Eric Lacot
email: eric.lacot@ujf-grenoble.fr

Wilfried Glastre

Olivier Jacquin

Olivier Hugon

Hugues Guillet

Hugues Guillet De Chatellus

Optimization of an autodyne laser interferometer for high-speed confocal imaging

INTRODUCTION

When a frequency shift is introduced between the two beams of an interferometer, one realizes the so-called heterodyne interferometry. Resulting from this shift, the interference between the two waves produces an intensity modulation at the beat frequency, which can be measured by a photo-detector. In this paper, we refer only to autodyne laser interferometry where the heterodyne wave mixing takes place inside the cavity of the laser source and is finally indirectly detected by a photodiode.

Since the development of the first laser in 1960, laser heterodyne interferometry has become a useful technique on which many high accuracy measurement systems for scientific and industrial applications are based [START_REF] Yoshizawa | Handbook of optical metrology: Principles and Applications[END_REF]. Since the pioneer work of K. Otsuka, on self-mixing modulation effects in class-B laser [START_REF] Otsuka | Effects of external perturbations on LiNdP 4 0 12 Lasers[END_REF] the sensitivity of laser dynamics to frequency-shifted optical feedback has been used in autodyne interferometry and metrology [START_REF] Otsuka | Self-Mixing Thin-Slice Solids-State Laser Metrology[END_REF], for example in self-mixing laser Doppler velocimetry [START_REF] Otsuka | Highly sensitive measurement of Doppler-shift with a microchip solid-state laser[END_REF][START_REF] Okamoto | Ultrahighly sensitive laser-Doppler velocity meter with a diode-pumped Nd:YVO4 microchip laser[END_REF][START_REF] Kawai | Ultrahigh-Sensitivity Self-Mixing Laser Doppler Velocimetry with Laser-Diode-Pumped Microchip LiNdP 4 O 12 Lasers[END_REF][START_REF] Suddo | Determination of velocity of self-mobile phytoplankton using a self thin-slice solid-state laser[END_REF], vibrometry [START_REF] Otsuka | Real-time nanometer vibration measurement with self-mixing microchip solid-state laser[END_REF][START_REF] Muzet | Experimental comparison of shearography and laser optical feedback imaging for crack detection in concrete structures[END_REF][START_REF] Lacot | Phase-sensitive laser detection by frequency-shifted optical feedback[END_REF], near field microscopy [START_REF] Gilles | Near-field amplitude and phase measurements using heterodyne optical feedback on solid-state lasers[END_REF][START_REF] Blaize | Phase sensitive optical near-field mapping using frequency-shifted laser optical feedback interferometry[END_REF] and laser optical feedback imaging (LOFI) experiments [START_REF] Lacot | Laser optical feedback tomography[END_REF][START_REF] Witomski | Synthetic aperture laser optical feedback imaging using galvanometric scanning[END_REF][START_REF] Hugon | Cell imaging by coherent backscattering microscopy using frequency shifted optical feedback in a microchip laser[END_REF][START_REF] Hugon | Coherent microscopy by laser optical feedback imaging (LOFI) technique[END_REF]. Compared to conventional optical heterodyne detection, frequency-shifted optical feedback shows an intensity modulation contrast higher by several orders of magnitude and the maximum of the modulation is obtained when the shift frequency is resonant with the laser relaxation oscillation frequency [START_REF] Lacot | Coherent laser detection by frequency-shifted optical feedback[END_REF]. In this condition, an optical feedback level as low as -170 dB (i.e. 10 17 times weaker than the intracavity power) has been detected [START_REF] Okamoto | Ultrahighly sensitive laser-Doppler velocity meter with a diode-pumped Nd:YVO4 microchip laser[END_REF].

In previous papers [START_REF] Lacot | Coherent laser detection by frequency-shifted optical feedback[END_REF][START_REF] Lacot | Comparative study of autodyne and heterodyne laser interferometry for imaging[END_REF][START_REF] Jacquin | Expérimetal comparison of autodyne and heterodyne laser interferometry using an Nd:YVO4 microchip laser[END_REF], we have demonstrated that in autodyne interferometry, the main advantage of the resonant gain (defined by the ratio between the cavity damping rate and the population-inversion damping rate of the laser) is to raise the laser quantum noise over the detector noise in a relatively large frequency range around to the laser relaxation frequency.

Moreover if the detection bandwidth is narrower than the laser dynamical relaxation width, the signal to noise ratio (SNR) of a LOFI setup is frequency independent, and more importantly, shot noise limited. We have also established that to maximize the dynamical range of a LOFI setup, the best value of the shift frequency is not the relaxation frequency, but the frequency at which the amplified laser quantum noise is equal to the detection noise level [START_REF] Lacot | Comparative study of autodyne and heterodyne laser interferometry for imaging[END_REF][START_REF] Jacquin | Expérimetal comparison of autodyne and heterodyne laser interferometry using an Nd:YVO4 microchip laser[END_REF].

For high-speed imaging, we need to decrease the signal acquisition time of our LOFI setup and therefore to work with a detection bandwidth larger than the laser dynamical relaxation bandwidth. Under this condition the laser transient dynamic cannot be ignored and the LOFI SNR becomes frequency dependent. The main objective of this paper is to determine the best laser dynamical parameters and the best experimental conditions to obtain high quality images (i.e. shot-noise limited) as fast as possible.

This paper is organized as follows. Firstly, after a basic description of our LOFI set-up (i.e. our autodyne interferometer) for confocal imaging, we briefly recall the expression of the LOFI permanent signal induced by the beating inside the laser cavity. Secondly, we determine the stationary LOFI SNR for different values of the experimental acquisition time compare to the laser dynamical response time. Thirdly, for autodyne imaging, we calculate the level of the transient LOFI signal appearing when during the laser scanning (i.e. from one image pixel to the next one), the target under investigation presents discontinuous physical properties. Finally, we determine the best laser parameters for high speed autodyne imaging with a shot-noise limited detection (i.e. highest quality image as fast as possible). In each section, analytical predictions are confirmed by numerical simulations. ) in the kilohertz range [START_REF] Jacquin | Expérimetal comparison of autodyne and heterodyne laser interferometry using an Nd:YVO4 microchip laser[END_REF][START_REF] Zaykowski | Single -frequency microchip Nd lasers[END_REF]. The laser is therefore a class-B laser (

AUTODYNE SIGNAL

A. LOFI set up

R R F  1 
). The laser beam is sent on the target, through a frequency shifter. A part of the light diffracted and/or scattered by the target is then reinjected inside the laser cavity after a second pass through the frequency shifter. Therefore, the optical frequency of the reinjected light is shifted by e F . This frequency can be adjusted and is typically of the order of the laser relaxation frequency R F . For the geometrical point of view, the laser beam waist and the laser focal spot on the target under investigation are optically conjugated. At this point, one can already notice that, compared to a conventional heterodyne setup, the autodyne setup shown here does not require complex alignment. Indeed, the LOFI setup is even always self-aligned because the laser simultaneously fulfills the function of the source (i.e. photons-emitter) and of the photo-detector (i.e. photons-receptor).

The optical feedback is characterized by the electric field complex reflectivity ) takes into account the target albedo, the numerical aperture of the collection optics, the frequency shifters efficiencies and the transmission of all optical components (except for the beam splitter which is addressed separately) and the overlap of the retro-diffused field with the Gaussian cavity beam (confocal feature). The coherent interaction (beating) between the lasing electric field and the frequency-shifted reinjected field leads to a modulation of the laser output power. For the detection purpose, a fraction of the output beam of the microchip laser is sent to a photodiode by means of a beam splitter characterized by a power reflectivity bs R . The photodiode is assumed to have a quantum efficiency of 100%. The voltage delivered by the photodiode is finally analyzed by a lock-in amplifier which gives the LOFI signal (i.e. the magnitude and the phase of the retro-diffused electric field) at the demodulation frequency e F [START_REF] Hugon | Cell imaging by coherent backscattering microscopy using frequency shifted optical feedback in a microchip laser[END_REF][START_REF] Hugon | Coherent microscopy by laser optical feedback imaging (LOFI) technique[END_REF]. The lock-in amplifier is characterized by its integration time int T . Experimentally, the LOFI images are obtained pixel by pixel (i.e.

B. LOFI Modelling

In the case of weak (

1  e R
) frequency shifted optical feedback, the dynamical behavior of a reinjected solid-state laser can be described by the following set of equations [START_REF] Lacot | Phase-sensitive laser detection by frequency-shifted optical feedback[END_REF][START_REF] Lacot | Coherent laser detection by frequency-shifted optical feedback[END_REF][START_REF] Lacot | Comparative study of autodyne and heterodyne laser interferometry for imaging[END_REF]:

  t F E BN N N dt dN N c     2 1 0 1   , ( 1a
)         t F t F E R R E BN dt dE c E e e c bs e c c c c           2 cos 1 2 1 , (1b)
where, N is the population inversion,

  t F N and   t F c E
, which have a zero mean value and a white noise type correlation function [START_REF] Kolobov | Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources[END_REF][START_REF] Bramati | Effects of pump fluctuations on intensity noise of Nd:YVO4 microchip lasers[END_REF][START_REF] Khanin | Principles of laser dynamics[END_REF].

The laser model presented above can be applied to three levels or four levels lasers with the condition that the lifetime of the upper level of the pumping transition is very short compared to the lifetime of the upper level of the laser transition. For example, this is condition is satisfied in a three levels laser such as erbium lasers as well as for a four levels laser such as neodymium laser. In the set of Eqs. (1), the feedback time delay ( ), linked to the optical round trip between the laser and the target is completely neglected. It means, that we only consider the case where the round trip time is shorter than the frequency shift (

1 2    e F ).

C. LOFI stationary signal

In the set of Eqs. (1), the periodic functions express the beating (i.e. the coherent interaction) between the lasing and the feedback electric fields. The net gain of the laser is then modulated by the re-injected light at the optical shift frequency e F . In the linear regime, the photon output rate

  2 c c out t E ) t ( p  
(number of photons per second) is therefore periodically modulated [START_REF] Lacot | Coherent laser detection by frequency-shifted optical feedback[END_REF]:

      e e out e bs e out e e out t F p R R F G p R F t p       2 cos 1 ) ( 2 , , , (2)
where  

1 1   r B p c out  
is the mean value the photon output rate with

                2 2 1 2 2 2 2 2 2 2 1 ) 2 2 ( 1 2 2 2 2 2 2 2 ) ( R e R c F e R e R e R c e F F F F F F F G R R                     (3)
where for a class-B laser (

R R F  1  ),     2 ) 1 ( 2 1     r F c R R
is the frequency of the laser relaxation oscillations and

  r R 1 2   
is the damping time of the relaxation oscillations [START_REF] Khanin | Principles of laser dynamics[END_REF]. At this point one can notice that for a class-B laser, R  which is very long compared to the photon lifetime in the cavity (

c R   1 
), is therefore the laser characteristic response time [START_REF] Khanin | Principles of laser dynamics[END_REF].

In a LOFI interferometer, a particularly interesting situation is the resonance case (

R e F F  )
where the LOFI signal gain (i.e. the autodyne gain) becomes:

2

) ( R c R F G    . ( 4
)
For a microchip laser, this ratio is typically of the order of 10 4 -10 6 [START_REF] Lacot | Coherent laser detection by frequency-shifted optical feedback[END_REF][START_REF] Jacquin | Expérimetal comparison of autodyne and heterodyne laser interferometry using an Nd:YVO4 microchip laser[END_REF] and the main advantage of the LOFI detection technique seems to come from this resonant amplification of the optical wave mixing [START_REF] Otsuka | Self-Mixing Thin-Slice Solids-State Laser Metrology[END_REF][START_REF] Lacot | Laser optical feedback tomography[END_REF][START_REF] Lacot | Coherent laser detection by frequency-shifted optical feedback[END_REF].

