Sidharta Chatterjee
email: sidharta123@yahoo.com

An Analysis of Threats to Marine Biodiversity and Aquatic Ecosystems

Keywords: Marine biodiversity, ecosystem services, habitat destruction, coral bleaching, coral reefs, IUCN

Marine ecosystems and aquatic habitats face a number of threats from humans. Serious conservation attention and eorts should be drawn and directed towards restoration of fragmented marine habitats and estuarine ecosystems. Threats are manifold: i.e., primarily from overexploitation of marine resources, overshing, climate warming, sewage disposal, industrial chemical discharge, oil spills, invasive species, and dredging. The Corals are particularly fragile animals and very sensitive to changes in temperature and salinity. Coral reefs are coastal habitats build by corals that are also one of the most productive habitats within the marine ecosystems, but they are however, at great danger of going extinct from any drastic changes in sea surface temperature and salinity. Ecosystem services provided by coastal areas helps to sustain coastal people whose livelihood depends on shing and sh products, aquaculture and exploitation of biologically diverse resources that the seas and the oceans provide them. For this reason, it is essential to monitor marine biodiversity and prevent further degradation of marine habitats. This paper attempts to address this critical issue to stimulate conservation eorts among policy makers and ecologists. Besides, it is also relevant to understand how ecosystem gets degraded, and what measures are best and should be undertaken to prevent further degradation of marine habitats.

Introduction

Biodiversity is the sumtotal of richness and variation of all living things. It represents variability among living organisms that include marine, terrestrial and aquatic ecosystems. Of these, the marine ecosystem represents the most biologically diverse bionetwork of ora and fauna which is still largely unexplored. Such diversity is a provider of economic services which comes as a blessing to coastal people (MoE Report 2011). However, marine ecosystems are under pressure and the global bio-diversity is threatened. These presents as broad-scale threats to marine biodiversity [START_REF] Dieter | Nature in the Balance: The Economics of Biodiversity[END_REF][START_REF] Mapstone | National Climate Change Adaptation Research Plan for Marine Biodiversity and Resources, National Climate Change Adaptation Research Facility[END_REF][START_REF] Imtiyaz | Threats to Marine Biodiversity[END_REF]. Broadly, such threats can be categorized as follows [START_REF] Nevill | Threats to marine biodiversity[END_REF] The threats to global biodiversity (See also [START_REF] Sechrest | Biodiversitythreats[END_REF] and marine ecosystems are real [START_REF] Casper | Changing Ecosystems: Eects of Global Warming[END_REF], because marine habitats are fragile. Marine ecosystems are highly responsive to uctuations and changes in physical environments [START_REF] Pan | Coastal marine biodiversity challenges and threats[END_REF] while at the same time, they are resilient and adaptable to such changes [START_REF] Steele | Regime shifts in marine ecosystems[END_REF]. Like terrestrial ecosystems, marine ecosystems and habitats are susceptible to destruction (Walker and Kendrik 2006), pollution and climate warming (Craig 2012, Walker andKendrik 2006). Coastal ecosystems are also prone to destruction due to high population pressure and construction of port facilities, marinas and other logistic amenities (Burke et al., WRI Report 2001, Ólafsson 2017). Unquestionably, expansion of human activities threatens ocean-wide biodiversity [START_REF] Dulvy | Extinction risk and conservation of the world's sharks and rays[END_REF]. Continuous dredging causes damage to riverine and estuarine ecosystems. Furthermore, marine wildlife and ecosystems of the polar regions are susceptible to the impacts of climate warming and enhanced anthropogenic activities. In simple words, coastal biodiversity is at risk and marine environments are threatened [START_REF] Imtiyaz | Threats to Marine Biodiversity[END_REF] from heightened human activities [START_REF] Dulvy | Extinction risk and conservation of the world's sharks and rays[END_REF]. It may not be reasonable to reduce coastal human activities, but it may be feasible to think about means and methods to reduce further damage to coastal ecosystems. The goal of this paper, therefore, is to address these critical issues regarding conservation of coastal and marine ecosystems and advocate eective strategies, and convert those into actions. Furthermore, this research analyzes the importance of habitats and why habitats are essential for propagation and survival of species. The analytic model simply reinforces the signicance of habitat quality and the eect of habitat protection on the survival of species.

Ecosystem Services

Marine ecosystems provide many essential servicesas for example, food resources for millions of people. Ecosystem services [START_REF] Cowling | An operational model for mainstreaming ecosystem services for implementation[END_REF] There are other services provided by marine ecosystems for instance; deep sea oil exploration and drilling, mining for minerals, etc. In a nutshell, biodiversity plays a positive role in maintaining the ecosystem services which supports a growing human population [START_REF] Worm | Impacts of Biodiversity Loss on Ocean Ecosystem Services[END_REF]). However, marine ecosystems are facing accelerating loss of aquatic species and they are threatened by various anthropogenic activities that are damaging to marine habitats. Habitat loss can lead to decline in ecosystem services [START_REF] Dobson | Habitat loss, trophic collapse, and the decline of ecosystem services[END_REF]. Although the immediate consequences are not known, such accelerated loss of marine species could impact smooth ow of ecosystem services in the long term. Damage to coastal ecosystems has economic implications as well. The economic value of coastal ecosystems depends on the marine biological diversity. Ocean biodiversity loss aects ecosystem services [START_REF] Worm | Impacts of Biodiversity Loss on Ocean Ecosystem Services[END_REF]. Marine species are largely dominated by arthropods (phylum crustaceans), molluscs and sh. Economic value of shes are well acknowledgedthey are the sources of food and nutrition for millions of people. Besides, the phylum crustaceans which include crabs, craysh, shrimp, prawns and lobsters have immense economic value. They are mostly found along the coastlines of seas and oceans. The coastlines of seas and oceans are therefore, biologically active areas of productivity, and they represent a substantial share of economic activities for the coastal people. The business of mariculture (seaweeds) is also an important contributor to ecosystem services. Beyond doubt, erosion of biodiversity has denite impact on humanity (MA 2005[START_REF] Cardinale | Biodiversity loss and its impact on humanity[END_REF], Worm et al. 2009). Therefore, the ongoing erosion of marine biodiversity could have economic and ecological implications in the long run.

Marine Biodiversity under Threat

It is now evident that there are many marine species including some marine animals and mammals which are threatened, endangered or vulnerable to climate change and susceptible to being severely aected by augmented human activities. Some marine species have already become extinct (See Estes, Burdin and Doak, 2015), while some othersfor instance, polar bear (Ursus maritimus), is facing risk of extinction from the melting Arctic ocean ice due to climate warming [START_REF] Hunter | Climate change threatens polar bear populations: a stochastic demographic analysis[END_REF]. The World Conservation Union (IUCN) monitors species diversity across regions. It maintains a listThe IUCN Red List [START_REF] Rodríguez | A practical guide to the application of the IUCN Red List of Ecosystems criteria[END_REF] which species the degree of abundance or scarcity of plants and animal species by assigning measurable categories to describe their relative occurrence in a particular region. To put it simply, IUCN Red List is an approach for monitoring and assessing the status of global biodiversity 1 . This has important implications; e.g., the Threatened species are candidates for protection and conservation, whereas the Endangered species are candidates for close observation as well as protection.

Biodiversity loss threatens human well-being [START_REF] Díaz | Biodiversity loss threatens human well-being[END_REF]. With an unprecedented rate of biodiversity loss, the uninterrupted provision of goods and services from marine ecosystems are threatened. So it is important to evaluate the loss of biodiversity at the ecosystem level. Of importance to note, biodiversity is a dominant driver of ecosystem productivity [START_REF] Tilman | Biodiversity impacts ecosystem productivity as much as resources, disturbance, or herbivory[END_REF]. Marine ecosystems are highly productive, and there is a positive relationship between diversity and ecosystem functions and services [START_REF] Worm | Impacts of Biodiversity Loss on Ocean Ecosystem Services[END_REF]. However, valuation of biodiversity is a dicult and tricky job (See Nunes and van den Bergh 2001). It is easy to assume the impact of biodiversity on human welfare, but it is rather dicult to perform a valuation analysis of all the possible indicators of ecosystem services.

