Ueli Meier
email: meierue@ipgp.jussieu.fr

Nikolai M Shapiro

Florent Brenguier

Detecting seasonal variations in seismic velocities within Los Angeles basin from correlations of ambient seismic noise

Keywords: Seismic noise, Interferometry, Crustal structure

We analyze 3 years of continuous seismic records from broadband stations of the Caltech Regional Seismic Network (CI) in vicinity of the Los Angeles basin. Using correlations of ambient seismic noise, relative velocity variations in the order of 0.1 % can be measured between all inter-station pairs. It is the first time that such an extensive study between 861 inter-station pairs over such a large area has been carried out. We perform these measurements using the 'stretching' technique, assuming that one of the two waveforms is merely a stretched version of the other. Obviously this assumption is always violated and the two waveforms are generally decorrelated due to temporal changes in the Earth crust, due to different sources or simply because the cross-correlations are not fully converged. We investigate the stability of these measurements by repeating each measurement over various time-windows of equal length.

On average between all inter-station pairs in the Los Angeles basin a seasonal signal in the relative velocity variation is observed, with peaks and troughs during winter and summer time respectively. Generally the observed signal decreases with increasing inter-station distance and relative travel-time perturbations can only be measured up to an inter-station distance of 60 km.

Furthermore the travel-time perturbations do not depend on azimuth of station pairs, suggesting that they are not related to seasonal variations of the noise sources. Performing a simple

Meier et al.

regionalization by laterally averaging measurements over a subset of stations we found the sedimentary basin showing the most consistent signal and conclude that the observed seasonality might be induced either by changes in the ground-water aquifer or thermo-elastic strain variations that persist down to a depth of 15-22 km.

INTRODUCTION

It has become common ground in seismology that the cross-correlation of a random wave field sensed at two different seismic stations yields the Green's function between the two stations (e.g. [START_REF] Weaver | On the emergence of the Green's function in the correlations of a diffuse field[END_REF][START_REF] Larose | Correlation of random wavefields: An interdisciplinary review[END_REF][START_REF] Gouédard | Cross-correlation of random fields: mathematical approach and applications[END_REF]. An immediate consequence of this theorem, is that any seismic instrument can be used as a potential source, resulting in increased lateral resolution compared to conventional active seismic imaging. It has been demonstrated that surface wave Green functions can be extracted by cross-correlation of multiply scattered seismic coda [START_REF] Campillo | Long-Range Correlations in the Diffuse Seismic Coda[END_REF], as well as by cross-correlation of ambient seismic noise [START_REF] Shapiro | Emergence of broadband Rayleigh waves from correlations of the ambient seismic noise[END_REF]Sabra et al., 2005a,b;[START_REF] Shapiro | High-Resolution Surface-Wave Tomography from Ambient Seismic Noise[END_REF]. This new data source has been exploited extensively to perform conventional surface wave dispersion measurements on local and regional scale (e.g. [START_REF] Brenguier | 3-D surface wave tomography of the Piton de la Fournaise volcano using seismic noise correlations[END_REF][START_REF] Lin | Ambient noise rayleigh wave tomography of new zealand[END_REF][START_REF] Moschetti | Surface wave tomography of the western Unite States from ambient seismic noise: Rayleigh wave group velocity maps[END_REF][START_REF] Stehly | Tomography of the alpine region from observations of seismic ambient noise[END_REF].

In all this studies the surface waves were retrieved from continuous ambient seismic noise records of a given duration and treated as being stationary. Green's functions retrieved from ambient seismic noise records over different time periods are strictly speaking not stationary, and the temporal variations contain useful information. Exploring these temporal variations, it is for example possible to detect instrumental errors [START_REF] Stehly | Traveltime measurements from noise correlation: stability and detection of instrumental time-shifts[END_REF][START_REF] Sens-Schönfelder | Synchronizing seismic networks with ambient seismic noise[END_REF] or to estimate slight changes in the seismic velocity structure. Temporal variations of the media seismic velocity are generally too small to be detected by direct waves. The longer the waves travel through the medium the more sensitive they become to velocity variations. Latter arriving phases, generally attributed to multiply scattered waves, are therefore of great potential to detect relative temporal variations in the subsurface. Estimating relative changes in the mean shear wave velocity from comparison of multiply scattered coda waves dates back to [START_REF] Poupinet | Monitoring Velocity Variations in the Crust Using Earthquake Doublets: An Application to the Calaveras Fault, California[END_REF], who measured temporal variations of crustal velocities from seismograms recorded from a pair of micro-earthquakes occurring at different dates with nearly identical hypocenter and magnitude.

Such events are however very rare, which explains the limited applicability of this technique. Experiments with repeated active seismic sources are also a possibility to obtain similar waveforms but are in general rather expensive (e.g. [START_REF] Nishimura | Temporal changes of the crustal structure associated with the M6.1 earthquake on September 3, 1998 and the volcanic activity of mount Iwate, Japan[END_REF][START_REF] Wegler | Increase of shear wave velocity before the 1998 eruption of Merapi volcano (Indonesia)[END_REF]. In contrast, continuous ambient seismic noise records from permanent or semi-permanent networks are more cost effective and furthermore the position of the receivers and hence the apparent sources does not change with time. [START_REF] Sens-Schönfelder | Passive image interferometry and seasonal variations of seismic velocities at Merapi Volcano, Indonesia[END_REF] demonstrated that Green's functions retrieved from ambient seismic noise records contain multiply scattered waves, which can be analyzed to monitor small temporal variations of the subsurface. Indeed, this technique has been successfully applied to detect small velocity variations in volcanoes (e.g. [START_REF] Sens-Schönfelder | Passive image interferometry and seasonal variations of seismic velocities at Merapi Volcano, Indonesia[END_REF]. Similarly, [START_REF] Wegler | Fault zone monitoring with passive image interferometry[END_REF] and Brenguier et al. (2008a) detected a decrease of relative seismic velocities that coincidence with the occurrence of an earthquake and Brenguier et al. (2008b) detected a decrease in relative velocity prior to volcanic eruptions. The same principle has also been successfully applied in laboratory experiments to detect small variations in the medium of interest (e.g. [START_REF] Snieder | Coda Wave Interferometry for Estimating Nonlinear Behaviour in Seismic Velocity[END_REF][START_REF] Grêt | Time-lapse monitoring of rock properties with wave interferometry[END_REF][START_REF] Leroy | Temperature-dependent diffusing acoustic wave spectroscopy with resonant scatterers[END_REF][START_REF] Larose | Monitoring stress related velocity variation in concrete with a 2.10 -5 relative resolution using diffuse ultrasound[END_REF][START_REF] Hadziioannou | Stability of Monitoring Weak Changes in Multiply Scattering Media with Ambient Noise Correlation: Laborartory Experiments[END_REF].

In this paper we extend this technique to broadband data from the Caltech Regional Network (CI) and analyze 3 years of continuous seismic records from 42 broadband stations. It is for the first time that such an extensive study, analyzing relative velocity changes from 861 inter-station pairs covering an area of approximately 90 × 190 km, has been carried out. We measured relative traveltime perturbations between all the inter-station pairs in various frequency bands between 0.1 to 2 Hz in the Los Angeles basin. We mainly opted for this frequency band because it includes the micro-seismic peak [START_REF] Stehly | A study of the seismic noise from its longrange correlation properties[END_REF]. We perform repeated measurements over various time-windows and computed corresponding uncertainties from the variance of those repeated measurements. In principle these uncertainties should depend on the amount of decorrelation as described by the correlation coefficient, central frequency, bandwidth and length of the time win-dow over which the measurements are performed [START_REF] Hadziioannou | Stability of Monitoring Weak Changes in Multiply Scattering Media with Ambient Noise Correlation: Laborartory Experiments[END_REF][START_REF] Weaver | On the precision of noisecorrelation interferometry[END_REF].

Our uncertainty estimates do however not depend on the maximum correlation coefficient and we therefore decided to discard measurements from two waveforms whose maximum correlation coefficient after stretching is below a certain threshold. In this way we simply discard bad quality measurements from further analysis. Using a simple regionalization scheme, laterally averaging measurements from a subset of stations, we were able to identify the sedimentary basin as having the most consistent signal. These are however still preliminary and qualitative results. In order to further improve on these results and quantitatively map relative velocity changes in the Earth's crust on regional scale a better understanding of the spatial sensitivities of these measurements is required. A problem that to the best of our knowledge is still an open issue.

DATA

We analyzed 3 years of continuous seismic records from 42 broadband stations of the Caltech Regional Network (CI) in the vicinity of the Los Angeles basin (Fig. 1). Not only are we using many more stations than in previous studies but also cover a much bigger area with a maximum inter-station distance of ≈ 190 km parallel to the coast (DGR-MOP) and ≈ 90 km perpendicular to the coast (BFS-RPV). The data were obtained from the Southern California Earthquake Data Center (SCEDC). We only analyzed the vertical component and performed standard data processing: 1) removed the instrument response, 2) down-sampled the data to a sampling rate of 0.05 s, 3) performed spectral whitening in the frequency band of 0.0064-4 Hz, 4) applied one-bit normalization and 5) computed 1-day cross-correlations for all the 861 inter-station pairs and 42 auto-correlations for a time-lag of ±600 s. Stacking the 1 day correlations over the whole 3 year time period results in the reference stack. In order to measure the temporal evolution of relative traveltime perturbations with respect to the reference stack, we computed 60 day-long stacks over the whole 3 year period in a five day moving window. In Figure 2 a 60-day stack together with the reference stack is shown for the station-pair DEC-USC (A), as well as all 60-day stacks over the whole 3-year period for the same station-pair (B). Note that slight changes in coherent phases arriving after ±25 s over the 3-year period are already visible. Furthermore seasonal variations in amplitudes of the 60-day stacks are also visible.

MEASURING RELATIVE VELOCITY PERTURBATIONS

By measuring relative velocity perturbations (δv/v) from the coda of two similar waveforms it is generally assumed that δv/v is spatially homogeneous. This implies that the relative time shift (δτ /τ ) between the two waveforms is independent of the lapse time (τ ) at which it is measured and δv/v = -δτ /τ = const. [START_REF] Snieder | Coda Wave Interferometry for Estimating Nonlinear Behaviour in Seismic Velocity[END_REF]. In principle it would be sufficient to measure δτ at one particular lapse time τ to estimate δτ /τ . Mainly because the relative velocity perturbations are only approximately homogeneous but also due to uncertainties in measuring δτ a statistical approach is generally chosen and δτ i is measured at various laps times τ i .

There are two similar techniques to measure δτ i within time windows of length T centered around τ i . [START_REF] Snieder | Coda Wave Interferometry for Estimating Nonlinear Behaviour in Seismic Velocity[END_REF] estimated δτ i as the time shift corresponding to the maximum of the cross-correlation of the two waveforms in each window of length T centered around τ i . δτ /τ is then approximated as the mean of all δτ i /τ i measured over a window of length L, where L > T .

Another technique estimates δτ i from the phase of the cross-spectrum of the two waveforms in each window of length T centered around τ i . The time shift δτ i is then obtained by fitting a line, starting at the origin, to the phase of the cross-spectrum φ(f ) = 2πδτ i f [START_REF] Poupinet | Monitoring Velocity Variations in the Crust Using Earthquake Doublets: An Application to the Calaveras Fault, California[END_REF].

