Jonathan R Gair
email: jgair@ast.cam.ac.uk

Alberto Sesana

Emanuele Berti

Marta Volonteri

Constraining properties of the black hole population using LISA

LISA will detect gravitational waves from tens to hundreds of systems containing black holes with mass in the range 10 4 M -10 7 M . Black holes in this mass range are not well constrained by current electromagnetic observations, so LISA could significantly enhance our understanding of the astrophysics of such systems. In this paper, we describe a framework for combining LISA observations to make statements about massive black hole populations. We summarise the constraints that LISA observations of extreme-mass-ratio inspirals might be able to place on the mass function of black holes in the LISA range. We also describe how LISA observations can be used to choose between different models for the hierarchical growth of structure in the early Universe. We consider four models that differ in their prescription for the initial mass distribution of black hole seeds, and in the efficiency of accretion onto the black holes. We show that with as little as 3 months of LISA data we can clearly distinguish between these models, even under relatively pessimistic assumptions about the performance of the detector and our knowledge of the gravitational waveforms.

Introduction

Current measurements of black hole (BH) masses are almost exclusively for systems with mass above ∼ 10 6 M . The shape of the mass function for less massive BHs is expected to retain a signature of the initial mass distribution of the seeds from which BHs grow, which is erased at the high mass end by the effects of accretion [1]. In addition, it is not known if the tight correlations observed between the properties of high mass BHs and their host galaxies extend down to lower masses [2], which has important consequences for our understanding of the co-evolution of galaxies and BHs. Probing BHs in the mass range 10 4 M < M < 10 7 M is thus crucial to our understanding of the growth of structure, and LISA [START_REF] Danzmann | Lisa: Pre-phase a report[END_REF] is one of the few instruments that has the potential to observe such systems. LISA is expected to see a few tens of massive BH mergers (MBHMs) per year [START_REF] Sesana | [END_REF][START_REF] Arun | Classical and Quantum[END_REF] and as many as several hundred extreme-mass-ratio inspirals (EMRIs) of stellar-mass compact objects into massive BHs in the centres of galaxies [5]. The MBHMs can be observed throughout the Universe, while the EMRIs will only be seen at low redshift, z 1, but LISA will be able to measure the parameters of both kinds of event to unprecedented precision [START_REF] Arun | Classical and Quantum[END_REF]7,8].

The mass function of BHs in the LISA range is uncertain due to the lack of direct observations. If the BH population traces the active galaxy population, then the mass function should turn over for M 3 × 10 6 M [9]. However, if the L-σ and M-σ relations derived for more massive galaxies can be extrapolated to lower masses, the observed galaxy luminosity function would imply a flat BH mass function in this range [9]. Applying corrections to Sloan Digital Sky Survey measurements of the velocity dispersion instead yields a mass function that increases toward lower masses [10]. Overall the present uncertainty in the slope of the quiescent BH mass function for M 10 7 M is at least ±0.3 [11]. LISA EMRI observations could therefore play an important role in pinning down this slope in the low-redshift Universe.

LISA MBHM events will occur following mergers of the host galaxies of the BHs, and thus trace the hierarchical growth of structure. Models of structure formation are tuned to match existing observations, and therefore make similar predictions for mergers at the high mass end of the BH mass function, but differ significantly for lower masses. In particular, both "light seed" [12] and "heavy seed" [13] models are consistent with existing data. It is unlikely that observations in the electromagnetic spectrum will rule out either class of models in the next decade, so LISA could make important contributions to our understanding of the early epoch of galaxy formation.

LISA will be able to make very precise measurements of the parameters of individual EMRI and MBHM systems that are observed [7,8,[START_REF] Arun | Classical and Quantum[END_REF]. Precise measurements for single systems are very important for fundamental physics [START_REF] Amaro-Seoane | Classical and Quantum[END_REF], but it is the full set of events that are seen which will carry the most important information for astrophysics. In this paper, we describe a method to combine this information in order to make astrophysical statements, which is based on a Bayesian framework using a parametric model for the probability distribution of observed events. LISA model selection using MBHMs was also considered in [15] using the non-parametric Kolmogorov-Smirnov test to compare parameter distributions. While our conclusions are broadly similar, the framework presented here is more general and can be easily extended to other problems.

This paper is organised as follows. In Section 2 we describe our approach to using LISA to place constraints on astrophysical models. This includes a discussion of the statistical framework used for model selection and a description of how we model LISA instrumental effects, i.e., the completeness of LISA observations and the parameter estimation errors that arise from noise in the detector. In Section 3.1 we summarise the constraints that LISA EMRI observations could place on the shape of the BH mass function in the LISA range. These results were previously described in [11] but we briefly review them here for completeness. In Section 3.2 we describe how LISA can be used to choose between different models for the hierarchical growth of structure. We illustrate this using four different models that differ in their seed mass distributions and accretion prescriptions. These results are new and appear here for the first time. In Section 4 we discuss our results and possible extensions of this work.

Methods

Our aim is to make inferences about the population of BHs in the Universe based on LISA observations. Bayes theorem relates the posterior probability, p( λ|D, M), for the parameters λ of a model M given observed data D, to the likelihood p(D| λ, M) of seeing that data under model M with parameters λ, and the prior, π( λ) for the parameters λ:

p( λ|D, M) = p(D| λ, M ) π( λ) Z , where Z = p(D| λ, M ) π( λ)d N λ (1)
In this context, the model M is a description of the population of BHs in the Universe, the data are the parameters of the sources that LISA detects and the uncertainty in the distributions comes from the fact that mergers occur stochastically in the Universea given model will predict the rate at which LISA events occur but cannot predict the exact systems that LISA will observe.

To compute the likelihood, p(D| λ, M), we can imagine dividing the parameter space of possible signals into bins, labelled by i. The data, D, is the number of events, n i , observed in each bin i. A particular model will predict the rate, r i ( λ), at which events in a particular parameter bin occur in the Universe. As the events start at random times, the number of events occurring in a given bin during the LISA mission will be drawn from a Poisson probability distribution with rate r i ( λ). Events in different bins are independent and so if we temporarily ignore LISA selection effects and parameter estimation errors, the likelihood of seeing the set of events D = {n i }, is

p(D| λ, M ) = K i=1 (r i ( λ)) n i e -r i ( λ) n i ! (2)
It is possible to take the continuum limit of this expression by letting the bin volume approach zero. The above expression then becomes a product of the point probabilities of the events observed, normalised by the total number of events the model predicts.

For model selection, i.e., to choose the model that provides the best description of the observed data, we use the evidence, which is the quantity Z appearing in the denominator of Bayes Theorem, Eq. (1). To compare models A and B, we compute the odds ratio (see, for example, [START_REF] Mackay | Information Theory, Inference and Learning Algorithms[END_REF])

O AB = Z A P (A) Z B P (B) (3)
in which P (X) denotes the prior probability assigned to model X.

If O AB 1 (O AB 1)
, model A (model B) provides a much better description of the data. The fact that we use an imperfect detector to make the observations introduces two complications into the analysis. Firstly, the data set is not complete -LISA sees only a certain subset of the events that occur during the mission. Secondly, the presence of noise in the detector introduces errors into the parameter estimates.

The incompleteness of the LISA observation means that only a certain fraction of the events that occur in a given bin in the parameter space will be detected. If the completeness is known as a function of the source parameters, then this can be included in the likelihood computation by replacing r i ( λ) by the effective observed rate, ri ( λ) = C i r i ( λ), in Eq. ( 2), where C i is the completeness in bin i. In our analysis, we assume a simple signal-to-noise ratio (SNR) cut model for completeness. We assume that 100% of events with matched-filtering SNR ρ > ρ thresh are detected, and that 0% of events with ρ < ρ thresh are detected. This is a reasonable model if ρ thresh is set moderately high, since if events with SNR less than ρ thresh were actually detected, they could just be excluded from the analysis. The parameters of events with lower SNR tend to be more poorly determined and thus do not contribute much to model discrimination, so there is little change in the results when such events are excluded.

Instrumental noise leads to imperfect parameter measurement. In the analysis of actual LISA data, we will derive a posterior probability distribution (pdf) for the parameters of the sources. Using this, the likelihood can be obtained by integrating the continuum version of Eq. ( 2) over the pdf [11]. An equivalent approach, which is more convenient for scoping out LISA's potential, is to suppose each source is assigned to the bin in which the maximum a posteriori probability lies, and to fold the parameter uncertainty into the effective rate of observed events, ri ( λ), in each bin. In practice, ri ( λ) can be computed by generating a large number of realisations of the set of events that LISA observes. For each event in each realisation the parameter uncertainty may be modelled as a Gaussian, p( λ) ∝ exp(-Γ ij (λ i -λ i 0 )(λ j -λ j 0 )/2), centred on the true parameters, λ 0 . A fractional rate δr i = B i p( λ)d N λ is assigned to each bin B i . This is similar in principle to the LISA "error kernel" described in [START_REF] Plowman | [END_REF]. The parameter error comes primarily from detector noise, but additional errors arise from weak lensing, which changes the apparent luminosity distance of sources at higher redshift, and from uncertainties in the cosmological parameters used to convert from luminosity distance to redshift. These can be included in the redshift-redshift component of Γ -1 by writing

Γ -1 zz = (Γ N ) -1 zz + ∆z 2 W L + ∆z 2 cos
, where N , W L, cos denote the contributions from noise, weak-lensing and cosmological parameter uncertainty. We model the weak-lensing error using results in [18] and the cosmological error following [19]:

∆z 2 cos = ∂D L ∂z -2 ∆D 2 L D 2 L + ∆H 2 0 H 2 0 D 2 L + ∆Ω 2 Λ ∂D L ∂Ω Λ , (4)
and we assume that ∆H 0 /H 0 = ∆Ω Λ /Ω Λ = 0.01 by the time LISA flies.

Results

Extreme-mass-ratio inspirals

This preceding framework was used in [11] to explore the constraints that LISA EMRI observations could place on the BH mass function at low redshift. This analysis made various assumptions -an SNR cut of ρ thresh = 30 was used to define the completeness function; all EMRIs were assumed to be on circular and equatorial orbits, which meant the completeness could be determined from the observable lifetime of a particular EMRI, as defined in [5]; parameter estimation errors were ignored in the analysis, but included in the generation of realisations of the LISA data; the data was taken to be measurements of the central BH mass M and source redshift z only; and the scaling of the intrinsic EMRI rate per BH was assumed to be known and given by results in [20]. Assuming a simple power-law mass function, A 0 (M/M * ) α 0 , and that all BHs have spin a = 0.9, it was found that LISA could measure the parameters to a precision ∆A 0 ≈ 0.5 10/N obs and ∆α 0 ≈ 0.2 10/N obs if N obs events were observed. This compares very well to the current precision, ∼ ±0.3, on the slope of the BH mass function, particularly given that these models typically predict ∼ 100s of observable EMRI events. Using a redshift-dependent ansatz for the mass function, A 0 (1+z) A 1 (M/M * ) α 0 -α 1 z , it was found that LISA would not be able to place reasonable constraints on the evolution parameters A 1 , α 1 . This is because the majority of EMRI events will be detected at low redshift, z 1. The main caveat in these results is the assumption that the mass-dependence of the EMRI rate will be known by the time LISA flies. These results can also be interpreted as the precision with which the convolution of the mass function with the EMRI rate can be determined. More work is required to determine if combined EMRI and MBHM observations can decouple these effects and perhaps measure evolution in the mass function.

Comparable mass black hole mergers

LISA observations of MBHMs can be used to choose between different models for the growth of structure. There are various models for the hierarchical assembly of galaxies, but these have been tuned to fit existing electromagnetic observations which do not constrain BHs in the mass range of interest to LISA. The models therefore make quite different predictions for the expected set of LISA events, which means that LISA has the potential to discriminate between them. We consider four different models, which differ in the prescription for the masses of the initial seeds from which BHs grow and in the prescription for accretion onto the BHs. We consider two "light seed" models [12], in which BH seeds of mass ∼ 100M form as remnants of metal-free stars at redshift z 20, and two "heavy seed" models [13], in which seeds with mass ∼ 10 5 M form directly from the collapse of massive protogalactic disks in the redshift range 10 z 15. In each case, we consider two accretion prescriptions [21]: (i) "coherent" accretion, in which material accreting onto the black hole tends to have similar angular momentum [22,23]

Table 1. Description of the four models used in this analysis. The last column gives the expected number of events observed by LISA in one year. These were computed under the optimistic assumptions about the detector, (iii), described in the text, and the bracketed numbers were computed under the most pessimistic assumptions about the detector, (ii) in the text. While the SNRs in this paper were computed using nonspinning waveform templates, the numbers here agree well with those quoted in [START_REF] Arun | Classical and Quantum[END_REF], which for spinning waveform templates including higher harmonic corrections.

which could occur if the large scale structure of the feeding material is in a disc-like configuration [24,25]; (ii) "chaotic" accretion, in which there are many short accretion episodes with different angular momentum spin axes in each one [26]. The four models are summarised in Table 1. We chose these four models to allow easier comparison to the literature. The same four models were used to explore LISA parameter estimation [START_REF] Arun | Classical and Quantum[END_REF] and in previous work on using LISA for model selection [15]. The accretion model primarily leads to different expectations for the black hole spins (intermediate-high, a ∼ 0.6 -0.9, in the coherent case; low, a < 0.2, in the chaotic case). In this work we ignore black hole spin, but the accretion prescription also leaves an imprint on the component masses. The models assume that the mass-to-energy conversion efficiency, , depends on black hole spin only, so the two models predict different average efficiencies of ∼ 20% and ∼ 10% respectively. The mass-to-energy conversion directly affects mass growth, with high efficiency implying slow growth, since for a black hole accreting at the Eddington rate, the black hole mass increases with time as

M(t) = M(0) exp 1 -t t Edd (5)
where t Edd = 0.45Gyr. The "coherent" versus "chaotic" models thus allow us to study how different growth rates affect LISA observations. We will again use an SNR cut, ρ thresh , to characterise whether an event is detectable or not, and we will make both optimistic and pessimistic assumptions about LISA in terms of the number of data streams that are available for data analysis. At low frequency two independent data streams can be constructed from the data stream of a single LISA constellation, but one of these data streams could be lost if there is a failure on one of the three satellites in the constellation. We will therefore consider four possible scenarios: (i) 1 independent data stream, ρ thresh = 8; (ii) 1 independent data stream, ρ thresh = 20; (iii) 2 independent data streams, ρ thresh = 8; (iv) 2 independent data streams, ρ thresh = 20. Scenarios (ii)/(iii) are the most pessimistic/optimistic. Additionally, we include only systems that merge within the LISA observation window in the analysis, and we consider five different possible lengths of the LISA data set used in the analysis (3 months, 6 months, 1 year, 18 months and 2 years). It is unlikely that LISA will only take data for a few months if it works at all, but these results illustrate what we will be able to say after 3 months of observation, after 6 months and so on.