It is because ecosystem functions [START_REF] Odum | Fundamentals of ecology[END_REF]) are varied, and so it may not be easy to perform a valuation of life support functions that ecosystems provide which include provision of clean air, healthy environment, and carbon storage (Nunes et al. 2000). However, valuation of biodiversity loss could be an alternative approach as an economic indicator of the loss of biological products and services ows. It is particularly relevant to Coral biodiversity and coastal ecosystems. Local degradation and loss of habitat for marine ora and fauna may reduce the overall ecosystem services being provided by the coastlines. The value of Ecosystem serviceswhich according to some estimate [START_REF] Boesch | The role of science in ocean governance[END_REF], amounts to around $21 trillion a year, wherein, coastal systems contribute 63% of this total amount. According to another estimate, world's coral reefs produces goods and services worth of economic value $29.8 billion per year. Of this amount, tourism accounts for $9.0 billion, and sheries $5.7 billion (Conservation International, 2008). So, loss of coral reefs would also result in loss of substantial ecosystem services. Obliteration of coral reefs may also result in loss of aesthetic beauty of many shallow shorelines. Destruction of coral reefs follow coral bleaching, and scientists know how exactly this happens. Corals have low physiological tolerance to high temperature. Therefore, one of the biggest enemy of corals is global warming. Coral bleaching 2 results from high stress factors such as rising or falling temperatures, intense light, low salinity (ocean acidication) all of which causes loss of endosymbionts (zooxanthellae). 1.1 C can initiate the bleaching eect (see Fig. 1) which if prolonged, will result in death of corals (See [START_REF] Veron | The coral reef crisis: The critical importance of< 350ppm CO 2[END_REF]. In fact, the ecosystems most threatened by climate change are coral reefs (McClenachan et al. 2007). Several incidents of coral bleaching (Brown 1997[START_REF] Eakin | Global coral bleaching 20142017: status and an appeal for observations[END_REF]) are on record (see for example, Global bleaching episode, NOAA 3 1998, 2010, and 2015). Although bleaching itself does not always lead to mass mortality of corals [START_REF] Eakin | Caribbean corals in crisis: record thermal stress, bleaching, and mortality in 2005[END_REF], nevertheless, climate anomalies have been linked to coral bleaching (Glynn 1990) that resulted in local extinction of coral populations (See Cahill et al. 2012). The fact is, coral reefs around the world are threatened [START_REF] Eakin | Global coral bleaching 20142017: status and an appeal for observations[END_REF]. For instance, the Coral Triangle (Hoegh-Guldberg 2009) is home to the greatest number of marine species, and contains 76% of world's coral species. It is a global centre of marine biodiversity. Coral triangle constitutes much of the Indian Ocean and encompasses Indonesia, Malaysia, The Philippines, Papua New Guinea, Solomon Islands and Timor Leste. However, this epicentre of marine biodiversity faces several threats and challenges. Coral reefs are constituents of coastal ecosystems. They provide critical ecological services. So, it is very important to preserve and device methods to limit further damage to the coastal ecosystems. Besides, the marine biodiversity of India also faces a multitude of challenges since according to the IUCN (Joshi, Varsha and Sruthi 2015), several species are either endangered or threatened. Signicant among them is Hawksbill turtle (Eretmochelys imbricata) which is critically endangered, and the well known Bengal Tiger (Panthera tigris tigris), which is endangered. There are several endangered species associated with the Mangroves ecosystems of India (See for instance, Joshi, Varsha and Sruthi 2015).

Importance of Marine Biodiversity

Marine ecosystems are valuable but vulnerable. They are biologically so diverse that they are home to 97% of all species on earth (Lefèvre, IUCN Report 2007). It is factual that the oceans and seas support a large number of biologically diverse species which contribute immensely to ecological diversity. Marine species diversity is immense; estimates of known and unknown marine species diversity could be obtained from a compilation by a consortium of scientists, ecologists, conservationists which have been published in Current Biology by [START_REF] Appeltans | The magnitude of global marine species diversity[END_REF]. They report a total number of 226,000 eukaryotic marine species, and speculate that there could be another one third or two thirds of marine species that may be undescribed. Of the total 222,000230,000 marine species described, nearly 200,000 belongs to Animalia [START_REF] Appeltans | The magnitude of global marine species diversity[END_REF]. Figure 2 above depicts the number of estimated marine animal species as a component of marine biodiversity. It shows the relative contribution of dierent animal phyla and their relative known or unknown species diversity as computed from Appeltans et al. 2014. Thus, it may not come as a surprise to consider the fact that such a diversity is a ready storehouse of a wide variety of biological resources (products) that could hold immense medicinal or economic value. In fact, marine ecology provides ecosystem services that have economic, biological and non-economic value. These include Fish and sh products, mollusks, seaweeds and algae (sources of phycocolloids 4 ) which are among the primary sources of ecosystem services. Phycocolloids are polysaccharides derived from seaweeds (Gonçalves et al. 2014) that have applications in food, cosmetic and chemical industries. Therefore, conservation of biodiversity [START_REF] Hourigan | Conserving ocean biodiversity: Trends and challenges[END_REF]) should be a priority to enable smooth and sustainable ow of ecosystem services, without overexploiting the nature. This requires periodic asessment of marine biodiversity and the impact of human activities on marine ecosystems. The United Nation's The Millennium Ecosystem Assessment (MA) and the Ministry of Environment (MoE 2011) of Japan periodically analyzes the anthropogenic impact on the structure of marine and terrestrial ecosystems.

4. Phycocolloids include algin, alginates, carrageenan etc. These are gelatinous chemicals produced by seaweeds that are used in food processing, as emulsier, stabilizer and toppings, and in the production of dairy products. Other phycocolloids like for example, ulvan, chitosan, fucoidan, laminarin have many applications. See for instance, Marine Biology : Castro & Huber, chapter 6: Multicellular Primary Producers: Seaweeds and Plants, 7th edition pg. 109. It is necessary to conserve and protect marine biodiversity as much as it is essential to preserve terrestrial and estuarine ecosystems. Identifying the factors that aect marine population dynamics, habitation composition, and stress eect (or constraints) is crucial for eective conservation of marine species. Another important aspect is to monitor how species respond to constraints posed in dierent types of aquatic ecosystems, for instance, freshwater, estuarine and marine ecosystems. Marine ecosystem is the home to many animals and mammals which also include molluscs, and the largest biomass of green plants occurs in the oceans which happen to be the microscopic algae (phytoplankton 5 ). However, marine ecology is threatened by elevated sea surface temperature (SST) which could adversely aect coral survival. Survival-extinction dynamics and its outcome depends on the adaptive capacity of a species under stress, as well as on various other factors. Since several exotic and charismatic marine species are threatened with extinction, dierent conservation focus is required in marine ecosystems [START_REF] Mcclenachan | Extinction risk and bottlenecks in the conservation of charismatic marine species[END_REF]) that include oceans and sea. Table 1 shows several marine species that are categorized according to their current status of abundance or scarcity. Table 1 also depicts three extinct species that have been enumerated alongside those that are critically endangered or threatened. The reader is referred to the IUCN Red List 6 for more detailed information on threatened species.

Critically endangered Threatened

Extinct

Methods

Analysis of Ecosystem Threats

Threats facing aquatic ecosystems and marine habitats are manifold (vide section 1). Owing to extensive human activities, marine biodiversity is at crisis. Illegal overshing has resulted in a decline in marine animal population. Degradation of marine habitat has resulted in obliteration of coral reefs. This has been compounded by climate warming which has furthermore compromised marine habitats. This section undertakes a systematic analysis of threats to marine biodiversity and the ecosystem. We attempt to model such threats using a simple mathematical representation to study and analyze the current threats to marine ecosystems and their eects on marine life and habitats.

5. Phytoplanktons are microscopic oating plants which are mostly algae. See Encyclopedia of Marine sciences; (Eds.) pg. 201. 6. IUCN Red List website: http://www.iucnredlist.org

The Model

We construct a simple model to examine and illustrate the importance of habitat for species survival. For our model, we dene the variables as follows: = extinction risk; # 1 = species diversity, x = threats, = habitat, = population, and = error term. Habitats are characterized by distribution, range, depth, physiography, zone (i.e., benthic), salinity, etc. The system of equation representing the model can be represented as;

= # 1 (1 ¡ ) x + (1)
By solving equation ( 1), we derive,

+ (1 ¡ ) x # 1 (2)
By plotting equation ( 2) below, we derive a graphical relationship between species survival and habitat threshold: The area under curve (AUC) in Fig. 4 depicts the optimum habitat threshold for a given species. Habitats determine species diversity. Survival of a species is dependent on habitat quality which is more specic for habitat specialists like for example, corals. Equation (2) denes a scenario of max(optimum) level of population of dierent species that could explain the diversity of a given habitat. It also explains that the population density is highly dependent on the habitat quality which strictly aects species diversity of a particular region. There is a certain limit (or extent) to which a reduction in population is sustainable and which do not lead to extinction. A drastic reduction in habitat, however, negatively impacts population density and no other factor could compensate for habitat destruction. So, preservation of habitat and restoration of a fragmented habitat should be the overriding precedence to avoid species loss through extinction. This explains why conservation of habitats are so essential. And nally, to asess whether if a species is threatened or not, there is need for collection of better data on population size and evaluation of the habitats.

Results and Discussion

Diverse threats threaten marine biodiversity. It is essential to understand the nature and dynamics of such threats to marine biodiversity and undertake proactive measures in order to manage such threats. Goals should be directed towards conservation of marine biodiversity (Jone, Srinivasan and Almany 2007) by undertaking activities that are benecial to marine habitats. The analytic model depicted above reinforces the importance of habitat conservation and management of threats to marine biodiversity. Although the model is simply linear, and employs very few variables, it nevertheless, provides a straightforward analysis of the importance of habitat protection. It also reinforces the importance of collective measures to prevent degradation of marine habitats that are home to 97% of global biodiversity. Ecosystem degradation is a gradual ongoing process; various factors contribute to degradation of marine habitats. So, eective monitoring is required to identify and protect keystone species. Given that the marine ecology provides ecosystem services, overexploitation of marine resources is an important issue to consider. In fact, overshing has been implicated as the principle cause of decline of many species of sh, shellsh (Pauly et al. 2002[START_REF] Rosenberg | Managing to the margins: the overexploitation of sheries[END_REF], marine mammals, The coral ecology for example, is a source of rich and precious ecosystem services. However, corals are so sensitive to changes in temperature and salinity, that a rise in SST [START_REF] Burke | Pilot Analysis of Global Ecosystems: Coastal Ecosystems, World Resources Institute[END_REF] of less than a few degrees centrigrade (>1.1 C) leads to coral bleaching and ultimately causes death of corals 7 . Unusual rise in ocean water temperature (OWT) due to marked El Niño phenomenon has alarmed scientists who have observed intense coral bleaching event that is likely to extend well beyond 2017 (Ezzat and Courtial 2016). Rise in OWT or SST has been implicated to climate change which has threatened the future of the world's coral reefs (Guldberg 1999, Guldberg and[START_REF] Hoegh-Guldberg | Coral Bleaching Hits Great Barrier Reef As Global Temperatures Soar[END_REF]. In a nutshell, almost all and every ecosystem are under threat; ie., equatorial, temperate, marine, estuarine, riverine, and freshwater ecosystems are reeling under stress from a multitude of anthropogenic activities. The coral bleaching event is not restricted to a single region; it spreads over three major ocean basins (Pacic, Atlantic and Indian). In fact, widespread assault on marine and terrestrial ecosystems have threatened global biodiversity. This paper attempts to analyze and understand the basis of threats to marine and aquatic ecosystems and reinforces the importance of habitat protection and conservation, as well as restoration of fragmented and threatened habitats. Habitats of marine ecosystems are under threat, as well as the ora and fauna that ourishes under such habitats.