With this technique the accuracy of the individual measurements is not limited by the sampling rate. δτ /τ and hence δv/v is then obtained by a linear regression of the various δτ i measurements over a window of length L, where L > T . Both techniques require to choose an appropriate value for the window length T , which should be long enough to obtain meaningful correlations and cross-spectra but also short enough to ensure that the distortion of the two waveforms is small within these windows. [START_REF] Sens-Schönfelder | Passive image interferometry and seasonal variations of seismic velocities at Merapi Volcano, Indonesia[END_REF] proposed instead to estimate δτ /τ as the factor by which the time axis on one waveform has to be stretched or compressed to obtain the best correlation with the other waveform over the time-window of length L. An obvious advantage to the above mentioned techniques, is that there is no need to set an appropriate value for T and more importantly, a direct estimate of δτ /τ is obtained. In a recent study [START_REF] Hadziioannou | Stability of Monitoring Weak Changes in Multiply Scattering Media with Ambient Noise Correlation: Laborartory Experiments[END_REF] discussed advantages and drawbacks of both above mentioned techniques.

Uniform scaling of the time axis (Stretching)

More formally, let's assume we have two waveforms, a reference waveform f ref (t) and a current waveform f cur (t). A stretched version of f cur (t) is defined as:

f cur (t) = f cur (t(1 + )) (1)
where is the stretching coefficient. When then look for that maximizes:

C( ) = f cur (t)f ref (t)dt ( f cur (t) 2 dt)( f ref (t) 2 dt) (2)
where integration is over a specific time window t 1 <= t <= t 2 of length L = t 2 -t 1 and the relative time shift of f cur with respect to f ref is simply given by: δτ /τ =max .

If f cur is a stretched version of f ref , we will find C max = 1 at some max = 0. In general the two waveforms f cur and f ref will slightly differ due to temporal variations in the Earth crust, due to different sources or because the correlations are not fully converged (i.e. C max < 1). Strong distortion and weak stretching might furthermore indicate a change of structure rather than a global velocity change. In other words, the assumptions on which eq. ( 2) are based are always going to be violated, leading to errors in the estimation of max . It is therefore crucial to investigate the stability of the measurements as well as trying to come up with uncertainty estimates that serve as a measure of how reliable a particular measurement is. So far it has been shown that increasing the window length L and/or the bandwidth can reduce the effect of differing waveforms [START_REF] Hadziioannou | Stability of Monitoring Weak Changes in Multiply Scattering Media with Ambient Noise Correlation: Laborartory Experiments[END_REF]. It is however still an unresolved issue how slightly differing waveforms are affecting the estimation of the stretching coefficient [START_REF] Weaver | On the precision of noisecorrelation interferometry[END_REF].

Here we take a pragmatic approach in order to investigate the stability of the performed measurements and perform repeated measurements in shifted but overlapping time-windows of equal length. If one of the two waveforms is merely a stretched version of the other, all repeated measurements should yield identical results. The variance of the repeated measurements can therefore be seen as an error estimate of the δτ /τ measurements. Additionally we also keep track of the maximum value of the correlation coefficient and discard measurements whose best stretching co-efficient results in a correlation coefficient below a certain threshold. In this way measurements from waveforms that differ too much are simply excluded.

In order to measure relative time-shifts in the coda of a 60-day stack with respect to the reference stack an appropriate time window has to be chosen. We computed cross-correlations for a time-lag of ±600 s. In order to obtain meaningful measurements of δτ /τ this is obviously to long and since we are interested in comparing measurements for different inter-station distances up to 189 km, an adaptive starting time (t 1 ) of the time window has to be chosen. Since we are not interested in direct surface waves we set for all inter-station pairs the starting time

t 1 = d i /v min ,
where d i is the i th inter-station pair distance and v min = 1 k/s is the surface wave velocity. In order to adapt the window length to different frequency ranges the time window length is set to L = 10T max , where T max is the maximum period of the considered bandwidth. When then repeat the measurement k times in a smaller time window of length L = 5T max and increased starting time

t start = t 1 +(k -1)T max ,
where k = 1, .., 5 (Fig. 3). From the five independent measurements in smaller time windows we compute error estimates of max around the values estimated from the measurement performed over the long time window.

In Figure 4 the cross-correlation as a function of the stretching coefficient (eq. 2) is shown for a 60-day stack starting at 01/05/2001 with respect to the 3-year reference stack for station-pair BRE-CHN in the negative (A) and positive (B) time windows, respectively. The dashed lines are computed in overlapping time-windows of length L = 50s with starting times t 1 = ±35, ±45, .., ±85s, and the solid line is computed over a time window of length L = 100s starting at t 1 = ±35. A zoom around the maximum is shown in (C) and (D) with the stretching coefficient that maximizes eq. 2 (solid vertical line), and the uncertainties (dashed line) given by the standard deviation of the repeated measurements over the shorter time windows (dashed curves in A,B). The value of the correlation coefficient gives a measure of how well the two waveforms match after stretching, and we would intuitively guess that the lower the maximum correlation coefficient the lower the quality of the measurements. In section 4 we will demonstrate how only considering measurements with a correlation coefficient bigger than a certain threshold increases the quality of the result. In this example we evaluated C( ) for stretching coefficients ranging from [-0.1 : 0.1] with an increment of ∆ = 10 -4 , the resulting max has therefore an accuracy of 10 -4 and 2001 evaluations of C( ) are required to find the maximum. This is obviously far from being efficient and considering that we ultimately perform 207 measurements for 861 inter-station pairs in positive and negative time windows a more efficient algorithm to maximize C( ) is required. We opted for a line search algorithm [START_REF] Nabney | Netlab: Algorithms for Pattern Recognition[END_REF] that allows us to estimate C max with an accuracy of 10 -4 requiring much less evaluations of max . The increased efficiency came however at the price, that we risk to find only a local maximum. It is therefore crucial to chose appropriate initial bounds on . If the initial search bound is too broad, there are likely to be various maxima in the correlation coefficient function C( ), if the initial search bound is however too narrow, there might be no maximum at all and our results might be biased. In other words, the initial search bound reflects the maximum traveltime perturbations between a 60-day and the reference stack we are expecting a priori. After evaluating and plotting C( ) for various inter-station pairs in the different frequency bands considered, we set the initial search range to -0.01 <= <= 0.01 for the frequency ranges 0.1-0.2, 0.1-1 and 0.1-2 Hz and to -0.005 <= <= 0.005 for the frequency range 0.5-2 Hz respectively.

RESULTS

We measured δτ /τ from all the 207, 60-day stacks with respect to the reference stack for all 861 inter-station pairs in a particular time window of length L starting at ±t 1 . Additionally we computed the corresponding standard deviations as the variance of the repeated measurements. In total we performed 6 × 178227 measurements in positive and negative time windows. Performing that many measurements requires an automated quality control of the 60-day stacked crosscorrelations. There are various problems that might arise, limited data availability during some days, missing information about the instrument response on other days, or unknown instrumental time-shifts for some stations, just to name a few. If one of the above mentioned problems arose during a particular time-period, the resulting 60-day stacks are likely to be corrupted. In order to detect those spurious correlations we first check if the two waveforms are correctly aligned and only perform measurements if the maximum of the cross-correlation in the considered time window occurs at time lags smaller than ±5 times the sampling rate. In this way we reject spurious waveforms that for one reason or another are not suitable to perform the measurements and ultimately would degrade the results. For illustrative purposes we start presenting our results with a comparison from one particular inter-station pair in various frequency bands. Note how the uncertainties on the measurements decrease with increasing bandwidth. Low C max values are an indication that the 60-day stack is not merely a stretched/compressed version of the reference stack, and that the two waveforms are highly decorrelated. In this case the corresponding estimates of δτ /τ are obviously less precise and ultimately meaningless. This observation is most evident in the frequency range between 0.5-2 Hz, where the C max values in the positive time window (K, blue) are very low. Looking at the waveforms (L, right panel) this is not surprising because there are no coherent phases in this frequency range. Since the inter-station pair BRE-CHN is perpendicular to the coast, the positive time window corresponds to waves traveling towards the ocean, a direction which is known to have very weak noise sources. The same observation can be made for the other three frequency ranges, but to a lesser extent. Furthermore it is worth noting that δτ /τ in the positive and negative time windows show a seasonal variation with peaks and troughs during winter and summer time respectively. The amplitude of this variations is in the order of ±0.5% in the frequency range between 0.1-0.2 Hz (A) and in the order of ±0.2% in the frequency ranges 0.1-1 Hz (D) and 0.5-2 Hz (G) respectively.

Laterally averaged distribution of δτ /τ for all 861 inter-station pairs

In a next step we computed averages and standard deviations of δτ /τ from all the 861 inter-station pairs. Instead of only plotting mean and standard deviations we decided to plot the whole Gaussian distribution for each time step over the whole 3-year period (Fig. 6), merely for illustrative purposes. Each vertical line on these plots corresponds to a Gaussian probability density function pdf (i.e. integrates to unity) with mean and standard deviation computed as described in section (3.1). This figure can be seen as a summary of all the performed measurements. The mean (black line) shows a clear seasonal variation. Note however that standard deviations are rather big especially for the frequency range between 0.1-0.2 Hz (Fig. 6A). This might be due to the fact that we averaged measurements over various inter-station pair distances, ranging from 6-190 km, various azimuths of station pairs and considered all paths that correspond to varying geological regimes.

The more likely reason for this high standard deviations is however the small bandwidth that makes the measurements less stable to distortions of the waveforms [START_REF] Hadziioannou | Stability of Monitoring Weak Changes in Multiply Scattering Media with Ambient Noise Correlation: Laborartory Experiments[END_REF].

And indeed as can be seen for the other frequency bands the broader the bandwidth the more stable the measurements are. Another problem is that we considered all the measurements, some of this measurements are potentially from waveforms that differ too much even after stretching.

Consequently the maximum correlation coefficient of those measurements is low and indeed if all measurements with C max < 0.9 are excluded the resulting standard deviation of the distribution of δτ /τ is much smaller (Fig. 7).We conclude that seasonal variations of δτ /τ measurements with peaks and troughs during winter and summer time respectively are a robust feature in the studied area and that in order to obtain more robust results the bandwidth can be increased. Furthermore it is crucial to exclude measurements from waveforms that differ too much as described by C max .

In what follows we will focus further discussion of the results on the seasonal variations and will only consider measurements performed over the frequency range 0.1-2 Hz.

Relative time shifts as a function of inter-station pair distance

In order to further investigate the origin of the observed signal we decided to investigate the dependence of δτ /τ with respect to inter-station pair distance. In Figure 8 This suggests that if measurements from varying inter-station pair distances were to be combined, an adaptive way on how to choose the stacking duration would be required.

Relative time shifts as a function of azimuth

It is well documented that the origin of seismic noise sources shows a clear seasonal variation (e.g. [START_REF] Stehly | A study of the seismic noise from its longrange correlation properties[END_REF][START_REF] Yang | Ambient noise Rayleigh wave tomography across Europe[END_REF]. In order to investigate if our observations are related to seasonal variations of the seismic sources or rather related to seasonal changes in the subsurface we investigated the dependence of δτ /τ on azimuth of station pairs. If we were measuring some process related to seasonal variations of the seismic sources, that show a clear dependence on azimuth, we would expect to see the same dependence in our δτ /τ measurements. In Figure 10, δτ /τ is plotted as a function of azimuth in the frequency range 0.1-2 Hz. Interestingly there seems to be no particular dependence on azimuth, for all six azimuth bins the same seasonal variations are observed. This suggests that our observations are indeed related to variations in the Earth crust and not due to seasonal variations in the noise sources.