Using Eq. ( 2), one can define the modulation contrast (MC) of the autodyne wave mixing:

    e bs e out e e out e e R R F G p R F p R F MC     1 ) ( 2 ) , ( , . (5)
In a LOFI experiment, because the laser simultaneously fulfills the functions of the source and of the detector, we assume to simply define the saturation level as the effective reflectivity corresponding to a maximum modulation of the laser output power (MC=1):

      2 2 1 1 1 4 1 bs e e Sat R F G F R     (6)
Finally, using a lock-in amplifier, the LOFI signal at the demodulation frequency e F is given by:

      
p R R F G R R F p R R F S     . ( 7
)

STATIONARY SNR OF AN AUTODYNE INTERFEROMETER

A. Stationary LOFI SNR

Without optical feedback (

0  e R
), the set of Eqs. ). Using the Wiener-Khintchine theorem, on can obtain the power density spectrum of the laser output power quantum fluctuations [START_REF] Lacot | Coherent laser detection by frequency-shifted optical feedback[END_REF][START_REF] Kolobov | Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources[END_REF][START_REF] Bramati | Effects of pump fluctuations on intensity noise of Nd:YVO4 microchip lasers[END_REF]:

                    F G t p F F F F p F PD out R R R c out Laser 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2             . ( 8
)
The LOFI noise power obtained after the photodiode detection (i.e. after the beam splitter reflection) and the lock-in amplifier filtering at the modulation frequency ( e F ) is then given by:

            dF T F F F F PD R T F N e Laser bs e Laser 2 int int int 2 , 2 , (9a)
where for an integration time int T :

    2 2 int 2 int 2 int int 2 1 1 1 , F T T T F F    (9b)
is assumed to be a first order power filter . By combining Eqs. ( 8) and ( 9), one finally obtains for a class-B laser (

R R F  1 
), the following analytical expression of LOFI noise induced by laser quantum noise:

                               2 2 int 2 int int int 2 2 2 1 1 1 1 2 1 , R e R c R R out bs e Laser F F T T T t p R T F N       , (10)
and by combining Eqs. ( 7) and ( 10), one finally obtains the stationary LOFI SNR:

      int int , , , , T F N R F S T R F SNR e Laser e e LOFI e e  . (11)
s photons p out / 10 2 . 3 17   ( mW P out 60  at nm 1064   ) kHz F R 356  and 14  R R F  ( 02 . 1  r , 1 9 10 5    s c  , 1 3 1 10 5    s  ).
For each integration time the continuous line shows the exact value of the LOFI SNR [Eq. ( 11)], while the dash line shows the corresponding LOFI shot-noise limit [Eq. ( 13)].

B. Acquisition with a fast response time laser (

R T   int )
If the lock-in integration time is long compare to the laser response time (

R T   int
), the noise [Eq. [START_REF] Lacot | Phase-sensitive laser detection by frequency-shifted optical feedback[END_REF]] is then simply given by:

      int 2 2 int 2 1 1 ) 2 2 ( 1 2 2 , T F F t p R T F N R e R c out bs R e Laser          (12a)
or identically, by introducing the expression of the gain given by Eq. ( 3):

      int int 2 1 2 , T F G t p R T F N e out bs R e Laser    . ( 12b
)
At this point, one can notice that the resonant amplification gain ) F ( G e present in the LOFI signal [see Eq. ( 7)], is also present in the LOFI noise and, as a result, the SNR of the LOFI setup is frequency independent:

    int int 1 , , T p R R R T R F SNR out bs bs e R e e      , ( 13
)
As mentioned above, the relaxation frequency seems to be of no particular importance [see Fig.

The LOFI set-up is therefore shot noise limited and the beam splitter reflectivity ( bs R ) appearing in Eqs. ( 7) and [START_REF] Lacot | Phase-sensitive laser detection by frequency-shifted optical feedback[END_REF] and finally in the right hand term of Eq. ( 14), can be interpreted as the quantum efficiency of the LOFI detection where the laser modulation produced inside the laser cavity signal is finally indirectly detected by the photodiode after the beam splitter.

Consequently, and throughout the rest of the manuscript, Eq. ( 13) is what we will call the LOFI shot noise limit.

C. Acquisition with a slow response time laser (

R T   int )
If now the lock-in integration time is short compared to the laser response time (

R T   int
), the laser quantum noise [Eq. [START_REF] Lacot | Phase-sensitive laser detection by frequency-shifted optical feedback[END_REF]] is then approximately given by:

      int int 2 2 int int 2 1 2 2 4 2 2 , T T F F T t p R T F N R R e c out bs R e Laser                  (15)
Now, to analyze the LOFI SNR, three different cases need to be studied depending on the interval between the shift frequency and the relaxation frequency.

Firstly, if we work very far away from resonance

(   R R e T F F  1 1 int   
), then the LOFI noise [Eq. [START_REF] Hugon | Cell imaging by coherent backscattering microscopy using frequency shifted optical feedback in a microchip laser[END_REF]] and LOFI signal [Eq. ( 7)] can respectively and approximately be given by: [START_REF] Lacot | Coherent laser detection by frequency-shifted optical feedback[END_REF] and finally one obtains (far away from the resonance frequency), the following approximate expression of the LOFI SNR:

    int int int 2 1 2 2 4 2 , T T F F t p R T F N R R e c out bs R e Laser         (16)     R e c out e bs bs e e LOFI F F p R R R R F S      2 1 2 2 1 2 ,
    R out bs bs e R e e T T p R R R T R F SNR   int int int 1 , ,     (18)
A brief comparison of Eq. ( 18) and of Eq. ( 13) (which gives the LOFI shot noise limit), shows that for: R T   int , the LOFI SNR is lower than the LOFI shot noise limit and becomes frequency independent when working very far away from the resonance frequency. In agreement with the theoretical predictions, Fig. 2(c) shows that the LOFI SNR (continuous line) saturates and therefore becomes frequency independent at very low and very high frequencies. One can also observe that in these conditions, the LOFI SNR is smaller than the corresponding LOFI shot noise limit (dash line) by a multiplicative factor given by:

1 int  R T  . Secondly: if we now work at resonance ( R e F F 
), then the LOFI noise and the LOFI signal are given by: [START_REF] Zaykowski | Single -frequency microchip Nd lasers[END_REF] and finally one easily obtains the resonant value of the LOFI SNR:

    R R c out bs R R Laser t p R T F N     2 1 2 2 , int   (19)     2 2 1 2 , R c out e bs bs e R LOFI p R R R R F S    
      int int int 1 1 , , T T p R R R p R R R T R F SNR R out bs bs e R out bs bs e R e R           (21)
The middle part of Eq. [START_REF] Kolobov | Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources[END_REF] shows that the LOFI SNR is now independent from the integration time ( int T ). This surprising effect arises because the laser noise power spectrum [Eq. ( 8)] have a Lorentzian type profile leading to a limit noise value [See Eq. ( 20)] when the detection bandwidth is large compared to the resonance width (

R T  1 1 int 
). A brief comparison of Eq. ( 21) and Eq. [START_REF] Lacot | Laser optical feedback tomography[END_REF] shows that, for a laser with a slow response time (

R T   int
), the LOFI SNR at resonance is higher than the standard LOFI shot noise limit. In agreement with Eq. ( 21), Fig. 2(c) shows that the LOFI SNR (continuous line) at resonance is higher than the corresponding LOFI shot noise limit (dash line) by a multiplicative factor given by:

1 int  T R  .
Thirdly, in the intermediate situation, where:

  R R e F F T  1 1 int   
, one can determine the frequency shift ( e F ) for which the LOFI SNR is equal to the LOFI shot noise limit. By equalizing Eqs. (12a) and ( 15) one obtains: ) and smaller than the LOFI shot-noise limit far way from the relaxation frequency (i.e. when:

int 2 1 T F F R R e    
). More precisely the LOFI SNR is larger (by a factor given by

1 int  T R 
) than the LOFI shot noise limit when working at the resonance frequency and smaller (by a factor given by ), the numerical results shown on Fig. 3(a) are in good agreement with the analytical predictions given by Eq. ( 13). Indeed, using a laser with a fast response time

( R T   int
), the simulated LOFI SNR is almost frequency independent and the numerical simulations show values that are very close to the analytical predictions. Indeed, by using Eq. ( 13), with a laser output power of 1, which correspond to the LOFI shot noise limit. A closer look at the results of Fig. 3(a) shows a small frequency dependence which seems to be more important for the shortest integration time.

This small frequency dependence is also in agreement with the theoretical prediction [see Fig. Firstly, using Eq. ( 6), with a LOFI gain   Secondly, the LOFI SNR shows an anti-resonance phenomenon for the longest integration times and a resonance phenomenon for the shortest integration times. In Fig. 3 ) than the LOFI shot noise limit values given in Table 1

. b) a)   5 10 8   R F G   4 10 1   R F G R T  018 . 0 int  R T  06 . 0 int  R T  18 . 0 int  R T  8 . 1 int  R T  5 . 1 int  R T  5 int  R T  15 int  R T  150 int  µs R 4   µs R 333   R T  6 . 0 int  R T  50 int  1.

AUTODYNE IMAGING

A. Transient LOFI signal

Suppose now that due to the scanning of the laser beam on the target under investigation, the target properties suddenly change at a time t=0 (for example at the edge of the target) with:

 

C R F G t t F p R R F G p r F t p                       2 cos 1 ) ( 2 exp 2 cos 1 ) ( 2 , , 0 2 , 2 , 2 ,
      c e e e e C R R      cos cos cos 2 , 2 , 1 , 1 , (24a)         c R R c e R e e e R e e C F C F F R F F R        cos 2 1 sin sin sin 2 , 2 , 1 , 1 ,  (24b) For a class-B laser( R R F  1 
), Eqs. [START_REF]The saturation effect observed numerically can't be obtained analytically[END_REF] shows that, whatever the experimental conditions, a good approximation of the amplitude of the transient oscillations (i.e. the order of magnitude of C ) is given by: At the output of the lock-in amplifier, using an integration time int T and a first order filter, the transient LOFI signal at the demodulation frequency e F can be express in the time domain by:

  1 , 2 , 2 , 1 , 2 , 1 ,
                      0 int int int 2 exp 1 ) ( 2 2 exp 1 , , dt t F j t T p C R F G R T t T T C F T e R
    c R R R t t t T              cos exp 
, and where the averaging shape

         int int
       2 2 int int int 2 2 1 1 1 1 2 1 2 , , R e R out bs e bs e LOFI F F T T p C R F G R T C F T                 . (27a)
or identically by introducing the LOFI signal given by Eq. ( 7):

      2 2 int int 2 , 2 , int 2 2 1 1 1 1 , , , R e R e e e LOFI e LOFI F F T T R C R F S T C F T                (27b)
      2 2 int int 2 , 2 2 1 , , , R R e R R e LOFI e e LOFI F F T T C F T R F S          (28)
which can be reduce to the following inequality:

    R e LOFI e e LOFI T C F T R F S    int 2 , , , , (29)
Fig. 4(a) confirms that by using a laser with a fast response time, the stationary LOFI signal is generally higher than the transient LOFI signal, whatever the working frequency e F is.