Conclusion

Loss of habitat threatens habitat specialists with extinction risk. The stress on major ecosystems has resulted in erosion of biodiversity due to various anthropogenic activites. The drivers of marine biodiversity loss have been enumerated in this paper. This was followed by an analytic model of ecosystem risks that depicts a simple representation of threats to marine biodiversity which reinforces the importance of biodiversity conservation and ecosystem protection. This research also attempts to open up debates and discussions on the importance of conservation management, and further points to new directions on this frontier.

  : i. Climate change induced Global warming a) Physical changes: change in sea water surface temperature b) Chemical changes: change in salinity ii. Habitat destruction a) Commercial exploitation and dredging activities, b) Destruction of Coral habitats c) Habitat damage from bottom trawling iii. Problem of overshing and bycatch a) Illegal shing, bycatch problems b) Unregulated and overshing iv. Pollution from sewage disposals which include harmful chemicals a) Hazardous pollutants (includes toxic and radioactive chemicals etc.) b) Sewage litters (e.g., plastic, inorganic debris, cans etc.)

  provided by marine environments and coastal areas are varied in nature. These can be enumerated as (adapted from Burke 2001): i. Fish and sheries for food production ii. Maintenance of Biodiversity iii. Tourism and recreation iv. Marine transport and port facilities v. Shoreline stabilization vi. Water quality vii. Hydrologic functions

Figure 1 .

 1 Figure 1. Major threats to marine biodiversity

Figure 2 .

 2 Figure 2. Mechanism of Coral Bleaching

Figure 3 .

 3 Figure 3. Estimate of marine animal species (See Appeltans et al. 2014: How Many Species in the Ocean? )

  Figure 4.

Table 1 .

 1 Source: IUCN Red List

	Alabama cavesh	Rockhopper Penguins Steller's sea cow
	Yangtze sturgeon	Paddlesh	Short-barbeled barb
	Vaquita	Green sturgeon	Syr-Dar shovelnose sturgeon
	Extinct in the wild

National Oceanic and Atmospheric Administration (NOAA). See NOAA, 2015; NOAA Coral Reef Watch : 2015 Annual Summaries of Thermal Conditions Related to Coral Bleaching for NCRMP Jurisdictions.

Acknowledgments.

The author extends his thanks to the library sta of Dr. V.S. Krishna Central Library, Andhra University, for their kind assistance.

7. http://www.globalcoralbleaching.org


Florian Bouguet

Bertrand Cloez

Fluctuations of the Empirical Measure of Freezing Markov Chains

Keywords: results . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Markov chain, Long-time behavior, Piecewise-deterministic Markov process, Ornstein-Uhlenbeck process, Asymptotic pseudotrajectory, 60J25, 60F05

In this work, we consider a finite-state inhomogeneous-time Markov chain whose probabilities of transition from one state to another tend to decrease over time. This can be seen as a cooling of the dynamics of an underlying Markov chain. We are interested in the long time behavior of the empirical measure of this freezing Markov chain. Some recent papers provide almost sure convergence and convergence in distribution in the case of the freezing speed 1/n θ , with different limits depending on θ < 1, θ = 1 or θ > 1. Using stochastic approximation techniques, we generalize these results for any freezing speed, and we obtain a better characterization of the limit distribution as well as rates of convergence as well as functional convergence. Contents 1 Introduction 2 2 Freezing Markov chains 4 2.1 Notation . . . . . . . . . . . . . .

Introduction

Let (i n ) n≥1 be an inhomogeneous-time Markov chain with state space {1, . . . , D} with the following transitions when i = j: P (i n+1 = j|i n = i) = q n (i, j), q n (i, j) = p n (q(i, j) + r n (i, j)), where (p n ) n≥1 is a decreasing sequence converging toward some p ∈ [0, 1], the remainders r n (i, j) tend to 0 (fast enough) and q is the discrete generator of some {1, . . . , D}-valued ergodic Markov chain. This model is related to the simulated annealing algorithm, and the sequence (p n ) n≥1 can be interpreted as the cooling scheme of an underlying Markov chain generated by q. If p < 1, since lim n→+∞ q n (i, j) = pq(i, j), the probability of (i n ) n≥1 to move decreases over time, from which the appellation freezing Markov chain.

The behavior of (i n ) n≥1 is simple enough to understand, and depends on the summability of the sequence (p n ) n≥1 . The chain (i n ) n≥1 shall converge in distribution to the unique invariant probability ν associated to q if ∞ n=1 p n = +∞ (see Theorem 2.4 below). On the other hand, if ∞ n=1 p n < +∞, the Markov chain shall freeze along the way, as a consequence of the Borel-Cantelli Lemma. Then, we shall assume that ∞ n=1 p n = +∞, so that we can investigate the convergence of the empirical distribution

x n = 1 n n k=1 δ i k .
The problem of the convergence of this empirical measure can be traced back to the thesis of Dobrušin [Dob53], and several questions are still open, as pointed out in the recent article [START_REF] Engländer | Turning a coin over instead of tossing it[END_REF]. Some results can be obtained from the general theory developed in [START_REF] Sethuraman | A martingale proof of Dobrushin's theorem for non-homogeneous Markov chains[END_REF]Pel12], and [DS07, EV16] study the present model. In particular, convergence results are only obtained for a freezing rate of the form p n = a/n θ (and r n (i, j) = 0). More precisely,

• if θ < 1 then (x n ) n≥1 converges to ν in probability; see [DS07, Theorem 1.2].

• if θ < 1/2, then (x n ) n≥1 converges to ν a.s. This can be extended to 1/2 ≤ θ < 1 when the state space contains only two points; see [DS07, Theorem 1.2] and [EV16, Corollary 2].

• if θ < 1 and D = 2, then, up to an appropriate scaling, the empirical measure (x n ) n≥1 converges in distribution to a Gaussian distribution; see [EV16, Theorem 2].

• if θ = 1 then (x n ) n≥1 converges in distribution, and the moments of the limit probability are explicit. If q corresponds to the complete graph (see Section 4) then this limit probability is the Dirichlet distribution. When D = 2, this covers classical distribution such as Beta, uniform, Arcsine or Wigner distributions; see [DS07, Theorems 1.3 and 1.4] and [EV16, Theorem 1].

• when D = 2, some convergence results are established for (x n ) n≥1 for general sequences (p n ) n≥1 , under technical conditions that we find hard to check in practice; see [START_REF] Engländer | Turning a coin over instead of tossing it[END_REF]Theorem 3].

We shall refer to the case θ < 1 as standard, since it is related to classic laws of large numbers and central limit theorems. This case was called subcritical in [START_REF] Engländer | Turning a coin over instead of tossing it[END_REF], in comparison with the critical case θ = 1. Since we can slightly generalize this critical case here, the term non-standard will be preferred from now on. In the present article, we generalize the aforementioned results by proving that, in the standard case, if ∞ n=1 (p n n 2 ) -1 < +∞ then (x n ) n≥1 converges to ν a.s., and we also give weaker conditions for convergence in probability; this is the purpose of Theorem 2.11. Under slightly stronger assumptions and up to a rescaling, we obtain convergence of (x n ) n≥1 to a Gaussian distribution with explicit variance in Theorem 2.12. Finally, if p n ∼ a/n, then (x n ) n≥1 converges in distribution exponentially fast to a limit probability (see Theorem 2.9). This distribution is characterized as the stationary measure of a piecewisedeterministic Markov process (PDMP), possesses a density with respect to the Lebesgue measure and satisfies a system of transport equations; see Propositions 3.1 and 3.4. Furthermore, Corollary 3.9 links the standard and non-standard setting by providing a convergence of the rescaled stationary measure of the PDMP to a Gaussian distribution as the switching accelerates. We also investigate the complete graph dynamics in Section 4 and are able to derive explicit results in Propositions 4.1 and 4.2. Most of our convergence results are also provided with quantitative speeds and functional convergences.

In contrast with the Pólya Urns model (see for instance [START_REF] Gouet | Strong convergence of proportions in a multicolor Pólya urn[END_REF]), all these results of convergences in distribution are not almost sure. However, note that, by letting p n = 1 for all n ≥ 1, we can recover classical limit theorems for homogeneous-time Markov chains (see [Jon04]). Furthermore, the remainder term r n (i, j) enables us to deal with different freezing schemes (see Remark 2.3). In particular, the proofs in [DS07] and [START_REF] Engländer | Turning a coin over instead of tossing it[END_REF] are mainly based on the method of moments, which is why more stringent assumptions are considered there. Our approach is completely different, and is based on the theory of asymptotic pseudotrajectories detailed in [Ben99] and revisited in [START_REF] Benaïm | Ergodicity of inhomogeneous Markov chains through asymptotic pseudotrajectories[END_REF].