Crude regionalization

We perform a crude regionalization by simply averaging measurements from inter-station pairs that belong to three different regions. We decided to form three different clusters as indicated in Figure 11A. Cluster 1 (red) consists of stations in the north-western part of the study area mainly on hard-rock sites (Fig. 11B). Cluster 2 (blue) consists of stations in the north-eastern part of the study area, again mainly located on hard-rock sites (Fig. 11C). Whereas cluster 3 (green) consists of stations that are in the central part of the study area and are located within the sedimentary basin (Fig. 11D). All three clusters consist of roughly the same amount of stations and also cover a similar area. The average distributions of δτ /τ of cluster 1, cluster 2 and cluster 3 are plotted in Figure 11B, C and D respectively. Clearly the average over stations within the sedimentary basin shows the most consistent variation with a clear seasonality in δτ /τ . The average over stations within cluster 2 shows also a seasonal variation, but the variance is much higher and there also seem to exist additional variation over shorter time ranges. In order to further investigate if those short-term variations are simply due to instabilities in the measurements or are indeed related to processes in the upper crust a more detailed investigation would be required.

INTERPRETATION

As far as the interpretation of observed relative velocity changes are concerned, it is of great interest not only to monitor these changes, but also to provide an interpretation of what processes might have caused these changes. In labor experiments, under controlled conditions, it has been demonstrated that either changes in stress, temperature or water saturation result in slight velocity changes that can be detected in the coda (e.g. [START_REF] Weaver | Temperature dependence of diffuse field phase[END_REF][START_REF] Grêt | Time-lapse monitoring of rock properties with wave interferometry[END_REF][START_REF] Larose | Monitoring stress related velocity variation in concrete with a 2.10 -5 relative resolution using diffuse ultrasound[END_REF]. Identifying the processes that might have caused the observed velocity changes in the Earth's crust poses a more difficult problem. Recent studies presented interpretations of relative velocity changes observed at volcanoes and across the San Andreas fault (e.g. Brenguier et al., 2008b,a). These studies covered a rather small area and more importantly the candidate processes that might have caused the measured relative velocity changes were known beforehand (i.e. volcanic eruptions, earthquakes) and the seasonal variations were consider as background noise. [START_REF] Sens-Schönfelder | Passive image interferometry and seasonal variations of seismic velocities at Merapi Volcano, Indonesia[END_REF] on the other hand observed similar seasonal variations and proposed a depth dependent hydrological model that described the seasonality in the observed relative velocity changes solely based on precipitation. This model however was based on the analysis from a single station-pair.

In the following we discuss three different candidate processes that might have caused the observed seasonal variations within the velocity field of the Los Angeles: 1) changes in water content, 2) changes in temperature and 3) changes in stress/strain field. Seasonality of climatological variables, such as temperature, precipitation and oceanic wave height are potentially causing seasonal changes in the mechanical properties of the Earth's crust. Seasonal variations have also been observed using SAR interferometry and GPS measurements [START_REF] Bawden | Tectonic contraction across Los Angeles after removal of groundwater pumping effects[END_REF][START_REF] Watson | Satellite interferometric observations of displacements associated with seasonal groundwater in the Los Angeles basin[END_REF][START_REF] Argus | Interseismic strain accumulation and anthropogenic motion in metropolitan Los Angeles[END_REF]. In these studies, observed seasonal vertical displacements have been attributed to vertical motion induced by annual variations of the groundwater table [START_REF] Bawden | Tectonic contraction across Los Angeles after removal of groundwater pumping effects[END_REF]. This hypothesis has been confirmed by [START_REF] Watson | Satellite interferometric observations of displacements associated with seasonal groundwater in the Los Angeles basin[END_REF] who additionally analyzed GPS data and found that the Los Angeles basin becomes most inflated at the beginning of the year after the rainy season when the aquifer should be at a maximum. This inflation may partially explain reduction of seismic velocities that we observe. Moreover, the observed signal is stronger within the sedimentary basin which is also consistent with seasonal changes in hydrological parameters. At the same time, surface waves in the frequency range considered 0.1-2 Hz are sensitive down to a depth of 10 km, which may be significantly deeper than layers affected by variations of the aquifer. Therefore, we also consider other seasonal processes such as changes in surface temperature that might affect seismic velocities in the upper crust. It has been demonstrated by Ben-Zion & Leary (1986) and more recently by [START_REF] Prawirodirdjo | Observation and modeling of thermoelastic strain in Southern California Integrated GPS Network daily position time series[END_REF] that, while changes in surface temperature affect directly only very superficial parts of the crust, they induce thermo-elastic strain variations that persist down to a depth in the order of the surface temperature wavelength. This wavelength of the temperature field has been estimated to range from 15-22 km [START_REF] Prawirodirdjo | Observation and modeling of thermoelastic strain in Southern California Integrated GPS Network daily position time series[END_REF] in three different regions within our study area. According to [START_REF] Ben-Zion | Thermoelastic strain in a half-space covered by unconsolidated material[END_REF] the amplitude of the thermo-elastic strain is over ten times larger than the effect of ≈ 15m changes of water level in a reservoir. It seems more likely that the strongest variations are caused by induced thermo-elastic induced strain variations within the Earth's upper crust and that the seasonal variations of hydrological parameters simply further enhance these changes.

DISCUSSION

We computed broadband cross-correlations from 861 inter-station pairs in the Los Angeles basin and are able to monitor relative travel-time perturbations over a period of three years. It is to our knowledge for the first time that these measurements have been performed for that many interstation pairs covering such a big area. On average we find a clear seasonality in the relative velocity changes that might persist down to a depth of 10-20 km. Variations between different inter-station pairs have however high variance and in order to obtain stable results we had to average over close by inter-station pairs. The results are therefore still rather qualitative and make a detailed interpretation difficult. A major problem we encountered is the trade-off between temporal and lateral resolution. For bigger inter-station pair distances it seems that longer ambient seismic noise records have to be stacked in order to obtain meaningful measurements. An adaptive way to choose the duration over which the reference stack is built as a function of inter-station pair distance might resolve some issues regarding the stability of the measurements. This would however also introduce different time-resolutions for different measurements complicating the interpretation even further.

We demonstrated how high quality measurements of relative velocity changes on a regional scale, covering an area of approximately 90× 150 km, from 861 inter-station pairs can be obtained.

We were however only able to give an average interpretation of these measurements, averaged over three different regions. In order to provide more localized estimates of the changes in the Earth crust a better understanding of the spatial sensitivities of δτ/ τ measurements -ideally as a function of frequency -is required, as well as a physical model that relates travel-time perturbations to changes in material properties. This would allow us to perform a meaningful regionalization of our measurements, providing interesting insights into the dynamics of the mechanical properties of the Earth's upper crust.

Figure 1 .

 1 Figure 1. 42 broadband stations indicated by the red circles of the Caltech Regional Network (CI) considered in this study.

Figure 2 .

 2 Figure 2. A) Sixty-day stacked cross-correlation function starting at 01/05/2001 (blue) together with the 3-year reference stack (red) for station-pair DEC-USC. B) All 60-day stacks over the 3-year period for station-pair DEC-USC. The cross-correlations are filtered between 0.1-0.2 [Hz].

Figure 3 .

 3 Figure 3. Sixty-day stacked cross-correlation function filtered between 0.1-0.2 Hz starting at 01/05/2001 (blue) together with the 3-year reference stack (red) for station-pair BRE-CHN. The time window of length L = 10T max (black box) and the time window of length L = 5T max (black dashed box) that is shifted towards the right and over which the relative time shifts are measured repeatedly is also shown for positive lapse times.

Figure 4 .

 4 Figure 4. Correlation coefficient as a function of the stretching coefficient (eq. 2) computed over 100s long (solid) and 50s long (dashed) windows for the two waveforms shown in Fig. 3 for negative (A) and positive (B) lapse times. (c) and (D) show a zoom around the maximum of the cross correlation function of the 100s long window (solid) together with the maximum (vertical line) and the corresponding uncertainties (dashed) estimated from the 5 repeated measurements over 50s long windows. Note the different horizontal scale in (C) and (D).

4. 1

 1 Comparison of δτ /τ for BRE-CHN, in four different frequency ranges: 0We measured δτ /τ from 207, 60-day stacks with respect to the reference stack for the inter-station pair BRE-CHN and compare the obtained results from cross-correlations filtered between 0.1-0.2, 0.1-1, 0.1-2 and 0.5-2 Hz. In Figure5, the outcome of this comparison is summarized: δτ /τ measurements are shown together with the corresponding standard deviations in the positive (blue) and negative (red) time window (left column); the corresponding C max values are shown for positive (blue) and negative (red) time windows (middle column); and all the 207, 60-day stacks in the considered time windows are shown (right column). From top to bottom the corresponding crosscorrelations where filtered between 0.1-0.2 (A-C), 0.1-1 (D-F), 0.1-2 (G-I) and 0.5-2 Hz (J-L).

Figure 5 .

 5 Figure 5. A) Relative time shift measurements and corresponding standard deviations of all 207, 60-day stacks with respect to the reference stack, measured in the positive (blue) and negative (red) time window (L = 100 s) for the inter-station pair BRE-CHN in the frequency range between 0.1-0.2 Hz . B) Values of the maximum correlation coefficient C max in the positive (blue) and negative (red) time window. C) All the 207, 60-day stacks filtered between 0.1-0.2 Hz, only the time windows over which the measurements were performed are shown. D-F) Similar plots as A-C) only that the cross-correlations are filtered between 0.1-1 Hz. G-I) Similar plots as A-C) only that the cross-correlations are filtered between 0.1-2 Hz. J-L) Similar plots as A-C) only that the cross-correlations are filtered between 0.5-2 Hz, note the shorter window length of L = 50 s. Vertical axes are changing in left and middle columns.

  , the distribution of the average is plotted for 6 different inter-station pair distances, 0-20 (top,left), 20-40 (top,right), 40-60 (middle,left), 60-80 (middle,right), 80-100 (bottom,left) and for inter-station pair distances > 100 km(bottom,right). Note that between 40-60 km the signal becomes weaker and finally disappears completely. This suggests that in the frequency range considered (0.1-2 Hz), seasonal variations in δτ /τ can only be observed for inter-station pair distances up to 40-60 km. A possible explanation for this observation might be, that no coherent phases are present in the considered time-windows for long inter-station pair distances due to attenuation. Looking at the histogram of the maximum correlation coefficient as a function of inter-station pair distance (Fig.9) it is nicely visible that the bigger the inter-station-pair distance the smaller the corresponding maximum correlation coefficient of the stretched waveforms become. For long inter-station pair distances stacking only 60-days of ambient seismic noise records is probably not long enough. Consequently the 60-days stack didn't fully converge and differ too much with respect to the three year-long reference stack.

Figure 6 .

 6 Figure 6. Gaussian pdfs of δτ /τ with superimposed mean (black) ± one standard deviation (dashed) for all 861 inter-station pairs over a 3 year period in various frequency bands, 0.1-0.2 (A), 0.1-1 (B),0.1-2 (C) and 0.5-2 Hz (D).

Figure 7 .

 7 Figure 7. Gaussian pdfs of δτ /τ from high quality measurements(i.e. C max > 0.9) with superimposed mean (black) ± one standard deviation (dashed) for all 861 inter-station pairs over a 3 year period in various frequency bands, 0.1-0.2 (A), 0.1-1 (B), 0.1-2 (C) and 0.5-2 Hz (D).

Figure 8 .Figure 9 .

