and N M dimensional known output response T p1 , T p2 , ..., T pN M . The actual response to the input pattern by the network is represented as O p1 , O p2 , ..., O pN M .

As shown in Fig. 3, Y ji is the output from the i th neuron in layer j for p th pattern, W jik is the connection weight from k th neuron in layer j -1 to i th neuron in layer j, and δ ji is the error value associated with the i th neuron in layer j. Fig. 3 The i th neuron in layer j.

In this paper, with the fingerprinting database built during the training phase, we design a simple neural network with only one hidden layer as shown in Fig. 4, the fingerprint pattern with j different signatures are placed in parallel in the input and the coordinates of the target in 3D are placed in the output. In general, the number of the neurons in the hidden layer is not defined. The more neurons in the hidden layer, the more complexity the system is and of course the longer computation duration is. Usually, we can set the number of the neurons using one of the following rules:

• The number of hidden neurons should be between the size of the input layer and the size of the output layer.

• The number of hidden neurons should be 2/3 of the size of the input layer, and should be larger than the size of the output layer.

• The number of hidden neurons should be less than twice the size of the input layer.

In our work, we define the number of hidden neurons using the second rule. The incremental back-propagation algorithm is applied to calculate and adjust weights for each neuron in the hidden layer. It is a supervised learning method, where weights are updated after each presenting training pattern, which means weights are adjusted many times during a single epoch.

IV. UWB MEASUREMENT CAMPAIGN

The measurement campaign has been done in the framework of the FP7 WHERE project. It has been carried out by the CEA-LETI in the SIRADEL headquarter building in Rennes, France. The aim of this indoor UWB measurement campaign was to collect UWB impulse responses in a same local area. In order to assess small-scale fading, impulse response measurements are made on several square grids. The investigation of the measurement campaign is limited to a few selected rooms and offices on the first floor of the SIRADEL building. An overall number of collected profiles has been calculated and saved: 302 (Tx positions) x 4 (Rx positions) = 1208 profiles [START_REF]Measurements of locationdependent channel features[END_REF]. The LDPs have been extracted from the profiles and used to establish the fingerprinting database.

In order to ignore the fingerprints that are too noisy due to strong shadowing, Link Quality Indicator (LQI) is defined as the ratio between the maximum signal amplitude and the maximum noise amplitude in dB scale (Estimated during the first five nanoseconds corresponding to a common time slot without any signal). In this paper, we have decided to remove fingerprints extracted from the measurement campaign with LQI less than a threshold of 10dB. The total number of profiles reduces to 165 x 4 = 660 profiles.

V. RESULTS AND DISCUSSIONS

A. Establishment of the fingerprinting database

Fingerprinting databases with different sizes are built to evaluate the influence of the training points density on the positioning performance. Fig. 5 shows the different training and learned points. The training points are plotted with blue circles and represents 20% of all points (165 x 20% = 33 points). The remaining 80% points (in green x markers) with unknown locations are to be estimated using the learnt neural network. The red diamond markers represent the four anchors with which measurements of LDPs are performed. Fig. 5 Training and learnt points of the fingerprinting database.

B. Choice of the fingerprint pattern

Different fingerprint patterns with different LDPs could be used in the neural network. First, we compare the localization performances when using different no time referenced available fingerprints (i.e. E tot , E max , τ rms , and τ percent ). For this, we represent in Fig. 6 the different cumulative density functions (CDF) of fingerprinting using these no time referenced fingerprints. This figure shows that the fingerprint E tot can provide the best localization accuracy (90% of the points can be estimated with the error less than 2 meters). Table II summaries the performances by giving the values of the mean and the standard deviation (Std) of positioning error for all fingerprints. As discussed above, TOA is a time referenced fingerprint, which can be estimated by setting a proper threshold above the noise floor [START_REF] Dardari | Analysis of threshold-based toa estimators in uwb channels[END_REF], [START_REF] Guvenc | Threshold-based toa estimation for impulse radio uwb systems[END_REF]. Usually, TOA is accurately estimated with a higher precision especially in UWB networks. This high accuracy is very interesting to enhance the positioning accuracy of proposed fingerprinting technique. In order to enhance the positioning accuracy, the combined fingerprint pattern using both the best no time referenced available fingerprint (i.e. E tot ) and the TOA is applied to the neural networks.