Briefly, a sequence is an asymptotic pseudotrajectory of a flow if, for any given time window, the sequence and the flow starting from the same point evolve close to each other (see for instance [START_REF] Benaïm | Asymptotic pseudotrajectories and chain recurrent flows, with applications[END_REF]Ben99]). This definition can be formalized for dynamical systems and be extended to discrete sequences of probabilities and continuous Markov semi-groups. This theory allows us to derive the behavior of the sequence of empirical measures (x n ) n≥1 from the one of auxiliary continuous-time Markov processes. The interested reader may find illustrations of this phenomenon in [BBC16, Figures 3.1, 3.2 and 3.3], see also Figure 5.1. In the present paper, depending on whether we work in a standard or non-standard setting, these processes are either a diffusive process or a switching PDMP. The careful study of these limit processes is of interest per se, and is done in Section 3. More precisely, Gaussian distributions appear naturally since we deal with an Ornstein-Uhlenbeck process generated by

L O f (y) = -y • ∇f (y) + ∇f (y) Σ (p,Υ) ∇f (y), (1.1)
where

Σ (p,Υ) is a D × D real-valued matrix such that Σ (p,Υ) k,l = 1 1 + Υ D i=1 ν i   D j=1 q(i, j)(h l,j -h l,i )(h k,j -h k,i) ) -p (ν k -1 i=k ) (ν l -1 i=l )   , (1.2)
with p and h respectively defined in Assumption 2.1, and in (2.6). On the contrary, we shall see that, in a non-standard framework, the empirical measure is linked to a PDMP, called exponential zig-zag process, generated by

L Z f (x, i) = (e i -x) • ∇ x f (x, i) + j =i aq(i, j)[f (x, j) -f (x, i)].
(1.3)

These Markov processes shall be defined and studied more rigorously in Section 3. In particular, besides some classic long-time properties (regularity, invariant measure, rate of convergence. . . ), we prove in Theorem 3.3 the convergence of the exponential zig-zag process to the Ornstein-Uhlenbeck process when the frequency of jumps accelerates, i.e. when a → +∞.

The rest of this paper is organized as follows. In Section 2, we specifiy the notation and assumptions mentioned earlier, that will be used in the whole paper. We also state convergence results for (x n ) n≥1 , which are Theorems 2.9, 2.11 and 2.12. We study the long-time behavior of the two auxiliary Markov processes in Section 3 and investigate the case of the complete graph in Section 4, for which it is possible to get explicit formulas. The paper is then concluded with the proofs of the main theorems in Section 5.

Notation

We shall use the following notation throughout the paper:

• If d is a positive integer, a multi-index is a d-tuple N = (N 1 , . . . , N d ) ∈ ({0, 1, . . . } ∪ {+∞}) d ; the set of multi-indices is endowed with the order N ≤ N if, for all 1 ≤ i ≤ d, N i ≤ N i . We define |N | = d i=1
N i and and we identify an integer N with the multi-index (N, . . . , N ). Likewise, for any

x ∈ R d , let |x| = d i=1 |x i |. • For some multi-index N and an open set U ⊆ R d , C N (U ) is the set of functions f : U → R which
are N i times continuously differentiable in the direction i. For any f ∈ C N (U ), we define

f (N ) = ∂ N1 1 . . . ∂ N d d f, f (N ) ∞ = sup x∈R d |f (N ) (x)|.
When there is no ambiguity, we write C N instead of C N (U ), and denote by C N b and C N c the respective sets of bounded C N functions and of compactly supported C N functions.

• Let be the simplex of R D defined by

= (x 1 , . . . , x D ) ∈ R D : x i ∈ [0, 1], D i=1 x i = 1 ,
and E = × {1, . . . , D}.

• We denote by L (X) the probability distribution of a random vector X, and we identify the measures over {1, . . . , D} with the 1 × D real-valued matrices. Let L be the Lebesgue measure over R D .

• If µ, ν are probability measures and f is a function, we write µ(f ) = f (x)µ(dx). For a class of functions F , we define

d F (µ, ν) = sup f ∈F |µ(f ) -ν(f )|.
Note that, for every class of functions F considered in this paper, convergence in d F implies (and is often equivalent to) convergence in distribution (see [START_REF] Benaïm | Ergodicity of inhomogeneous Markov chains through asymptotic pseudotrajectories[END_REF]Lemma 5.1]). In particular, let

W (µ, ν) = sup |f (x)-f (y)|≤|x-y| |µ(f ) -ν(f )|, d TV (µ, ν) = sup f ∞≤1 |µ(f ) -ν(f )|
be respectively the Wasserstein distance and the total variation distance.

• For θ ∈ (0, +∞) D , let D(θ) be Dirichlet distribution over R D , i.e. the probability distribution with probability density function

x → Γ D k=1 θ k D k=1 Γ(θ k ) D k=1 x θ k -1 k 1 {x∈ } .
For θ 1 , θ 2 > 0, let β(θ 1 , θ 2 ) be the Beta distribution over R, i.e. the probability distribution with probability density function

x → Γ(θ 1 + θ 2 ) Γ(θ 1 )Γ(θ 2 ) x θ1 (1 -x) θ2 1 0<x<1 .
• Let x ∧ y := min(x, y) and x ∨ y := max(x, y) for any x, y ∈ R.

• We write, for n ≥ 1,

u n = O(v n ) if there exists some bounded sequence (h n ) n≥1 such that u n = h n v n . Moreover, if lim n→+∞ h n = 0, then we write u n = o(v n ).

Assumptions and main results

Let D be a positive integer and (i n ) n≥1 be a {1, . . . , D}-valued inhomogeneous-time Markov chain such that, ∀i = j, P (i n+1 = j|i n = i) = q n (i, j). The following assumption, which will be in force in the rest of the paper, describes the behavior of the transitions q n as time goes by.

Assumption 2.1 (Freezing speed). Assume that that the matrix Id +q is irreducible and, for n ≥ 1 and i = j, q n (i, j) = p n (q(i, j) + r n (i, j)) , (2.1)

where (p n ) is a sequence decreasing to p ∈ [0, 1] such that ∞ n=1 p n = +∞, and lim n→+∞ r n (i, j) = 0. For i = j, assume q(i, j) ≥ 0, q n (i, j) ≥ 0 and

q(i, i) = - j =i q(i, j), q n (i, i) = - j =i q n (i, j).
Note that we do not require (p n ) n≥1 to converge to 0. Of course, if p > 0, then the series n p n trivially diverges; as pointed out in the introduction, if this series converge then the problem is trivial. In fact, if p n = 1 and r n (i, j) = 0 for any integers i, j, n, then the freezing Markov chain (i n ) n≥1 is a classic Markov chain. When p = 0, the dynamics of Assumption 2.1 corresponds to the lazier and lazier random walk introduced in [BBC16].

Remark 2.2 (Irreducibility or indecomposability). The irreducibility of the transition matrix Id +q associated to q is a classic hypothesis when it comes to Markov chains, since otherwise we can split their state space into different recurrent classes. However, the result of the present article can be extended to indecomposable1 Markov chains, which is a weaker concept. For instance, the transition matrix

  0 1 0 1 0 0 1/3 1/3 1/3   is indecomposable but not irreducible.
Namely, Id +q is irreducible if it cannot be written as

P (Id +q)P = A 0 B A ,
where A, A are square matrices and P is a permutation matrix. We could allow such a decomposition, as long as B has a nonzero entry.

In any case, Id +q possesses a unique absorbing class of states on which it is irreducible. Using Perron-Frobenius Theorem (see [START_REF] Gantmacher | The Theory of Matrices[END_REF]Theorem 2p.53]), the matrix Id +q possesses a unique invariant measure ν , and the associated chain converges toward it under aperiodicity assumptions (see also Remark 3.3). Note that aperiodicity hypotheses are not relevant for the freezing Markov chain whenever p < 1, since the freezing scheme automatically provides aperiodicity to the Markov chain. ♦ Under Assumption 2.1, Id +q possesses a unique invariant distribution ν , which writes ν q = 0; let ν ∈ be its associated vector.

Remark 2.3 (Interpretation of the term r n (i, j)). The remainder r n (i, j) in (2.1) can either model small perturbations of the main freezing speed p n q(i, j), or a multiscale freezing scheme with p n being the slowest freezing speed. For instance, the case

q n = -n -θ n -θ n -(θ+ θ) -n -(θ+ θ) , θ, θ > 0 is covered by Assumption 2.1, with q = -1 1 0 0 , p n = n -θ .

♦

The following result characterizes the long-time behavior of the inhomogeneous Markov chain (i n ) n≥1 .

Theorem 2.4 (Convergence of the freezing Markov chain). Under Assumption 2.1, if either p < 1, or p = 1 and Id +q is aperiodic,

lim n→+∞ i n = ν in distribution.
Now, let us define (e 1 , . . . , e D ) the natural basis of R D and introduce two different scaling rates

γ n = 1 n , α n = p n γ n , (2.2)
and the associated rescaled vectors

x n = γ n n k=1 e i k , y n = α n (x n -ν). (2.3) It is clear that (2.3) writes x n+1 = γ n+1 γ n x n + γ n+1 e in+1 , (2.4)
that the vector x n belongs to the simplex and that (x n , i n ) ∈ E = × {1, . . . , D}. We highlight the fact that, in general, the sequence (x n ) n≥1 is not a Markov chain by itself, but (x n , i n ) n≥1 is.

Remark 2.5 (Interpretation of ). The transpose x → x is a natural bijection between and the set of probability measures over {1, . . . , D}. Then, the sequence (x n ) n≥1 can be viewed as the sequence of empirical measures of the Markov chain (i n ) n≥1 . From that viewpoint, we highlight the fact that the L 1 norm over can be interpreted (up to a multiplicative constant) as the total variation distance: indeed, for any x, x ∈ ,

|x -x| = 1 2 d TV x , x = 1 2 d TV D i=1 x i δ i , D i=1
x i δ i .