 89 Figure 8. Gaussian pdfs of δτ /τ with superimposed mean (black) ± one standard deviation (dashed) for all 861 inter-station pairs as a function of inter-station pair distance in the frequency range 0.1-2 Hz.

Figure 10 .

 10 Figure 10. Gaussian pdfs of δτ /τ with superimposed mean (black) ± one standard deviation (dashed) for all 861 inter-station pairs as a function of azimuth for C max > 0.9 in the frequency range 0.1-2 Hz.

Figure 11 .

 11 Figure 11. A) Stations that form the three clusters whose average relative time shifts are shown in B) red stations, C) blue stations and D) green stations. Again in B,C,D) Gaussian pdfs of δτ /τ are shown with superimposed mean (solid) ± one standard deviation (dashed).

ACKNOWLEDGMENT

We thank E. Larose, P. Roux, M. Campillo and R.L. Weaver for helpful discussions, R.L. Weaver for communicating his results on the precision of noise-correlation interferometry and Y. Ben-Zion for discussions regarding the thermo-elastic strain variations. All the data were provided by the Southern California Earthquake Data Center (SCEDC). This work was supported by an ANR contract ANR-06-CEXC-005 (COHERSIS) and an EU contract WHISPER.


Clément Foucart
email: .clement.foucart@etu.upmc.fr

Generalized Fleming-Viot processes with immigration via stochastic flows of partitions

Keywords: Exchangeability, duality, stochastic flow, lookdown construction, immigration, Fleming-Viot process, measure-valued process, coalescent, partition-valued process, coming down from infinity Mathematics Subject classification (2010): 60J25 60G09 92D25

The generalized Fleming-Viot processes were defined in 1999 by Donnelly and Kurtz using a particle model and by Bertoin and Le Gall in 2003 using stochastic flows of bridges. In both methods, the key argument used to characterize these processes is the duality between these processes and exchangeable coalescents. A larger class of coalescent processes, called distinguished coalescents, was set up recently to incorporate an immigration phenomenon in the underlying population. The purpose of this article is to define and characterize a class of probability measure-valued processes called the generalized Fleming-Viot processes with immigration. We consider some stochastic flows of partitions of Z + , in the same spirit as Bertoin and Le Gall's flows, replacing roughly speaking, composition of bridges by coagulation of partitions. Identifying at any time a population with the integers N := {1, 2, ...}, the formalism of partitions is effective in the past as well as in the future especially when there are several simultaneous births. We show how a stochastic population may be directly embedded in the dual flow. An extra individual 0 will be viewed as an external generic immigrant ancestor, with a distinguished type, whose progeny represents the immigrants. The "modified" lookdown construction of Donnelly-Kurtz is recovered when neither simultaneous multiple births nor immigration are taken into account. In the last part of the paper we give a sufficient criterion for the initial types extinction.

Introduction

Originally, Fleming-Viot processes were defined in 1979 in [START_REF] Fleming | Some measure-valued Markov processes in population genetics theory[END_REF] to model the genetic phenomenon of allele drift. They now form an important sub-class of measure-valued processes, which have received significant attention in the literature. [START_REF] Donnelly | A countable representation of the Fleming-Viot measurevalued diffusion[END_REF], established in [START_REF] Donnelly | A countable representation of the Fleming-Viot measurevalued diffusion[END_REF] a duality between the classical Fleming-Viot process and the Kingman coalescent (defined in [START_REF] Kingman | The coalescent[END_REF]). Shortly after, the class of coalescent processes was considerably generalized by assuming that multiple coagulations may happen simultaneously. We refer to Chapter 3 of [START_REF] Berestycki | Recent progress in coalescent theory, volume 16 of Ensaios Matemáticos[END_REF] for an introduction to the Λ-coalescents and to the seminal papers [START_REF] Möhle | A classification of coalescent processes for haploid exchangeable population models[END_REF], [START_REF] Schweinsberg | Coalescents with simultaneous multiple collisions[END_REF] for the definition of the general exchangeable coalescent processes (also called Ξ-coalescents). An infinite particle representation set up in 1999, allowed Donnelly and Kurtz, in [START_REF] Donnelly | Particle representations for measure-valued population models[END_REF] to define a generalized Fleming-Viot process by duality with the Λ-coalescent. In 2003, Bertoin and Le Gall started from another point of view and introduced, in [START_REF] Bertoin | Stochastic flows associated to coalescent processes[END_REF], a stochastic flow of bridges which encodes simultaneously an exchangeable coalescent process and a continuous population model, the socalled generalized Fleming-Viot process. Finally, in 2009, Birkner et al. in [START_REF] Birkner | A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks[END_REF] have adapted the same arguments as Donnelly-Kurtz for the case of the Ξ-coalescent.

More recently, a larger class of coalescents called distinguished coalescents were defined, see [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF], in order to incorporate immigration in the underlying population. The purpose of this article is to define by duality the class of generalized Fleming-Viot processes with immigration. Imagine an infinite haploid population with immigration, identified by the set N. This means that each individual has at most one parent in the population at the previous generation; indeed, immigration implies that some individuals may have parents outside this population (they are children of immigrants). To encode the arrival of new immigrants, we consider the external integer 0 as the generic parent of immigrants. We shall then work with partitions of Z + , the so-called distinguished partitions. Our approach will draw both on the works of Bertoin-Le Gall and of Donnelly-Kurtz. Namely, in the same vein as Bertoin and Le Gall's article [START_REF] Bertoin | Stochastic flows associated to coalescent processes[END_REF], we define a stochastic flow of partitions of Z + , denoted by (Π(s, t), -∞ < s ≤ t < ∞). The dual flow ( Π(t), t ≥ 0) := (Π(-t, 0), t ≥ 0) shall encode an infinite haploid population model with immigration forward in time. Namely, for any individual i ≥ 1 alive at the initial time 0, the set Πi (t) shall represent the descendants of i, and Π0 (t) the descendants of the generic immigrant, at time t. We stress that the evolution mechanism involves both multiple reproduction and immigration. The genealogy of this population model is precisely a distinguished exchangeable coalescent. Our method is close to the "modified" lookdown construction of Donnelly-Kurtz, but differs from (and simplifies) the generalization given by Birkner et al. in [START_REF] Birkner | A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks[END_REF]. In the same vein as Fleming and Viot's fundamental article [START_REF] Fleming | Some measure-valued Markov processes in population genetics theory[END_REF], we consider the type carried by each individual at the initial time. Denote by U i the type of the individual i ≥ 1, and distinguish the type of the generic immigrant by fixing U 0 = 0. The study of the evolution of frequencies as time passes, leads us to define and study the so-called generalized Fleming-Viot process with immigration (GFVI for short) denoted in the sequel by (Z t , t ≥ 0). This process will be explicitly related to the forward partition-valued process ( Π(t), t ≥ 0). When the process ( Π(t), t ≥ 0) is absorbed in the trivial partition ({Z + }, ∅, ...), all individuals are immigrant children after a certain time. We shall discuss conditions entailing the occurrence of this event.

The rest of the paper is organized as follows. In Section 2 (Preliminaries), we recall some basic facts on distinguished partitions. We introduce a coagulation operator coag and describe how a population may be encoded by distinguished partitions forward in time. Some properties related to exchangeable sequences and partitions are then presented. We recall the definition of an exchangeable distinguished coalescent and define a stochastic flow of partitions using a Poisson random measure on the space of partitions. In Section 3, we study the dual flow and the embedded population. Adding initially a type to each individual, the properties of exchangeability of Section 2 allow us to define the generalized Fleming-Viot process with immigration. We show that any generalized Fleming-Viot process with immigration is a Feller process. Arguing then by duality we determine the generator of any GFVI on a space of functionals which forms a core. In Section 4, we give a sufficient condition for the extinction of the initial types. Thanks to the duality established in Section 3, the extinction corresponds to the coming down from infinity of the distinguished coalescent.

Preliminaries

We start by recalling some basic definitions and results about exchangeable distinguished partitions which are developed in [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF]. For every n ≥ 0, we denote by [ n ] the set {0, ..., n} and call P 0 n the set of partitions of [ n ]. The space P 0 ∞ is the set of partitions of [ ∞ ] := Z + . By convention the blocks are listed in the increasing order of their least element and we denote by π i the i-th block of the partition π. The first block of π, which contains 0, is thus π 0 and is viewed as distinguished. It shall represent descendants of an immigrant. An element of P 0 ∞ is then called a distinguished partition. We endow the space P 0 ∞ with a distance, that makes it compact, defined by

d(π, π ′ ) = (1 + max{n ≥ 0; π | [ n ] = π ′ | [ n ] }) -1 .
The notation i π ∼ j means that i and j are in the same block of π. An exchangeable distinguished partition is a random element π of P 0 ∞ such that for all permutations σ satisfying σ(0) = 0, the partition σπ defined by

i σπ ∼ j if and only if σ(i) π ∼ σ(j),
has the same law as π. If π is an exchangeable distinguished partition, the asymptotic frequency

|π i | := lim n→∞ #(π i ∩ [ n ]) n
exists for every i ≥ 0, almost-surely. We denote by |π| ↓ the sequence of asymptotic frequencies (|π i |, i ≥ 0) after a decreasing rearrangement apart from |π 0 |. Thanks to Kingman's correspondence, the law of |π| ↓ determines completely that of π (see Theorem 2.1 in [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF]).

The operator coag is defined from

P 0 ∞ × P 0 ∞ to P 0 ∞ such that for (π, π ′ ) ∈ P 0 ∞ × P 0 ∞ , the partition coag(π, π ′ ) satisfies for all i ≥ 0, coag(π, π ′ ) i = j∈π ′ i π j .
The operator coag is Lipschitz-continuous and associative in the sense that for any π, π ′ and

π ′′ in P 0 ∞ coag(π, coag(π ′ , π ′′ )) = coag(coag(π, π ′ ), π ′′ ).
Moreover, the partition of Z + into singletons, denoted by 0 [ ∞ ] , may be viewed as a neutral element for the operator coag, indeed

coag(π, 0 [ ∞ ] ) = coag(0 [ ∞ ] , π) = π.
More generally, for every n ≥ 0, we denote by 0 [ n ] the partition of [ n ] into singletons. It should be highlighted that given two independent exchangeable distinguished partitions π, π ′ , the partition coag(π, π ′ ) is still exchangeable (see Lemma 4.3 in [2] and Remark 2.1 below).

Partitions of the population and exchangeability

We explain how the formalism of partitions may be used to describe a population with immigration as time goes forward. As in the Introduction, imagine an infinite haploid population with immigration, identified by the set N evolving forward in time. An additional individual 0 is added and plays the role of a generic immigrant. The model may be described as follows: let t 0 < t 1 ,

• at time t 1 the families sharing the same ancestor at time t 0 form a distinguished partition π (1) . The distinguished block π

(1) 0 comprises the children of immigrants,

• the indices of the blocks of π (1) are viewed as the ancestors living at time t 0 . In other words, for any j ≥ 0, the block π

(1) j

is the offspring at time t 1 of the individual j living at time t 0 . Consider a time t 2 > t 1 and denote by π (2) the partition of the population at time t 2 such that the block π k represents the descendants at time t 2 of the individual j at time t 0 . Therefore the partition coag(π (2) , π (1) ) encodes the descendants at time t 2 of the individuals living at time t 0 .

For any fixed π ∈ P 0 ∞ , define the map α π by α π (k) := the index of the block of π containing k.