To carry out model selection, we must compute the likelihood of the data under the various models, as given by Eq. ( 2). For the current analysis, we use only three parameters to characterise each system: the total mass, M, the mass ratio, q, and the source redshift, z. We compute the expected rate of observed events in each bin accounting for errors in the parameter estimation as described in Section 2. We compute parameter estimation errors using the Fisher matrix approximation for quasicircular, non-spinning BH binary inspirals modelled in the restricted post-Newtonian approximation, following [19]. Cosmological parameter uncertainties and weak lensing errors are folded in as described above. The fact that we are ignoring spins, eccentricity and higher harmonics in our analysis has two consequences. Firstly, our estimates for the signal-to-noise ratio for each source are pessimistic, because spins and higher harmonics of the signal (which are neglected in our model) usually increase the energy radiated and the mass reach of the detector. Secondly, spin encodes important information about the history of a particular BH, and in particular its accretion history [21]. This could help resolve models where BHs are "born equal" but grow via different mechanisms. In this sense our results should be considered conservative.

The four models we use do not have free parameters, so we cannot determine a posterior probability distribution on the model parameters. Instead, we can use Eq. ( 3) to decide which model provides the best description of the data. The models we are comparing have all been constructed to be consistent with existing constraints from observations in the electromagnetic spectrum, so there is presently no reason to prefer one model over the others. We therefore assume equal prior probabilities on all models, so the odds ratio reduces to the likelihood ratio

Λ AB = p(D|M = A) p(D|M = B) . (6)
It is clear that when Λ AB 1, model A should be preferred. What value of Λ AB is sufficient to make such a statement? This can be answered by looking at the distribution of Λ AB over many realisations of model A and model B. We generate a realisation of model X by drawing events from the underlying population, applying the appropriate SNR cut, and adding in parameter errors to each event. We can then compute the likelihoods for this event set under model A and model B and hence Λ AB . In Figure 1 we show the distribution of ln(Λ AB ) in 1000 realisations each of model A and model B.

As we would expect, when the realisation is drawn from model A, Λ AB tends to be greater than 1, while when it is drawn from model B, Λ AB tends to be less than 1. For a given choice of threshold on Λ AB , points in the model A histogram to the right of that threshold represent "detections", i.e., realisations in which model A would be chosen over model B when model A was correct. Points in the model B histogram are "false alarms", i.e., realisations in which model A is chosen over model B when in fact model B is correct. The histograms become better separated when using a We show the distribution of the logarithm of the likelihood ratio, ln(Λ AB ), for comparison of model SE to model SC using 3 months of LISA data (left panel) and using 1 year of data (right panel). The curves labelled "FAP distribution", where FAP stands for "false alarm probability", describe realisations drawn from the "wrong" model (SC). Those labelled "detection rate distribution" describe realisations drawn from the "right" model (SE). We note that the distribution for one year of data is somewhat broader than that for 3 months. This is not unsurprising, since more events are expected in one year of data, and both the mean and the variance of the underlying Poisson distribution increase with the total number of events. longer segment of LISA data, since we have more events in that case. Another way to represent this information is through a receiver-operator-characteristic (ROC) curve, which shows "detection" probability versus "false alarm" probability (FAP). For a given threshold on Λ AB , the detection probability is the fraction of realisations of model A that lie to the right of that threshold, while the false alarm probability is the fraction of realisations of model B that lie to the right. In Figure 2, we show the ROC curves for all possible comparisons between the four models, for a 3 months of LISA data and with the most pessimistic scenario, (ii), for the detector performance. The ROC curve is a frequentist way to represent the performance of an algorithm, but it encodes similar information to the Bayesian approach of assigning probabilities of p(D|M = A)/[p(D|M = A) + p(D|M = B)] and p(D|M = B)/[p(D|M = A) + p(D|M = B)] to models A and B respectively. Note that the false alarm and detection rates are per LISA observation: a FAP of 0.1 indicates that if a LISA observation of this duration was repeated independently 10 times, we would expect to incorrectly choose model A only once.

It is clear from Figure 2 that even using as little as 3 months of data, LISA can easily choose between the four models. With the exception of the SE/SC and LE/LC comparisons, each pair of models shows a detection rate in excess of 80% at a 1% false alarm rate. Models SE and SC are most difficult to distinguish with a detection rate of only 50% at a 10% false alarm rate, followed by models LE and LC with a detection rate of 60% at that FAP. This is to be expected, as these pairs of models have the same seed mass distribution and differ only in the accretion prescription, so the distribution of masses for the events are quite similar (this pessimistic conclusion would most likely change if we included spins in our model waveforms). In the other cases, the mass distributions are quite distinct so the accurate mass measurements that are possible with LISA allow discrimination of the models with only a handful of events. In the left panel of Figure 3 we consider the SE to SC comparison and detector scenario (ii) only and show how the ROC performance depends on the mission duration. We see that our ability to distinguish models increases rapidly with the duration of the observation. For a 1 year observation we can distinguish SE and SC with a rate of ∼ 80% for an FAP of 5%, which is comparable to what can be achieved using a 3 month observation for the other comparisons.

In the right panel of Figure 3 we again restrict to the SE to SC comparison, but now fix the observation to 3 months and show how the performance depends on the detector scenario. There is a relatively modest increase in performance for the more optimistic detector scenarios. At an FAP of 10%, the detection rate increases from ∼ 50% to ∼ 70% going from the most pessimistic to the most optimistic scenario. The detector performance has a relatively weak effect since many of the events with the greatest distinguishing power have very high SNR and can be seen under any scenario. The use of two data streams rather than just one both increases the SNR of an event with given parameters and reduces the parameter estimation errors that arise due to instrumental noise. It is clear from Figure 3 that both of these are important, since both of the 2 data stream curves lies above both of the 1 data stream curves. If the SNR increase alone was important, we would expect the 2 data stream, SNR = 20 curve to lie between the one data stream curves, as SNR = 20 in two data streams corresponds to SNR ≈ 14 in each data stream.

Reference [15] also studied MBHM model selection with LISA considering the same four models using the non-parametric Kolmogorov-Smirnov test to compare the distributions of one or two parameters between models. Their conclusions were broadly the same, i.e., that LISA can tell between models very easily. The approach described here has several advantages over theirs: we have a parametric model using the reasonable assumption of a Poisson distribution in each bin; we use the distribution of all model parameters simultaneously to compare models (3 in this case, but it is easily extendable to more); the present framework naturally extends to parameterised models for the underlying BH population, as used in the EMRI case; and, as described in Section 2, once actual LISA data is available, we will be able to fold the measured uncertainties in the source parameters into the analysis, rather than relying on theoretical estimates.

Discussion

We have described a framework for using the set of events observed by LISA to constrain models of the massive BH population in the Universe. Using EMRI events, we should be able to constrain the slope of a parametric model for the mass function of BHs in the LISA range to a precision of ∼ ±0.2 with just ten observed events and this improves with the number of observed events as N -1/2 obs . EMRI events alone will not be able to constrain any evolution of this mass function with redshift. LISA MBHM events can be used to choose between different models for the assembly of structure in the Universe. Assuming that the LISA events were drawn from one of four simple models, we have shown that we would be able to confidently identify the correct model after collecting as little as three months of LISA data. BHs in the LISA mass range, 10 4 M -10 7 M , are not well constrained by current data, and so LISA has the potential to significantly improve our understanding of such systems and how they were assembled.

These preliminary results could be extended in several ways. The parameter space considered in the EMRI case should be expanded to explore what we can learn from EMRI measurements of BH spins and the eccentricities, inclinations etc. of EMRI orbits. MBHM waveform models including the merger/ringdown signal [27] and spin precession dynamics should improve our ability to distinguish between models. It will also be important to explore whether LISA EMRI and low-z MBHM observations together can probe the redshift evolution of the mass function and break the degeneracy between the mass function and the mass-scaling of the EMRI rate.

In the MBHM model selection context, the four models considered were deliberately chosen in [START_REF] Arun | Classical and Quantum[END_REF] to be as different as possible in order to provide ranges for the parameter estimation accuracies that might be achieved by LISA. It is therefore perhaps unsurprising that LISA will be able to distinguish these models very easily. The real Universe is likely to be a hybrid between light and heavy seed models, so it will be informative to explore more realistic mixtures between the present (oversimplified) seeding and accretion prescriptions. Indeed, if parameters can be introduced into the models that characterise the input physics (e.g., the seed mass distribution and accretion efficiency), we could determine the precision with which LISA will be able to measure such parameters. The results described here provide a clear illustration of the significant astrophysics that can be done with LISA observations and indicate that it will be worthwhile to carry out such more sophisticated analyses.

Figure 1 .

 1 Figure 1.We show the distribution of the logarithm of the likelihood ratio, ln(Λ AB ), for comparison of model SE to model SC using 3 months of LISA data (left panel) and using 1 year of data (right panel). The curves labelled "FAP distribution", where FAP stands for "false alarm probability", describe realisations drawn from the "wrong" model (SC). Those labelled "detection rate distribution" describe realisations drawn from the "right" model (SE). We note that the distribution for one year of data is somewhat broader than that for 3 months. This is not unsurprising, since more events are expected in one year of data, and both the mean and the variance of the underlying Poisson distribution increase with the total number of events.

20 Figure 3 .

 203 Figure 3. As Figure 2 but now for a fixed model comparison (SE versus SC), but varying the length of the LISA observation for fixed detector scenario (ii) (left) and varying the detector scenario for a fixed LISA observation of 3 months (right).

  ,

	Model Seed mass prescription Accretion prescription LISA events per year
	SE	VHM (light seeds)	coherent	37 (18)
	SC	VHM (light seeds)	chaotic	40 (21)
	LE	BVR (heavy seeds)	coherent	24 (22)
	LC	BVR (heavy seeds)	chaotic	21

  Receiver operator characteristic curves for different model comparisons. These were computed as described in the text for a fixed LISA mission duration of 3 months and assuming detector scenario (ii), i.e., one data channel only and ρ thresh = 20.

		1
		0.8
	Detection rate	0.4 0.6
			SE to SC
		0.2	SE to LE SE to LC
			SC to LE
			SC to LC
		0	LE to LC
		0.01	0.1	1
			FAP
	Figure 2.

Acknowledgments

JG's work is supported by the Royal Society. EB's research was supported by NSF grant PHY-0900735. MV was supported by NASA ATP Grant NNX07AH22G and a Rackham faculty grant.


). These flow patterns are presented on figures 1, 2 and 3 respectively, with their parameter range. This paper aims at making a comprehensive review of the literature on the PRB flows and a synthesis of the qualitative and quantitative data which describe them. Notably, we complete Kelly's reviews [1,2] by analysing more than ninety new references on this subject, among which fifty are later than 1993. The temporal linear stability analyses of the Poiseuille flow vis-à-vis the R  and the R // give the diagrams presented on figures 4 and 5 for infinite and finite spanwise aspect ratio ducts, respectively. The results of these diagrams have been known since early eighties (cf. §4). They are qualitative results which only give a simplified idea of the stability of the PRB flows.

The spatio-temporal linear stability analyses (dating from early nineties), which imply the concepts of convective and absolute instabilities, are more appropriate to describe the stability of these flows (cf. §5). Furthermore, recent experimental, theoretical and numerical studies, carried out in the non-linear domain, have shown the existence of three-dimensional unsteady thermoconvective structures whose shape is more complex than the R  and the R // . A review of these complex structures and on their stability is proposed in §8. This variation is shown for R // flows on figure 16, where three zones are revealed.