C. Imaging with a slow response time laser (

      2 int 2 int 2 , 2 2 1 , , , T F F T C F T R F S R e R e LOFI e e LOFI        , ( 30
)
and two different cases need to be analyzed. Firstly, if we work near the resonance frequency,

with int 1 T F F R e  
, Eq. (30) reduces to: ), Eq. (30) gives:

    R R e LOFI e R e LOFI T C F F T R F F S      int 2 , , , , (31).
    R e LOFI e e LOFI T C F T R F S    int 2 , , , , (32)
and the transient dynamics can be ignored compared to the stationary dynamics. Fig. 4(c) confirms that by using a laser with a slow response time, the stationary LOFI signal is higher than the transient signal when working far way the relaxation frequency At this point one can also notice that by using a laser with a slow response time (

 

). In a previous experimental paper [START_REF] Jacquin | Expérimetal comparison of autodyne and heterodyne laser interferometry using an Nd:YVO4 microchip laser[END_REF] we have shown that the best working frequency called  , e F is the frequency at which the laser quantum noise is equal to the detector noise level. In the mentioned paper, the laser time response is

(     53 , , , int 2 , 2 ,    T R F T R F S e e LOFI e e LOFI
). Note also that these experimental conditions are also compatible with our starting hypothesis (

1 12 int    e F T
).

D. Numerical simulation

In the most general situation, the LOFI SNR is given by the ratio between the LOFI signal divided by the detector noise plus the laser quantum noise plus the transient fluctuation. Here, our main objective is to compare the laser quantum noise with the transient fluctuations which can become important for short integration times (i.e. fast imaging condition). We have therefore neglected the detector noise in the current numerical study.

To show the effect of the transient dynamics on LOFI imaging, we have compared 1D scans extracted from the measured output power modulation. These scans have been obtained from the numerical integration of the set of Eqs. (1) (with no averaging). The target under investigation is a symmetric reflectivity pyramid composed of four levels which allows the observation of the transient dynamics effects during the scan in the case of an effective reflectivity increase ( ). For the current numerical study, we have chosen very low values of the effective reflectivity to study the LOFI sensitivity under ultimate conditions where the transient dynamics can be of the same order of magnitude than the laser quantum noise. Fig. 5 shows the numerical results obtained with the two lasers already studied in the section 3 of the present manuscript. To visually separate the transient dynamics effects from the noise effects on the LOFI images, the solid curves are numerically realized without laser quantum noise (i.e. without the Langevin noise forces), while the curves with circles combine both effects (transient signal and quantum noise). For the laser having the lowest value of the LOFI gain (  

R R e s T F F a   5 20 , ) int    R R e s T F F b   5 20 , 5 . 1 ) int     R R e s T F F d   06 . 0 20 , 5 . 1 ) int     R R e s T F F c   06 . 0 20 , ) int   
       e R sat R F R
). The saturation effect is also visible in Table 3 where for both SNR and MC, the numerical results [obtained from Fig. 5(c)] are always lower than the analytical ones, obtained from Eq. ( 5) for the MC and from Eq. ( 11) for the calculation of the SNR [START_REF]The saturation effect observed numerically can't be obtained analytically[END_REF].

At this point one can also notice that due to the very high value of the transient LOFI signal In practical LOFI experiments, to be shot noise limited, the laser quantum noise needs to be just above the detector noise. So finally, this is the detector noise level which determines the lowest possible value of the LOFI gain [START_REF] Lacot | Comparative study of autodyne and heterodyne laser interferometry for imaging[END_REF][START_REF] Jacquin | Expérimetal comparison of autodyne and heterodyne laser interferometry using an Nd:YVO4 microchip laser[END_REF] and therefore the shortest possible value of the laser response time. For this particular time, the laser used in the LOFI experiment is optimized and allows to obtain image as fast as possible with a shot noise limited detection. In a previous paper [START_REF] Lacot | Comparative study of autodyne and heterodyne laser interferometry for imaging[END_REF], we have shown that for a detection noise level characterized by a noise equivalent power:

NEP (

Hz W

). The optimum value for le LOFI gain is given by:

  out bs opt c R opt p hc NEP R r F G    2 1 ) ( 1           (34)
which allows to determine the optimum value of the laser response time :

  out c bs opt opt R p hc NEP R r     2 2 2 1 ,           (35)
For example, for a laser with an output power ), a cavity damping rate and above all requires a rapid electronic detection (with a gigahertz bandwidth) which is technically much more complicated to carry out than an electronic setup with a megahertz bandwidth.

Even if a laser diode seems to be an interesting candidate for LOFI experiments, the theoretical and experimental study needs to be done to verify this possibility. Indeed the laser rate equations used in this paper are not completely correct to describe the dynamical behavior of a laser diode. Indeed, our modeling doesn't take into account of the phase amplitude coupling of the laser electric field (i.e. the Henry factor) occurring inside the laser cavity of a laser diode and also doesn't take into account of the optical feedback time delay which cannot be neglected for a laser diode with a laser relaxation frequency in the gigahertz range. Indeed, for Therefore even if the laser diode seems to be a good candidate, the comparison between a laser diode and a solid-state laser for autodyne interferometry cannot be made so directly and therefore needs to be made carefully.

CONCLUSION

In a LOFI setup, the beating between the reference beam and the signal beam takes place inside the laser cavity and therefore the laser fulfills simultaneously the roles of emitter and detector of photons. In these conditions, the laser relaxation oscillations play a leading role both in the laser and a very high value of the relaxation frequency (in the gigahertz range) seems to be an interesting candidate compared to microchip laser with a response time of several tens of microsecond and a relaxation frequency in the megahertz range.

AF

  schematic diagram of the LOFI experimental setup (i.e. the autodyne experimental interferometer) is shown in Fig. 1. Typically the laser is a CW microchip with an output power out P of several milliwatts and a typical relaxation oscillation frequency R in the megahertz range and a damping rate of the relaxation oscillation ( R  1

  target, where the phase e  describes the optical round trip between the laser and the target, while the effective power reflectivity (

Fig. 1 .

 1 Fig. 1. Schematic diagram of the LOFI interferometer setup for scanning microscopy. L 1 , L 2 and L 3 : Lenses, BS: Beam Splitter with a power reflectivity R bs , GS: Galvanometric Scanner, FS Frequency Shifter with a round trip frequency-shift F e , PD: Photodiode with a white noise spectrum. The lock-in amplifier is characterized by its integration time int T . The laser is characterized by its output power out p (photons/s), its

  point by point, line after line) by a full 2D galvanometric scanning and the necessary time needs to obtain an image composed of N pixels is roughly given by: int T N  . For high speed imaging (i.e. high cadence imaging), one needs to use a value of int T as small as possible. To determine the SNR of the obtained LOFI images, int T needs to be compared with the response time of the class-B laser ( R  ). In this paper, whatever the temporal values of int T (in the millisecond or microsecond range), we refer to a fast response time laser when:

cE

  is the slowly varying amplitude of the laser electric field, B is related to the Einstein coefficient, 1  is the decay rate of the population inversion, c  is the laser cavity decay rate and 0 1 N  is the pumping rate. Regarding the noise, the laser quantum fluctuations are described by the conventional Langevin noise functions

  the amplification gain of the autodyne waves mixing with:

( 1 )

 1 allows us to study the laser quantum fluctuations induced by the Langevin noise terms (

Fig. 2

 2 Fig.2shows the evolution of the stationary LOFI SNR (

2

  (a) for comparison]. At this point, one can notice that the condition:

  conclude this section, by reminding that the stationary LOFI SNR ( independent [see Fig.2(a)] and above all shot noise limited for a laser with fast response time ( the other hand, for a laser with slow response time ( stationary LOFI SNR is frequency dependent [Fig.2(c)], larger than the LOFI shot-noise limit near the relaxation frequency (i.e. when:

RF

  very far away from the resonance frequency. D. Numerical simulation of the stationary LOFI SNR By using a Runge-Kutta method, we have numerically solved the set of differential equations (1) to determine the stationary LOFI SNR for different experimental conditions (i.e. for different values of int T compared to R  ). More precisely, we have compared the dynamical behavior of two lasers having the same output power ( out p ) and the same relaxation frequency ( ), but having two different values of the LOFI gain ( two different values of the laser response time ( R  ). Here, our aim is to determine the best laser for high quality imaging (i.e. large LOFI SNR).

Fig. 3

 3 Fig.3shows a comparison of the stationary LOFI SNR obtained with the two lasers, for different

  the LOFI SNR the values given in Table

  2(b) for comparison].For the laser having the longest time response ( results of the numerical simulations shown on Fig.3(b) are now frequency dependent and much more complicated to analyze.

.F

  The LOFI signal is therefore saturated at the relaxation frequency R and non-saturated for shift frequencies with:

  by the analytical results shown on Fig. 2(c), with nevertheless a resonance height less important due to the saturation of the LOFI signal.

  .[START_REF] Lacot | Comparative study of autodyne and heterodyne laser interferometry for imaging[END_REF] one can observe on Fig.3(b) that the LOFI SNR is always smaller (

  assume that the transient signal is an oscillating signal centered at the relaxation frequency R F . The constants C and c  can be determined by ensuring the continuity of the signal and of its first derivative.

,

  Eqs. (25) show that the amplitude of the transient signal ( transient signal is at maximum of the same order of magnitude than the stationary signal (

  e. no frequency mismatch), one obtains:

Fig. 4

 4 Fig.4shows the normalized (

  Fig. 4. Normalized ( 1 2 ,  C R e

  the most general case), Eq. (27b) becomes:

Fig. 4 (

 4 Fig.4(c) confirms that by using a laser with a slow response time, the stationary LOFI signal and

  minimization of the transient signal far away from the resonance is made to the detriment of the LOFI gain. Indeed for of usable LOFI gain [Eq. (3)] is limited by the following inequality: conclude this section by reminding that the transient signal can be neglected compared to the stationary signal if one uses a laser with a fast response time ( frequency is, and also with slow response time laser (

.

  Under these conditions, the transient dynamics can be neglected if the following inequality is verified:

Fig. 5 .

 5 Fig. 5. Numerical 1D scan obtained from the measured laser output power MC of a LOFI set-up, when the laser is scanned on a symmetric reflectivity pyramid composed of 4 levels. Experimental conditions: mW P out

Table 2 .Table 3 .