♦

Remark 2.6 (Weighted means). Note that one could consider weighted means of the form

x n = 1 n k=1 ω k n k=1 ω k e i k ,
|x n -ν| ≤ C exp -v n k=1 γ k .
♦ Following [Ben99, BBC16], and given sequences (γ n ) n≥1 , ( n ) n≥1 , we define the following parameter which rules the speed of convergence in the context of standard fluctuations:

λ(γ, ) = -lim sup n→+∞ log(γ n ∨ n ) n k=1 γ k .
(2.5) Finally, we need to introduce a fundamental tool in the study of the standard fluctuations: the matrix h, which is solution of the multidimensional Poisson equation

j =i q(i, j)(h j -h i ) = ν -e i , or equivalently j =i q(i, j)(h k,j -h k,i ) = ν k -1 i=k (2.6)
for all 1 ≤ i, k ≤ D, where we denoted by h i the i-th column vector of the matrix h. This solution is classically defined by

h i = - +∞ 0
e i e t(Id +q) -ν dt.

With the help of Perron-Frobenius Theorem (see [START_REF] Gantmacher | The Theory of Matrices[END_REF]Theorem 2p.53]), it is easy to see that h is well-defined.

Throughout the paper, we shall treat two different cases, which entail different limit behaviors for the fluctuations of (x n ) n≥1 or (y n ) n≥1 . Each of these cases corresponds to one of the two following assumptions.

Assumption 2.7 (Non-standard behavior). Assume that

p n ∼ n→+∞ a n .
Note that, under Assumption 2.7, the sequences (γ n ) n≥1 and (p n ) n≥1 are equivalent up to a multiplicative constant and the scaling (α n ) n≥1 is trivial, hence we are not interested in the behavior of

(y n ) n≥1 . Assumption 2.8 (Standard behavior). i) Assume that lim sup n→+∞ γ n p n = 0. ii) Assume that p n+1 p n = 1 + Υ n + o 1 n , lim n→+∞ R n √ p n γ n = 0, with R n = sup i j =i |r n (i, j)|.
Now, we have all the tools needed to study the behavior of the empirical measure (x n ) n≥1 .

Theorem 2.9 (Non-standard fluctuations). Under Assumptions 2.1 and 2.7,

lim n→+∞ (x n , i n ) = π in distribution,
where π is characterized in Propositions 3.1 and 3.4.

Moreover, if there exist positive constants

A ≥ 1, θ ≤ 1 such that max j =i (|r n (i, j)|) ≤ A n θ ,
then, denoting by ρ the spectral gap of Id +q, for any

u < θ A + θ 1 + 1 aρ ,
there exist a class of functions F defined in (5.4) and a positive constant C such that

d F (L (x n , i n ), π) ≤ Cn -v .
It should be noted that our approach for the study of the long-time behavior of (x n , i n ) n≥0 also provides functional convergence for some interpolated process (X t , I t ) t≥0 defined in (5.3) (see Lemma 5.1, from which Theorem 2.9 is a straightforward consequence). Moreover, note that the speed of convergence provided by Theorem 2.9 writes, for any function f : × {1, . . . , D} → R, two times differentiable in the first variable, there exists a constant C f such that

|E[f (x n , i n )] -π(f )| ≤ C f n -v .
Remark 2.10 (Is it possible to generalize Assumption 2.7?). This remarks leans heavily on the proof of Theorem 2.9 and may be omitted at first reading. It is interesting to wonder whether it is possible to obtain non-standard fluctuations for a more general freezing speed (p n ) n≥1 . To that end, let us try to mimic the computations of the proof of Lemma 5.1 with ( x n , i n ) n≥1 with

x n = γ n n k=1 e i k ,
for any vanishing sequences (γ n ) n≥1 and ( γ n ) n≥1 . Our method being based on asymptotic pseudotrajectories, the limit of the rescaled process of (x n , i n ) n≥1 belongs to a certain class of PDMPs which can be attained if, and only if,

lim n→+∞ p n γ n = C 1 , lim n→+∞ γ n γ n = C 2 , lim n→+∞ 1 γ n γ n+1 γ n -1 = -C 3 , (2.7) with C 1 , C 2 , C 3 > 0. Without loss of generality, one can choose γ n = γ n and C 2 = C 3 = 1. Then, the third term of (2.7) entails γ n = (n + o(1)) -1 as n → +∞, which in turn implies p n = C 1 n -1 + o(n -1 )
when injected in the first term of (2.7).

Also, note that assuming A < 1 or θ > 1 in Theorem 2.9 would not provide better speeds of convergence, since one would obtain a speed of the form

u < θ ∧ 1 A ∨ 1 + θ ∧ 1 1 + 1 aρ .

♦

Theorem 2.11 (Standard convergence of the empirical measure). Under Assumptions 2.1 and 2.8.i),

lim n→+∞ x n = ν in probability, or equivalently in L 1 . Moreover, if ∞ n=1 γ 2 n p -1 n < +∞, then lim n→+∞ x n = ν a.s. Moreover, if = λ(γ, γ/p) ∧ λ(γ, R) > 0, then, for any v < there exists a (random) constant C > 0 such that |x n -ν| ≤ C n v a.s.
Theorem 2.12 (Standard fluctuations). Under Assumptions 2.1 and 2.8, (y n ) n≥1 converges in distribution to the Gaussian distribution N 0, Σ (p,Υ)

The precise proofs of the main results are deferred to Section 5. As pointed out in the introduction, our proofs of Theorems 2.9 and 2.12 rely on comparing (x n ) n≥1 and (y n ) n≥1 with auxiliary continuoustime Markov processes, using the theory of asymptotic pseudotrajectories and the SDE method. Then, these discrete Markov chains will inherit some properties of the Markov processes that we shall prove in Section 3. In particular, the results we use provide functional convergence of the rescaled interpolating processes to the auxiliary Markov processes (see [BBC16, Theorem 2.12] and [Duf96, Théorème 4.II.4]).

Remark 2.13 (Examples of freezing rates). For the sake of simplicity, consider r n (i, j) = 0 for all i, j, n. Assumption 2.8 covers sequences (p n ) n≥1 of the form p n = n -θ for any 0

< θ < 1, since γ 2 n p -1 n = n θ-2 . In this case, = λ(n -1 , n θ-1 ) = 1 -θ > 0.
But we can also consider more exotic freezing rates, for instance

p n = log(n) ζ n -1 , for some ζ ≥ 1. Then, γ 2 n p -1 n = n -1 log(n) -ζ . If ζ > 1,
then the series converges and = 1. Our results do not provide almost sure convergence in the case ζ = 1, however, but only convergence in probability.

It should be noted that assuming that (p n ) n≥1 is decreasing, lim n→+∞ p n = 0 and ∞ n=1 p n = +∞ do not imply in general that p n+1 ∼ p n . A slight modification of the proof shows that, if p n+1 is not equivalent to p n , we have to assume the existence of a sequence (β n ) n≥1 such that

lim n→+∞ p n γ 2 n β 2 n β n β n-1 (1 -γ n ) -1 = -1, ∞ n=1 γ 2 n β 2 n p n = +∞, lim n→+∞ β n γ n = -1
and such that the sequences (γ 2 n β 2 n p -1 n ) n≥1 and (β n γ n ) n≥1 are decreasing; then the conclusion of Theorem 2.12 holds. ♦

The auxiliary Markov processes

In this section, we study the ergodicity of the processes arising as limits of the freezing Markov from Section 2. We also study their invariant measure, and provide explicit formulas when it is possible.

The exponential zig-zag process

In this section, we investigate the asymptotic properties of the exponential zig-zag process, which arise from the non-standard scaling of the Markov chain (i n ) n≥1 . To this end, let (X t , I t ) t≥0 be the strong solution of the following SDE (see [IW89]), with values in E:

(X t , I t ) = (X 0 , I 0 ) + t 0 A(X s -, I s -) + e I s -ds + D j=1 t 0 B I s -,j (X s -, I s -)N I s -,j (ds), (3.1)
where the N i,j are independent Poisson processes of intensity aq(i, j)1 {i =j} and

A =       -1 0 • • • 0 0 . . . . . . . . . . . . . . . -1 0 0 • • • 0 0       , B i,j      0 (0) . . . 0 0 • • • 0 i -j      . (3.2)
Thus, the infinitesimal generator of this process is

L Z defined in (1.3) (see e.g. [EK86, Dav93, Kol11]).
Actually, the exponential zig-zag process is a PDMP; the interested reader can consult [START_REF] Davis | Markov Models & Optimization[END_REF][START_REF] Benaïm | Qualitative properties of certain piecewise deterministic Markov processes[END_REF] for a detailed construction of the process (X, I). Let us describe briefly its dynamics: setting I 0 = i, the process possesses a continuous component X which is exponentially attracted to the vector e i . The discrete component I t is piecewise-constant, and jumps from i to j following the epochs of the processes N i,j , which in turn leads the continuous component to be attracted to e j (see Figure 3.1 for sample paths of the exponential zig-zag process, and Figure 4.2 for a typical path in the framework of Section 4.2).

The following result might be seen as a direct consequence of [BLBMZ12, Theorem 1.10] or [CH15, Theorem 1.4], although these articles do not provide explicit rates of convergence, which are useful for instance in the proof of Corollary 3.9. Figure 3.1: Sample paths on [0, 5] of the exponential zig-zag process for X 0 = (1/3, 1/3, 1/3), q(i, j) = 1 and a = 0.5, a = 2, a = 20 (from left to right).