Equations (11-12), equation ( 13) and equation ( 16) are correlation laws respectively for the coordinates x 1 of the beginning of zone ( 2 17 and in the nomenclature, l e is determined from the stationary envelope Wmax(x). The curve L e =f(V g ) drawn on figure 18, representing the variation of the reduced growth length L e as a function of the reduced group velocity V g , is shown to be a universal curve in the sense that it is independent of the Prandtl number, as long as Re and Ra are not too large [68,78]. The divergence of the curve L e =f(V g ) at V g =2 corresponds to the transition between the absolutely unstable R  (V g <2) and the convectively unstable R  (V g >2). The values of Ra and Re at which l e diverge are shown to be a very precise criterion to define the transition curve Ra  conv (Re) (compare figures 10 and 17). §7 -In this section, the fine spatial and temporal structure of the R  is analysed. In §7.1, the influence on this structure of the open boundary conditions (OBC) that must be imposed at the outlet boundary of the computational domains for the numerical simulations is reviewed. Among all the tested OBC, the Orlanski type boundary conditions (equation ( 17)) is shown to be the one which perturbs the least the R  flows. The influence of the inlet boundary conditions (IBC) on the R  development in the numerical simulations and in the experiments is reviewed in §7.2.

The IBC influence on the R  is shown to be more important than the one of the OBC.

  PRB flows between two infinite platesA.1.1 k x =0, k y 0 : formation of R // A.1.2 k x 0, k y =0 : formation of R  A.1.3 k x 0,k y 0 : formation of any structures and Squire transformation A.2 Case of the PRB flows in finite lateral extension channels A.2.1 k x =0 : formation of R // A.2.2 k x 0 : formation of 3D structures Abrigded English version  §1 -The Poiseuille-Rayleigh-Bénard (PRB) flows are laminar mixed convection flows in horizontal rectangular ducts uniformly heated from below and uniformly cooled from above. The three most common configurations in PRB flows are the Poiseuille flow and two thermoconvective flow configurations : the transversal rolls (R  ) and the longitudinal rolls (R //

  figure 10 for the convective/absolute transition curve Ra  conv computed from this

  figures 27 and 28 for this third case). In §8.1.1, the seventeen very recent

  The industrial applications of the PRB flows are presented in §2 and the pioneer experiments are presented in §3. A review on the heat transfers and on the characteristics growth length of the R  and of the R // is proposed in §6, and a synthesis of the quantitative data on the spatial and temporal characteristics of the R  and of the R // is given in §7. §2 -The two main industrial applications of the PRB flows concern, first, the development of the horizontal rectangular chemical vapor deposition (CVD) reactors which are used to produce thin and uniform films of inorganic matter (cf. §2.1) and, second, the cooling of the electronic equipments (cf. §2.2). A general presentation of the CVD process is done in §2.1.1. In §2.1.2, it is shown that the presence of R  flows, of R // flows or of a steady transversal roll located just above the entrance of the heated zone (cf. figure6 and table1 where this roll is noted ''R  fixe'') must be eliminated of the CVD reactors because these thermoconvective structures cause non-Re of the critical Rayleigh number for the R  (Ra  *) is reported. Three analytical formulae, extracted from[67, 68], giving Ra  * and the critical time pulsation and wave number for the R  as a function of Re and Pr, are also reported (cf. equations (1-3)). In §4.2.3, we briefly comment on the results of the 3D temporal In §5.1 and §5.2, we remind the reader of the general notions about the convective and absolute intabilities (cf. figure 14 in the case of 2D PRB flows) and

	stability and the type of the thermoconvective structures appearing above the neutral
	stability threshold are mentioned. In §4.2.1, the studies dealing with the stability of
	the Poiseuille flow vis-à-vis the first type and second type R // (cf. figure 7) are
	reviewed. In §4.2.2, a long discussion about the experimental verification of the
	increase with about the amplitude and Ginzburg-Landau equations. The criteria of stability and of
	absolute and convective instability for the Ginzburg-Landau equation (5) in its
	linearised version are given by equations (6-8) respectively. In §5.3, the two main
	contributions to the analyse of the convective and absolute instabilities in PRB flows
	uniform deposits. Recent studies [15, 18-20] show that the aperiodic or chaotic flows are reviewed. Müller et al. [67, 68, 78, 91] analyse the transitions between Poiseuille
	observed at high Rayleigh and low Reynolds numbers should be used to get uniform flow and R
	deposits in CVD reactors. A synthesis of the CVD numerical studies for
	configurations close to the PRB flows is presented in §2.1.3 and in table I. The
	studies on the cooling of electronic equipments, close to the PRB flow configuration,
	are summed up in §2.2.
	§3 -In this paragraph, we present a review of the pioneer experiments which
	took place from 1920 (with the first studies of Idrac [35, 36]) to 1938. They are
	mainly concerned with applications to meteorology. Their results were synthesized by
	Sir D. Brunt in 1951 [3].
	§4 -From 1962 to 1984, the PRB flow studies were mainly based on linear
	stability analyses of the Poiseuille flow, completed by experiments to verify the
	results of the theoretical models. Some of these models (based on temporal linear
	stability analyses) and the associated experiments concentrated on the fully-
	established thermoconvective flows (cf. §4.2), while the others (based on spatial
	linear stability analyses) and the associated experiments concentrated on the thermal
	entrance zone (cf. §4.3). The whole theoretical analyses are summarized in table II, in
	which the parameter range, the form of the normal modes, the criteria of neutral

linear stability analysis in finite lateral extension ducts

[52, 53, 54]

. Some of these results are presented in figures 5, 8-10. The mathematical formulation of the temporal linear stability analysis is given in annexe A. In §4.2.4, the weaknesses of this temporal linear stability analysis are brought out : first, it predicts either an odd or an even number of R // to appear depending on the aspect ratio B (cf. figure

8

), whereas all the experiments and numerical simulations predict an even number of R // for fullyestablished flows at Re>0 and for B>1.1 (cf. [76] for the explanation) ; second, it rightly predicts the critical Rayleigh number Ra // * for the appearance of the R // , but fails to predict the critical Rayleigh number for the appearance of the R  and the critical Reynolds number Re*  determined experimentally and numerically (cf.

figure 10 and table III).

To remedy this, the space and time growth of the perturbations has to be analysed, i.e. a convective/absolute stability analysis is necessary (cf. §5). In §4.3, three studies on the PRB flow stability in the entrance zone of infinite lateral extension ducts are analysed (cf. table

II

). The used stability criteria are schematically represented in figure

12

. Figure

11

shows that the R  are more unstable than the R // just at the entrance of the heated zone for small Reynolds numbers, which is widely verified by several recent papers. As shown in figure

13

, the stability criterion used in

[87]

permits to find the experimental results again with a good agreement. §5 - in 2D PRB flows by 2D numerical simulations and with the Ginzburg-Landau equation (9) (cf. table IV for the coefficient values of this equation and cf.

  ), x 2 of Nu max in zone (2) and x 3 at which Nu exceeds the forced convection value of zone (1) by 3%. Other correlation laws (equations (14-15)) for the appearance length and the establishment length of the R // , determined from LDA experiments, are reported in §6.3.1. The results and the variation laws for the space and time average Nusselt number in fully-established

PRB flows are reviewed in §6.2 and table V. This Nusselt number is shown to keep constant whatever Re in the case of R // flows, but it is shown to decrease when Re increases in the case of R  flows. Most of the heat transfer studies are carried out in ducts of very large or infinite (2D) cross section. Very few data on the spanwise confinement influence on heat transfers are available. §6.3.2 is dedicated to the determination of the characteristic growth length l e of the R  . As shown in figure

Acknowledgements

Dimensionless parameters

Nu


L Galtchouk
email: leonid.galtchouk@math.unistra.fr

S Pergamenshchikov
email: serge.pergamenchtchikov@univ-rouen.fr

Geometric ergodicity for classes of homogeneous Markov chains *

Keywords: 60F10, 60J05 Homogeneous Markov chain, Geometric ergodicity, Convergence rate, Coupling renewal processes, Lyapunov function, Renewal theory, Non asymptotic exponential upper bound, Ergodic diffusion processes

The paper deals with non asymptotic computable bounds for the geometric convergence rate of homogeneous ergodic Markov processes. Some sufficient conditions are stated for simultaneous geometric ergodicity of Markov chain classes. This property is applied to nonparametric estimating in ergodic diffusion processes.

Introduction

In this paper we study a class of homogeneous Markov chains (Φ) n≥0 with values in some measurable space (X , B(X )) defined by a parametrized family of transition probabilities (P ϑ x ) x∈X , ϑ∈Θ ,

where Θ is a parametric set for this family. For each ϑ ∈ Θ the sequence Φ = (Φ n ) n≥0 is a homogeneous Markov chain defined on the measurable space (X , B(X )) with a transition probability P ϑ , i.e.

P(Φ 1 ∈ Γ|Φ 0 = x) = P ϑ x (Γ) . Our main goal is to state geometric ergodicity for this class simultaneously over all values of the parameter ϑ ∈ Θ.

Geometric ergodicity is studied in a number of papers (see, for example, [START_REF] Baxendale | Renewal theory and computable convergence rate for geometrically ergodic Markov chains[END_REF], [START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF]- [START_REF] Nummelin | Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory[END_REF]). We remind that a chain (Φ n ) n≥0 on the space (X , B(X )) with a invariant measure π is called geometrically ergodic if there exist a X → [1, ∞[ function V (x) and some constants R > 0, κ > 0 such that, for any n ≥ 1,

sup x∈X sup 0≤g≤V 1 V (x) |E x g(Φ n ) -π(g)| ≤ R e -κn . (1.2)
As we shall see later (see, Definition 6.1 below) the function V , providing the drift condition, is given by the Lyapunov functions (see, e.g. [START_REF] Khas'minskii | Stochastic Stability of Differential Equations[END_REF] in the case of diffusion processes and [START_REF] Feigin | Random coefficient autoregressive processes : a Markov chain analysis of stationary and finiteness of moments[END_REF], [START_REF] Klüppelberg | Renewal theory for functionals of a Markov chain with compact state space[END_REF], [START_REF] Klüppelberg | The tail of the stationary distribution of a random coefficient AR(q) process with applications to an ARCH(q) process[END_REF] for Markov chains). For this reason, in the sequel, we shall call such functions by the Lyapunov functions.

The property (1.2) is useful in applied problems related to the identification of stochastic systems, described by stochastic processes with dependent values, in particular, governed by stochastic difference or stochastic differential equations. Necessity of simultaneous geometric ergodicity appears in statistics, when one studies a minimax risk with respect to some family of distributions related to a statistical experiment. In particular, in this paper we shall apply simultaneous geometric ergodicity to nonparametric estimating the drift coefficient in the stochastic differential equation (see [START_REF] Galtchouk | Nonparametric sequential minimax estimation of the drift coefficient in diffusion processes[END_REF]):

dy t = S(y t ) dt + σ(y t )dW t , 0 ≤ t ≤ T , (1.3)
where S and σ are unknown functions and S has to be estimated from observations (y t ) 0≤t≤T . In studying minimax risks for kernel estimators we need to use geometric ergodicity for the process (1.3) simultaneously over all coefficients S and σ from some functional class (see (2.7) below). In this case, ϑ = (S, σ) is the class parameter.

It is clear that to apply the property (1.2) to some distribution family we have to find some explicit expressions for the parameters R > 0 and κ > 0. It is a well-known problem in the Markov Chain Monte Carlo (MCMC) theory, when a stopping rule for simulations is based on the accuracy of nstep approximations. Therefore, we need to find computable bounds in (1.2). Note that some explicit expressions for R and κ were calculated in [START_REF] Baxendale | Renewal theory and computable convergence rate for geometrically ergodic Markov chains[END_REF], [START_REF] Meyn | Computable bounds for geometric convergence rates of Markov chains[END_REF], [START_REF] Roberts | Convergence properties of perturbed Markov chains[END_REF] and [START_REF] Rosenthal | Minorization conditions and convergence rates for Markov chain Monte Carlo[END_REF] for ψ-irreducible homogeneous Markov chains. These results are not applicable in our case because it is not clear what does it mean ψirreducibility for a class of parametrized Markov chains. Note that in [START_REF] Baxendale | Renewal theory and computable convergence rate for geometrically ergodic Markov chains[END_REF] and [START_REF] Meyn | Computable bounds for geometric convergence rates of Markov chains[END_REF] the parameters R and κ were obtained by making use of the Kendall theorem. In [START_REF] Rosenthal | Minorization conditions and convergence rates for Markov chain Monte Carlo[END_REF] these parameters are obtained through the direct coupling method for the Markov chain. To this end the authors impose in [START_REF] Roberts | Convergence properties of perturbed Markov chains[END_REF]- [START_REF] Rosenthal | Minorization conditions and convergence rates for Markov chain Monte Carlo[END_REF] some additional assumptions which are not satisfied in the case of the diffusion process (1.3) (see Remark 3.1 in Section 3). Moreover, it should be noted that the upper bound in (1.2) given in [START_REF] Roberts | Convergence properties of perturbed Markov chains[END_REF] is calculated under the assumption that the minorization condition holds on the whole state space. This assumption is never true for the model (1.3).

In the paper we apply the coupling method to the renewal process generated by entrance times of the process into the minorization set. Note, that Meyn and Tweedie use the same approach in order to obtain convergence results (see, [START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF], chapter 13). Their results imply the power convergence rate. To obtain the geometric rate we make use of the Lyapunov functions method for the related coupling process.