 23 numerical results shown on Figs. 5(a) and 5(b) are in good agreement with the analytical predictions of the section 3. Indeed, the transient dynamics fluctuations are always smaller than the laser quantum noise fluctuations, which confirms the fact that by using a laser with a fast response time ( transient dynamics is negligible whatever the working frequency is. Note also that when the effective reflectivity is multiplied by a factor 100 (level n°2 to level n°4), the MC increases by a factor 10, while when the effective reflectivity is multiplied by a factor 4 (level n°3 to level n°4) the MC increases by a factor 2.Moreover, in Table2, the MC and the SNR, numerically determined using the curves with the circles of Figs. 5(a) and 5(b) and analytically calculated [from Eqs. (5) and (11)], are very close.One can notice that the good agreement comes from the fact that for this laser, the optical feedback level is below the saturation level of the laser ( transient fluctuations are negligible in the amount of noise. With this laser, one also observes that by increasing the shift frequency, the values of the MC is smaller but that the SNR remains approximately unchanged, which confirms again the fact that by using a laser with a fast time response ( MC and SNR of the LOFI images [Figs. 5(a) and 5(b)] obtained with the laser having the lower value LOFI gain ( results are in bold while the analytical results (Eq.11) are written in italics between parentheses. In the numerical results the noise is composed of the transient fluctuation plus the laser quantum noise while in the analytical results, the noise corresponds only to the laser quantum noise. MC and SNR of the LOFI images [Figs. 5(c) and 5(d)] obtained with the laser having the higher value LOFI gain ( results are in bold while the analytical results [Eq.(11)] are written in italics between parentheses. In the numerical results the noise is composed of the transient fluctuation plus the laser quantum noise while in the analytical results, the noise corresponds only to the laser quantum noise. Now, if the value of the LOFI gain ( can observe [compare Figs. 5(c) and 5(a)] that the MC is higher but that the SNR is lower with the laser having the highest LOFI gain (i.e. the longest response time). The degradation of the SNR comes from the high value of the transient LOFI signal [see Eq. (31)], characterized on Fig. 5(c) by transient dynamic fluctuations as high as the the laser quantum noise fluctuations. The degradation of the LOFI SNR also comes from the

(

  Fig. 5(c), one needs to work far away from the resonance frequency. If we now compare Figs.

  photons

  minimum acquisition time compatible with the shot-noise condition ( also take a factor 10 and the laser relaxation frequency needs to as high as: high speed imaging combined to shot-noise limited detection, a laser diode with:, . However the use of a laser diode with a relaxation frequency state microchip laser) requires the use an electro-optics modulator to generate the frequency shift (instead of an acousto optics modulator)

  to laser-target distance of 1.5m)

  quantum noise which determines the SNR and in the laser transient dynamics which determines the response time of the LOFI setup. In the present study, we have theoretically compared the stationary LOFI SNR and the LOFI response time of two lasers having the same output power, the same relaxation frequency, but having two different values of the LOFI gain induced by two different values of the laser response time.Firstly, we have determined that the stationary LOFI SNR is frequency independent and above all shot noise limited when the response time of the laser is shorter than the lock-in integration time ( stationary LOFI SNR is frequency dependent and is higher than the LOFI shot-noise limit at the laser relaxation frequency and smaller than the LOFI shot-noise limit far away from the laser relaxation frequency. Secondly, we have shown that the transient LOFI signal can be neglected compared to the stationary LOFI signal either by using a laser with a fast response time ( the frequency shift is far away from the relaxation frequency. Therefore to obtain a shot noise limited detection without any perturbation from the laser transient dynamics we need to work under the condition: study, we have numerically confirmed that for a fixed integration time ( int T ), the best LOFI images (images with the best SNR) are always obtained when using the laser with the lowest LOFI gain, (i.e. the shortest laser time response R  ) and that the detection is shot noise limited if the following condition: imaging, we need a time value of int T as short as possible whereas to be shot noise limited we need to use a laser with a fast response time ( LOFI imaging the best laser is therefore a class-B laser with the shortest possible value of R  , allowing the use of relatively short integration time int T . Therefore, for high speed imaging combined with a shotnoise limited detection, a laser diode with a very short response time (in the nanosecond range)


L Decreusefond

TIME REVERSAL OF VOLTERRA PROCESSES DRIVEN STOCHASTIC DIFFERENTIAL EQUATIONS

Keywords: 2010 Mathematics Subject Classification. 60G18, 60H10, 60H07, 60H05 fractional Brownian motion, Malliavin calculus, stochastic differential equations, time reversal

We consider stochastic differential equations driven by some Volterra processes. Under time reversal, these equations are transformed into past dependent stochastic differential equations driven by a standard Brownian motion. We are then in position to derive existence and uniqueness of solutions of the Volterra driven SDE considered at the beginning.

Introduction

Fractional Brownian motion (fBm for short) of Hurst index H ∈ [0, 1] is the Gaussian process which admits the following representation: For any t ≥ 0,

B H (t) = t 0 K H (t, s) dB(s)
where B is a one dimensional Brownian motion and K H is a triangular kernel, i.e. K H (t, s) = 0 for s > t, the definition of which is given in [START_REF] Decreusefond | Flow properties of differential equations driven by fractional Brownian motion[END_REF]. Fractional Brownian motion is probably the first process which is not a semi-martingale and for which it is still interesting to develop a stochastic calculus. That means we want to define a stochastic integral and solve stochastic differential equations driven by such a process. From the very beginning of this program, two approaches do exist. One approach is based on the Hölder continuity or the finite p-variation of the fBm sample-paths. The other way to proceed relies on the gaussiannity of fBm. The former is mainly deterministic and was initiated by Zähle [START_REF] Zähle | Integration with respect to fractal functions and stochastic calculus[END_REF], Feyel, de la Pradelle [START_REF] Feyel | On fractional Brownian processes[END_REF] and Russo, Vallois [START_REF] Russo | The generalized covariation process and Itô formula[END_REF][START_REF] Russo | Itô formula for C 1 -functions of semimartingales[END_REF]. Then, came the notion of rough paths introduced by Lyons [START_REF] Lyons | Differential equations driven by rough signals[END_REF], whose application to fBm relies on the work of Coutin, Qian [START_REF] Coutin | Stochastic analysis, rough path analysis and fractional Brownian motions[END_REF]. These works have been extended in the subsequent works [START_REF] Coutin | Good rough path sequences and applications to anticipating stochastic calculus[END_REF][START_REF] Decreusefond | Flow properties of differential equations driven by fractional Brownian motion[END_REF][START_REF] Friz | Approximations of the Brownian rough path with applications to stochastic analysis[END_REF][START_REF] Friz | A note on the notion of geometric rough paths[END_REF][START_REF] Gradinaru | m-order integrals and generalized Itô's formula: the case of a fractional Brownian motion with any Hurst index[END_REF][START_REF] Lejay | On (p, q)-rough paths[END_REF][START_REF] Lyons | An extension theorem to rough paths[END_REF][START_REF] Neuenkirch | Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion[END_REF][START_REF] Nourdin | Schémas d'approximation associés à une équation différentielle dirigée par une fonction höldérienne; cas du mouvement brownien fractionnaire[END_REF][START_REF] Nourdin | On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion[END_REF][START_REF] Nourdin | Some linear fractional stochastic equations[END_REF]. A new way of thinking came with the independent but related works of Feyel, de la Pradelle [START_REF] Feyel | Curvilinear integrals along enriched paths[END_REF] and Gubinelli [START_REF] Gubinelli | Controlling rough paths[END_REF]. The integral with respect to fBm was shown to exist as the unique process satisfying some characterization (analytic in the case of [START_REF] Feyel | Curvilinear integrals along enriched paths[END_REF], algebraic in [START_REF] Gubinelli | Controlling rough paths[END_REF]). As a byproduct, this showed that almost all the existing integrals throughout the literature were all the same as they all satisfy these two conditions. Behind each approach but the last too, is a construction of an integral defined for a regularization of fBm, then the whole work is to show that under some convenient hypothesis, the approximate integrals converge to a quantity which is called the stochastic integral with respect to fBm. The main tool to prove the convergence is either integration by parts in the sense of fractional deterministic calculus, either enrichment of the fBm by some iterated integrals proved to exist independently or by analytic continuation [START_REF] Unterberger | An explicit rough path construction for continuous paths with arbitrary Hölder exponent by Fourier normal ordering[END_REF][START_REF] Unterberger | A Lévy area by Fourier normal ordering for multidimensional fractional Brownian motion with small Hurst index[END_REF].

In the probabilistic approach [START_REF] Alòs | Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than 1/2[END_REF][START_REF] Alòs | Stochastic calculus with respect to Gaussian processes[END_REF][START_REF] Decreusefond | Stochastic integration with respect to Gaussian processes[END_REF][START_REF] Decreusefond | Stochastic integration with respect to fractional Brownian motion[END_REF][START_REF] Decreusefond | Stochastic calculus with respect to Volterra processes[END_REF][START_REF] Decreusefond | Stochastic analysis of the fractional Brownian motion[END_REF][START_REF] Hu | Differential equations driven by Hölder continuous functions of order greater than 1/2[END_REF][START_REF] Nualart | Differential equations driven by fractional Brownian motion[END_REF][START_REF] Nualart | Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion[END_REF], the idea is also to define an approximate integral and then prove its convergence. It turns out that the key tool is here the integration by parts in the sense of Malliavin calculus.

In dimension greater than one, with the deterministic approach, one knows how to define the stochastic integral and prove existence and uniqueness of fBm driven SDEs for fBm with Hurst index greater than 1/4. Within the probabilistic framework, one knows how to define a stochastic integral for any value of H but one cannot prove existence and uniqueness of SDEs whatever the value of H. The primary motivation of this work is to circumvent this problem.

In [START_REF] Decreusefond | Stochastic calculus with respect to Volterra processes[END_REF][START_REF] Decreusefond | Stochastic analysis of the fractional Brownian motion[END_REF], we defined stochastic integrals with respect to fBm as a "damped-Stratonovitch" integral with respect to the underlying standard Brownian motion. This integral is defined as the limit of Riemann-Stratonovitch sums, the convergence of which is proved after an integration by parts in the sense of Malliavin calculus. Unfortunately, this manipulation generates non-adaptiveness: Formally the result can be expressed as

t 0 u(s) • dB H (s) = δ(K * t u) + trace(K * t ∇u),
where K is defined by

Kf (t) = d dt t 0 K H (t, s)f (s) ds and K * t is the adjoint of K in L 2 ([0, t], R).
In particular, there exists k such that

K * t f (s) = t s k(t, u)f (u) du
for any f ∈ L 2 ([0, t], R) so that even if u is adapted (with respect to the Brownian filtration), the process (s → K * t u(s)) is anticipative. However, the stochastic integral process (t → t 0 u(s)• dB H (s)) remains adapted, hence, the anticipative aspect is, in some sense, artificial. The motivation of this work is to show, that up to time reversal, we can work with adapted process and Itô integrals. The time-reversal properties of fBm were already studied in [START_REF] Darses | Time reversal for drifted fractional Brownian motion with Hurst index H > 1/2[END_REF] in a different context: It was shown there that the time-reversal of the solution of an fBm-driven SDE of the form

dY (t) = u(Y (t)) dt + dB H (t)
is still a process of the same form. With a slight adaptation of our method to fBm-driven SDEs with drift, one should recover the main theorem of [START_REF] Darses | Time reversal for drifted fractional Brownian motion with Hurst index H > 1/2[END_REF].

In what follows, there is no restriction about the dimension but we need to assume that any component of B H is an fBm of Hurst index greater than 1/2. Consider that we want to solve the equation ( 1)

X t = x + t 0 σ(X s ) • dB H (s), 0 ≤ t ≤ T
where σ is a deterministic function whose properties will be fixed below. It turns out that it is essential to investigate the more general equations:

(A) X r, t = x + t r σ(X r, s ) • dB H (s), 0 ≤ r ≤ t ≤ T.
The strategy is then the following: We will first consider the reciprocal problem:

(B) Y r, t = x - t r σ(Y s, t ) • dB H (s), 0 ≤ r ≤ t ≤ T.
The first critical point is that when we consider {Z r, t := Y t-r, t , r ∈ [0, t]}, this process solves an adapted, past dependent, stochastic differential equation with respect to a standard Brownian motion. Moreover, because K H is lower-triangular and sufficiently regular, the trace term vanishes in the equation defining Z. We have then reduced the problem to an SDE with coefficients dependent on the past, a problem which can be handled by the usual contraction methods. We do not claim that the results presented are new (for instance see the brilliant monograph [START_REF] Friz | Multidimensional stochastic processes as rough paths[END_REF] for detailed results obtained via rough paths theory) but it seems interesting to have purely probabilistic methods which show that fBm driven SDEs do have strong solutions which are homeomorphisms. Moreover, the approach given here shows the irreducible difference between the case H < 1/2 and H > 1/2 : The trace term only vanishes in the latter situation, so that such an SDE is merely a usual SDE with past-dependent coefficients. This representation may be fruitful for instance, to analyze the support and prove the absolute continuity of solutions of (1).