As expected, Fig. 7 shows that TOA outperforms E tot and that the fusing TOA with E tot fingerprint pattern enhances the localization accuracy. More than 95% of the estimated points are localised with a positioning error less than 1 meter. Table . III summarises these performances. One can see that the adding TOA to E tot enhances much better the positioning accuracy than adding E tot to TOA. This is explained by the fact that TOA is usually much more precise than received signal strength indicators (i.e. E tot or E max ). Both actual and estimated points using the combined fingerprint pattern of E tot and TOA are shown in Fig. Since it is interesting for operators to reduce complexity and resource consumptions, exploitation of TOA estimates must be reduced as much as possible because they consume much resources and cause overhead [START_REF] Laaraiedh | Comparison of hybrid localization schemes using rssi, toa, and tdoa[END_REF]. Besides, RSSI related fingerprints are usually available without any additional costs. For these reasons, a promising localisation scheme consists in using all available RSSI based fingerprints (i.e. E tot ) and reducing the number of TOA. This scheme seems to be a very interesting opportunistic localization scenario.

In order to show the effect of fusion of RSSI with TOA on positioning accuracy, we assume all four E tot fingerprints are available and we gradually increase the number of additional TOAs. Fig. 9 plots the CDFs of positioning error over the area with respect to the number of added TOAs. The evolution of average positioning error with respect to the number of added TOAs is shown in Fig. 10. These two figures show a gradual enhancement provided by increasing the number of used TOA. Nevertheless, adding some TOA may deteriorate the localization accuracy. For example, the fusion of E tot with only one TOA (i.e. RSSI+1TOA) case which is represented in red line. This can be explained by the fact that some fingerprints are very imprecise or that they come from a misplaced device with respect to other devices. fingerprint pattern, Fig. 11 plots the CDFs of the positioning error with respect to the different sizes of the fingerprinting database. It shows that generally, the accuracy of localization tends to be better with the increasing of the size of the fingerprinting database. However, when the fingerprinting database reaches a given size, the technique tends to perform the same positioning accuracy, with no significant improvement. Table .  IV presents the statistical positioning errors using different sizes of the database.

Fig. 1

 1 Fig. 1 Sigmoid logistic transfer function.

Fig. 2

 2 Fig. 2 Back-propagation network.

Fig. 4

 4 Fig. 4 Neural networks architecture.

Fig. 6

 6 Fig. 6 CDF of positioning error with different no time referenced available fingerprints.

Fig. 7 CDF

 7 Fig. 7 CDF of positioning error for simple and combined fingerprint patterns.

Fig. 8

 8 Fig. 8 Actual vs. estimated positions using E tot and TOA.

Fig. 9 Fig. 10

 910 Fig. 9 Effect of gradually adding TOA to RSSI fingerprints.

Fig. 11

 11 Fig. 11 CDF of positioning error using different sizes of the fingerprinting database.

TABLE I Location

 I

-dependent parameters.

TABLE II Summary

 II of fingerprinting performances for different fingerprints.

	Fingerprint	Mean (m)	Std (m)
	Etot	1.055	0.765
	Emax	1.691	1.290
	τrms	3.893	3.291
	τpercent	7.122	4.952

TABLE III

 III

		Summary of fingerprinting performances for
	simple and combined fingerprint patterns.
	Fingerprint	Mean (m)	Std (m)
	Etot	1.055	0.765
	T OA	0.455	0.297
	Etot + T OA	0.377	0.255

TABLE IV

 IV Summary of fingerprinting performances with respect to the database size. UWB measurement campaign. The feedforward neural network with incremental back-propagation algorithm is used to learn the database and to locate the targeted positions. Different types of fingerprints and different sizes of the fingerprinting database are considered to evaluate the positioning performance. Simulations have shown that RSSI-based fingerprint can provide good localization accuracy and that the fusion of very precise TOA fingerprints can drastically improve the positioning accuracy. In addition, we have shown that increasing the number of learnt points does not mean usually to enhance performances. A fair database size is sufficient to reach the best performances. Future work consists in using Motley Keenan Multiwall model and ray tracing techniques to replace measured by simulated databases .

	Size of the database	Mean (m)	Std (m)
	10%	1.373	1.266
	20%	1.055	0.765
	25%	0.994	0.614
	50%	0.962	0.596
	extracted from an

ACKNOWLEDGMENT

This work has been performed in the framework of the ICT project ICT-248894 WHERE2, which is partly funded by the European Union.