In order to explain the novelty of the method introduced in the paper, we give the scheme of proving the property (1.2). The first step consists in passing to a splitting chain, which yields a chain with an accessible atom. Then, one makes use of the Regenerative Decomposition for splitting chains in order to evaluate the convergence rate (see [START_REF] Kingman | Regenerative Phenomena[END_REF], [START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF]). Let us remind that the principal term in this decomposition gives the deviation in the renewal theorem, which may be evaluated thanks to the Kendall renewal theorem that provides a geometric convergence rate. In our case the same convergence rate is obtained thanks to making use of the Lyapunov functions method for the coupling renewal process (see Theorem 4.1 in Section 4). This upper bound enables us to find the explicit non asymptotic exponential upper bound in the ergodic theorem for which we can find the supremum over the transition probability family in (1.1).

In this paper we find some sufficient conditions which provide simultaneous geometric ergodicity of the family (1.1) over all values of the parameter ϑ. We check these conditions for the diffusion model (1.3). As corollary, we obtain explicit upper bounds for geometric convergence rate in the ergodic theorem for diffusion processes. These bounds may be used in the Monte Carlo technique to calculate some functionals of ergodic diffusion processes. In that case one can replace these functionals by the corresponding integrals with respect to the invariant density which has a simple explicit form. The accuracy of this approximation is given by the explicit non asymptotic bounds in the geometric convergence rate for diffusion processes.

The paper is organized as follows. In the next section the main results are stated. Section 3 provides the explicit formulas for parameters in the geometric convergence rate. Section 4 is devoted to related coupling renewal processes. In section 5 geometric ergodicity is proved for a parametrized class of homogeneous Markov chains. In section 6 we apply this property to stochastic differential equations. Some basic results on homogeneous Markov chains are given in the Appendix.

Main results

Assume now, that the family of transition probabilities (P ϑ ) ϑ∈Θ satisfies the following conditions H 1 ) There exist 0 < δ < 1, some set C ∈ B ( X ) and some probability measure

ν on B(X ) with ν(C) = 1 such that, for any A ∈ B(X ) with ν(A) > 0, inf x∈C inf ϑ∈Θ P ϑ (x, A) -δν(A) > 0 .
(

For the sequel we denote

η = inf x∈C inf ϑ∈Θ P ϑ (x, C) -δ . (2.2) H 2 ) There exist X → [1, ∞) function V , some constants 0 < ρ < 1, D ≥ 1,
and a set C from B(X ) such that

V * = sup x∈C V (x) < ∞ and, for any x ∈ X , sup ϑ∈Θ E ϑ x (V (Φ 1 )) ≤ (1 -ρ)V (x) + D1 C (x) . (2.3)
Here E ϑ x means the expectation with respect to the transition probability P ϑ (x, •).

Remark 2.1. Condition H 2 ) is called the uniform drift condition and that of H 1 ) is the uniform minorization condition.

Theorem 2.1. Assume the conditions H 1 )-H 2 ) hold true with the same set C ∈ B(X ). Then, for each θ ∈ Θ, the chain Φ admits an invariant distribution π ϑ on B(X ). Moreover, for any n ≥ 2,

sup ϑ∈Θ sup x∈X sup 0<f ≤V 1 V (x) E ϑ x f (Φ n ) - X f (z) π ϑ (dz) ≤ R * e -κ * n , (2.4)
where the parameters R * = R * (ρ, δ, D, η, V * ) and κ * = κ * (ρ, δ, D, η 1 , V * ) are given in (3.5).

Apply now to the process (1.3). To this end we have to introduce some functional class of functions ϑ = (S, σ). First, for some x * ≥ 1, M > 0 and

L > β > 0, we denote by V 1 the class of functions S from C 1 (R) such that sup |x|≤x * |S(x)| + | Ṡ(x)| ≤ M and -L ≤ inf |x|≥x * Ṡ(x) ≤ sup |x|≥x * Ṡ(x) ≤ -β .
Second, for some fixed reals σ min > 0 and σ max > σ min , we denote by V 2 the class of functions σ from C 2 (R) such that, for all x ∈ R,

σ min ≤ min (|σ(x)| , | σ(x)| , |σ(x)|) ≤ max (|σ(x)| , | σ(x)| , |σ(x)|) ≤ σ max .
Finally, we set

Θ = V 1 × V 2 . (2.5)
Note that (see, for example, [START_REF] Gihman | Stochastic differential equations[END_REF]), for any function ϑ from Θ, the equation (1.3) admits a unique strong solution, which is an ergodic process having an invariant measure π ϑ with the invariant density q ϑ defined as

q ϑ (x) = σ -2 (x) exp{ x 0 S 1 (v)dv} +∞ -∞ σ -2 (z) exp{ z 0 S 1 (v)dv}dz , (2.6)
where

S 1 (v) = 2S(v)/σ 2 (v).
Theorem 2.2. For any 0 < ǫ ≤ 1/2 and t > 0,

sup ϑ∈Θ sup x∈R sup 0<g≤1 E ϑ x g(y t ) -R g(x)q ϑ (x)dx (1 + x 2 ) ǫ ≤ R ǫ e -κ ǫ t , (2.7)
where the parameters R ǫ > 0 and κ ǫ > 0 are given in (3.13).

Remark 2.2. Note that the property (2.7) is called simultaneous geometric ergodicity. As is shown in Section 6, the function (1 + x 2 ) ǫ is a Lyapunov function. It should be said that in [START_REF] Roberts | Quantitative bounds for convergence rates of continuous time Markov processes[END_REF], for the process (1.3) with a constant diffusion coefficient (i.e. for σ = 1), an exponential bound for deviation (2.7) was obtained. The proof of that result was based on the coupling method applied directly to the diffusion process (1.3) provided the existence of a Lyapunov function. In contrast with [START_REF] Roberts | Quantitative bounds for convergence rates of continuous time Markov processes[END_REF], in our paper an explicit family of Lyapunov functions is given that is of help in applications.

Remark 2.3. It should be noted that the inequality (2.7) may be applied to Monte Carlo calculation of the expectation E ϑ x g(y t ). Indeed, the previous expectation can be replaced with the integral of g with respect to the invariant density (2.6). The precision of such approximation is given in (2.7).

Computable bounds for geometric convergence rate

In this section we introduce the parameters R * and κ * which make explicit the upper bound for geometric ergodicity in (2.4). For any 0 < γ < 1, we denote

B * = Ǔ * 1 + Ǔ * V * 1 -(1 -δη 1 ) γ , (3.1)
where η 1 = η/(1 -δ) and

Ǔ * = max 1 -ρ + D (1 -δ)(1 -ρ) 1-γ (1 -(1 -ρ) γ ) , V * (1 -ρ) 1-γ .
We remind that the parameter η appears in the condition H 1 ). Moreover, we put

           r * = (1 -γ) 2 | ln(1 -ρ)| | ln(1 -δη 1 )| ln( Ǔ * V * ) + | ln(1 -δη 1 )| ; l = 2 + ln(V * B * ) r * + ln q(1 -e -r * ) -1 2r * , (3.2)
where .

q = 1 -B *
Here [a] means the integer part of a reel number a > 0. Next,

A = V * B * + 1 (1 -e -r * )(1 -e -γr * )
and

̺ 1 = (1 -γ) 2 r | ln(1 -ǫ)| ln A + r * l + | ln(1 -ǫ)| , (3.3)
where r = | ln(1 -q)| and ǫ = δη 1 (1 -δ) l-2 . Now we introduce the following parameter

A 3 = 1 -γ γ   3 + 2 e r * A 1 + Ae r * l (1 -(1 -ǫ) γ ) (e r * -1)   V * B * + e κ (3.4)
and

κ = (1 -γ) ̺ 1 r * ̺ 1 + ln V * B * .
We define the parameters R * and κ * in the Theorem 2.1 as

         κ * = κ * (ρ, δ, D, V * ) = (1 -γ) ̺ 1 r * 2( ̺ 1 + ln V * B * ) ; R * = R * (ρ, δ, D, V * ) = 2 ( A 2 + 1)e κ * + 1 e κ * -1 V * (B * ) 2 .
(3.5)

Further we define the upper bound in the ergodic Theorem 2.2. First of all, we take the set C in the minorization condition H 1 ) as the interval C = [-K, K] for some K > 0 and we chose the measure ν as the uniform distribution on C, i.e. for any measurable set A from B(R),

ν(A) = 1 2K mes(A ∩ [-K , K]) , (3.6)
where mes(•) is the Lebesgue measure on R.

In order to define the threshold δ > 0 we need the quadratic function

Ω * (z) = ω * 1 z 2 + ω * 2 z + ω * 3 (3.7) with ω * 1 = 4(Lσ * ) 2 + Lσ * + 1, ω * 2 = Lσ * σ max + H * 0 and ω * 3 = H * 1 + 2(H * 0 ) 2
, where

H * 0 = M + σ * σ min , H * 1 = M(1 + σ * ) + L + 2σ 2 *
and σ * = σ max σ min .

We chose the parameter δ as

δ K = 3Ke -Ω * ( K) 2 √ 2πσ max , (3.8)
where K = K/σ min . Now we set

K 0 = M 1 + 8σ min + 8σ * + 4 σ * (M 1 + M 2 ) + 16σ 2 * , (3.9)
where σ * = σ 2 max /(β(1 -e -β )),

M 1 = (M + βx * ) 2 + σ 2 max β β 2
and M 2 = M 1 (1 -e -β ) .

We set

η K = 1 -δ K - 4σ * (K 2 + M 2 ) (K 2 -M 1 ) 2 . (3.10)
Note, that for any

K ≥ K 0 δ K ≤ 3 Ke -K 2 2 √ 2π ≤ 3 4 √ eπ and η K ≥ 3( √ eπ -1) 4 √ eπ .
Therefore, Propositions A.7-A.8 imply the condition H 1 ) with the parameters δ K and η K defined in (3.8) and (3.10) for any K ≥ K 0 . To check the condition

H 2 ) we set b * 0 = β(2 + x * )(1 + x * ) + (M + 3)(3 + x * ) and b * 1 = b * 0 2β . (3.11)
Propositions 6.3-6.4 imply, that for any 0 < ǫ ≤ 1/2, the diffusion process

(1.3) satisfies the drift condition H 2 ) with V (x) = (1 + x 2 ) ǫ , C = [-K ǫ , K ǫ ], ρ ǫ = (1 -ǫ) (1 -e -2ǫβ ) and D ǫ = V * ǫ e -2ǫβ + b * 1 (1 -e -2ǫβ ) , (3.12)
where

K ǫ = K 0 + b * 1 ǫ 1/ǫ -1 and V * ǫ = (1 + K 2 ǫ ) ǫ .

Now we define

κ ǫ = κ * (ρ ǫ , δ ǫ , D ǫ , η ǫ , V * ǫ ) and R ǫ = R * (ρ ǫ , δ ǫ , D ǫ , η ǫ , V * ǫ ) , (3.13)
where the functions κ * and R * are defined in (3.5),

δ ǫ = δ K ǫ , η ǫ = η K ǫ and D ǫ = D K ǫ .
Remark 3.1. Note that we can not apply the bounds for geometric convergence rate from the paper [START_REF] Rosenthal | Minorization conditions and convergence rates for Markov chain Monte Carlo[END_REF] (Theorem 12) and [START_REF] Roberts | Convergence properties of perturbed Markov chains[END_REF] (Theorem 8) since there the bounds were obtained under the condition

(1 + K 2 ) ǫ > 2D K ρ ǫ . (3.14)
This condition is not satisfied in our case for sufficiently small values of the parameter ǫ > 0. In fact, we apply the bounds (2.7) in [START_REF] Galtchouk | Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions[END_REF] to point-wise estimating the drift coefficient S under observations of the process (1.3) at discrete times (t k ) k≥1 . The question of interest is the behavior of kernel estimators that are nonlinear functionals of observations. In order to study the non asymptotic estimating precision one needs of some concentration inequalities [START_REF] Galtchouk | Uniform concentration inequality for ergodic diffusion processes observed at discrete times[END_REF] based on the bounds (2.7) with the parameter ǫ > 0 no matter how small.

In order to illustrate the behavior of the geometric rate, we suppose that the parameters satisfy the following conditions:

       lim δ→0 η = 1 , lim δ→0 1 -η δ = +∞ , lim ρ→1 D V * = 0 and lim ρ→1 ln V * | ln(1 -ρ)| = 1 . (3.15)
It should remark that these conditions hold true for the process (1.3) with the parametric set (2.5) as L > β → ∞ and ǫ → 0 for some fixed ǫ > 0.

It should be noted, that the geometric rate in (1.2) obtained in [START_REF] Meyn | Computable bounds for geometric convergence rates of Markov chains[END_REF] satisfies

κ * = O(δ 8 ) as δ → 0 .
In [START_REF] Rosenthal | Minorization conditions and convergence rates for Markov chain Monte Carlo[END_REF] under the condition (3.14) this rate is "best", i.e.

κ * = O(δ) as δ → 0 .
Under the conditions (3.15) the coefficient κ * defined in (3.5)

κ * = (c 1 + o(1)) δ 13/2+µ 0 (γ) | ln δ| 2 as δ → 0 , ρ → 1 ,
where c 1 > 0 and µ 0 (γ) → 0 as γ → 0.