This paper is organized as follows: After some preliminaries on fractional Sobolev spaces, often called Besov-Liouville space, we address, in Section 3, the problem of Malliavin calculus and time reversal. This part is interesting in its own since stochastic calculus of variations is a framework oblivious to time. Constructing such a notion of time is achieved using the notion of resolution of the identity as introduced in [START_REF] Üstünel | The construction of filtrations on abstract Wiener spaces[END_REF][START_REF] Li | Un traitement unifié de la représentation des fonctionnelles de Wiener[END_REF]. We then introduce the second key ingredient which is the notion of strict causality or quasi-nilpotence, see [START_REF] Zakai | When does the Ramer formula look like the Girsanov formula ?[END_REF] for a related application. In Section 4, we show that solving Equation (B) reduces to solve a past dependent stochastic differential equation with respect to a standard Brownian motion, see Equation (C) below. Then, we prove existence, uniqueness and some properties of this equation. Technical lemmas are postponed to Section 5.

Besov-Liouville Spaces

Let T > 0 be fix real number. For a measurable function f :

[0, T ] → R n , we define τ T f by τ T f (s) = f (T -s) for any s ∈ [0, T ].
For t ∈ [0, T ], e t f will represent the restriction of f to [0, t], i.e., e t f = f 1 [0, t] . For any linear map A, we denote by A * T , its adjoint in L 2 ([0, T ]; R n ). For η ∈ (0, 1], the space of η-Hölder continuous functions on [0, T ] is equipped with the norm

f Hol(η) = sup 0<s<t<T |f (t) -f (s)| |t -s| η + f ∞ .
Its topological dual is denoted by Hol(η) * . For f ∈ L 1 ([0, T ]; R n ; dt), (denoted by L 1 for short) the left and right fractional integrals of f are defined by :

(I γ 0 + f )(x) = 1 Γ(γ) x 0 f (t)(x -t) γ-1 dt , x ≥ 0, (I γ T -f )(x) = 1 Γ(γ) T x f (t)(t -x) γ-1 dt , x ≤ T,
where γ > 0 and I 0 0 + = I 0 T -= Id . For any γ ≥ 0, p, q ≥ 1, any f ∈ L p and g ∈ L q where p -1 + q -1 ≤ γ, we have :

(2)

T 0 f (s)(I γ 0 + g)(s) ds = T 0 (I γ T -f )(s)g(s) ds.
The Besov-Liouville space I γ 0 + (L p ) := I + γ,p is usually equipped with the norm : (3)

I γ 0 + f I + γ,p = f L p . Analogously, the Besov-Liouville space I γ T -(L p ) := I - γ,p
is usually equipped with the norm :

I -γ T -f I - γ,p = f L p .
We then have the following continuity results (see [START_REF] Feyel | On fractional Brownian processes[END_REF][START_REF] Samko | Fractional Integrals and Derivatives[END_REF]) :

Proposition 2.1. i. If 0 < γ < 1, 1 < p < 1/γ, then I γ 0 + is a bounded operator from L p into L q with q = p(1 -γp) -1 .
ii. For any 0 < γ < 1 and any p ≥ 1, I + γ,p is continuously embedded in Hol(γ -1/p) provided that γ -1/p > 0.

iii. For any 0 < γ < β < 1, Hol(β) is compactly embedded in I γ,∞ . iv. For γp < 1, the spaces I + γ,p and I - γ,p are canonically isomorphic. We will thus use the notation I γ,p to denote any of this spaces.

Malliavin calculus and time reversal

Our reference probability space is Ω = C 0 ([0, T ], R n ), the space of R n -valued, continuous functions, null at time 0. The Cameron-Martin space is denoted by H and is defined as

H = I 1 0 + (L 2 ([0, T ])).
In what follows, the space L 2 ([0, T ]) is identified with its topological dual. We denote by κ the canonical embedding from H into Ω. The probability measure P on Ω is such that the canonical map W : ω → (ω(t), t ∈ [0, T ]) defines a standard n-dimensional Brownian motion. A mapping φ from Ω into some separable Hilbert space H is called cylindrical if it is of the form φ(w)

= d i=1 f i ( v i,1 , w , • • • , v i,n , w )x i where for each i, f i ∈ C ∞ 0 (R n , R) and (v i,j , j = 1, • • • , n) is a sequence of Ω * .
For such a function we define ∇ W φ as

∇ W φ(w) = i,j=1 ∂ j f i ( v i,1 , w , • • • , v i,n , w )ṽ i,j ⊗ x i ,
where ṽ is the image of v ∈ Ω * by the map (I 1 0 + • κ) * . From the quasi-invariance of the Wiener measure [START_REF] Üstünel | An Introduction to Analysis on Wiener Space[END_REF], it follows that ∇ W is a closable operator on L p (Ω; H), p ≥ 1, and we will denote its closure with the same notation. The powers of ∇ W are defined by iterating this procedure. For p > 1, k ∈ N, we denote by D p,k (H) the completion of H-valued cylindrical functions under the following norm

φ p,k = k i=0 (∇ W ) i φ L p (Ω; H⊗L p ([0,1]) ⊗i ) .
We denote by L p,1 the space D p,1 (L p ([0, T ]; R n )). The divergence, denoted δ W is the adjoint of ∇ W : v belongs to Dom p δ W whenever for any cylindrical φ,

|E T 0 v s ∇ W s φ ds | ≤ c φ L p
and for such a process v,

E T 0 v s ∇ W s φ ds = E φ δ W v .
We introduced the temporary notation W for standard Brownian motion to clarify the forthcoming distinction between a standard Brownian motion and its time reversal. Actually, the time reversal of a standard Brownian is also a standard Brownian motion and thus, both of them "live" in the same Wiener space. We now precise how their respective Malliavin gradient and divergence are linked. Con-

sider B = (B(t), t ∈ [0, T ]) an n-dimensional standard Brownian motion and BT = (B(T ) -B(T -t), t ∈ [0, T ]
) its time reversal. Consider the following map

Θ T : Ω -→ Ω ω -→ ω = ω(T ) -τ T ω,
and the commutative diagram

L 2 τT ----→ L 2 I 1 0 +     I 1 0 + Ω ⊃ H ----→ ΘT H ⊂ Ω Note that Θ -1 T = Θ T since ω(0) = 0. For a function f ∈ C ∞ b (R nk ), we define ∇ r f (ω(t 1 ), • • • , ω(t k )) = k j=1 ∂ j f (ω(t 1 ), • • • , ω(t k ))1 [0, tj ] (r) and ∇r f (ω(t 1 ), • • • , ω(t k )) = k j=1 ∂ j f (ω(t 1 ), • • • , ω(t k ))1 [0, tj ] (r).
The operator ∇ = ∇ B (respectively ∇ = ∇ B ) is the Malliavin gradient associated with a standard Brownian motion (respectively its time reversal). Since,

f (ω(t 1 ), • • • , ω(t k )) = f (ω(T ) -ω(T -t 1 ), • • • , ω(T ) -ω(T -t k )), we can consider f (ω(t 1 ), • • • , ω(t k ))
as a cylindrical function with respect to the standard Brownian motion. As such its gradient is given by

∇ r f (ω(t 1 ), • • • , ω(t k )) = k j=1 ∂ j f (ω(t 1 ), • • • , ω(t k ))1 [T -tj , T ] (r).
We thus have, for any cylindrical function F , (4)

∇F • Θ T (ω) = τ T ∇F (ω).
Since Θ * T P = P and τ T is continuous from L p into itself for any p, it is then easily shown that the spaces D p, k and Ďp, k (with obvious notations) coincide for any p, k and that (4) holds for any element of one of these spaces. Hence we have proved the following theorem: Theorem 3.1. For any p ≥ 1 and any integer k, the spaces D p, k and Ďp, k coincide. For any F ∈ D p, k for some p, k,

∇(F • Θ T ) = τ T ∇(F • Θ T ), P a.s..
By duality, an analog result follows for divergences. Theorem 3.2. A process u belongs to the domain of δ if and only if τ T u belongs to the domain of δ and then, the following equality holds:

(5) δ(u(ω))(ω) = δ(τ T u(ω))(ω) = δ(τ T u • Θ T )(ω).
Proof. For u ∈ L 2 , for cylindrical F , we have on the one hand:

E F (ω) δu(ω) = E ( ∇F (ω), u) L 2 ,
and on the other hand,

E ( ∇F (ω), u) L 2 = E [(τ T ∇F • Θ T (ω), u) L 2 ] = E [(∇F • Θ T (ω), τ T u) L 2 ] = E [F • Θ T (ω)δ(τ T u)(ω)] = E [F (ω)δ(τ T u)(ω)] .
Since this is valid for any cylindrical F , (5) holds for u ∈ L 2 . Now, for u in the domain of divergence (see [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF][START_REF] Üstünel | An Introduction to Analysis on Wiener Space[END_REF]),

δu = i (u, h i ) L 2 δh i -(∇u, h i ⊗ h i ) L 2 ⊗L 2 , where (h i , i ∈ N) is an orthonormal basis of L 2 ([0, T ]; R n ). Thus, we have δ(u(ω))(ω) = i (u(ω), h i ) L 2 δh i (ω) -( ∇u(ω), h i ⊗ h i ) L 2 ⊗L 2 = i (u(ω), h i ) L 2 δ(τ T h i )(ω) -(∇u(ω), τ T h i ⊗ h i ) L 2 ⊗L 2 = i (τ T u(ω), τ T h i ) L 2 δ(τ T h i )(ω) -(∇τ T u(ω), τ T h i ⊗ τ T h i ) L 2 ⊗L 2 ,
where we have taken into account that τ T in an involution. Since (h i , i ∈ N) is an orthonormal basis of L 2 ([0, T ]; R n ), identity ( 5) is satisfied for any u in the domain of δ.

Causality and quasi-nilpotence.

In anticipative calculus, the notion of trace of an operator plays a crucial role, we refer to [START_REF] Dunford | Linear operators. Part II[END_REF] for more details on trace.

Definition 3.1. Let V be a bounded map from L 2 ([0, T ]; R n ) into itself. The map V is said to be trace-class whenever for one CONB (h n , n ≥ 1) of L 2 ([0, T ]; R n ), n≥1 |(V h n , h n ) L 2 | is finite.
Then, the trace of V is defined by

trace(V ) = n≥1 (V h n , h n ) L 2 .
It is easily shown that the notion of trace does not depend on the choice of the CONB.

Definition 3.2. A family E of projections (E λ , λ ∈ [0, 1]) in L 2 ([0, T ]; R n ) is called a resolution of the identity if it satisfies the conditions (1) E 0 = 0 and E 1 = Id. (2) E λ E µ = E λ∧µ . (3) lim µ↓λ E µ = E λ for any λ ∈ [0, 1) and lim µ↑1 E µ = Id .
For instance, the family

E = (e λT , λ ∈ [0, 1]) is a resolution of the identity in L 2 ([0, T ]; R n ). Definition 3.3. A partition π of [0, T ] is a sequence {0 = t 0 < t 1 < . . . < t n = T }.

Its mesh is denoted by |π| and defined by |π|

= sup i |t i+1 -t i |.
The causality plays a crucial role in what follows. The next definition is just the formalization in terms of operator of the intuitive notion of causality.