Thus, in the population above, α π (1) (k) corresponds to the ancestor at time t 0 of the individual k in the population at time t 1 . We have, by definition of the operator coag, the key equality

α coag(π,π ′ ) = α π ′ • α π .
Therefore, the ancestor living at time t 0 of the individual k at time t 2 is

α π (1) • α π (2) (k) = α coag(π (2) ,π (1) ) (k).
We give in the sequel some properties of the map α π when π is an exchangeable distinguished partition. They will be useful to define a generalized Fleming-Viot process with immigration. We denote by M 1 the space of probability measures on [0, 1]. Let ρ be a random probability measure on [0, 1]. We say that the exchangeable sequence (U i , i ≥ 1) has de Finetti measure ρ, if conditionally given ρ the variables (U i , i ≥ 1) are i.i.d. with law ρ.

We make the following key observation. Lemma 1 Let (U i , i ≥ 1) be an infinite exchangeable sequence taking values in [0, 1], with de Finetti measure ρ and fix U 0 = 0. Let π be an independent exchangeable distinguished partition. Then the infinite sequence (U απ(k) , k ≥ 1) is exchangeable. Furthermore, its de Finetti measure is (1

- i≥0 |π i |)ρ + i≥1 |π i |δ U i + |π 0 |δ 0 .
Proof. The proof requires rather technical arguments and is given in the Appendix.

Remark 2.1 Let (U i , i ≥ 1) be an i.i.d. sequence with a continuous distribution ρ. Observing that

k π ∼ l ⇐⇒ U απ (k) = U απ (l) ,
we can use the previous lemma to recover the (distinguished) paint-box structure of any (distinguished) exchangeable random partition, see Kingman [11] or Theorem 2.1 in [START_REF] Bertoin | Random fragmentation and coagulation processes[END_REF] for the case with no distinguished block. Moreover, Lemma 1 yields another simple proof for the exchangeability of coag(π, π ′ ) provided that π and π ′ are independent and both exchangeable.

Distinguished coalescents and flows of partitions

We start by recalling some basic facts on distinguished coalescents which are developed in Section 3 of [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF]. In particular, we recall the definition of a coagulation measure and its decomposition.

Distinguished coalescents and coagulation measure

Consider an infinite haploid population model with immigration, the population being identified with N. Recall that a generic immigrant 0 is added to the population. We denote by Π(s) the partition of the current population into families having the same ancestor s generations earlier.

As explained before, individuals issued from the immigration form the distinguished block Π 0 (s) of Π(s). When some individuals have the same ancestor at a generation s, they have the same ancestor at any previous generation. The following statement makes these ideas formal. We stress that time goes backward. An exchangeable distinguished coalescent is a Markov process

(Π(t), t ≥ 0) valued in P 0 ∞ such that given Π(s) Π(s + t) Law = coag(Π(s), π),
where π is an exchangeable distinguished partition independent of Π(s), with a law depending only on t.

A distinguished coalescent is called standard if Π(0) = 0 [ ∞ ]
. We emphasize that classical coagulations and coagulations with the distinguished block may happen simultaneously.

A distinguished coalescent is characterized by a measure µ on P 0 ∞ , called the coagulation measure, which fulfills the following conditions:

• µ is exchangeable, meaning here invariant under the action of the permutations σ of Z + with finite support (i.e permuting only finitely many points), such that σ(0) = 0

• µ({0 [ ∞ ] }) = 0 and for all n ≥ 0, µ(π ∈ P 0 ∞ : π | [ n ] = 0 [ n ] ) < ∞.
To be more precise, let (Π(t), t ≥ 0) be a distinguished coalescent; the coagulation measure µ is defined from the jump rates of the restricted processes (Π | [ n ] (t), t ≥ 0) for n ≥ 0. For every π ∈ P 0 n , let q π be the jump rate of (Π | [ n ] (t), t ≥ 0) from 0 [ n ] to π and P 0 ∞,π be the set

P 0 ∞,π := {π ′ ∈ P 0 ∞ ; π ′ | [ n ] = π}.
We have by definition µ(P 0 ∞,π ) = q π . We will denote by L * n the generator of the continuous Markov chain (Π | [ n ] (t), t ≥ 0). Let φ be a map from P 0 n to R and π ∈ P 0 n , then

L * n φ(π) = π ′ ∈P 0 n q π ′ [φ(coag(π, π ′ )) -φ(π)].
Conversely for any coagulation measure µ, by the same arguments as in [START_REF] Bertoin | Random fragmentation and coagulation processes[END_REF] for the genuine coalescents, a distinguished coalescent with coagulation measure µ, is constructed using a Poisson random measure on the space R + × P 0 ∞ with intensity dt ⊗ µ (see Proposition 3.2 in [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF]). We mention that Theorem 3.1 in [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF] yields a decomposition of a coagulation measure into a "Kingman part" and a "multiple collisions part". Let µ be a coagulation measure, then there exist c 0 , c 1 non-negative real numbers and a measure ν on

P m := s = (s 0 , s 1 , ...); s 0 ≥ 0, s 1 ≥ s 2 ≥ ... ≥ 0, i≥0 s i ≤ 1 such that µ = c 0 1≤i δ K(0,i) + c 1 1≤i<j δ K(i,j) + ˆs∈Pm ρ s (.)ν(ds)
where K(i, j) is the simple partition (meaning with at most one non-singleton block) with doubleton {i, j} and ρ s denotes the law of an s-distinguished paint-box (see Definition 2.3 in [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF]). The measure ν satisfies the condition

ˆPm (s 0 + i≥1 s 2 i )ν(ds) < ∞.

Stochastic flow of partitions

We define a stochastic flow of partitions and give a construction from a Poisson random measure.

The following definition may be compared with that of Bertoin and Le Gall's flows [START_REF] Bertoin | Stochastic flows associated to coalescent processes[END_REF].

Definition 1 A flow of distinguished partitions is a collection of random variables (Π(s, t), -∞ < s ≤ t < ∞) valued in P 0 ∞ such that:
(i) For every t ≤ t ′ , the distinguished partition Π(t, t ′ ) is exchangeable with a law depending only on t ′t.

(ii) For every t < t

′ < t ′′ , Π(t, t ′′ ) = coag(Π(t, t ′ ), Π(t ′ , t ′′ )) almost surely (iii) if t ′ 1 < t ′ 2 < ... < t ′ n , the distinguished partitions Π(t ′ 1 , t ′ 2 ), ..., Π(t ′ n-1 , t ′ n ) are independent. (v) Π(0, 0) = 0 [ ∞ ] and Π(t, t ′ ) → 0 [ ∞ ] in probability when t ′ -t → 0.
The process (Π(t), t ≥ 0) := (Π(0, t), t ≥ 0) is by definition a distinguished exchangeable coalescent. Given a coagulation measure µ, we introduce and study next a stochastic flow of partitions constructed from a Poisson random measure on R × P 0 ∞ with intensity dt ⊗ µ. Instead of composing bridges as Bertoin and Le Gall in [START_REF] Bertoin | Stochastic flows associated to coalescent processes[END_REF], we coagulate directly the partitions replacing thus the operator of composition by the operator coag. For any partitions π (1) , ..., π (k) , we define recursively the partition coag k (π (1) , ..., π (k) ) by coag 0 = 0 [ ∞ ] , coag 1 (π (1) ) = π (1) and for all k ≥ 2, coag k (π (1) , ..., π (k)

) := coag coag k-1 π (1) , ..., π (k-1) , π (k) = coag k-1 π (1) , ..., π (k-2) , coag π (k-1) , π (k) .
Introduce a Poisson random measure N on R × P 0 ∞ with intensity dt ⊗ µ and for each n ∈ N, let N n be the image of N by the map

π → π | [ n ] . The condition µ(π | [ n ] = 0 [ n ]
) < ∞ ensures that for all s < t there are finitely many atoms of

N n in ]s, t] × P 0 n \ {0 [ n ] }.
We denote by {(t 1 , π (1) ), (t 2 , π (2) ), ..., (t K , π (K) )} these atoms with 1) , ..., π (K) .

K := N n (]s, t] × P 0 n \ {0 [ n ] }) and define Π n (s, t) := coag K π (
It remains to establish the compatibility of the sequence of random partitions

(Π n (s, t), n ∈ N), which means that for all m ≤ n, Π n | [ m ] (s, t) = Π m (s, t).
All non-trivial atoms of N m are plainly non-trivial atoms of N n , and moreover the compatibility property of the operator coag with restrictions implies that

coag K π (1) , ..., π (K) | [ m ] = coag K π (1)
| [ m ] , ..., π (K) | [ m ] .
Two cases may occur, either π

(i) | [ m ] = 0 [ m ]
and does not affect the coagulation, or π

(i) | [ m ] = 0 [ m ] and is actually an atom of N m on ]s, t] × P 0 m \ {0 [ m ] }.
We then have the following identity:

Π m (s, t) = Π n | [ m ] (s, t).
This compatibility property allows us to define a unique process

(Π(s, t), -∞ < s ≤ t < ∞) such that for all s ≤ t, Π | [ n ] (s, t) = Π n (s, t). The collection (Π(s, t), -∞ < s ≤ t < ∞)
is by construction a flow in the sense of Definition 1. Obviously, the process (Π(t), t ≥ 0) := (Π(0, t), t ≥ 0) is a standard distinguished coalescent with coagulation measure µ. Interpreting a classical coagulation as a reproduction and a coagulation with the distinguished block as an immigration event, a coagulation measure µ may be viewed as encoding the births and the immigration rates in some population when time goes forward.

In the same vein as Bertoin and Le Gall's flows, a population model is embedded in the dual flow, as we shall see.

The dual flow

Let N be the image of N by the time reversal t → -t and consider the filtration ( Ft , t ≥ 0) := (σ( N[0,t]×P 0 ∞ ), t ≥ 0). For all s ≤ t, we denote by Π(s, t) the partition Π(-t, -s). The process ( Π(s, t), -∞ < s ≤ t < ∞) is called the dual flow. By a slight abuse of notation the process ( Π(0, t), t ≥ 0) will be denoted by ( Π(t), t ≥ 0). Plainly the following cocycle property holds for every s ≥ 0, Π(t + s) = coag( Π(t, t + s), Π(t)).

We stress that the partition Π(t, t + s) is exchangeable, independent of Ft and has the same law as Π(s). The cocycle property yields immediately that ( Π(t), t ≥ 0) and its restrictions ( Π| [ n ] (t), t ≥ 0) are Markovian. The following property ensures that ( Π(t), t ≥ 0) is actually strongly Markovian.

Proposition 2

The semigroup of the process ( Π(t), t ≥ 0) verifies the Feller property. For any continuous map

φ from P 0 ∞ to R, the map π → E[φ(coag( Π(t), π))] is continuous and E[φ(coag( Π(t), π))] → t→0 φ(π).
Proof. This is readily obtained thanks to the continuity of coagulation maps.

Remark 2.2 We stress that the process ( Π(t), t ≥ 0) is not a coalescent process. However, since the Poisson random measures N and N have the same law, the process ( Π(t), t ≥ 0) has the same one-dimensional marginals as a standard coalescent with coagulation measure µ.