Remark 3.2. Note that the condition L, β → ∞ on the drift function of the process (1.3) concerns the behavior of the function only outside of the interval [-x * , x * ], i.e. outside of the informative part of the function S. Remind (see, [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF]), that the class (2.5) is used to bound the function S on the interval [-x * , x * ] and outside of the interval the conditions are imposed to preserve ergodicity.

Remark 3.3. It should remark that, unfortunately, the bounds from the papers [START_REF] Baxendale | Renewal theory and computable convergence rate for geometrically ergodic Markov chains[END_REF], [START_REF] Meyn | Computable bounds for geometric convergence rates of Markov chains[END_REF] are not applicable, in general case, to classes of Markov chains. Indeed, irreducibility is one of conditions providing the geometrical rate in [START_REF] Meyn | Computable bounds for geometric convergence rates of Markov chains[END_REF]. Therefore, in the case of a parametric Markov chain class, irreducibility measure should depend on a class parameter. It is not clear, what will going on this measure when one takes the supremum over the class parameter and how one needs to change the irreducibility condition in order to obtain uniform bounds over the class parameter ϑ ∈ Θ by using the proof in [START_REF] Meyn | Computable bounds for geometric convergence rates of Markov chains[END_REF].

Coupling Renewal Method

In this Section we shall obtain a non asymptotic upper bound with explicit constants in the renewal theorem by making use of the coupling method. The notions used here can be found in [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF], [START_REF] Klüppelberg | Renewal theory for functionals of a Markov chain with compact state space[END_REF], [START_REF] Lindvall | Lectures on the Coupling Method[END_REF].

Let (Y j ) j≥0 and (Y ′ j ) j≥0 be two independent sequences of random variables taking values in N. Assume that the initial random variables Y 0 and Y ′ 0 have distributions a = (a(k)) k≥0 and b = (b(k)) k≥0 , respectively, i.e. for any k ≥ 0,

P(Y 0 = k) = a(k) and P(Y ′ 0 = k) = b(k) .
The sequences (Y j ) j≥1 and (Y ′ j ) j≥1 are supposed to be the i.i.d. sequences with the same distribution p = (p(k)) k≥0 , i.e. for any k ≥ 0, For any n ≥ 0, we define the following stopping times

P(Y 1 = k) = P(Y ′ 1 = k) = p(k) . We assume also that p(0) = P(Y 1 = 0) = 0, i.
t n = inf{k ≥ 0 : k i=0 Y i > n} and t ′ n = inf{k ≥ 0 : k i=0 Y ′ i > n} .
Further, we set

W n = t n j=0 Y j -n and W ′ n = t ′ n j=0 Y ′ j -n . (4.2)
It is easy to see that the sequences (W n ) n≥1 and (W ′ n ) n≥1 are homogeneous Markov chains taking values in N * such that, for any n, k and l from N * ,

P W n = k|W n-1 = l = P W ′ n = k|W ′ n-1 = l = p(k)1 {l=1} + 1 {k=l-1} 1 {l≥2} . (4.3)
Firstly we study the entrance times (s k ) k≥0 of the chain (W n ) n≥0 to the state {1} which are defined as s -1 = 0 and for k ≥ 0

s k = inf{l ≥ s k-1 + 1 : W l = 1} . (4.4)
One can check directly that the stopping times s k , k ≥ 1, can be represented as

s k = s 0 + k j=1 ς j . (4.5)
One can check directly that in this case (ς j ) j≥1 are i.i.d. random variables independent of s 0 and, for any l ∈ N * ,

P(ς 1 = l) = P(s 0 = l|W 0 = 1) = p(l) , (4.6)
i.e. the random variables (ς j ) j≥1 have the same distribution as Y 1 . Now we study the properties of the stopping time s 0 .

Proposition 4.1. Assume that the condition C) holds. Then E e rs 0 ≤ 3 e υ * (r) .

Proof. First of all, note that, for k ≥ 2 and l ≥ 0,

P(s 0 = l|W 0 = k) = 1 {l=k-1} .
Moreover, taking into account that, for any k ≥ 1,

P(W 0 = k) = a(0)p(k) + a(k) , (4.7)
we obtain that r) .

E e rs 0 = P(W 0 = 1) E e rY 1 + ∞ k=2 e r(k-1) P(W 0 = k) ≤ E e rY 1 + a(0)e -r E e rY 1 + e -r E e rY 0 ≤ 3 e υ * (
Hence Proposition 4.1. Now, we introduce the embedded Markov chain (Z k ) k≥0 by

Z k = W ′ s k (4.8)
and the corresponding entrance time to the state {1}:

̟ = inf{k ≥ 1 : Z k = 1} . (4.9)
In order to study the property of this stopping time, we need of the following notations

l * = l * (r) = 2 + ln e 2υ * (r) (1 -e -r ) -1 q(r) 2r , (4.10)
where q(r) = (1 -e υ 1 (r) )/2 and the parameter υ 1 (r) < 0 will be specified below. Moreover, for any 0 < γ , ǫ * < 1, we set

A * 1 (r) = A * (r) 1 + A * (r)e rl * 1 -(1 -ǫ * ) γ , (4.11)
where

A * (r) = 1 + Q * (r)e rl * +υ * (r) (1 -q(r)) 1-γ (1 -(1 -q(r)) γ ) and Q * (r) = e υ * (r) 1 -e -r .
Proposition 4.2. Assume the condition C). Then, for any 0 < γ < 1, for any ǫ * > 0 and υ 1 (r) ≤ -r for which

0 < ǫ * ≤ min 1≤j≤l * -1 p(j) and ln Ee -rY 1 ≤ υ 1 (r) , (4.12)
one has

E e ̺ * ̟ ≤ Q * 1 (r) A * 1 (r) , where Q * 1 (r) = e υ * (r) + e υ ′ * (r) + Q * (r)
and

̺ * = ̺ * (r) = (1 -γ) 2 | ln(1 -q(r))| | ln(1 -ǫ * )| ln A * (r) + rl * + | ln(1 -ǫ * )| . (4.13)
Proof. Firstly we note that the sequence (4.8) is a homogeneous Markov chain with values in N * such that, for any m and l from N * and for any k ≥ 1,

P(Z k = m|Z k-1 = l) = P(W ′ ς 1 -l = m|Y ′ 0 = 0) . (4.14) Note, that W ′ k = -k for k < 0 under the condition Y ′ 0 = 0. Therefore, for any positive function V , E [V (Z 1 )|Z 0 = l] = E V (l -ς 1 ) 1 {ς 1 <l} + E Q(ς 1 -l)1 {ς 1 ≥l} , (4.15)
where

Q(n) = E (V (W n )|Y 0 = 0). Using the distribution (4.3) yields Q(n) ≤ E V (W n-1 -1)|Y 0 = 0 + EV (Y 1 )P W n-1 = 1|Y 0 = 0 .
Choosing now V (x) = e rx one has

Q(n) ≤ e -r Q(n -1) + EV (Y 1 ) .
From the last inequality, taking into account that Q(0) = EV (Y 1 ), it follows that, for any n ≥ 1,

Q(n) ≤ e -nr Q(0) + EV (Y 1 ) n j=1 e -(n-j)r ≤ Q * (r) , (4.16)
where the upper bound Q * (r) is defined in (4.10). This implies that the last term in (4.15) can be estimated as

E Q(ς 1 -l)1 {ς 1 ≥l} ≤ Q * (r)e υ * (r)-rl .
Therefore,

E (V (Z 1 )|Z 0 = l) V (l) ≤ e υ 1 (r) + Q * (r)e υ * (r)-2rl .
By making use of the definition of l * in (4.10), we obtain

sup l≥l * E (V (Z 1 )|Z 0 = l) V (l) ≤ 1 + e υ 1 (r) 2 = 1 -q(r) < 1 . (4.17)
Moreover, for any 1 ≤ l ≤ l * , Therefore, by Proposition A.2 for a * = -(1 -γ) ln(1 -q(r)), one gets

E (V (Z 1 )|Z 0 = l) ≤ (1 -ρ)V (l) + e υ 1 (r) V (l) + Q * (r)e υ * (
sup l≥1 U C (l, a * , V ) V (l) ≤ A * (r) ,
where the upper bound A * (r) is given in (4.11). Moreover, from (4.12) and (4.14) we get that, for 2 ≤ l ≤ l * -1,

P(Z 1 = 1|Z 0 = l) ≥ p(l -1) ≥ ǫ * .
Therefore, putting in Proposition A.3 k * = ǫ * , a = ̺ * defined in (4.13) and the set B = {1} we obtain that, for any l ≥ 1,

E (e ̺ * ̟ |Z 0 = l) ≤ V (l)A * 1 (r)
, where the parameters ̺ * and A * 1 (r) are defined in (4.11) and (4.13). This upper bound implies

E (e ̺ * ̟ ) ≤ A * 1 (r) E V (Z 0 ) . Moreover, note now that E V (Z 0 ) = ∞ j=1 E V (Z 0 )|W ′ 0 = j P(W ′ 0 = j) .
Similarly to (4.15) we obtain

E V (Z 0 )|W ′ 0 = j = E V (j -s 0 ) 1 {s 0 <j} + E Q(s 0 -j)1 {s 0 ≥j} .
Using here the inequality (4. [START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF]) yields

E V (Z 0 )|W ′ 0 = j ≤ V (j) + Q * (r) .
Moreover, similarly to (4.7) we obtain that, for any j ≥ 1,

P(W ′ 0 = j) = b(0)p(j) + b(j) . Thus, EV (Z 0 ) ≤ b(0)EV (Y 1 ) + EV (Y ′ 0 ) + Q * (
r) and we come to the inequality (4.12). Hence Proposition 4.2. Proposition 4.3. Assume the condition C). Then, for any ǫ * > 0 satisfying the condition (4.2) and for any 0 < γ < 1, there exists κ > 0 such that

Ee κs ̟ ≤ A * 2 (r) , (4.18)
where

A * 2 (r) = (1 -γ) 3e 2υ * (r) + Q * 1 (r)A * 1 (r) γ and κ = κ(r) = (1 -γ)̺ * r ̺ * + υ * (r) ,
the coefficients A * 1 (r) and ̺ * are defined in (4.11) and (4.13). Proof. Indeed, we have

E e κs ̟ = κ ∞ 0 e κt P(s ̟ > t) dt ≤ κ ∞ 0 e κt (P(s N > t) dt + P(̟ > N)) dt ,
where N = N(t) = 1 + [ϑt], and ϑ is some positive parameter which will be chosen later. Note now that, for 0 < ϑ < r/υ * (r),

P(s N > t) ≤ 3e υ * (r)(N +1)-rt ≤ 3e 2υ * (r) e -(r-υ * (r)ϑ)t .
Moreover, due to Proposition 4.2

P(̟ > N) ≤ Q * 1 (r) A * 1 (r) e -N ̺ * ≤ Q * 1 (r) A * 1 (r) e -ϑ̺ * t . Therefore, denoting ι * (ϑ) = min ((r -υ * (r)ϑ) , ϑ̺ * ) , one gets E e κs ̟ ≤ κ 3e 2υ * (r) + Q * 1 (r) A * 1 (r) ∞ 0 e -(ι * (ϑ)-κ)t dt . Maximizing now ι * (ϑ) yields max 0<ϑ<υ * (r)/r ι * (ϑ) = ι * (ϑ max ) = r̺ * ̺ * + υ * (r) , ϑ max = r ̺ * + υ * (r)
.

Therefore, choosing now ϑ = ϑ max and κ = (1 -γ)ι * (ϑ max ), we come to the inequality (4.18). Hence Proposition 4.3.

Let us define the renewal sequence (u(n)) n≥0 as follows

u(n) = ∞ j=0 p * j (n) , (4.19)
where p * j denotes the jth convolution power. For j = 0 we set p 0 (n) = 1 for n = 0 and p 0 (n) = 0 for n ≥ 1. We remind that, for two sequences (a(j)) j≥0 and (u(j)) j≥0 , the convolution sequence (a * u(j)) j≥0 is defined for any j ≥ 0 as

a * u(j) = j i=0 a(i)u(j -i) .
Proposition 4.4. Assume that the condition C) holds and there exists ǫ * > 0 satisfying the inequality (4.12). Then, for any 0 < γ < 1 and n ≥ 2,

|∆(n)| ≤ A * 2 e -κn , where ∆(n) = a * u(n) -b * u(n), the coefficients κ and A * 2 are given in (4.18). Proof. Obviously, that for n ≥ 1, a * u(n) = P ∪ j=0 j i=0 Y i = n = P(W n-1 = 1) and b * u(n) = P ∪ j=0 j i=0 Y ′ i = n = P(W ′ n-1 = 1) .
Therefore,

∆(n) = P(W n-1 = 1 , W ′ n-1 ≥ 2) -P(W ′ n-1 = 1 , W n-1 ≥ 2)
. Now, we introduce the "coupling" stopping time τ as

τ = inf{k ≥ 1 : (W k , W ′ k ) = (1, 1)} .
Note that, for any n ≥ 2, by the Markov property for the chain (W k , W ′ k ) k≥1 , one has

P(W n = 1 , W ′ n ≥ 2, τ ≤ n -1) = n-1 k=1 P(W n = 1 , W ′ n ≥ 2, τ = k) = n-1 k=1 P(τ = k) v n-k .
where

v k = P(W k = 1 | W 0 = 1)P(W k ≥ 2 | W 0 = 1
). Similarly, one gets

P(W ′ n = 1 , W n ≥ 2, τ ≤ n -1) = n-1 k=1 P(τ = k)v n-k .
This implies that

∆(n) = α 1 (n -1) -α 2 (n -1)
,

where α 1 (n) = P(W n = 1 , W ′ n ≥ 2 , τ > n) and α 2 (n) = P(W ′ n = 1 , W n ≥ 2 , τ > n) . Therefore, for any n ≥ 2, |∆(n)| ≤ max (α 1 (n -1) , α 2 (n -1)) ≤ P(τ > n) .
Taking into account that τ ≤ s ̟ a.s., we obtain Then, for any 0 < γ < 1, n ≥ 2 and ǫ * > 0 satisfying the inequality (4.12),

|∆(n)| ≤ P(s ̟ > n) ≤ e -κn Ee κs ̟ .
u(n) - 1 E Y 1 ≤ A * 3 (r) e -κn ,
where

A * 3 (r) = 1 -γ γ 3 + 2 e r A * (r) 1 + A * (r)e rl * (1 -(1 -ǫ * ) γ ) (e r -1) e υ * (r) (4.20)
and the parameter κ > 0 is defined in (4.18).