Definition 3.4. A continuous map V from L 2 ([0, T ]; R n ) into itself is said to be E-

causal if and only if the following condition holds:

E λ V E λ = E λ V for any λ ∈ [0, 1]. For instance, an operator V in integral form V f (t) = T 0 V (t, s)f (s) ds is causal if and only if V (t, s) = 0 for s ≥ t, i.e.
, computing V f (t) needs only the knowledge of f up to time t and not after. Unfortunately, this notion of causality is insufficient for our purpose and we are led to introduce the notion of strict causality as in [START_REF] Feintuch | System theory[END_REF]. Definition 3.5. Let V be a causal operator. It is a strictly causal operator whenever for any ε > 0, there exists a partition π of [0, T ] such that for any

π ′ = {0 = t 0 < t 1 < . . . < t n = T } ⊂ π, (E ti+1 -E ti )V (E ti+1 -E ti ) L 2 < ε, for i = 0, • • • , n -1.
Note carefully that the identity map is causal but not strictly causal. Indeed, if V = Id, for any s < t,

(E t -E s )V (E t -E s ) L 2 = E t -E s L 2 = 1 since E t -E s is a projection. However, if V is hyper-contractive,
[0, 1]) or E = (Id -e (1-λ)T , λ ∈ [0, 1]). If V is an E-causal map continuous from L 2 into L p for some p > 2 then V is strictly E-causal.
Proof. Let π be any partition of [0, T ]. Assume E = (e λT , λ ∈ [0, 1]), the very same proof works for the other mentioned resolution of the identity. According to Hölder formula, we have: For any 0 ≤ s < t ≤ T ,

(E t -E s )V (E t -E s )f L 2 = t s |V (f 1 (s, t] )(u)| 2 du ≤ (t -s) 1-2/p V (f 1 (s, t] ) L p/2 ≤ c (t -s) 1-2/p f L 2 .
Then, for any ε > 0, there exists η > 0 such that |π| < η implies (E ti+1 -

E ti )V (E ti+1 -E ti )f L 2 ≤ ε for any {0 = t 0 < t 1 < . . . < t n = T } ⊂ π and any i = 0, • • • , n -1.
The importance of strict causality lies in the next theorem we borrow from [START_REF] Feintuch | System theory[END_REF].

Theorem 3.4. The set of strictly causal operators coincides with the set of quasinilpotent operators, i.e., trace-class operators such that trace(V n ) = 0 for any integer n ≥ 1.

Moreover, we have the following stability theorem.

Theorem 3.5. The set of strictly causal operators is a two-sided ideal in the set of causal operators. Definition 3.6. Let E be a resolution of the identity in L 2 ([0, T ]; R n ). Consider the filtration F E defined as

F E t = σ{δ W (E λ h), λ ≤ t, h ∈ L 2 }. An L 2 -valued random variable u is said to be F E -adapted if for any h ∈ L 2 , the real valued process < E λ u, h > is F E -adapted. We denote by D E p,k (H) the set of F E -adapted random variables belonging to D p,k (H). If E = (e λT , λ ∈ [0, 1]
), the notion of F E adapted processes coincides with the usual one for the Brownian filtration and it is well known that a process u is adapted if and only if ∇ W r u(s) = 0 for r > s. This result can be generalized to any resolution of the identity.

Theorem 3.6 (Proposition 3.1 of [42]). Let u belongs to L p,1 . Then u is F E - adapted if and only if ∇ W u is E-causal.
We then have the following key theorem: Theorem 3.7. Assume the resolution of the identity to be E = (e λT , λ ∈ [0, 1]) either E = (Id -e (1-λ)T , λ ∈ [0, 1]) and that V is an E-strictly causal continuous operator from L 2 into L p for some p > 2. Let u be an element of D E 2,1 (L 2 ). Then, V ∇ W u is of trace class and we have trace(V ∇ W u) = 0.

Proof. Since u is adapted, ∇ W u is E-causal. According to Theorem 3.5, V ∇ W u is strictly causal and the result follows by Theorem 3. [START_REF] Coutin | Stochastic analysis, rough path analysis and fractional Brownian motions[END_REF].

In what follows, E 0 is the resolution of the identity in the Hilbert space L 2 defined by e λ T f = f 1 [0, λT ] and Ě0 is the resolution of the identity defined by ěλT f = f 1 [(1-λ)T,T ] . The filtration F E 0 and F Ě0 are defined accordingly. Next lemma is immediate when V is given in the form V f (t) = t 0 V (t, s)f (s) ds. Unfortunately such a representation as an integral operator is not always available. We give here an algebraic proof to emphasize the importance of causality.

Lemma 3.8. Let V be a map from L 2 ([0, T ]; R n ) into itself such that V is E 0 - causal. Let V * be the adjoint of V in L 2 ([0, T ]; R n ). Then, the map τ T V * T τ T is Ě0 -causal.
Proof. This is a purely algebraic lemma once we have noticed that [START_REF] Decreusefond | Stochastic integration with respect to Gaussian processes[END_REF] τ T e r = (Id -e T -r )τ T for any 0 ≤ r ≤ T.

For, it suffices to write

(7) τ T e r f (s) = f (T -s)1 [0, r] (T -s) = f (T -s)1 [T -r, T ] (s) = (Id -e T -r )τ T f (s), for any 0 ≤ s ≤ T.
We have to show that e r τ T V * T τ T e r = e r τ T V * T τ T or equivalently e r τ T V τ T e r = τ T V τ T e r , since e * r = e r and τ * T = τ T . Now, ( 7) yields e r τ T V τ T e r = τ T V τ T e r -e T -r V τ T e r .

Use [START_REF] Decreusefond | Stochastic integration with respect to fractional Brownian motion[END_REF] again to obtain

e T -r V τ T e r = e T -r V (Id -e T -r )τ T = (e T -r V -e T -r V e T -r )τ T = 0, since V is E-causal.

Stratonovitch integrals.

In what follows, η belongs to (0, 1] and V is a linear operator. For any p ≥ 2, we set:

Hypothesis I (p, η). The linear map V is continuous from L p ([0, T ]; R n ) into the Banach space Hol(η).
Definition 3.7. Assume that Hypothesis I(p, η) holds. The Volterra process associated to V , denoted by W V is defined by

W V (t) = δ W V (1 [0, t] ) , for all t ∈ [0, T ].
For any subdivision π of [0, T ], i.e., π = {0 = t 0 < t 1 < . . . < t n = T }, of mesh |π|, we consider the Stratonovitch sums:

(8) R π (t, u) = δ W ti∈π 1 θ i ti+1∧t ti∧t V u(r) dr 1 [ti, ti+1) + ti∈π 1 θ i [ti∧t,ti+1∧t] 2 V (∇ W
r u)(s) ds dr.

Definition 3.8. We say that u is V -Stratonovitch integrable on [0, t] whenever the family R π (t, u), defined in [START_REF] Decreusefond | Stochastic calculus with respect to Volterra processes[END_REF], converges in probability as |π| goes to 0. In this case the limit will be denoted by t 0 u(s) • dW V (s). Example 1. The first example is the so-called Lévy fractional Brownian motion of Hurst index H > 1/2, defined as

1 Γ(H + 1/2) t 0 (t -s) H-1/2 dB s = δ(I H-1/2 T - (1 [0, t] )).

This amounts to say that

V = I H-1/2 T - . Thus Hypothesis I(p, H -1/2 -1/p) holds provided p(H -1/2) > 1.

Example 2. The other classical example is the fractional Brownian motion with stationary increments of Hurst index H > 1/2, which can be written as

t 0 K H (t, s) dB(s), where (9) K H (t, r) = (t -r) H-1 2 Γ(H + 1 2 ) F ( 1 2 -H, H - 1 2 , H + 1 2 , 1 - t r )1 [0,t) (r).
The Gauss hyper-geometric function F (α, β, γ, z) (see [START_REF] Nikiforov | Special Functions of Mathematical Physics[END_REF]) is the analytic continuation on C × C × C\{-1, -2, . . .} × {z ∈ C, Arg|1 -z| < π} of the power series We know from [START_REF] Samko | Fractional Integrals and Derivatives[END_REF] that K H is an isomorphism from L p ([0, 1]) onto I + H+1/2,p and

+∞ k=0 (α) k (β) k (γ) k k! z k ,
K H f = I 1 0 + x H-1/2 I H-1/2 0 + x 1/2-H f. Consider K H = I -1 0 + • K H . Then it is clear that t 0 K H (t, s) dB(s) = t 0 (K H ) * T (1 [0,t] )(s) dB(s),
hence that we are in the framework of Definition 3.8 provided that we take V = (K H ) * T . Hypothesis I(p, H -1/2 -1/p) is satisfied provided that p(H -1/2) > 1. The next theorem then follows from [START_REF] Decreusefond | Stochastic calculus with respect to Volterra processes[END_REF]. Theorem 3.9. Assume that Hypothesis I(p, η) holds. Assume that u belongs to L p,1 . Then u is V -Stratonovitch integrable, there exists a process which we denote by D W u such that D W u belongs to L p (P ⊗ ds) and (10)

T 0 u(s) • dW V (s) = δ W (V u) + T 0 D W u(s) ds.
The so-called "trace-term" satisfies the following estimate:

(11) E T 0 |D W u(r)| p dr ≤ c T pη u p Lp, 1 ,
for some universal constant c. Moreover, for any r ≤ T , e r u is V -Stratonovitch integrable and

r 0 u(s) • dW V (s) = T 0 (e r u)(s) • dW V (s) = δ W (V e r u) + r 0 D W u(s) ds
and we have the maximal inequality:

(12) E . 0 u(s) • dW V (s) p Hol(η) ≤ c u p Lp,1 ,
where c does not depend on u.

The main result of this Section is the following theorem which states that the time reversal of a Stratonovitch integral is an adapted integral with respect to the time reversed Brownian motion. Due to its length, its proof is postponed to Section 5.1.

Theorem 3.10. Assume that Hypothesis I(p, η) holds. Let u belong to L p,1 and let

VT = τ t V τ T . Assume furthermore that V is Ě0 -causal and that ǔ = u • Θ -1 T is F Ě0 -adapted. Then, (13) T -r T -t τ T u(s) • dW V (s) = t r VT (1 [r, t] ǔ)(s) d BT (s), 0 ≤ r ≤ t ≤ T,

where the last integral is an Itô integral with respect to the time reversed Brownian motion BT (s) = B(T ) -B(T -s) = Θ T (B)(s).

Remark 3.1. Note that at a formal level, we could have an easy proof of this theorem. For instance, consider the Lévy fBm, a simple computations shows that VT = I H-1/2 0 + for any T. Thus, we are led to compute trace(I H-1/2 0 + ∇u). If we had sufficient regularity, we could write trace(I

H-1/2 0 + ∇u) = T 0 s 0 (s -r) H-3/2 ∇ s u(r) dr ds = 0,
since ∇ s u(r) = 0 for s > r for u adapted. Obviously, there are many flaws in these lines of proof: The operator I H-1/2 0 + ∇u is not regular enough for such an expression of the trace to be true. Even more, there is absolutely no reason for VT ∇u to be a kernel operator so we can't hope such a formula. These are the reasons that we need to work with operators and not with kernels.

driven SDEs

Let G the group of homeomorphisms of R n equipped with the distance: We introduce a distance d on G by

d(ϕ, φ) = ρ(ϕ, φ) + ρ(ϕ -1 , φ -1 ), where ρ(ϕ, φ) = ∞ N =1 2 -N sup |x|≤N |ϕ(x) -φ(x)| 1 + sup |x|≤N |ϕ(x) -φ(x)| • Then, G is a complete topological group. Consider the equations (A) X r, t = x + t r σ(X r, s ) • dW V (s), 0 ≤ r ≤ t ≤ T. (B) Y r, t = x - t r σ(Y s, t ) • dW V (s), 0 ≤ r ≤ t ≤ T.
As a solution of (A) is to be constructed by "inverting" a solution of (B), we need to add to the definition of a solution of (A) or (B) the requirement of being a flow of homeomorphisms. This is the meaning of the following definition.