As explained in Section 2.1, the partition-valued process ( Π(t), t ≥ 0) may be viewed as a population model forward in time. For every t ≥ s ≥ 0 and every k ∈ N, we shall interpret the block Πk (s, t) as the descendants at time t of the individual k living at time s and the distinguished block Π0 (s, t) as the descendants of the generic immigrant. Thanks to the cocycle property, for all 0 ≤ s ≤ t, we have

Πk (t) = j∈ Πk (s) Πj (s, t)
and the ancestor living at time s of any individual j at time t is given by α Π(s,t) (j). The random distinguished partition Π(t) is exchangeable and possesses asymptotic frequencies. For all i ≥ 0, we shall interpret | Πi (t)| as the fraction of the population at time t which is descendent from i. We stress that as in Donnelly-Kurtz's construction [START_REF] Donnelly | Particle representations for measure-valued population models[END_REF] and the generalisation [START_REF] Birkner | A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks[END_REF], the model is such that the higher the individual is, the faster his descendants will die. Namely, for all t ≥ 0 and all j ≥ 0 we have α Π(t) (j) ≤ j and then for all individuals i < j, the descendants of i will always extinct after that of j.

Remark 2.3 When neither simultaneous multiple births nor immigration is taken into account, the measure µ is carried on the simple partitions (meaning with only one non-trivial block) with a distinguished block reduced to {0} and we recover exactly the "modified" lookdown process of Donnelly-Kurtz for the Λ-Fleming-Viot process (see p195-196 of [START_REF] Donnelly | Particle representations for measure-valued population models[END_REF]) . Moreover, the partitionvalued process ( Π(t), t ≥ 0) corresponds in law with those induced by the dual flow of Bertoin and Le Gall ( Bt , t ≥ 0) using the paint-box scheme, see [START_REF] Bertoin | Stochastic flows associated to coalescent processes[END_REF].

Let us study the genealogical process of this population model. We will recover a distinguished coalescent. Let T > 0 be a fixed time and consider the population at time T . By definition, the individuals k and l have the same ancestor at time Tt if and only if k and l belong to the same block of Π(Tt, T ). Moreover by definition of the dual flow,

( Π(T -t, T ), t ∈ [0, T ]) = (Π(-T, -T + t), t ∈ [0, T ])
which is a distinguished coalescent with coagulation measure µ on the interval [0, T ].

In the same way as Donnelly and Kurtz in [START_REF] Donnelly | Particle representations for measure-valued population models[END_REF], we associate initially to each individual a type represented by a point in a metric space E ∪ {∂}, where ∂ is an extra point not belonging to E representing the distinguished type of the immigrants. The choice of E does not matter in our setting and for the sake of simplicity, we choose for E the interval ]0, 1], and for distinguished type ∂ = 0. The generic external immigrant 0 has the type U 0 := 0. At any time t, each individual has the type of its ancestor at time 0. In other words, for any k ∈ N the type of the individual k at time t is U α Π(t) (k) . The exchangeability properties of Section 2.2 will allow us to define and characterize the generalized Fleming-Viot process with immigration.

Remark 2. [START_REF] Birkner | A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks[END_REF] We assume in this work only one source of immigration but several sources may be considered by distinguishing several blocks and types. In this work no mutation is assumed on the types. Birkner et al. defined in [START_REF] Birkner | A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks[END_REF] a generalization of the lookdown representation and the Ξ-Fleming-Viot process with mutations. Assuming that neither immigration nor mutation is taken into account, we will recover a process with the same law as a Ξ-Fleming-Viot process (the measure Ξ is defined by Ξ := c 1 δ 0 + i≥1 s 2 i ν(ds)). However, we stress that our method differs from that of Birkner et al.

Generalized Fleming-Viot processes as de Finetti measures

We define and characterize in this section a measure-valued process which represents the frequencies of the types in the population at any time. This process will be called the generalized Fleming-Viot process with immigration and will be explicitly related to the forward partition process ( Π(t), t ≥ 0). In the same way as in [START_REF] Bertoin | Stochastic flows associated to coalescent processes[END_REF] and [START_REF] Birkner | A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks[END_REF], a duality argument allows us to characterize in law the GFVIs.

Generalized Fleming-Viot processes with immigration

Let ρ ∈ M 1 , we assume that the initial types (U i , i ≥ 1) are i.i.d. with law ρ and independent of N . For all t ≥ 0, the random partition Π(t) is exchangeable and applying Lemma 1, we get that the sequence (U α Π(t) (l) , l ≥ 1) is exchangeable. We denote by Z t its de Finetti measure. Lemma 1 leads us to the following definition.

Definition 2

The process (Z t , t ≥ 0) defined by

Z t := | Π0 (t)|δ 0 + ∞ i=1 | Πi (t)|δ U i + (1 - ∞ i=0 | Πi (t)|)ρ,
starting from Z 0 = ρ, is called the generalized Fleming-Viot process with immigration.

Remark 3.1 For all t ≥ 0, the random variable Z t can be viewed as the Stieltjes measure of a distinguished bridge (see [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF]). Definition 2 yields a paint-box representation of the population forward in time in the same vein as the dual flow of bridges of Bertoin and Le Gall.

Proposition 3

The process (Z t , t ≥ 0) is Markovian with a Feller semigroup.

Proof. The sequence (U α Π(s+t) (l) , l ≥ 1) is exchangeable with de Finetti measure Z s+t . By the cocycle property of the dual flow, we have Π(s + t) = coag( Π(s, s + t), Π(s)). Therefore, for all l ≥ 1,

U α Π(s+t) (l) = U α Π(s) •α Π(s,s+t) (l) .
By Lemma 1, we immediately get that for all t ≥ 0 and s ≥ 0

Z s+t = (1 - j≥0 | Πj (s, s + t)|)Z s + j≥1 | Πj (s, s + t)|δ U α Π(s) (j) + | Π0 (s, s + t)|δ 0 .
We recall that Π(s, s + t) is independent of Fs and hence of Z s . By Theorem 2, conditionally on Z s , the variables (U α Π(s) (j) , j ≥ 1) are i.i.d. with distribution Z s . The process (Z t , t ≥ 0) is thus Markovian and its semigroup denoted by R t can be described as follows. For every ρ ∈ M 1 , R t (ρ, .) is the law of the random probability measure

(1 - i≥0 | Πi (t)|)ρ + i≥1 | Πi (t)|δ U i + | Π0 (t)|δ 0
where the variables (U i , i ≥ 1) are i.i.d. distributed according to ρ and independent of Π(t).

We then verify that R t enjoys the Feller property. If f is a continuous function from M 1 to R, the convergence in probability when t → 0 of Π(t) to 0 [ ∞ ] implies that | Π(t)| ↓ tends to 0. We then have the convergence of R t f to f when t → 0. Plainly, for any sequence (ρ n , n ≥ 1) which weakly converges to ρ, R t (ρ n , .) converges to R t (ρ, .). The Feller property is then established.

Infinitesimal generator, core and martingale problem

As in the articles of Bertoin-Le Gall [START_REF] Bertoin | Stochastic flows associated to coalescent processes[END_REF], Donnelly-Kurtz [START_REF] Donnelly | Particle representations for measure-valued population models[END_REF] and Birkner et al [START_REF] Birkner | A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks[END_REF], the characterization in law of a GFVI will be obtained by a duality argument. Let f be a continuous function on [0, 1] p . Define a function from M 1 × P 0 p to R by

Φ f : (ρ, π) ∈ M 1 × P 0 p → ˆ[0,1] p+1 δ 0 (dx 0 )ρ(dx 1 )...ρ(dx p )f (x απ(1) , ..., x απ (p) ).
Let (Π(t), t ≥ 0) be a distinguished coalescent, then we have the following lemma, where the notations E ρ and E π refer respectively to expectation when

Z 0 = ρ and Π | [ p ] (0) = π. Lemma 4 E ρ [Φ f (Z t , π)] = E π [Φ f (ρ, Π | [ p ] (t))].
Proof of Lemma 4. Let (U i , i ≥ 1) be independent and identically distributed with law ρ and U 0 = 0 , we have

E ρ [Φ f (Z t , π)] = E ρ [ ˆδ0 (dx 0 )Z t (dx 1 )...Z t (dx p )f (x απ(1) , ..., x απ (p) )] = E[f (U α Π(t) (απ (1)) , ..., U α Π(t) (απ (p)) )]
= E[ ˆδ0 (dy 0 )ρ(dy 1 )...ρ(dy p )f (y α coag(π, Π(t)) (1) , ..., y α coag(π,

Π(t)) (p) )] = E π [Φ f (ρ, Π | [ p ] (t))].
The second equality holds because Z t is the de Finetti measure of (U α Π(t) (i) , i ≥ 1). Observing that α Π(t) • α π = α coag(π, Π(t)) , we get the third equality. Moreover, for t fixed the random partition Π(t) has the same law as a standard distinguished coalescent at time t which yields the last equality.

The Kolmogorov equations ensure that the generator L of the process (Z t , t ≥ 0) verifies for all continuous functions f on [0, 1] p , π ∈ P 0 p and ρ ∈ M 1 ,

LΦ f (., π)(ρ) = L * p Φ f (ρ, .)(π) (1)
where L * p is the generator of the continuous time Markov chain (Π | [ p ] (t), t ≥ 0). The process (Z t , t ≥ 0) is then characterized in law by a triplet (c 0 , c 1 , ν) according to the decomposition of the coagulation measure µ given in Subsection 2.2.1. Let G f be the map defined by

G f (ρ) = ˆ[0,1] p f (x 1 , ..., x p )ρ(dx 1 )...ρ(dx p ) = Φ f (ρ, 0 [ p ] ).
A classical way to characterize the law of a Fleming-Viot process is to study a martingale problem, see for example Theorem 3 of [START_REF] Bertoin | Stochastic flows associated to coalescent processes[END_REF] or Proposition 5.2 of [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF]. The following theorem claims that a generalized Fleming-Viot process with immigration solves a well-posed martingale problem. Let f be a continuous function f on [0, 1] p . According to (1), the operator L is such that

LG f (ρ) = π∈P 0 p q π ˆ[0,1] p [f (x απ(1) , ..., x απ(p) ) -f (x 1 , ..., x p )]δ 0 (dx 0 )ρ(dx 1 )...ρ(dx p ).
Theorem 5 The law of the process (Z t , t ≥ 0) is characterized by the following martingale problem. For every integer p ≥ 1 and every continuous function f on [0, 1] p , the process

G f (Z t ) -´t 0 LG f (Z s )ds is a martingale in ( Ft , t ≥ 0).
Remark 3.2 When µ is supported on the simple distinguished partitions, we recover the wellposed martingale problem of Section 5, Lemma 5.2 of [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF]. We then identify the processes obtained from stochastic flows of bridges in [START_REF] Bertoin | Stochastic flows associated to coalescent processes[END_REF] and [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF] and from stochastic flows of partitions.

Proof. Using (1), and applying Theorem 4.4.2 in [START_REF] Ethier | Markov processes[END_REF], we get that there is at most one solution to the martingale problem. Dynkin's formula implies that the process in the statement is a martingale.

The following theorem yields an explicit formula for the generator of (Z t , t ≥ 0). Its proof differs from that given in [START_REF] Birkner | A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks[END_REF] for the Ξ-Fleming-Viot process.

Theorem 6

The infinitesimal generator L of (Z t , t ≥ 0) has the following properties:

(i) For every integer p ≥ 1 and every continuous function f on [0, 1] p , we have

LG f = L c 0 G f + L c 1 G f + L ν G f where L c 0 G f (ρ) := c 0 1≤i≤p ˆ[0,1] p [f (x 0,i ) -f (x)]ρ ⊗p (dx) L c 1 G f (ρ) := c 1 1≤i<j≤p ˆ[0,1] p [f (x i,j ) -f (x)]ρ ⊗p (dx) L ν G f (ρ) := ˆPm {E[G f (sρ + s 0 δ 0 + i≥1 s i δ U i )] -G f (ρ)}ν(ds)
where x denotes the vector (x 1 , ..., x p ) and the vector x 0,i is defined by x 0,i k = x k , for all k = i and x 0,i i = 0, -the vector x i,j is defined by x i,j k = x k , for all k = j and x i,j j = x i , -the sequence (U i , i ≥ 1) is i.i.d. with law ρ and s is the dust of s meaning that s := 1 -i≥0 s i .