Proof. We obtain the inequality (4.20) through Proposition 4.4 in which we choose a(0) = 1 with a(j) = 0 for j ≥ 1. Moreover, we choose the distribution (b(j)) j≥0 as

b(j) = 1 EY 1 P(Y 1 > j) = 1 EY 1 ∞ i=j+1 p(i) .
It is easy to see directly that, for any

j ≥ 1, b * u(j) = 1 EY 1 .
Note now, that through the condition of this theorem we obtain

e rn b(n) = e rn P(Y 1 > n) EY 1 ≤ Ee rY 1 1(Y 1 > n) EY 1 → 0 , as n → ∞ .

Proof of Theorem 2.1

First we fix some 0 < γ < 1 and we set a 1 = -(1 -γ) ln(1 -ρ). We start with studying the properties of the function

U ϑ C (x, a 1 , V ) = E ϑ x τ C j=1 e a 1 j V (Φ j )
where τ C = inf{n ≥ 1 : Φ n ∈ C}. The condition H 2 ) and Proposition A.2 imply immediately that, for any 0 < γ < 1,

sup ϑ∈Θ sup x∈X U ϑ C (x, a 1 , V ) V (x) ≤ 1 -ρ + D (1 -ρ) 1-γ (1 -(1 -ρ) γ ) := U * . (5.1)
Now, we introduce a splitting chain family as in [START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF], p. 108 (see also [START_REF] Nummelin | Splitting technique for Harris Recurrent Markov Chains[END_REF]).

We set X = X × {0, 1}, X 0 = X × {0} and X 1 = X × {1}. Let B(X i ) be the σ-fields generated by the set A i = A × {i} with A ∈ B(X ), i = 0, 1. In the sequel we will denote by < x > i the ith component of x ∈ X . It is clear, that < x > 0 ∈ X and < x > 1 ∈ {0, 1}. Furthermore, we define the σ -field B( X ) as a σ-field generated by B(X 0 ) ∪ B(X 1 ) and for any measure λ on B(X ) we relate the measure λ * on B( X ) as

λ * (A 0 ) = (1 -δ)λ(A ∩ C) + λ(A ∩ C c ) and λ * (A 1 ) = δλ(A ∩ C) .
Now, for each ϑ ∈ Θ, we introduce a homogeneous Markov chain ( Φn ) n≥0 by the following transition probabilities

Pϑ (x, •) =            P ϑ (x, •) * , if x ∈ X 0 \ C 0 ; P ϑ (x, •) * -δν * (•) 1 -δ , if x ∈ C 0 ; ν * (•) , if x ∈ X 1 .
(5.2)

Note, that for any x ∈ X 1 ,

Pϑ (x, C 0 ∪ C 1 ) = ν * (C 0 ∪ C 1 ) = ν(C) = 1 . (5.3)
Obviously, that the set α = C 1 is an accessible atom for the chain ( Φn ) n≥1 , i.e. for any positive X → R function g Ěϑ x g( Φ1 ) = Ěϑ y g( Φ1 ) , for any x , y ∈ α . This implies directly that, for any nonnegative random variable ξ measurable with respect to the σ-field generated by the chain ( Φn ) n≥1 , one has

Ěϑ

x ξ = Ěϑ y ξ for any x , y ∈ α .

In the sequel we denote by Ěϑ α (•) the such expectations. Moreover, one can check directly that the chain ( Φn ) n≥1 is ν * -irreducible. Next, for any set Č from B( X ) we introduce the corresponding entrance time

τ Č = inf n ≥ 1 : Φn ∈ Č (5.4)
and the corresponding entrance function

Ǔϑ Č (x, a 1 , V ) = Ěϑ x τ Č j=1 e a 1 j V ( Φj ) , (5.5)
where V (x) = V (< x > 1 ). By Proposition A.5 we obtain that, for any x ∈ X ,

U ϑ C (x, a 1 , V ) = (1 -δ) Ǔϑ C 0 ∪C 1 (x 0 , a 1 , V ) 1 {x∈C} + Ǔϑ C 0 ∪C 1 (x 0 , a 1 , V ) 1 {x∈C c } + δ Ǔϑ C 0 ∪C 1 (x 1 , a 1 , V ) 1 {x∈C} ,
where x i = (x, i) for i = 0, 1. Note now, that due to the property (5.3),

Ǔϑ C 0 ∪C 1 (x 1 , a 1 , V ) = Ěϑ x 1 e a 1 V ( Φ1 ) ≤ e a 1 V * = V * (1 -ρ) 1-γ .
Therefore, using the upper bound (5.1) and the coefficient Ǔ * given in (3.1), we obtain

sup ϑ∈Θ sup x∈ X Ǔϑ C 0 ∪C 1 (x, a 1 , V ) V (x) ≤ Ǔ * .
Note now that, for x ∈ C 0 by the definition (5.2)

Pϑ (x, α) = Pϑ (x, C 1 ) = δ P ϑ (x, C) -δ 1 -δ ≥ δη 1 ,
where the parameter 0 < η 1 ≤ 1 is defined in (3.1). By making use of Proposition A.3 with

a * = a 1 = -(1 -γ) ln(1 -ρ) and k * = δη 1 , one gets sup ϑ∈Θ sup x∈ X Ǔϑ α (x, r * , V ) V (x) ≤ B * , (5.6)
where B * and r * are given in (3.1) and (3.2). Therefore, by Proposition A.1, the chain ( Φn ) n≥0 is ergodic for each ϑ ∈ Θ with the invariant measure given as

πϑ ( Γ) = 1 Ěϑ α τα Ěϑ α τα j=1 1 { Φj ∈ Γ} . (5.7)
Now, for any n ≥ 2, we define ι = max{1 ≤ j ≤ n -1 : Φj ∈ α} and we put ι = 0 if τα ≥ n. Moreover, note that, for any X → R function f and any n ≥ 2,

Ěϑ x f ( Φn ) 1 {τ α <n} = n-1 j=1 Ěϑ x f ( Φn ) 1 {τ α ≤j} 1 {ι=j} = n-1 j=1 Ěϑ x 1 {τ α ≤j} Ěϑ x f ( Φn ) 1 {ι=j} | Φ1 , . . . , Φj .
Now, taking into account that ( Φn ) n≥1 is a homogeneous Markov chain, we can calculate the last conditional expectation as follows

Ěϑ x f ( Φn ) 1 {ι=j} | Φ1 , . . . , Φj = 1 { Φj ∈α} Ěϑ α f ( Φn-j ) 1 { Φ1 / ∈α ,..., Φn-j-1 / ∈α} = 1 { Φj ∈α} g f,α (n -j) , (5.8)
where g f,α (k) = Ěϑ α f ( Φk ) 1 {τ α ≥k} . By convention, we set g f,α (0) = 0. Therefore,

Ěϑ

x f ( Φn )

1 {τ α <n} = n j=1 Pϑ x (τ α ≤ j) g f,α (n -j) = h x * g f,α (n) ,
where h x(0) = 0 and, for j ≥ 1,

h x(j) = Pϑ x (τ α ≤ j) = Pϑ x Φj ∈ α .
Moreover, for j ≥ 1,

h x(j) = j l=1 Pϑ x τα = l , Φj ∈ α = j l=1 a x(l) u(l -j) = a x * u(j) ,
where a x(l) = Pϑ x (τ α = l) and u(l) = Pϑ α Φl ∈ α .

(5.9)

It is clear that a x(0) = 0 and u(0) = 1, i.e. a x * u(0) = 0. This implies that

h x(j) = a x * u(j) ,
for all j ≥ 0. Finally, taking into account that a x * u * g f,α (1) = 0, we obtain that for any n ≥ 1,

Ěϑ x f ( Φn ) 1 {τ α <n} = a x * u * g f,α (n) .
(5.10)

Note that the sequence (u(n)) n≥0 is a renewal sequence, i.e.

u(n) = ∞ j=0 p * j (n) with p(k) = Pϑ α (τ α = k) .
Now, we set Y 0 = τα and Y j = inf{j ≥ Y j-1 + 1 : Φj ∈ α}. One can check directly that (Y j ) j≥1 is i.i.d. sequence with the distribution (p(k)) k≥1 , i.e. u(n) n≥0 is the renewal function for the sequence (Y j ) j≥1 . We denote

ω(n) = u(n) - 1 Ěϑ α Y 1 .
We estimate this term by Theorem 4.1. First we have to check the condition C 1 ) uniformly over the parameter ϑ ∈ Θ, i.e. to show that, for any j ≥ 1,

inf ϑ∈Θ Pϑ α (τ α = j) ≥ δ η 1 (1 -δ) j-1 . (5.11)
Let us check this property for j = 1. We remind that, by the condition H 1 ), one has ν(C) = 1. Thus, the definition (5.2) implies

Pϑ α (τ α = 1) = ν * (C 1 ) = δ ≥ δ η 1 .
To show the property (5.11) for j ≥ 2 note, that Pϑ α (X 0 ) = ν * (X 0 ) = 1 -δ. Moreover, taking into account that P ϑ (z, X 0 ) * ≥ 1 -δ we obtain the same lower bound for the splitting distribution (5.2), i.e. for any ž ∈ X 0

(ž, X 0 ) = 1 {ž∈C 0 } P ϑ (z, X 0 ) * -δν * (X 0 ) 1 -δ + 1 {ž∈X 0 \C 0 } P ϑ (z, X 0 ) * ≥ (1 -δ) .
Similarly, for any ž ∈ X 0 we obtain

Pϑ (ž, C 1 ) = 1 {ž∈C 0 } P ϑ (z, C 1 ) * -δν * (C 1 ) 1 -δ + 1 {ž∈X 0 \C 0 } P ϑ (z, C 1 ) * ≥ δ P ϑ (z, C) -δ 1 -δ ≥ δη 1 .
Now through the induction we can show, that for any j ≥ 1

Pϑ α Φ1 ∈ X 0 , . . . , Φj ∈ X 0 ≥ (1 -δ) j .
Therefore, taking into account that for any x ∈ X we have Pϑ (x, X 1 \ C 1 ) = 0, we obtain for j ≥ 2

Pϑ α (τ α = j) = Pϑ α Φ1 / ∈ α , . . . , Φj-1 / ∈ α , Φj ∈ α = Pϑ α Φ1 ∈ X 0 , . . . , Φj-1 ∈ X 0 , Φj ∈ C 1 ≥ δ η 1 Pϑ α Φ1 ∈ X 0 , . . . , Φj-1 ∈ X 0 .
This yields the lower bound (5.11). Similarly we can show

sup j≥1 sup ϑ∈Θ Pϑ α (τ α = j) (1 -δη 1 ) j-1 ≤ and sup ϑ∈Θ Ěϑ α e -r * Y 1 ≤ δ e r * -1 + δη 1 .
Moreover, taking into account that Ěϑ α e -r * Y 1 ≤ e -r * we obtain that

sup ϑ∈Θ Ěϑ α e -r * Y 1 ≤ B * 1 < 1 ,
where B * 1 is given in (3.2). Therefore, the sequence (Y j ) j≥0 satisfies the condition of Theorem 4.1 with r = r * , υ * (r) = ln (V * B * ), l * = l,

ǫ * = δ η 1 (1 -δ) l-2 and υ 1 (r) = ln B * 1 ,
where l is defined in (3.2). Therefore, for n ≥ 2,

ω(n) ≤ A 3 e -κn ,
where the parameters A 3 and κ are defined in (3.4). Taking into account that A 3 ≥ e κ we obtain that, for any n ≥ 0,

ω(n) ≤ A 3 e -κn .
Therefore, for any 0

< κ < κ ω(κ) = n≥0 e κn ω(n) ≤ A 3 e κ-κ e κ-κ -1 .
(5.12)

Now, taking into account that

πϑ (f ) = 1 Ěα Y 1 +∞ j=0 g f,α (j)
and that Ěα Y 1 ≥ 1, one obtains, for any n ≥ 1,

| Ěϑ

x f ( Φn )