Definition 4.1. By a solution of (A), we mean a measurable map

Ω × [0, T ] × [0, T ] -→ G (ω, r, t) -→ (x → X r,t (ω, x))
such that the following properties are satisfied :

(1) For any

0 ≤ r ≤ t ≤ T , for any x ∈ R n , X r, t (ω, x) is σ{W V (s), r ≤ s ≤ t}- measurable, (2) For any 0 ≤ r ≤ T , for any x ∈ R n , the processes (ω, t) → X r,t (ω, x) and
(ω, t) → X -1 r,t (ω, x) belong to L p,1 for some p ≥ 2.

(3) For any 0 ≤ r ≤ s ≤ t, for any x ∈ R n , the following identity is satisfied:

X r,t (ω, x) = X s,t (ω, X r,s (ω, x)).
(4) Equation (A) is satisfied for any 0 ≤ r ≤ t ≤ T P-a.s..

Definition 4.2. By a solution of (B), we mean a measurable map

Ω × [0, T ] × [0, T ] -→ G (ω, r, t) -→ (x → Y r,t (ω, x))
such that the following properties are satisfied :

(1) For any 0 ≤ r ≤ t ≤ T , for any 2) For any 0 ≤ r ≤ T , for any x ∈ R n , the processes (ω, r) → Y r,t (ω, x) and

x ∈ R n , Y r, t (ω, x) is σ{W V (s), r ≤ s ≤ t}- measurable, (
(ω, r) → Y -1 r,t (ω, x) belong to L p,1 for some p ≥ 2.

(3) Equation (B) is satisfied for any 0 ≤ r ≤ t ≤ T P-a.s.. [START_REF] Coutin | Stochastic analysis, rough path analysis and fractional Brownian motions[END_REF] For any 0 ≤ r ≤ s ≤ t, for any x ∈ R n , the following identity is satisfied:

Y r,t (ω, x) = Y r,s (ω, Y s,t (ω, x)).
At last consider the equation, for any 0

≤ r ≤ t ≤ T , (C) Z r, t = x - t r VT (σ • Z .,t 1 [r,t] )(s) d BT (s)
where B is a standard n-dimensional Brownian motion.

Definition 4.3. By a solution of (C), we mean a measurable map

Ω × [0, T ] × [0, T ] -→ G (ω, r, t) -→ (x → Z r,t (ω, x))
such that the following properties are satisfied :

(1) For any 0 ≤ r ≤ t ≤ T , for any x ∈ R n , Z r, t (ω, x) is σ{ BT (s), s ≤ r ≤ t}measurable, (2) For any 0 ≤ r ≤ t ≤ T , for any x ∈ R n , the processes (ω, r) → Z r,t (ω, x) and (ω, r) → Z -1 r,t (ω, x) belong to L p,1 for some p ≥ 2.

(3) Equation (C) is satisfied for any 0 ≤ r ≤ t ≤ T P-a.s.. Theorem 4.1. Assume that VT is an E 0 causal map continuous from L p into I α,p for α > 0 and p ≥ 4 such that αp > 1. Assume σ is Lipschitz continuous and sub-linear, see Eqn. [START_REF] Hu | Differential equations driven by Hölder continuous functions of order greater than 1/2[END_REF] for the definition. Then, there exists a unique solution to equation (C). Let Z denote this solution. For any (r, r ′ ),

E [|Z r,T -Z r ′ ,T | p ] ≤ c|r -r ′ | pη . Moreover, (ω, r) → Z r,s (ω, Z s,t (ω, x)) ∈ L p,1 , for any r ≤ s ≤ t ≤ T.
Since this proof needs several lemmas, we defer it to Section 5.2. Theorem 4.2. Assume that VT is an E 0 causal map continuous from L p into I α,p for α > 0 and p ≥ 2 such that αp > 1. For fixed T , there exists a bijection between the space of solutions of Equation (B) on [0, T ] and the set of solutions of Equation (C).

Proof. Set Z r,T (ω, x) = Y T -r,T (Θ -1 T (ω), x) or equivalently (14) Y r,T (ω, x) = Z T -r,T (Θ T (ω), x).
According to Theorem 3.10, Y is satisfies (B) if and only if Z satisfies (C). The regularity properties are immediate since L p is stable by τ T .

The first part of the next result is then immediate.

Corollary 4.3. Assume that VT is an E 0 causal map continuous from L p into I α,p for α > 0 and p ≥ 2 such that αp > 1. Then Equation (B) has one and only solution and for any 0 ≤ r ≤ s ≤ t, for any x ∈ R n , the following identity is satisfied:

(15) Y r,t (ω, x) = Y r,s (ω, Y s,t (ω, x)).
Proof. According to Theorem 4.2 and 4.1, (B) has at most one solution since (C) has a unique solution. As to the existence, point (1) to (3) are immediately deduced from the corresponding properties of Z and Equation ( 14).

According to Theorem 4.1, (ω, r) → Y r,s (ω, Y s,t (ω, x)) belongs to L p,1 hence we can apply the substitution formula and we get:

(16) Y r,s (ω, Y s,t (ω, x)) = Y s,t (ω, x) - s r σ(Y τ,s (ω, x)) • dW V (τ ) x=Ys,t(ω, x) = x - t s σ(Y τ,t (ω, x) • dW V (τ ) - s r σ(Y τ,s (ω, Y s,t (ω, x))) • dW V (τ ). Set R τ,t = Y τ,t (ω, x) for s ≤ τ ≤ t Y τ,s (ω, Y s,t (ω, x)) for r ≤ τ ≤ s.
Then, in view of ( 16), R appears to be the unique solution (B) and thus R s,t (ω, x) = Y s,t (ω, x). Point (4) is thus proved. Corollary 4.4. For x fixed, the random field (Y r,t (x), 0 ≤ r ≤ t ≤ T ) admits a continuous version. Moreover,

E [|Y r,s (x) -Y r ′ ,s ′ (x)| p ] ≤ c(1 + |x| p )(|s ′ -s| pη + |r -r ′ | pη ).
We still denote by Y this continuous version.

Proof. Without loss of generality, assume that s ≤ s ′ and remark that Y s, s ′ (x) thus belongs to σ{ BT u , u ≥ s}.

E [|Y r,s (x) -Y r ′ ,s ′ (x)| p ] ≤ c (E [|Y r,s (x) -Y r ′ ,s (x)| p ] + E [|Y r ′ ,s (x) -Y r ′ ,s ′ (x)| p ]) = c (E [|Y r,s (x) -Y r ′ ,s (x)| p ] + E [|Y r ′ ,s (x) -Y r ′ ,s (Y s,s ′ (x))| p ]) = c (E [|Z s-r,s (x) -Z s-r ′ ,s (x)| p ] + E [|Z s-r ′ ,s (x) -Z s-r ′ ,s (Y s,s ′ (x))| p ]) .
According to Theorem 5.6, ( 17)

E [|Z s-r,s (x) -Z s-r ′ ,s (x)| p ] ≤ c|r -r ′ | pη (1 + |x| p ).
In view of Theorem 3.10, the stochastic integral which appears in Equation (C) is also a Stratonovitch integral hence we can apply the substitution formula and say

Z s-r ′ ,s (Y s,s ′ (x)) = Z s-r ′ ,s (y)| y=Y s,s ′ (x) .
Thus we can apply Theorem 5.6 and we obtain

E [|Z s-r ′ ,s (x) -Z s-r ′ ,s (Y s,s ′ (x))| p ] ≤ cE [|x -Y s,s ′ (x)| p ] .
The right hand side of this equation is in turn equal to

E [|Z 0,s ′ -Z s ′ -s,s ′ (x)| p ] ,
thus, we get ( 18)

E [|Z s-r ′ ,s (x) -Z s-r ′ ,s (Y s,s ′ (x))| p ] ≤ c(1 + |x| p )|s ′ -s| pη
Combining ( 17) and [START_REF] Gradinaru | m-order integrals and generalized Itô's formula: the case of a fractional Brownian motion with any Hurst index[END_REF] gives

E [|Y r,s (x) -Y r ′ ,s ′ (x)| p ] ≤ c(1 + |x| p )(|s ′ -s| pη + |r -r ′ | pη ),
hence the result.

Thus, we have the main result of this paper.

Theorem 4.5. Assume that VT is an E 0 causal map continuous from L p into I α,p for α > 0 and p ≥ 4 such that αp > 1. Then Equation (A) has one and only one solution.

Proof. Under the hypothesis, we know that Equation (B) has a unique solution which satisfies [START_REF] Friz | Approximations of the Brownian rough path with applications to stochastic analysis[END_REF]. By definition of a solution of (B), the process Y -1 : (ω, s) → Y -1 st (ω, x) belongs to L p,1 hence we can apply the substitution formula. Following the lines of proof of the previous theorem, we see that Y -1 is a solution of (A).

In the reverse direction, two distinct solutions of (A) would give raise to two solutions of (B) by the same principles. Since this is definitely impossible in view of Theorem 4.3, Equation (A) has at most one solution.

5. Technical proofs 5.1. Substitution formula. The proof of 3.10 relies on several lemmas including one known in anticipative calculus as the substitution formula, cf. [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF]. Moreover,

E trace(V ∇ W u) p ≤ c u p Lp,1 . Proof. For each k, let (φ k, m , m = 1, • • • , 2 k ) be the functions φ k, m = 2 k/2 1 [(m-1)2 -k , m2 -k ) .
Let P k be the projection onto the span of the φ k, m , since ∇ W V u is of trace class, we have (see [START_REF] Simon | Trace ideals and their applications[END_REF])

trace(V ∇ W p t u) = lim k→+∞ trace(P k V ∇ W p t u P k ). Now, trace(P k V ∇ W u P k ) = k m=1 (V ∇ W p t u, φ k,m ⊗ φ k,m ) L 2 ⊗L 2 = k m=1 2 k m2 -k ∧t (m-1)2 -k ∧t m2 -k ∧t (m-1)2 -k ∧t V (∇ W r u)(s) ds dr.
According to the proof of Theorem 3.9, the first part of the theorem follows. The second part is then a rewriting of [START_REF] Dunford | Linear operators. Part II[END_REF].

For p ≥ 1, let Γ p be the set of random fields:

u : R m -→ L p,1
x -→ ((ω, s) → u(ω, s, x))

equipped with the semi-norms,

p K (u) = sup x∈K u(x) Lp,1
for any compact K of R m .

) T 0 u(s, F ) • dW V (s) = T 0 u(s, x) • dW V s x=F . 19
Proof. Simple random fields of the form

u(ω, s, x) = K l=1 H l (x)u l (ω, s)
with H l smooth and u l in L p,1 are dense in Γ p . In view of [START_REF] Feintuch | System theory[END_REF], it is sufficient to prove the result for such random fields. By linearity, can reduce the proof to random fields of the form H(x)u(ω, s). Now for any partition π,

δ W ti∈π 1 θ i ti+1∧t ti∧t H(F )V (u(ω, .))(r) dr 1 [ti, ti+1) = H(F )δ W ti∈π 1 θ i ti+1∧t ti∧t V (u(ω, .))(r) dr 1 [ti, ti+1) - ti∈π ti+1∧t ti∧t ti+1∧t ti∧t H ′ (F )∇ W s F V u(r) ds dr.
On the other hand,

∇ W s (H(F )u(ω, r)) = H ′ (F )∇ W s F u(r), hence ti∈π 1 θ i [ti∧t,ti+1∧t] 2 V (∇ W r H(F )u)(s) ds dr = ti∈π 1 θ i [ti∧t,ti+1∧t] 2 H ′ (F )∇ W s F V u(r) ds dr.
According to Theorem 3.9, Eqn. ( 19) is satisfied for simple random fields.

Definition 5.1. For any 0 ≤ r ≤ t ≤ T , for u in L p, 1 , we define

t r u(s) • dW V (s) as t r u(s) • dW V (s) = t 0 u(s) • dW V (s) - r 0 u(s) • dW V (s) = T 0 e t u(s) dW V (s) - T 0 e r u(s) • dW V (s) = δ W (V (e t -e r )u) + t r D W u(s) ds.
By the very definition of trace class operators, the next lemma is straightforward.