(ii) Let D stand for the domain of L. The vector space generated by functionals of the type G f forms a core of (L, D).

Proof. (i) We have

LG

f (ρ) = L * p Φ f (ρ, .)(0 [ p ] ). Therefore,
LG f (ρ) =

π∈P 0 p q π [Φ f (ρ, π) -Φ f (ρ, 0 [ p ] )]
with q π = µ(P 0 ∞,π ). The decomposition of µ with the triplet (c 0 , c 1 , ν) implies that

π∈P 0 p q π [Φ f (ρ, π) -Φ f (ρ, 0 [ p ] )] =c 0 1≤i≤p [f (x 0,i ) -f (x)]ρ ⊗p (dx) + c 1 1≤i<j≤p [f (x i,j ) -f (x)]ρ ⊗p (dx) + π∈P 0 p ˆPm ρ s (P 0 ∞,π )ν(ds)[Φ f (ρ, π) -Φ f (ρ, 0 [ p ] )].
It remains to establish the following equality

L ν G f (ρ) = π∈P 0 p ˆPm ρ s (P 0 ∞,π )ν(ds)[Φ f (ρ, π) -Φ f (ρ, 0 [ p ] )].
Let s ∈ P m , as already mentioned, we denote by s its dust. Let (U i , i ≥ 1) be i.i.d. random variables with distribution ρ. Denoting by Π an independent s-distinguished paint-box, the variables (U α Π (j) , j ≥ 1) are exchangeable with a de Finetti measure which has the same law as

sρ + i≥1 s i δ U i + s 0 δ 0 .
Thus, we get

E[G f (sρ + i≥1 s i δ U i + s 0 δ 0 )] = E[f (U α Π (1) , ..., U α Π (p) )].
Moreover,

E[f (U α Π (1) , ..., U α Π (p) )]-E[f (U 1 , ..., U p )] = π∈P 0 p P[Π | [ p ] = π] E[f (U απ(1) , ..., U απ(p) )] -E[f (U 1 , ..., U p )] . By definition, P[Π | [ p ] = π] = ρ s (P 0 ∞,π
) and we get by integrating on P m :

π∈P 0 p ˆPm ρ s (P 0 ∞,π )ν(ds) E[f (U απ (1) , ..., U απ(p) )] -E[f (U 1 , ..., U p )] = ˆPm E[f (U α Π (1) , ..., U α Π (p) ) -f (U 1 , ..., U p )]ν(ds) = ˆPm E[G f (sρ + i≥1 s i δ U i + s 0 δ 0 ) -G f (ρ)]ν(ds).
Therefore, the statement of (i) is obtained.

(ii) The previous calculation yields that the map ρ → LG f (ρ) is a linear combination of functionals of the type Φ f (ρ, π). Besides, for all π ∈ P 0 p , the map ρ → Φ f (ρ, π) can be written as G g (ρ) with g the continuous function on [0, 1] #π-1 defined by g(x 1 , ..., x #π-1 ) = f (x απ (1) , ..., x απ (p) ) with x 0 = 0. Therefore, denoting by D the vector space generated by the functionals of type G f , the space D is invariant under the action of the generator L. Considering the maps of the form f (x 1 , ..., x p ) = g(x 1 )...g(x p ), we get that D contains the linear combinations of functionals ρ → g, ρ p . By the Stone-Weierstrass theorem, these functionals are dense in the space of continuous functions on M 1 . Thanks to the Feller property of (Z t , t ≥ 0) and according to Proposition 19.9 of [START_REF] Kallenberg | Foundations of modern probability. Probability and its Applications[END_REF], the space D is a core. Thus, the explicit expression of L restricted to D, given in the statement, determines the infinitesimal generator L of (Z t , t ≥ 0).

Extinction of the initial types

Let (Z t , t ≥ 0) be a GFVI characterized in law by the triplet (c 0 , c 1 , ν). The extinction of the initial types corresponds to the absorption of (Z t , t ≥ 0) in δ 0 . It means for the forward partition-valued process ( Π(t), t ≥ 0) to be absorbed at the trivial partition ({Z + }, ∅, ...). We are interested in this section to determine under which conditions on (c 0 , c 1 , ν) the extinction occurs almost surely. We are only able to give a sufficient condition. However this condition is also necessary when the measure ν satisfies an additional assumption of regularity (as in Limic's article [START_REF] Limic | On the speed of coming down from infinity for Ξ-coalescent processes[END_REF]). We stress that as mentioned in Proposition 5.2 and Remark 5.3 of [START_REF] Birkner | A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks[END_REF], the absorption of a Ξ-Fleming-Viot process (without immigration) is closely related to the coming down from infinity of the Ξ-coalescent.

Extinction criterion

Define the following subspace of P m P f m := {s ∈ P m ; n i=0 s i = 1 for some finite n}. As in Section 5.5 of Schweinsberg's article [START_REF] Schweinsberg | Coalescents with simultaneous multiple collisions[END_REF], we consider the following cases:

• Suppose ν(P f m ) = ∞ then by a basic property of Poisson random measure, we know that T f := inf{t > 0; (t, π) is an atom of N such that #π < ∞} = 0 almost surely. We deduce that immediately after 0, there is only a finite number of types and then the extinction occurs almost surely in a finite time.

• Suppose 0 < ν(P f m ) < ∞ then T f is exponential with parameter ν(P f m ). At time T f only a finite number of types reproduces and extinction will occur almost surely. This allows us to reduce the problem to the case when ν(P f m ) = 0.

Suppose henceforth that ν(P f m ) = 0. We define

ζ(q) := c 1 q 2 /2 + ˆPm ∞ i=1
(e -qs i -1 + qs i ) ν(ds).

Theorem 7 If the following conditions are fulfilled:

i) c 0 + ν(s ∈ P m , s 0 > 0) > 0 ii) ´∞ a dq ζ(q)
< ∞ for some a > 0 (and then automatically for all a > 0)

then the generalized Fleming-Viot process with immigration (Z t , t ≥ 0) is absorbed in δ 0 almost surely.

Moreover, under the regularity condition (R):

ˆPm ( ∞ i=0 s i ) 2 ν(ds) < ∞,
the conditions i) and ii) are necessary.

Following the terminology of Limic in [START_REF] Limic | On the speed of coming down from infinity for Ξ-coalescent processes[END_REF], the assumption (R) is called the regular case.

Remark 4.1 Some weaker assumption than (R) may be found under which the conditions i) and ii) are still necessary (see Schweinsberg's article [START_REF] Schweinsberg | Coalescents with simultaneous multiple collisions[END_REF] for the case of Ξ-coalescents). For sake of simplicity, we will not focus on that question in this article.

Theorem 7 extends Theorem 4.1 in [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF]. Namely, the M-generalized Fleming-Viot processes always verify (R). Moreover assume that (R) holds and that the immigration and the reproduction never happen simultaneously. Therefore, the measure ν can be decomposed in the following way: ν = ν1 {s∈Pm;s=(s 0 ,0,...)} + ν1 {s∈Pm;s 0 =0} .

Let us define

• ν 0 := ν1 {s∈Pm;s=(s 0 ,0,...)} which is viewed as a measure on [0, 1], encoding immigration rate, such that ´1 0 s 0 ν 0 (ds 0 ) < ∞.

• ν 1 := ν1 {s∈Pm;s 0 =0} such that ´Pm i≥1 s 2 i ν 1 (ds) < ∞ The map ζ does not depend on ν 0 . We deduce that, in this setting, the immigration has no impact on the extinction occurrence. However, we stress that the measure ν may be carried on {s ∈ P m ; s 0 > 0, s 1 > 0}.

Remark 4.1 If the measure ν is carried on the set ∆ := {(s 0 , s 1 ) ∈ [0, 1] 2 ; s 0 +s 1 < 1} then the regularity assumption (R): ´∆(s 0 + s 1 ) 2 ν(ds) < ∞ is still satisfied. Indeed for all (s 0 , s 1 ) ∈ ∆, (s 0 + s 1 ) 2 ≤ 3s 0 + s 2 1 . In this setting, Theorem 7 gives a necessary and sufficient condition for extinction.

By duality, the extinction occurs if and only if there is an immigration and the embedded distinguished coalescent (Π(t), t ≥ 0) comes down from infinity, meaning that its number of blocks becomes instantaneously finite. We shall investigate this last question in the following subsection.

Proof of Theorem 7: coming down from infinity for a distinguished coalescent

Let (Π(t), t ≥ 0) be a distinguished coalescent with triplet (c 0 , c 1 , ν).

Definition 3

We say that a distinguished coalescent comes down from infinity if P(#Π(t) < ∞ for all t > 0) = 1.

In the same manner as Schweinsberg's article [START_REF] Schweinsberg | Coalescents with simultaneous multiple collisions[END_REF] Section 5.5, we will get a sufficient condition which will be necessary in a so-called regular case (in the same sense as in Limic's article [START_REF] Limic | On the speed of coming down from infinity for Ξ-coalescent processes[END_REF]). The arguments used in [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF] to study the coming down for an M-coalescent may be adapted in this more general framework.

Lemma 8 Consider the process (#Π(t), t ≥ 0) of the number of blocks in a distinguished exchangeable coalescent. Provided that ν(P f m ) = 0, there are two possibilities for the evolution of the number of blocks in Π: either P(#Π(t) = ∞ for all t ≥ 0) = 1 or P(#Π(t) < ∞ for all t > 0) = 1.

Proof. This is an easy adaptation of Lemma 31 in [START_REF] Schweinsberg | Coalescents with simultaneous multiple collisions[END_REF].

Let π ∈ P 0 n , with #(π 0 \ {0}) = k 0 , #π 1 = k 1 , .
.., #π r = k r where r ≥ 0, k 0 ≥ 0 and k i ≥ 2 for all i ∈ [r] and r i=0 k i ≤ n. We will denote by λ n,k 0 ,...,kr the jump rate q π . The decomposition of the coagulation measure µ provides an explicit formula for λ n,k 0 ,...,kr , however its expression is rather involved and we will not use it here. We stress that by exchangeability the quantity λ n,k 0 ,...,kr does not depend on the order of the integers k 1 , ..., k r . From a partition with n blocks, k 0 -tuple, k 1 -tuple,...,k r -tuple merge simultaneously with rate λ n,k 0 ,...,kr . The k 0 -tuple represents the blocks coagulating with the distinguished one.

Define N(n, k 0 , ..., k r ) to be the number of different simultaneous choices of a k 0 -tuple, a k 1tuple,.., and a k r -tuple from a set of n elements with k 0 ≥ 0, k i ≥ 2 for i ∈ [r]. The exact expression may be found but is not important in the rest of the current analysis. Denoting by Π * (t) = (Π 1 (t), ..., ), we determine the generator G [n] of (#Π * |[n] (t), t ≥ 0). Let f be any map from [n] to R,

G [n] f (l) = ⌊l/2⌋ r=0 k 0 ,{k 1 ,...kr}; r i=0 k i ≤l N(l, k 0 , ..., k r )λ l,k 0 ,...,kr [f (l -(k 0 + ... + k r ) + r) -f (l)]
As already mentioned, for each fixed k 0 , we do not have a separate term for each possible ordering of k 1 , ..., k r . That is why the inner sum extends over k 0 ≥ 0 and the multiset {k 1 , ..., k r } such that r i=0 k i ≤ l.