Proposition 3.1 (Ergodicity). The exponential zig-zag process (X t , I t ) t≥0 admits a unique stationary distribution π. If ρ is the spectral gap of q, then for any for any v < aρ(1 + aρ) -1 , there exists a constant

C > 0 such that ∀t ≥ 0, W ((X t , I t ), π) ≤ C e -vt . Moreover, if L (I 0 ) = ν , then ∀t ≥ 0, W ((X t , I t ), π) ≤ W ((X 0 , I 0 ), π) e -t .
Note that the speed of convergence provided in Proposition 3.1 can be improved when D = 2, since we are able to use more refined couplings (see Proposition 4.5).

Proof of Proposition 3.1: The pattern of this proof follows [START_REF] Benaïm | Quantitative ergodicity for some switched dynamical systems[END_REF]. Let (X t , I t , X t , I t ) t≥0 be the coupling for which the discrete components I and I are equal forever once they are equal once. Let t > 0 and α ∈ (0, 1). Firstly, note that, if I αt = I αt , then the processes always have common jumps and

|X t -X t | = |X αt -X αt | e -t ≤ 2 e -t .
(3.3)

From the Perron-Frobenius theorem (see [START_REF] Gantmacher | The Theory of Matrices[END_REF]SC97]), for any ε > 0, there exists C > 0 such that

d TV (I t , I t ) ≤ C e -(aρ-ε)t .
Then there exists a coupling of the random variables I αt and I αt such that

P(I αt = I αt ) ≤ C e -(aρ-ε)αt . (3.4) Now, combining (3.3) and (3.4), E |(X t , I t ) -( X t , I t )| ≤ E |(X t , I t ) -( X t , I t )| I αt = I αt P(I αt = I αt ) + E |(X t , I t ) -( X t , I t )| I αt = I αt P(I αt = I αt ) ≤ 2P(I αt = I αt ) + 2 e -(1-α)t ≤ 2 C e -(aρ-ε)αt +2 e -(1-α)t .
One can optimize this speed of convergence by taking α = (1 + aρ -ε) -1 , and get

W (X t , I t ), ( X t , I t ) ≤ C e -vt (3.5) with C = 2 C + 2 and v = (aρ -ε)(1 + aρ -ε) -1 . Then, (L ((X t , I t )
) is a Cauchy sequence and converges to a (stationary) distribution π. Letting L ( X 0 , I 0 ) = π in (3.5), achieves the proof in the general case. Now, if L (I 0 ) = ν , then L (I 0 ) = L ( I 0 ); we can let I 0 = I 0 , and then it suffices to use (3.3) with α = 0.

If Assumption 2.1 is in force, there exists a unique invariant measure π, which satisfies

E L Z f (x, i)π(dx, di) = 0,
for any function f smooth enough. Now, let us establish the absolute continuity of this invariant distribution with respect to the Lebesgue measure L.

Lemma 3.2 (Absolute continuity of the exponential zig-zag process). Let K ⊂ ˚ be a compact set. There exist constants t 0 , c 0 > 0 and a neighborhood V of K such that, for any (x, i) ∈ E and for all t ≥ t 0 , P(X t ∈ •,

I t = j|X 0 = x, I 0 = i) ≥ c 0 L(• ∩ V (y)). (3.6)
Remark 3.3 (When Id +q is only indecomposable). This remark echoes Remark 2.2 and describes the behavior of the Markov chain (x n , i n ) n≥1 when Id +q is reducible but indecomposable. In that case, Proposition 3.1 holds as well. However, Id +q possesses a unique recurrent class which is strictly contained in {1, . . . , D}, the vector ν possesses at least one zero and belongs to the frontier of the simplex , and π( ˚ ) = 0. It is then impossible to obtain an equivalent to Proposition 3.1 with a convergence in total variation; when Id +q is irreducible, this is possible using techniques inspired from [BMP + 15, Proposition 2.5].

If Id +q is indecomposable, one can obtain equivalents of Lemma 3.

F i : x → e i -x and then, if D ≥ 3, ∀x ∈ , G 0 (x) = Vect{F i (x) -F j (x) : i = j} = Vect{e i -e j : i = j} = R D ,
where Vect A denotes the vector space spanned by

A ⊆ R D . If D = 2, then G 1 (x) = R 2 .
As a consequence, the strong bracket condition of [BLBMZ15, Definition 4.3] is satisfied. In particular, using [BLBMZ15, Theorems 4.2 and 4.4], we have that, for every x ∈ , y ∈ ˚ , there exist t 0 (x), c 0 (x) > 0 and open sets U 0 (x), V (x, y), such that for all x 0 ∈ U 0 (x), i, j ∈ {1, . . . , D}, A ⊆ and t > t 0 (x),

P(X t ∈ A, I t = j|X 0 = x 0 , I 0 = i) ≥ c 0 (x)L(A ∩ V (x, y)). Now, = ∪ x∈ U 0 (x)
and is compact, so there exist x 1 , ..., x n such that = ∪ n k=1 U 0 (x k ) . In particular, setting V (y) = ∪ n k=1 V (x k , y), c 0 = min 1≤k≤n c 0 (x k ), t 0 = max 1≤k≤n t 0 (x k ), we have, for all x 0 ∈ , i, j ∈ {1, . . . , D}, A ⊆ and t > t 0 ,

P(X t ∈ A, I t = j|X 0 = x 0 , I 0 = i) ≥ c 0 L(A ∩ V (y)),
Once again, K is compact so we can extract a finite family from the open sets (V (y)) y∈K . Using the Markov property, this holds for every t ≥ t 0 , which entails (3.6).

Proposition 3.4 (System of transport equations for π). The distribution π introduced in Proposition 3.1 admits the following decomposition:

π = D i=1 ν i π i ⊗ δ i , π i (dx) = ϕ(x, i)dx.
(3.7)

where the function ϕ satisfies, for any (x, i) ∈ E,

(D -1)ϕ(x, i) + D k=1 x k ∂ k ϕ(x, i) -∂ i ϕ(x, i) + D j=1 ν j ν i aq(j, i)ϕ(x, j) = 0. (3.8)
Once we will have proved that π admits the decomposition (3.7), the next step is the characterization of ϕ. Indeed, since it satisfies

D i=1 ν i L Z f (x, i)ϕ(x, i)dx = 0, (3.9)
for every smooth enough function f , all we have to do is compute the adjoint operator of L Z . For general switching model, it would not possible to characterize ϕ as a solution of a simple system of PDEs like (3.8). However, the present form of the flow enables us to derive a simple expression for the adjoint operator of L Z . Before turning to the proof of Proposition 3.4, let us present the following formula of integration by parts over the simplex .

Lemma 3.5 (Integration by parts over ). For all f, g ∈ C 1 c ˚ , and k, l ∈ {1, . . . , D}, we have

g(x)(∂ k -∂ l )f (x)dx = -(∂ k -∂ l )g(x)f (x)dx.
Proof of Lemma 3.5: Fix l = 1 and let 1 = x 2 , . . . , x D ∈ [0, 1] :

D i=2 x i ≤ 1 . Then, g(x)∂ k f (x)dx 1 . . . dx D = 1 g 1 - D i=2 x i , x 2 , ... ∂ k f 1 - D i=2 x i , x 2 , ... dx = 1 ∂ k g 1 - D i=2 x i , x 2 , ... f 1 - D i=2
x i , x 2 , ...

+ ∂ 1 g 1 - D i=2 x i , x 2 , ... f 1 - D i=2 x i , x 2 , ... + g 1 - D i=2 x i , x 2 , ... ∂ 1 f 1 - D i=2
x i , x 2 , ...

-∂ k g 1 - D i=2 x i , x 2 , ... f 1 - D i=2
x i , x 2 , ... dx 1 . . . dx D .

Now, as g(0, x 2 , . . . ) = f (0, x 2 , . . . ) = 0 and ∂ k 1 = 0, use a (classic) multidimensional integration by parts to establish that

1 ∂ k g 1 - D i=2 x i , x 2 , ... f 1 - D i=2
x i , x 2 , ... dx 1 . . . dx D = 0, which entails Lemma 3.5.

Proof of Proposition 3.4: Integrating (3.6) with respect to the unique invariant measure π, we obtain that π admits an absolutely continuous part (note that uniqueness comes from Proposition 3.1). Since π cannot have an absolutely continuous part and a singular one (see [BH12, Theorem 6]), π admits a density with respect to the Lebesgue measure, which entails (3.7). Now, let us characterize the function ϕ. We have

D i,k=1 ν i (-x k + 1 i=k )ϕ(x, i)∂ k f (x, i)dx = D i=1 - D k=1 ν i x k ϕ(x, i)∂ k f (x, i)dx + ν i ϕ(x, i)∂ i f (x, i)dx
and, using Lemma 3.5, for any

1 ≤ i ≤ D, - D k=1 x k ϕ(x, i)∂ k f (x, i)dx + ϕ(x, i)∂ i f (x, i)dx = - k =i x k ϕ(x, i)∂ k f (x, i)dx - x i ϕ(x, i)∂ i f (x, i)dx + ϕ(x, i)∂ i f (x, i)dx = k =i ∂ k (x k ϕ(x, i)] f (x, i)dx - x k ϕ(x, i)∂ i f (x, i)dx - ∂ i (x k ϕ(x, i)) f (x, i)dx - x i ϕ(x, i)∂ i f (x, i)dx + ϕ(x, i)∂ i f (x, i)dx = k =i ∂ k (x k ϕ(x, i)) f (x, i)dx - ∂ i (x k ϕ(x, i)) f (x, i)dx = k =i ∂ k (x k ϕ(x, i)) f (x, i)dx -(1 -x i ) ∂ i ϕ(x, i)f (x, i)dx.
Hence, (3.9) writes

D i=1 ν i f (x, i)   k =i ∂ k (x k ϕ(x, i)) -(1 -x i )∂ i ϕ(x, i) + D j=1 ν j ν i q(j, i)ϕ(x, j) - D j=1 q(i, j)ϕ(x, i)   dx = 0.
It follows that ϕ is the solution of (3.8).