∈ L 2,1 such that ∇ W ⊗ τ T V u and ∇ W ⊗ V τ T u are of trace class. Then, τ T ∇ W ⊗ V u and ∇ W τ T ⊗ V u are of trace class. Moreover, we have: trace(∇ W ⊗ τ T V u) = trace(τ T ∇ W ⊗ V u) and trace(∇ W ⊗ (V τ T )u) = trace(∇ W τ T ⊗ V u).
Proof of 3.10. We first study the divergence term. In view of 3.2, we have

δ B (V (e T -r -e T -t )τ T ǔ • Θ T ) = δ B (V τ T (e t -e r )ǔ • Θ T ) = δ B (τ T VT (e t -e r )ǔ • Θ T ) = δ( VT (e t -e r )ǔ)(ω) = t r VT (1 [r, t] ǔ)(s) dB T (s).
According to Theorem 3.8, ( VT ) * is Ě0 causal and according to 3.3, it is strictly Ě0 causal. Thus, Theorem 3.7 implies that ∇V (e t -e r )ǔ is of trace class and quasi-nilpotent. Hence Lemma 5.4 induces that τ T VT τ T ⊗ τ T ∇τ T (e t -e r )ǔ is trace-class and quasi-nilpotent. Now, according to Theorem 3.1, we have

τ T VT τ T ⊗ τ T ∇τ T (e t -e r )ǔ = V (∇τ T (e T -r -e T -t )ǔ • Θ T ).
According to Theorem 3.9, we have proved (13).

The forward equation. Lemma 5.5. Assume that Hypothesis I(p, η) holds and that σ is Lipschitz continuous. Then, for any

0 ≤ a ≤ b ≤ T , the map VT • σ : C([0, T ], R n ) -→ C([0, T ], R n ) φ -→ VT (σ • ψ 1 [a,b] )
is Lipschitz continuous and Gâteaux differentiable. Its differential is given by:

(20) d VT • σ(φ)[ψ] = VT (σ ′ • φ ψ).
Assume furthermore that σ is sub-linear, i.e., Then, for any

ψ ∈ C([0, T ], R n ), for any t ∈ [0, T ], | VT (σ • ψ)(t)| ≤ cT η+1/p (1 + t 0 |ψ(s)| p ds) ≤ cT η+1/p (1 + ψ ∞ ).
Proof. Let ψ and φ be two continuous functions, since

C([0, T ], R n ) is continuously embedded in L p , VT (σ • ψ -σ • φ) belongs to Hol(η). Moreover, sup t≤T | VT (σ • ψ 1 [a,b] )(t) -VT (σ • φ 1 [a,b] )(t)| ≤ c VT ((σ • ψ -σ • φ) 1 [a,b] ) Hol(η) ≤ c (σ • ψ -σ • φ) 1 [a,b] L p ≤ c φ -ψ L p ([a, b]) ≤ c sup t≤T |ψ(t) -φ(t)|, since σ is Lipschitz continuous. Let ψ and ψ two continuous functions on [0, T ]. Since σ is Lipschitz continuous, we have σ(ψ(t) + εφ(t)) = σ(ψ(t)) + ε 1 0 σ ′ (uψ(t) + (1 -u)φ(t)) du.
Moreover, since σ is Lipschitz, σ ′ is bounded and

T 0 1 0 σ ′ (uψ(t) + (1 -u)φ(t)) du p dt ≤ c T.
This means that (t →

Thus, lim

ε→0 ε -1 ( VT (σ • (ψ + εφ)) -VT (σ • ψ)) exists,
and VT • σ is Gâteaux differentiable and its differential is given by [START_REF] Hirsch | Propriété d'absolue continuité pour les équations différentielles stochastiques dépendant du passé[END_REF]. Since σ • ψ belongs to C([0, T ], R n ), according to Hypothesis I, we have:

| VT (σ • ψ)(t)| ≤ c t 0 s ηp |σ(ψ(s))| p ds 1/p ≤ cT η t 0 (1 + |ψ(s)| p ) ds 1/p ≤ cT η+1/p (1 + ψ p ∞ ) 1/p ≤ cT η+1/p (1 + ψ ∞ ).
The proof is thus complete.

Following [START_REF] Uppman | Sur le flot d'une équation différentielle stochastique[END_REF], we then have the following non trivial result. Theorem 5.6. Assume that Hypothesis I(p, η) holds and that σ is Lipschitz continuous. Then, there exists one and only one measurable map from Ω×[0, T ]×[0, T ] into G which satisfies the first two points of Definition (C). Moreover,

E [|Z r, t (x) -Z r ′ , t (x ′ )| p ] ≤ c(1 + |x| p ∨ |x ′ | p ) (|r -r ′ | pη + |x -x ′ | p )
and for any x ∈ R n , for any 0 ≤ r ≤ t ≤ T, we have

E [|Z r,t (x)| p ] ≤ c(1 + |x| p )e cT ηp+1 .
Note even if x and x ′ are replaced by σ{ BT (u), t ≤ u} measurable random variables, the last estimates still holds.

Proof. Existence, uniqueness and homeomorphy of a solution of (C) follow from [START_REF] Uppman | Sur le flot d'une équation différentielle stochastique[END_REF]. The regularity with respect to r and x is obtained as usual by BDG inequality and Gronwall Lemma. For x or x ′ random, use the independence of σ{ BT (u), t ≤ u} and σ{ BT (u), r ∧ r ′ ≤ u ≤ t}. Theorem 5.7. Assume that Hypothesis I(p, η) holds and that σ is Lipschitz continuous and sub-linear. Then, for any x ∈ R n , for any 0 ≤ r ≤ s ≤ t ≤ T , (ω, r) → Z r,s (ω, Z s,t (x)) and (ω, r) → Z -1 r,t (ω, x) belong to L p,1 . Proof. According to [START_REF] Hirsch | Propriété d'absolue continuité pour les équations différentielles stochastiques dépendant du passé[END_REF]Theorem 3.1], the differentiability of ω → Z r,t (ω, x) is ensured. Furthermore,

∇ u Z r,t = -VT (σ • Z .,t 1 [r, t] )(u) - t r
VT (σ ′ (Z .,t ).∇ u Z .,t 1 [r, t] )(s) d B(s), where σ ′ is the differential of σ. For M > 0, let Now, since 0 ≤ r ≤ s ≤ t ≤ T , Z s,t (x) is independent of Z r,s (x), thus the previous computations still hold and (ω, r) → Z r,s (ω, Z s,t (x)) belong to L p,1 .

ξ M = inf{τ, |∇ u Z τ, t | p ≥ M }
According to [START_REF] Sugita | On a characterization of the Sobolev spaces over an abstract Wiener space[END_REF], to prove that Z -1 r,t (x) belongs to D p,1 , we need to prove (1) for every h ∈ L 2 , there exists an absolutely continuous version of the process (t → Z -1 r,t (ω + th, x)), (2) there exists DZ -1 r,t , an L 2 -valued random variable such that for every h ∈ L Let Θ v = sup u≤v |∂ x Z u, t (x)|. The same kind of computations as above entails that (for the sake of brevity, we do not detail the localisation procedure as it is similar to the previous one): Hence,

E Θ 2q v ≤ c + c E t u Θ 2(q-1)
E Θ 2q v ≤ c 1 + t v E Θ 2q s ds ,
and [START_REF] Lejay | On (p, q)-rough paths[END_REF] follows by Fatou and Gronwall lemmas. Since Z r,t (ω, Z -1 r,t (ω, x)) = x, the implicit function theorem imply that Z -1 r, t (x) satisfies the first two properties and that ∇Z r,t (ω, Z -1 r,t (x)) + ∂Z r,t ∂x (ω, Z -1 r,t (x)) ∇Z -1 r,t (ω, x). It follows by Hölder inequality and Equation ( 22) that DZ -1 r,t (x)) p,1 ≤ c Z r,t (x)) 2p,1 (∂ x Z r, t (x)) -1 2p , hence Z -1 r, t belongs to L p, 1 .

  we have the following result: Lemma 3.3. Assume the resolution of the identity to be either E = (e λT , λ ∈

and (a) 0 = 1

 1 and (a) k = Γ(a + k) Γ(a) = a(a + 1) . . . (a + k -1).

Theorem 5 . 1 . 0 D

 510 Assume that Hypothesis I(p, η) holds. Let u belong toL p,1 . If V ∇ W u is of trace class, then T W u(s) ds = trace(V ∇ W u).

Lemma 5 . 3 .Corollary 5 . 4 .

 5354 Let A and B be two continuous maps fromL 2 ([0, T ]; R n ) into itself. Then, the map τ T A ⊗ B (resp. Aτ T ⊗ B)is of trace class if and only if the mapA ⊗ τ T B (resp. A ⊗ Bτ T ) is of trace class. Moreover, in such a situation, trace(τ T A ⊗ B) = trace(A ⊗ τ T B), resp. trace(Aτ T ⊗ B) = trace(A ⊗ Bτ T ).The next corollary follows by a classical density argument. Let u

( 21 )

 21 |σ(x)| ≤ c(1 + |x|), for any x ∈ R n .

1 0 1 0σ

 11 σ ′ (uψ(t)+(1-u)φ(t)) du) belongs to L p . Hence, according to Hypothesis I, VT ( ′ (uψ(.) + (1 -u)φ(.)) du) C ≤ cT.

  Corollary 5.2 (Substitution formula). Assume that Hypothesis I(p, η) holds. Let {u(x), , x ∈ R m } belong to Γ p . Let F be a random variable such that ((ω, s) → u(ω, s, F )) belongs to L p,1 . Then,

	(

  and Z M τ, t = Z τ ∨ξM , t . Since VT is continuous from L p in to itself and σ is Lipschitz, according to BDG inequality, for r ≤ u,E [|∇ u Z r,t | p ] ≤ c 1 + E The integrability of E [|∇ u Z r,t | p ]with respect to u follows.

	E |∇ u Z M r,t | t r	u pη	r	u	|Z τ,t | p dτ du + E	r	t	s pη	r	s	|∇ u Z M τ,t | p dτ ds
	≤ c 1 + E
					r	t	|∇ u Z M τ,t | p dτ	.
	Then, Gronwall Lemma entails that
		E |∇ u Z M r,t | p ≤ c 1 + E		r	t	|Z τ,t | p dτ	,
	hence by Fatou lemma,
									t	|Z τ,t | p dτ	.
									r

p ≤ cE | VT (σ • Z M .,t 1 [r, t] )(u)| p + c E t r | VT (σ ′ (Z M .,t ) ∇ u Z M .,t 1 [r, t] )(s)| p ds ≤ c 1 + E t r |Z τ,t | p (t pη+1 -τ pη+1 ) dτ + E t r |∇ u Z M τ,t | p (t pη+1 -τ pη+1 ) dτ ≤ ct pη+1 1 + E t r |Z τ,t | p dτ + E

  DZ -1 r,t belongs to L 2 (Ω, L 2 ).

	2 ,
		1 t	(Z -1 r,t (ω + th, x) -Z -1 r,t (ω, x))	t→0 ---→	0	T	DZ -1 r,t (s)h(s) ds,
	where the convergence holds in probability,
	(3) We first show that
	(22)			E	∂Z r,t ∂x	(ω, Z -1 r,t (x))	-p	is finite.
	Since	∂Z r,t ∂x	(ω, x) = Id +

t r VT (σ ′ (Z .,t (x)) ∂Z .,t (ω, x) ∂x )(s) d B(s),