We define the map Φ such that Φ(n) is the total rate of decrease in the number of blocks in (Π * (t), t ≥ 0) when the current configuration has n blocks. We get

Φ(n) = ⌊n/2⌋ r=0 k 0 ,{k 1 ,...kr}; r i=0 k i ≤n N(n, k 0 , ..., k r )λ n,k 0 ,...,kr [k 0 + ... + k r -r].
Lemma 9 i) A more tractable expression for Φ is given by the following

Φ(q) = c 0 q + c 1 2 q(q -1) + ˆPm (qs 0 + ∞ i=1 (qs i -1 + (1 -s i ) q )ν(ds). ii) Define Ψ(q) := c 0 q + c 1 q 2 /2 + ˆPm qs 0 + ∞ i=1
(e -qs i -1 + qs i ) ν(ds).

There exist C and C ′ two nonnegative constants such that CΨ(q) ≤ Φ(q) ≤ C ′ Ψ(q).

iii) The map q → Ψ(q)/q is concave.

Proof. Proof of i): We have N(q, 1) = q 1 = q and N(q, 0, 2) = q 2 . Using the binomial formula, the first two terms are plain. We focus now on the integral term. Let s ∈ P m and π be a s-distinguished paint-box (see Definition 2.3 in [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF]). To compute the total rate of decrease in the number of blocks from a configuration with q ≥ 1 blocks, let us consider

Y (q) l (π) := #{k ∈ [q]; α π (k) = l}.
Conditionally given |π l |, the variable Y (q) l (π) is binomial with parameters (|π l |, q) (degenerated in the case of |π l | = 0) and we have

Φ(q) = c 0 q + c 1 2 q(q -1) + ˆPm E[Y (q) 0 (π)] + ∞ l=1 E[Y (q) l (π) -1 {Y (q) l (π)>0} ] ν(ds) = c 0 q + c 1 2 q(q -1) + ˆPm qs 0 + ∞ l=1 [qs l -1 + (1 -s l ) q ] ν(ds).
Note that these computations are exactly the same as those pages 224-225 in [START_REF] Limic | On the speed of coming down from infinity for Ξ-coalescent processes[END_REF] using the "coloring procedure". Proof of ii) Same calculations as in Lemma 8 in Limic's article [START_REF] Limic | On the speed of coming down from infinity for Ξ-coalescent processes[END_REF]. Proof of iii) We remark that Ψ is the Laplace exponent of a spectrally positive Lévy process. Therefore the map h : q → Ψ(q)/q is the Laplace exponent of a subordinator which is concave.

The following theorem may be compared with Schweinsberg's criterion in [START_REF] Schweinsberg | A necessary and sufficient condition for the Λ-coalescent to come down from infinity[END_REF] and Theorem 4.1 in [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF].

Theorem 10 The convergence of the series n≥1 1 Φ(n) implies the coming down from infinity of the distinguished coalescent.

Under the regularity condition (R):

ˆPm ( ∞ i=0 s i ) 2 ν(ds) < ∞,
the convergence of the series is necessary.

We refer to Example 34 p40 in [START_REF] Schweinsberg | Coalescents with simultaneous multiple collisions[END_REF] and Section 3.2 p231 in [START_REF] Limic | On the speed of coming down from infinity for Ξ-coalescent processes[END_REF] for a coalescent which is not regular, comes down from infinity, with a divergent series. We shall follow the proof of Theorem 4.1 in [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF] and study some (super)martingales.

Lemma 11 Let (Π(t), t ≥ 0) be a distinguished coalescent with triplet (c 0 , c 1 , ν). Assume that ν(P f m ) = 0. Let us define the fixation time

ζ := inf{t ≥ 0 : Π(t) = {Z + , ∅, ...}}.
The expectation of fixation time is bounded by

E[ζ] ≤ ∞ n=1 1/Φ(n).
As a consequence, if the series in the right-hand side converges, the fixation time is finite with probability one.

Proof. Assuming the convergence of the sum ∞ n=1 1/Φ(n), we define

f (l) = ∞ k=l+1 1/Φ(k).
It is easy to check directly from i) in Lemma 9 that the map Φ is increasing (see alternatively Lemma 28 in [START_REF] Schweinsberg | Coalescents with simultaneous multiple collisions[END_REF]). We thus have

f (l -(k 0 + ... + k r ) + r) -f (l) = l k=l-(k 0 +...+kr)+r+1 1 Φ(k) ≥ k 0 + ... + k r -r Φ(l) . Therefore G [n] f (l) ≥ ⌊l/2⌋ r=0 k 0 ,{k 1 ,...,kr} r i=0 k i ≤l λ l,k 0 ,...,kr N(l, k 0 , ..., k r ) k 0 + ... + k r -r Φ(l) = 1. The process f (#Π * | [ n ] (t)) -´t 0 G [n] f (#Π * | [ n ] (s))
ds is a martingale. The quantity

ζ n := inf{t ≥ 0 : #Π * | [ n ] (t) = 0}
is a finite stopping time. Let k ≥ 1, applying the optional sampling theorem to the bounded stopping time ζ n ∧ k, we get :

E[f (#Π * | [ n ] (ζ n ∧ k))] -E[ ˆζn∧k 0 G [n] f (#Π * | [ n ] (s))ds] = f (n) With the inequality G [n] f (l) ≥ 1, we deduce that E[ζ n ∧ k] ≤ E[f (#Π * | [ n ] (ζ n ∧ k))] -f (n).
By monotone convergence and Lebesgue's theorem, we have

E[ζ n ] ≤ f (0) -f (n).
Passing to the limit in n, we have ζ n ↑ ζ ∞ := inf{t; #Π(t) = 1} and f (n) -→ 0, thus

E[ζ ∞ ] ≤ f (0) = ∞ k=1 1/Φ(k).
To prove that convergence of the series is necessary for coming down from infinity we follow the same steps as in Section 6 of [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF]. Assuming that (R) holds, that the coalescent comes down from infinity and that the series is infinite, we may define a supermartingale (thanks to Lemmas 12, 13 and 14 below) and find a contradiction by applying the optional sampling theorem (Lemma 15). The proofs of these lemmas are easy adaptations of those in Section 6 of [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF]. We simply give their statements and the corresponding references in [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF].

The following technical lemma allows us to estimate the probability for the sum of n independent binomial variables to be larger than n/2. de Finetti measure ρ and fix U 0 = 0. Let π be an independent exchangeable distinguished partition. Then the infinite sequence (U απ(k) , k ≥ 1) is exchangeable. Furthermore, its de Finetti measure is (1 -

i≥0 |π i |)ρ + i≥1 |π i |δ U i + |π 0 |δ 0 .
Proof. By de Finetti's theorem, without loss of generality we may directly assume that the sequence (U i , i ≥ 1) is i.i.d. with a distribution ρ ∈ M 1 . We show that for all n ≥ 1, the random vector (U απ (1) , ..., U απ (n) ) is then exchangeable. Let f 1 , ..., f n be n measurable functions on [0, 1] and [n] be the set {1, ..., n},

E[f 1 (U απ (1) )...f n (U απ (n) )|π] =   i∈π 0 ∩[n] f i (0)     k≥1 ˆ1 0 i∈π k ∩[n] f i (u)ρ(du)   .
Let σ be a permutation of Z + such that σ(0) = 0, and η be a permutation such that σ -1 (π i ) = σπ η(i) . We stress that η(0) = 0, and we have which allows us to conclude that (U απ(1) , ..., U απ (n) ) is exchangeable.

E[f 1 (U απ (σ(1)) )...f n (U απ (σ(n)) )|π] =   i∈σ -1 (π 0 )∩[n] f i (0)     k≥1 ˆ1 0 i∈σ -1 (π k )∩[n] f i (u)ρ(du)   =   i∈σπ 0 ∩[n] f i (0)     k≥1 ˆ1 0 i∈σπ η(k) ∩[n]
We have now to identify the de Finetti measure of this exchangeable sequence. This is an easy adaptation of the proof of Lemma 4.6 of [START_REF] Bertoin | Random fragmentation and coagulation processes[END_REF]. For the sake of completeness, we give the proof in detail. The exchangeability is given by Lemma 1, and so to prove the statement it suffices, by de Finetti's theorem, to study the limit when m → ∞ of 1 m m j=1 δ U απ (j) .

We denote by S the set of singletons in the partition π. That is

S := i≥0,|π i |=0 π i .
In the sequel, we will denote by S c the complement of S. The paint-box structure of an exchangeable partition tells us that for all t ≥ 0, S is empty or infinite. We decompose the sum in the following way Combining all these results, we get

lim sup m→∞ f ; 1 m m j=1 δ U απ (j) ≤ f ; (1 - i≥0 |π i |)ρ + |π 0 | δ 0 + i≥1 |π i | δ U i .
We then obtain the statement of the proposition.

( 2 )

 2 k comprises the descendants at time t 2 of the individual k, living at time t 1 . Obviously, the set k∈π[START_REF] Berestycki | Recent progress in coalescent theory, volume 16 of Ensaios Matemáticos[END_REF]

Lemma 12 + 1 -

 121 Let s ∈ P m . Let π be an s-distinguished paint-box and the variables Y(n) l(π) defined as in Lemma 9. For every n 0 ≥ 4, provided that ∞ i=0 s i is sufficiently small, there is the bound P[∃n ≥ n 0 ; s l .

  1 (U απ(σ(1)) )...f n (U απ(σ(n)) )|π] = E[f 1 (U ασπ(1) )...f n (U ασπ(n) )|π].The exchangeability of the partition π ensures thatE[f 1 (U ασπ(1) )...f n (U ασπ(n) )] = E[f 1 (U απ (1) )...f n (U απ (n) )],

  By definition, we have1 -|S| -j∈J(η) |π j | → η→0 j ⊂J(η) |π j |f (U j ) → η→0 j≥0|π j |f (U j ).

Acknowledgments. I would like to thank my advisor Jean Bertoin for his very useful advice, and the anonymous referees for their careful reading and helpful suggestions.

Proof. Easy adaptation of the arguments of Lemma 6.2 in [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF].

Lemma 13 Assume that the coalescent comes down from infinity. With probability one, we have

Moreover, if we define

}, then the sequence of stopping times τ n converges to τ almost surely.

Proof. Using the assumption (R) and the above Lemma 12, we get

The same arguments as in Lemma 6.3 of [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF] yield the statement.

Lemma 14 Assuming (R), the coming down from infinity and that n≥1 1

) t≤τn is a non-negative supermartingale.

Proof. With the part iii) of Lemma 9 and the assumption (R), this is an easy adaptation of the arguments of Lemma 6.4 in [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF].

Proof. Assume that the coalescent comes down from infinity. Exactly as in Lemma 6.5 in [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF], we may apply the optional sampling theorem to the previous supermartingale and find a contradiction.

Thanks to part (ii) of Lemma 9, we have the following equivalence

In order to establish Theorem 7, it suffices to show that

Plainly, the quantity q ζ(q) is bounded. Therefore for some constants c and C,

Finally, we get that the conditions i) and ii) of Theorem 7 are necessary under the assumption (R).

Appendix

We restate and prove here Lemma 1.

Lemma 1 Let (U i , i ≥ 1) be an infinite exchangeable sequence taking values in [0, 1], with