The Ornstein-Uhlenbeck process

In this short section, we recall a classic property of multidimensional Ornstein-Uhlenbeck processes, which is useful to characterize the behavior of (y n ) n≥1 in a standard setting. Thus, we define (Y t ) t≥0 as the strong solution of the following SDE, with values in R D :

Y t = Y 0 - t 0 Y s -ds + √ 2 t 0 (Σ (p,Υ) ) 1/2 dW s , (3.10)
where W is a standard D-dimensional Brownian motion and (Σ (p,Υ) ) 1/2 is the square root of the positivedefinite symmetric matrix Σ (p,Υ) , i.e. (Σ (p,Υ) ) 1/2 ((Σ (p,Υ) ) 1/2 ) = Σ (p,Υ) . The process Y is a classic Ornstein-Uhlenbeck process with infinitesimal generator L O defined in (1.1). Such processes have already been thoroughly studied, so we present only the following proposition, which quantifies the speed of convergence of Y to its equilibrium. Moreover,

W Y t , N 0, Σ (p,Υ) = W (Y 0 , π) e -t .
Proof of Proposition 3.6: First, since

N 0, Σ (p,Υ) (dx) = C exp x (Σ (p,Υ) ) -1 x 2 dx,
a straightforward integration by parts shows that, for any f ∈ C 2 c , N 0, Σ (p,Υ) (L O f ) = 0 so that N 0, Σ (p,Υ) is an invariant measure for the Ornstein-Uhlenbeck process Y.

It is well-known and easy to check that (Y t ) t≥0 writes

Y t = Y 0 e -t + √ 2(Σ (p,Υ) ) 1/2 t 0 e -(t-s) dW s ,
where W is a standard Brownian motion. Consequently, if we consider Y another Ornstein-Uhlenbeck process generated by L O and driven by the (same) Brownian motion W ,

E |Y t -Y t | = E |Y 0 -Y 0 | e -t .
(3.11)

Taking the infimum over all the couplings gives a contraction in Wasserstein distance. Now, if L ( Y 0 ) = N 0, Σ (p,Υ) and (Y 0 , Y 0 ) is the optimal coupling between L (Y 0 ) and N 0, Σ (p,Υ) with respect to W , then (3.11) writes

W Y t , N 0, Σ (p,Υ) = W Y 0 , N 0, Σ (p,Υ) e -t ,
which entails the uniqueness of the invariant probability distribution as well as the exponential ergodicity of the process.

Acceleration of the jumps

The current section links the Sections 3.1 and 3.2 in the following sense:

Markov chain

(i n ) n≥1
Exponential zig-zag process (X t , I t ) t≥0

Ornstein-Uhlenbeck process (Y t ) t≥0

Slow freezing

Fast freezing

Acceleration of the jumps Indeed, we prove in Theorem 3.7 the convergence of the (rescaled) exponential zig-zag process to a diffusive process as the jump rates go to infinity. Such results are fairly standard and are already known in the cases of (linear) zig-zag processes (see [START_REF] Fontbona | Quantitative estimates for the long-time behavior of an ergodic variant of the telegraph process[END_REF][START_REF] Bierkens | Limit theorems for the Zig-Zag process[END_REF]) or of particle transport processes (see [CK06]). Heuristically, since there are more frequent jumps, the process tends to concentrate around its mean ν, and the effect of the discrete component fades away. This phenomenon can be seen on Figure 3.1. We shall end this section with Corollary 3.9, which provides the convergence of the stationary distribution of the exponential zig-zag process toward a Gaussian distribution.

To this end, let (a n ) n≥1 be a sequence of positive numbers such that a n → +∞ as n → +∞ and, for any integer n, let (X

(n) t , I (n)
t ) t≥0 be a Markov process with values in E generated by

L (n) f (x, i) = (e i -x) • ∇ x f (x, i) + a n j =i q(i, j)[f (x, j) -f (x, i)]. We define Y (n) t = √ a n (X (n) -ν) and denote by Y (n) t (k) and X (n) t (k) the respective k th component of Y (n) t and X (n) t .
Theorem 3.7 (Convergence of the processes). If (Y (n) 0 ) n≥1 converges in distribution to a probability distribution µ, then the sequence of processes (Y (n) ) n≥1 converges in distribution to the diffusive Markov process generated by L O f (y) = -y • ∇f (y) + ∇f (y) Σ (0,1) ∇f (y)

with initial condition µ.

Proof of Theorem 3.7: We shall use a diffusion approximation and follow the proof of [FGM12, Proposition 1.1]. For now, we drop the superscript (n), and let, for any 1 ≤ k, l ≤ D,

ϕ k (x, i) = √ a n (x k -ν k ) + 1 √ a n h k,i , ψ k,l (x, i) = ϕ k (x, i)ϕ l (x, i).
Then,

Lϕ k (x, i) = √ a n (ν k -x k ), Lψ k,l (x, i) = √ a n ((1 i=k -x k )ϕ l (x, i) + (1 i=l -x l )ϕ k (x, i)) + a n ((x k -ν k )(ν l -1 i=l ) + (x l -ν l )(ν k -1 i=k )) + j =i q(i, j) (h k,j h l,j -h k,i h l,i ) .
Then, by Dynkin's formula, for fixed n, the processes (M t (k)) t≥0 and (N t (k, l)) t≥0 are local martingales with respect to the filtration generated by (X (n) , I (n) ), where

M t (k) = Y t (k) - √ a n t 0 (ν k -X s (k))ds + 1 √ a n h k,It , N t (k, l) = Y t (k)Y t (l) + Y t (k)h l,It + Y t (l)h k,It + 1 a n h k,It h l,It - t 0 -2Y s (l)Y s (k) + h k,Is (1 {Is=l} -X s (l)) + h l,Is (1 {Is=k} -X s (k)) + j =Is q(I s , j) (h k,j h l,j -h k,Is h l,Is ) ds.
Remark that, for any

1 ≤ i ≤ D, if σ k,l (i) = D j=1 q(i, j)(h l,j -h l,i )(h k,j -h k,i) ), D j=1 q(i, j) (h k,j h l,j -h k,i h l,i ) = D j=1 q(i, j) (h l,j -h l,i ) (h k,j -h k,i ) + h l,i (ν k -1 i=k ) + h k,i (ν l -1 i=l ) = σ k,l (i) + h l,i (ν k -1 i=k ) + h k,i (ν l -1 i=l ) .
Then, denoting by

Z s (k) = t 0 Y s (k)ds, N t (k, l) = Y t (k)Y t (l) + 2 t 0 Y s (k)Y s (l)ds - t 0 σ k,l (I s )ds + 1 a n h k,It h l,It + 1 √ a n h k,It (Y t (l) + Z t (l)) + 1 √ a n h l,It (Y t (k) + Z t (k)) , and
M t (k)M t (l) = N t (k, l) + Y t (k)Z t (l) + Y t (l)Z t (k) + Z t (k)Z t (l) -2 t 0 Y s (k)Y s (l)ds + t 0 σ k,l (I s )ds + 1 √ a n h k,It Z t (l) + h l,It Z t (k) - t 0 h k,Is Y s (l)ds - t 0 h l,Is Y s (k)ds .
By integration by parts,

Y t (k)Z t (l) = t 0 Z s (l)dM s (k) - t 0 Z s (l)Y s (k)ds + t 0 Y s (k)Y s (l)ds + 1 √ a n t 0 h k,Is Y s (l)ds -h k,It Z s (l) , hence M t (k)M t (l) = N t (k, l) + t 0 Z s (k)dM s (l) + t 0 Z s (l)dM s (k) + t 0 σ k,l (I s )ds.
Finally, for any 1 ≤ k, l ≤ D, the processes

M (n) (k) -B (n) (k) and M (n) (k)M (n) (l) -A (n) (k, l) are local martingales, with A (n) t (k, l) = t 0 σ k,l (I (n) s )ds, B (n) t (k) = - t 0 Y (n) s ds + 1 √ a n h k,I (n) t .
Note that I (n) is a Markov process on its own, generated by

L (n) I f (i) = a n j =i q(i, j)[f (j) -f (i)].
In other words, for any t > 0, we can write I

(n) t = I ant a.s., for some pure-jump Markov process (I t ) t≥0 generated by

L I f (i) = j =i q(i, j)[f (j) -f (i)].
Using the ergodicity of (I t ) t≥0 together with lim n→+∞ a n = +∞, we have

lim n→+∞ A (n) t (k, l) = lim n→+∞ t 0 σ k,l (I ans )ds = lim n→+∞ 1 a n ant 0 σ k,l (I u )du = t D i=1 ν i σ k,l (i) = tν(σ k,l ).
Thus, the processes

Y (n) (k), B (n) (k), A (n) (k, l
) satisfy the assumptions of [EK86, Chapter 7, Theorem 4.1], which entails Theorem 3.7.