1 {τ α <n} -πϑ (f )| ≤ a x * ω * g f,α (n) + Q x * g f,α (n) + G f,α (n) , where Q x(n) = Pϑ x (τ α > n) and G f,α (n) = j=n+1 g f,α (j) .
Therefore, for any n ≥ 0,

∆ x(n) = | Ěϑ x f ( Φn ) -πϑ (f )| ≤ a x * ω * g f,α (n) + Q x * g f,α (n) + g f,x (n) + G f,α (n) , (5.13)
where Therefore, from (5.13) we obtain for any 0

g f,x (n) = Ěϑ x f (
< κ < r * , ∆ x(κ) ≤ a x(κ) ω(κ) g f,α (κ) + Q x(κ) g f,α (κ) + g f,x (κ) + G f,α (κ) . (5.14)
It is clear

g f,x (κ) = Ǔϑ α (x, κ, f ) with f (x) = f (< x > 1 ) . Moreover, note that a x(κ) = Ěx e κτ α ≤ Ǔϑ α (x, κ, f ) , Q x(κ) = Ěx e κτ α -1 e κ -1 ≤ Ǔϑ α (x, κ, f ) e κ -1 and
G f,α (κ) = g f,x (κ) -g f,x (0) e κ -1 ≤ Ǔϑ α (α, κ, f ) e κ -1 .
Therefore, taking into account that κ ≤ r * , we obtain, that for any function 1 ≤ f ≤ V and for any 0 < κ < κ,

∆ x(κ) ≤ ω(κ) + e κ + 1 e κ -1 Ǔϑ α (x, κ, V ) Ǔϑ α (α, κ, V ) ≤ A 3 e κ-κ e κ-κ -1 + e κ + 1 e κ -1 Ǔϑ α (x, κ, V ) Ǔϑ α (α, κ, V ) .
Now, putting here κ = κ * = κ/2 and taking into account the inequality (5.6) we get, for any x ∈ X , that

∆ x(κ * ) ≤ ( A 3 + 1)e κ * + 1 e κ * -1 V (< x > 1 ) V * (B * ) 2 .
Moreover, note that the chain (Φ n ) n≥1 is ergodic with the invariant measure π ϑ defined in (5.7) and (A.12). By applying Proposition A.5 with λ equals to the Dirac measure at x, we obtain that, for any function 0

< f ≤ V , E ϑ x f (Φ n ) -π ϑ (f ) = (1 -δ) Ěϑ x 0 f ( Φn ) -πϑ ( f ) 1 {x∈C} + δ Ěϑ x 1 f ( Φn ) -πϑ ( f ) 1 {x∈C} + Ěϑ x 0 f ( Φn ) -πϑ ( f ) 1 {x∈C c } .
Therefore, for any x ∈ X , one gets

|E ϑ x f (Φ n ) -π ϑ (f )| ≤ | Ěϑ x 0 f ( Φn ) -πϑ ( f )| + | Ěϑ x 1 f ( Φn ) -πϑ ( f )| , i.e. n≥0 e κ * n |E ϑ x f (Φ n ) -π ϑ (f )| ≤ ∆ x 0 (κ * ) + ∆ x 1 (κ * ) .
From here it follows the inequality (2.4). Hence Theorem 2.1.

Application to diffusion processes

In order to study geometric ergodicity for the process (1.3) we start with the chain (Φ y n ) n≥0 , where Φ y n = y n . Proposition 6.1. For any ϑ ∈ Θ, the sequence (Φ y n ) n≥0 is a homogeneous Markov chain aperiodic and ψ-irreducible, where ψ is the Lebesgue measure on B(R).

Proof. Taking into account (see, for example, [START_REF] Gihman | Stochastic differential equations[END_REF]) that the solution of the equation (1.3) is a homogeneous Markov process, we obtain immediately that (Φ y n ) n≥0 is a homogeneous Markov chain. In this case (see [START_REF] Gihman | Stochastic differential equations[END_REF]), for any ϑ, the process (y t ) t≥0 admits the transition density υ ϑ (t, x, y) as follows :

υ ϑ (t, x, y) = Υ(t, x, y) √ 2πtσ(y) exp ς(y) ς(x) H ϑ (u)du - (ς(y) -ς(x)) 2 2t , (6.1)
Here,

H ϑ (z) = S(ς(z)) σ(ς(z)) - σ(ς(z)) 2σ 2 (ς(z) , ς(x) = x 0 σ -1 (u) du
and ς(•) is the inverse function of ς, i.e. it is the unique solution of the equation z = ς(ς). Moreover,

Υ(t, x, y) = E exp - 1 2 t 0 H ϑ (w * u,t (x, y))du , H ϑ (x) = Ḣϑ (x) + H 2 ϑ (x) and w * u,t (x, y) = ς(x) + u t (ς(y) -ς(x)) + w u - u t w t .
It means that, for any n ≥ 1, for any A ∈ B(R) and for any x ∈ R,

P ϑ (Φ y n ∈ A|Φ y 0 = x) = A υ ϑ (n, x, z)dz . (6.2)
Thus, the chain (Φ y n ) n≥0 is ψ-irreducible, where ψ is the Lebesgue measure on B(R). Moreover, in this case the chain is aperiodic. Now, we check the minorization condition H 1 ) for the chain (Φ y n ) n≥0 .

Proposition 6.2. For any K ≥ 8σ min , the chain (Φ y n ) n≥0 satisfies the minorization condition H 1 ) with C = [-K, K], δ = δ K and η = η K defined in (3.10), and the probability measure ν K defined in (3.6).

Proof. We start with studying the properties of the function H ϑ defined in (6.1). From definition of the class Θ we find immediately that, for any z ∈ R,

|ς(z)| ≤ |z| σ max .
Moreover, note that, for any ϑ from Θ and for any y from R,

| Ṡ(y)| ≤ M + L and |S(y)| ≤ M + L|y| . (6.3)
Therefore, sup

|z|≤z * sup ϑ∈Θ |H ϑ (z)| ≤ H * 0 + Lσ * z * ,
where H * 0 is given in (3.6). Note now, that the derivative of H ϑ can be represented as Ḣϑ (z) = F ϑ (ς(z)) with is given in (3.6). Moreover, we note that on the set

F ϑ (y) = Ṡ(y) - S(y) σ(y) σ(y) - σ (
Γ K = { sup 0≤u≤1 |w u | ≤ K/2} with K = K/σ min ,
the process (w * v,t (x, y)) 0≤v≤t≤1 is bounded :

sup x,y∈C sup 0≤u≤t≤1 |w * u,t (x, y)| ≤ 2 K .
Therefore, for any x, y from C,

Υ(1, x, y) ≥ P(Γ K ) e -1 2 H * (2 K) .
By making use of the Doob inequality we obtain

P(Γ K ) ≥ 1 - 4E sup 0≤t≤1 w 2 t K 2 1 ≥ 1 - 16σ 2 min K 2 ,
i.e. for K ≥ 8σ min ,

P(Γ K ) ≥ 3/4 and Υ(1, x, y) ≥ 3 4 e -1 2 H * (2 K) .
Moreover, for any x, y ∈ C,

sup ϑ∈Θ ς(y) ς(x) H ϑ (u)du ≤ H * 0 + Lσ * K K .
This implies that inf

x,z∈C

υ ϑ (1, x, z) ≥ 3(4 √ 2πσ max ) -1 e -Ω * ( K) ,
where Ω * (z) is introduced in (3.10). Therefore, taking into account that, for any A from B(X ), • for some constants γ > 0, b * > 0 and for any x ∈ R,

P ϑ (Φ y 1 ∈ A|Φ y 0 = x) = A υ ϑ (1,
sup ϑ∈Θ A ϑ (V )(x) ≤ -γV (x) + b * ; (6.5)
• lim x→∞ V (x) = ∞ and there exists m > 0 such that Therefore, we get, for any ϑ ∈ Θ,

sup x∈R V (x) + | V (x)| 1 + |x| m < ∞ . ( 6
A ϑ (V )(x) ≤ 2ǫV (x)xS(x) 1 + x 2 + ǫσ 2 max ≤ 2ǫ V (x)xS(x) 1 + x 2 1 {|x|≥x * } + ǫ (1 + x 2 * ) ǫ M + σ 2 max ≤ -2ǫV (x)β + b * .
Hence Proposition 6.3.

Proposition 6.4. Let V (x) be a uniform over ϑ ∈ Θ Lyapunov function for equation (1.3) from definition 6.1 with constants γ and b * . Then, for any K > 0 and 0 < ε < 1 for which

inf |x|≥K V (x) ≥ b * εγ ,
the chain (Φ y n ) n≥0 satisfies the following inequality

sup ϑ∈Θ E ϑ x V (Φ y 1 ) ≤ (1 -ρ)V (x) + D K 1 {|x|≤K} , (6.7
)

where ρ = (1-ε)(1-e -γ ), D K = V * K e -γ +b * (1-e -γ )/γ and V * K = sup |x|≤K V (x).
Proof. By the Ito formula, one gets

V (y t ) = V (y 0 ) + t 0 A ϑ (V )(y s )ds + t 0 V (y s )σ(y s )dw s .
In Proposition A.7, we have proved that the moments of the solution of equation (1.3) are bounded. This implies that the stochastic integral is a martingale in the above Ito formula. Therefore, by setting

Z(t) = E ϑ x V (y t ), one has Ż(t) = E ϑ x A ϑ (V )(y t ) = -γZ(t) + ψ t ,
where ψ t = E ϑ x (A ϑ (V )(y t ) + γy t ). The inequality (6.5) gives ψ t ≤ b * . Resolving this differential equation, we obtain, that for 0 ≤ t ≤ 1

Z(t) = Z(0)e -γt + t 0 e -γ(t-s) ψ s ds ≤ Z(0)e -γt + b * 1 -e -γ γ = V (x)e -γt + b * 1 -e -γ γ .
Therefore,

sup ϑ∈Θ E ϑ x V (Φ y 1 ) ≤ V (x)e -γ + b * 1 -e -γ γ .
From here we obtain the inequality (6.7). Hence Proposition 6.4.

Proof of Theorem 2.2

First note, that thanks to Propositions 6.1 the diffusion process (1.3) with the parameter ϑ = (S, σ) from Θ introduced in (2.5), satisfies the condition H 1 ) with the set C = [-K, K] for K ≥ K 0 given (3.7) and the measure ν defined in (3.6). Moreover, Propositions 6.3 -6.4 imply that for any 0 < ǫ ≤ 1/2 this process satisfies the condition H 2 ) with the parameters (3.12). Moreover, for any t ≥ 1 and any R →]0, 1] function g, we set

g(x) = E ϑ x g(y t ) = E ϑ x g(y {t} ) .
Moreover, taking into account that π(g) = π(g), one has

E ϑ x g(y t ) -π(g) = E ϑ x g(Φ y [t] ) -π(g) .
Therefore by applying Theorem 2.1 to the chain (Φ y n ) n≥0 , we come to the upper bound (2.7). Hence Theorem 2.2.

Definition A.3. The Markov ψ-irreducible chain (Φ n ) n≥0 is called periodic of period d if there exist disjoint sets Γ 1 , . . . , Γ d in B(X ) with ψ ∩ d j=1 Γ c j = 0
such that, for 1 ≤ i ≤ d -1 and for any x ∈ Γ i , one has

P x (Φ 1 ∈ Γ i+1 ) = 1 and for x ∈ Γ d one has P x (Φ 1 ∈ Γ 1 ) = 1. The chain is aperiodic if d = 1.
Definition A.4. We will say that the chain (Φ n ) n≥0 satisfies the minorization condition if, for some δ > 0, some set C ∈ B(X ) and some probability measure ν with ν

(C) = 1, one has inf A∈B(X ) inf x∈C P (x, A) -δ ν(A) ≥ 0 . (A.1)
Definition A.5. A set α ∈ B + (X ) is called accessible atom if, for any x and y from α, P(x, Γ) = P(y, Γ) , ∀ Γ ∈ B(X ) .

In order to study the ergodicity property, we associate to any set C ∈ B(X ) the stopping time τ C = inf{k ≥ 1 : Φ k ∈ C} .

Proposition A.1. Suppose that the Markov chain Φ is ψ-irreducible and contains an accessible atom α such that

E α τ α < ∞ .
Then the chain is ergodic with the invariant probability measure π defined as

π(Γ) = 1 E α τ α E α τ α j=1 1 {Φ j ∈Γ} .
Proof. Indeed, by the definition of π, for any set Γ ∈ B(X ), one has

X π(dz) P(z, Γ) = 1 E α τ α E α ∞ j=1 1 {j≤τ α } E α 1 {Φ j+1 ∈Γ} |Φ 1 , . . . , Φ j = 1 E α τ α   E α τ α j=2 1 {Φ j ∈Γ} + P α Φ τ α +1 ∈ Γ   .
Proposition A.3. Assume that for some a = a * > 0 the Markov chain (Φ n ) n≥1 satisfies the property (A.4) with the C -bounded function V , i.e.

V * = sup x∈C V (x) < ∞ . (A.5)
Let B be a set from B(X ) such that, for some k * > 0,

inf x∈C\B P(x, B) ≥ k * . (A.6)
Then, for any 0 < γ < 1 and

0 < a ≤ (1 -γ)| ln(1 -k * )| ln V * U * (a * ) + | ln(1 -k * )| a * , (A.7) one has sup x∈X 1 V (x) U B (x, a, V ) ≤ U * 1 (a) , (A.8)
where

U * 1 (a) = U * (a) 1 + U * (a)V * 1 -(1 -k * ) γ .
Proof. First, we introduce the sequence of stopping times (τ C (n)) n≥0 as follows : τ C (0) = 0 and, for n ≥ 1,

τ C (n) = inf{k ≥ τ C (n -1) + 1 : Φ k ∈ C} .
Obviously, that τ C (1) = τ C . Moreover, the condition (A.4) implies that E x τ C (n) < ∞ for any n ≥ 1 and x ∈ X . Using this sequence we obtain that for 0 < a ≤ a *

U B (x, a, V ) = ∞ n=0 E x τ C (n+1) j=τ C (n)+1 e aj V (Φ j ) 1 {τ B ≥j} ≤ U C (x, a, V ) + ∞ n=1 E x 1 {τ B >τ C (n)} e aτ C (n) U C (z n , a, V ) ,
where z n = Φ τ C (n) . Moreover, taking into account the inequalities (A.4) and (A.5), we get that, for a ≤ a * , n) .

U B (x, a, V ) ≤ U * (a)V (x) + U * (a)V * ∞ n=1 Υ n (x, a) , where Υ n (x, a) = E x 1 {τ B >τ C (n)} e aτ C (
Note now that, for n = 1,

Υ 1 (x, a) ≤ V (x)U * (a) .
Let now n ≥ 2. We have

Υ n (x, a) = E x 1 {τ B >τ C (n)} e aτ C (n) = E x 1 {τ B >τ C (n-1)} e aτ C (n-1) E z n-1 e aτ C 1 {τ B >τ C } . Therefore, for n ≥ 2, Υ n (x, a) ≤ Υ n-1 (x, a) sup z∈C\B E z e aτ C 1 {τ B >1} .
By the Hölder inequality and the condition (A.5), we get

sup z∈C\B E z e aτ C 1 {τ B >1} ≤ (V * U * (a * )) a/a * (1 -k * ) 1-a/a * := g(a) .
Therefore, for any a satisfying the condition (A.7) we obtain

g(a) ≤ (1 -k * ) γ
and, for any n ≥ 2, 1) .

Υ n (x, a) ≤ Υ 1 (x, a) g(a) n-1 ≤ V (x)U * (a) (1 -k * ) γ(n-
This implies directly the bound (A.8). Hence Proposition A.3.

A.3 Properties of splitting chains

Now, we study some property of the splitting chain ( Φn ) n≥1 constructed in Section 4 , which we represent as Φn = ( φn , ιn ) , (A.9)

where φn ∈ X and ιn ∈ {0, 1}.

Proposition A.4. For any measure λ on B(X ) and any set Γ ∈ B( X ), First, we check this equality for n = 1. Note that, for any X → R function g and for any x ∈ X , one has Hence, the Proposition A.5.

X Pϑ (x, Γ) λ * (dx) = λ * 1 ( Γ) , (A.10) where λ 1 (•) = X P ϑ (x, •) λ(dx) .
Proposition A.6. Assume that the splitting chain ( Φn ) n≥1 has an invariant probability measure π. Then, the chain (Φ n ) n≥1 has the invariant probability measure π on B(X ) which is given as π(Γ) = π(Γ 0 ) + π(Γ 1 ) . (A.12)

Moreover, π = π * .

Proof. First we check directly that π = π * . Moreover, for any Γ ∈ B(X )

π(Γ) = π(Γ 0 ∪ Γ 1 ) = X π(dž) Pϑ (ž, Γ 0 ∪ Γ 1 ) = X π * (dž) Pϑ (ž, Γ 0 ∪ Γ 1 ) .
Therefore, applying here Proposition A.5 we obtain that π(Γ) = where M * = (M + βx * ) 2 /β + 2σ 2 max . Proof. To obtain this inequality we make use of the method proposed in ([9], p.20) for linear stochastic equation. First of all note that thanks to Theorem 4.7 from [START_REF] Liptser | Statistics of random processes[END_REF], for any T > 0, there exists some ǫ > 0 such that for each ϑ ∈ Θ and x ∈ R sup The Chebyshev inequality implies now the bound (A.16). Hence Proposition A.8.

1 =

 1 min e -r * , δ e r * -1 + δη 1

  e. the sequences (Y j ) j≥1 and (Y ′ j ) j≥1 take values in N * = N \ {0}. Moreover, we suppose that the distributions a, b and p satisfy the following condition C) There exists a real number r > 0 such that ln max (E e rY 0 , E e rY 1 ) ≤ υ * (r) and ln E e rY ′ 0 ≤ υ ′ * (r) .

  r)-rl , i.e. the chain (Z k ) k≥1 satisfies the condition (A.2) in the Appendix with C = {1, . . . , l * -1} and D = e rl * +υ * (r) Q * (r) .

Proposition 4 . 4 Theorem 4 . 1 .

 4441 3 implies the upper bound (4.18). Hence Proposition 4.Assume that there exists r > 0 such that ln E e rY 1 ≤ υ * (r) .

Therefore, the summing by parts yields j≥0 e

 j≥0 rj b(j) = E e rY 1 -1 (e r -1)EY 1 ≤ e υ * (r) -1 e r -1 := e υ ′ * (r) .and Proposition 4.4 implies the inequality (4.20). Hence Theorem 4.1.

  Φn ) 1 {τ α ≥n} . Now for any sequence (b(n)) n≥0 we denote by b(κ) the Laplace transformation, i.e. b(κ) = n≥0 e κn b(n) .

. 6 ) 6 . 3 .

 663 Proposition For any 0 < ǫ ≤ 1/2, the function V (x) = (1+x 2 ) ǫ satisfies the inequality (6.5) with γ = 2ǫβ and b * = ǫb * 0 , where b * 0 is given in (3.11). Proof. The definition of the space Θ implies directly that, for |x| ≥ x * , xS(x) ≤ |x|(M + βx * ) -βx 2 .

Proof.

  Indeed, by the definition of the operation and of the transition probability Pϑ (•, •) we obtainX Pϑ (x, Γ) λ * (dx) = X P ϑ (x, Γ) * λ(dx) = λ * 1 ( Γ) .Proposition A.5. For any n ≥ 1, any measurable positive X n → R function G and for any measure λ on B(X ), one hasX E ϑ x G n (Φ 1 , . . . , Φ n ) λ(dx) = X Ěϑ x G n ( φ1 , . . . , φn ) λ * (dx) . (A.11)Proof. It is clear, that it suffices to check this equality for positive functions of the formG n (x 1 , . . . , x n ) = n j=1g j (x j ) .

X g(< y > 1 )g 1 g

 11 P * (x, dy) = X g(y) P(x, dy) , where < y > 1 denotes the first component of the y ∈ X = X × {0, 1}. Making use of this equality implies easy (A.11) for n = 1. Assume now that the equality (A.11) is true until n -1. We check it for n. Indeed, we haveĚϑxG n ( φ1 , . . . , φn ) = Ěϑx n j=1g j (x j ) = Ěϑ x g 1 ( φ1 ) T ( Φ1 ) , (y) λ 1 (dy) ,where the measure λ 1 (•) is defined in (A.10). Therefore, taking into account Proposition A.4 , we can represent the integral on the right hand side of the equality (A.11) asX Ěϑx G n ( φ1 , . . . , φn ) λ * (dx) = X T (y)µ * (dy) .By the induction assumption, one has X T (y)µ * (dy) = j+1 (Φ j ) µ(dy) = X E x G n (Φ 1 , . . . , Φ n ) λ(dx) .

XP

  (z, Γ) π(dz) , i.e. π is the invariant measure for the chain (Φ n ) n≥1 . Hence Proposition A.6.A.4 Moment inequality for the process (1.3)Proposition A.7. Let (y t ) t≥0 be a solution of the equation (1.3). Then, for any m ≥ 1 and for any x ∈ R,sup t≥0 sup ϑ∈Θ E ϑ x (y t ) 2m ≤ (2m -1)!! (x 2 + M * /β) m , (A.13)

0≤t≤T E ϑ x e ǫy 2 te 0 e 2 +E ϑ y 2 t ≤ x 2 + M 2 and P ϑ x y 2 1 ≥ K 2 ≤= 1 0ϑ x ζ 2 = 1 0e

 202221211 < ∞ . (A.14) Applying the Ito formula to y 2m t denoting B ϑ (y) = 2yS(y) + σ 2 (y) + βy 2 , yield dy 2m t = -mβy 2m t dt + my 2(m-1)t B ϑ (y t ) + 2(m -1)σ 2 (y t ) dt + 2my 2m-1 t σ(y t )dW t .Therefore, taking into account that y 0 = x we can represent the last equation in the following integral formy 2m t = e -mβt x 2m + m t 0 e -mβ(t-s) y 2(m-1) s B ϑ (y s ) + 2(m -1)σ 2 (y s ) ds + 2m t 0 e -mβ(t-s) y 2m-1 s σ(y s )dW s . (A.15)One can check directly thatsup ϑ∈Θ sup y∈R |B ϑ (y)| ≤ (M + βx * ) 2 β + σ 2 max .Moreover, the property (A.14) yields that, for any m ≥ 1, -mβ(t-s) y 2m-1 s σ(y s )dW s = 0 .Therefore, setting z t (m) = E ϑ x y 2m t , we obtainz t (m) ≤ x 2m + m(2m -1)M * t -mβ(t-s) z s (m -1) ds .The induction implies directly the bound (A.13). Hence Proposition A.7.Proposition A.8. Let (y t ) t≥0 be a solution of the equation (1.3). Then, for anyK > M 1 , M 2 ) β(1 -e -β ) (K 2 -M 1 ) 2 , (A.16)where M 2 and M 1 are given in (3.9).Proof. First, putting in (A.15) m = 1, we obtainsup t≥0 P 2ζ ≥ K 2 -M 1 (1 -e -β ), where ζ e -β(1-s) y s σ(y s )dW s . Now, taking into account that for |x| ≤ K E -2β(1-s) E ϑ x y 2 s σ 2 (y s )ds ≤ σ 2 max K 2 + M 2 1 -e -β β .

  (z) = H * 1 + 2(H * 0 ) 2 + Lσ * σ max z + 2(Lσ * ) 2 z 2and the coefficient H *

					1
		y) 2σ(y)	+	( σ(y)) 2 σ 2 (y)	.
	By making use of the upper bounds (6.3), we obtain
	sup ϑ∈Θ	|F ϑ (y)| ≤ H * 1 + Lσ * |y|
	and, therefore, for any z * > 0,
	sup	sup
	|z|≤z *

ϑ∈Θ | H ϑ (z)| ≤ H * (z * ) ,

where H *

  Definition 6.1. Any R → [1, ∞) twice continuously differentiable function V is called uniform over ϑ ∈ Θ Lyapunov function for the equation (1.3) if the following conditions fulfill:

			x, z)dz ,
	yields the inequality (2.1) with δ K and ν K (•) defined in (3.10) and (3.6) respec-
	tively. Moreover, the inequality (2.2) follows directly from Proposition A.8.
	Hence Proposition 6.2.
	Now, for any C 2 (R) → R function V , we introduce the generator
	A ϑ (V )(x) = V (x)S(x) +	1 2	σ 2 (x) V (x) .	(6.4)

The second author is partially supported by the RFFI-Grant 09-01-00172-a.

A Appendix

A.1 Homogeneous Markov chains with atoms

We follow the Meyn-Tweedie approach (see [START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF]). We remind some definitions from [START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF] for a homogeneous Markov chains (Φ n ) n≥0 defined on a measurable state space (X , B(X )). Denote by P (x, •) , x ∈ X , the transition probability of this chain, i.e. for any A ∈ B(X ), x ∈ X ,

The n-step transition probability is

We remind that a measure π on B(X )) is called invariant for this chain if, for any A ∈ B(X ),

If there exists an invariant positive measure π with π(X ) = 1 then the chain is called positive.

Definition A.1. The chain (Φ n ) n≥0 is ϕ-irreducible if there exists a nontrivial measure ϕ on B(X ) such that, whenever ϕ(A) > 0, one has

One can show that, for any ϕ-irreducible chain, there exists a "maximal" irreducible measure which is noted as ψ and the chain is called ψ-irreducible. A irreducible measure ψ is maximal if and only if ψ(A) = 0 implies

In the sequel, we denote

Definition A.2. The chain (Φ n ) n≥0 is Harris recurrent if it is ψ-irreducible and, for any A ∈ B + (X ), one has

Moreover, it is easy to see that

This implies the relationship X π(dz) P(z, Γ) = π(Γ) , i.e. the measure π is invariant. Obviously, that π(X ) = 1, i.e. π is a probability measure.

A.2 Lyapunov functions method for Markov chains

We start with the definition of a "Lyapunov function". For this we impose the following drift condition to the chain (

In this case we call V the Lyapunov function. Now, for any X → [1, +∞) function f and any set A ∈ B(X ), we set

Proposition A.2. Assume, that the condition D) holds. Then, for any 0 < a < -ln(1 -ρ), one has Taking into account that V (x) ≥ 1, we obtain the inequality (A.4).