Remark 3.8 (Heuristics for a direct Taylor expansion of the generator). As for many limit theorems for Markov processes, one would like to predict the convergence of the exponential zig-zag process to the Ornstein-Uhlenbeck diffusion from a Taylor expansion of the generator. Let us describe here a quick heuristic argument based on [CK06], which justifies the particular choice of functions ϕ k in the proof of Theorem 3.7. For the sake of simplicitylet us work in the setting of Section 4.2, that is the generator of (X t , I t ) t≥0 is of the form

L Z f (x, i) = g i (x)∂ x f (x, i) + aθ 3-i [f (x, 3 -i) -f (x, i)]
where

g i : x → (1 {i=1} -x). For some smooth function f : R D → R, we have L Z f (x, i) = g i (x) • ∇ x f (x)
which cannot be rescaled to converge to some diffusive operator. We need an approximation f a of f in a sense that lim a→+∞ f a = f and L Z f a has the form of a second order operator. Then, let

f a : (x, i) → f (x) + a -1 h(x, i) • ∇ x f (x)
where h(x, i) is the solution of the multidimensional Poisson equation associated to the transitions of the flows

j =i q(i, j)[h(x, j) -h(x, i)] = θ 3-i [h(x, 3 -i) -h(x, i)] = j ν j g j (x) -g i (x) = ν 1 -1 {i=1} .
Then,

L Z f a (x, i) = 1 a g i (x) • ∇ x (h • ∇ x f )(x, i) + ∇ x f (x) j ν j g j (x).
Here, j ν j g j (x) -g i (x) = ν 1 -1 {i=1} does not depend on x, neither does the function h, which is thus defined by (2.6). Furthermore, h(x, i)

= (θ 1 + θ 2 ) -1 1 i=1 . Moreover lim a→+∞ g i (ν + y/ √ a) = e i -ν, so lim a→+∞ L Z f a (x, i) = L O f (x) up to renormalization. ♦
From Proposition 3.1, for any fixed n ≥ 1, the process (X

(n) t , I (n)
t ) t≥0 admits and converges to a unique invariant distribution π (n) , characterized in (3.7) as

π (n) = D i=1 ν i π (n) i ⊗ δ i , π (n) i (dx) = ϕ (n) (x, i)dx.
Let π(n) be the first margin of the invariant measure of the Markov process (Y

(n) t , I (n) t ) t≥0 , i.e. the probability distribution over R D defined by π(n) (dy) = D i=1 ν i √ a n ϕ (n) y √ a n + ν, i dy.
Corollary 3.9 (Convergence of the stationary distributions). The sequence of probability measures (π (n) ) n≥1 converges to N 0, Σ (0,1) .

Proof of Corollary 3.9: Let n ≥ 1, t ≥ 0 and

F = f ∈ C 2 c (R D ) : f ∞ ≤ 1, |f (x) -f (y)| ≤ |x -y| .
Up to a constant, d F is the Fortet-Mourier distance and metrizes the weak convergence. Fix t ≥ 0 and let X (n) 0 = ν and L (I

(n) 0 ) = ν . From Theorem 3.7, lim n→+∞ d F Y (n) t , Y t = 0,
where Y is an Ornstein-Uhlenbeck process with generator L O and initial condition 0. Using the definition of d F and Proposition 3.1,

d F Y (n) t , π(n) ≤ W (Y (n) t , I (n) t ), π (n) ≤ W δ 0 ⊗ ν, π (n) e -t = W δ 0 , π(n) e -t .
Let us check that the term W δ 0 , π(n) is uniformly bounded. To that end, let

f (n) (x, i) = x 2 k + 2 a n h k,i x k + 2ν k x k , so that L (n) f (n) (x, i) = -2x 2 k + 2 a n 1 {i=k} -x k h k,i + 2ν k 1 {i=k} . Since π (n) L (n) f (n) = 0, E x 2 k π (n) (dx, di) -ν 2 k = 1 a n h k,k ν k - E x k h k,i π (n) (dx, di) . Hence, with C = D k=1 h k,k ν k -min i,j h i,j , and since E x k π (n) (dx, di) = ν k , E x -ν 2 2 π (n) (dx, di) = D k=1 E (x k -ν k ) 2 π (n) (dx, di) = D k=1 E x 2 k -2ν k x k + ν 2 k π (n) (dx, di) = D k=1 E x 2 k π (n) (dx, di) -ν 2 k = 1 a n D k=1 h k,k ν k - E x k h k,i π (n) (dx, di) ≤ 1 a n D k=1 h k,k ν k -min i,j h i,j ≤ C a n .
By Hölder's inequality,

W δ 0 , π(n) = R |y|π (n) (dy) = E √ a n |x -ν|π (n) (dx, di) ≤ √ C.
Consequently to Proposition 3.6,

d F π(n) , N 0, Σ (0,1) ≤ d F π(n) , Y (n) t + d F Y (n) t , Y t + d F Y t , N 0, Σ (0,1) ≤ 2 √ Ce -t + d F Y (n) t , Y t .
Then,

lim sup t→+∞ d F π(n) , N 0, Σ (0,1) ≤ 2 √ Ce -t ,
which goes to 0 as t → +∞.

Complete graph

In this section, we consider a particular case of freezing Markov chain, where all the states are connected, and the jump rate to a state does not depend on the position of the chain. This example of Markov chain has already been studied in the literature, for instance in [DS07]. Section 4.1 deals with the general D-dimensional case, for which most of the results of Section 3 can be written explicitly, notably the invariant measure of the exponential zig-zag process, which is a mixture of distributions (see Figure 4.1). Section 4.2 studies more deeply the case D = 2, where we can refine the speed of convergence provided in Proposition 3.1.

General case

Throughout this section, following [DS07], we assume that there exists a positive vector θ ∈ (0, +∞) D such that, for any 1 ≤ i, j ≤ D,

q(i, j) = θ j -|θ|1 i=j , |θ| = D j=1 θ j , (4.1)
and we will recover [DS07, Theorem 1.4]. If D = 2, let us highlight that an irreducible matrix Id +q automatically satisfies (4.1) (if Id +q is indecomposable then this is true as soon as q(1, 2)q(2, 1) = 0). Proposition 4.1 (Limit distribution for the complete graph in the non-standard setting). Under Assumptions 2.1 and 2.7, and if q satisfies (4.1), then ν i = θ i |θ| -1 and

lim n→+∞ (x n , i n ) = D i=1 ν i D(aθ + e i ) ⊗ δ i in distribution.
In particular, lim

n→+∞ x n = D(aθ) in distribution, lim n→+∞ i n = ν in distribution.
Proof of Proposition 4.1: If q satisfies (4.1), it is straightforward that its invariant distribution ν is given by ν i = θ i |θ| -1 for any 1 ≤ i ≤ D. The convergence of (i n ) n≥1 to ν and of (x n , i n ) n≥1 to some distribution π are direct corollaries of Theorems 2.4 and 2.9. Moreover, Proposition 3.4 holds, hence π satisfies (3.7) and it is clear that

ϕ(x, i) = Γ(|θ| + 1) Γ(θ i + 1) j =i Γ(θ j ) x θi i j =i x aθj -1 j = ν i Γ(|θ|) D j=1 Γ(θ j ) x θi i j =i x aθj -1 j
is the unique (up to a multiplicative constant) solution of (3.8), which entails that

π = D i=1 ν i D(aθ + e i ) ⊗ δ i . Finally, if L (X, I) = π, it is clear that L (I) = ν and that L (X)(dx) = D i=1 ν i ϕ(x, i)dx = Γ(|θ|) D j=1 Γ(θ j ) j =i x aθj -1 j dx = D(θ)(dx).
In the framework of (4.1), it is also possible to obtain explicitly the solution of the Poisson equation related to q as well as the covariance matrix of the limit distribution in the standard setting. This is the purpose of the following result, whose proof is straightforward using Theorem 2.12 together with the expressions (1.2) and (2.6).

Proposition 4.2 (Limit distribution for the complete graph in the standard setting). Under Assumptions 2.1 and 2.8, and if q satisfies (4.1), then ν = |θ| -1 θ and h i = |θ| -1 e i and

lim n→+∞ y n = N 0, Σ (p,Υ) in distribution, with Σ (p,Υ) k,l = -2-p 1+Υ ν k ν l if k = l 2-p 1+Υ ν k (1 -ν k ) if k = l .
Finally, let us emphasize the fact that Corollary 3.9 provides an interesting convergence of rescaled Dirichlet distributions, when considered in the particular case of the complete graph.

Corollary 4.3 (Convergence of the rescaled Dirichlet distribution to a Gaussian law). For any vector

θ ∈ (0, +∞) D , if (X n ) n≥1 is a sequence of independent random variables such that L (X n ) = D(a n θ), then lim n→+∞ √ a n (X n -ν) = N 0, diag(ν) -νν in distribution.

The turnover algorithm

In this subsection, we consider the turnover algorithm introduced in [START_REF] Engländer | Turning a coin over instead of tossing it[END_REF]. This algorithm studies empirical frequency of heads when a coin is turned over with a certain probability, instead of being tossed as usual. The authors provide various convergences in distribution for this proportion, depending on the asymptotic behavior of the turnover probability, which corresponds to (p n ) n≥1 in the present paper. However, this turnover algorithm can be seen as a particular case of freezing Markov chain, and can then be written as the stochastic algorithm defined in (2.4), in the special case D = 2. Since x n (1) = 1-x n (2), there is only one relevant variable in this section, which belongs to [0, 1]:

x n = x n (1) = γ n n k=1 1 {i k =1} . (4.2)
Note that we are in the framework of Section 4.1, with θ 1 = q(2, 1) and θ 2 = q(1, 2), and that Propositions 4.1 and 4.2 hold. In particular, we have ν i = θ i (θ 1 + θ 2 ) -1 . Then, for any y ∈ R and (x, i) ∈ [0, 1] × {1, 2}, the infinitesimal generators defined in (1.1) and (1.3) write

L O f (y) = -yf (y) + 2 -p 1 + Υ ν 1 (1 -ν 1 )f (y) (4.3) and L Z f (x, i) = (1 {i=1} -x)∂ x f (x, i) + aθ 3-i [f (x, 3 -i) -f (x, i)]. (4.4)
Remark 4.4 (Comparison with [START_REF] Engländer | Turning a coin over instead of tossing it[END_REF]). In the present paper, we recover [EV16, Theorems 1 and 2] as direct consequences of Theorems 2.9 and 2.12. The aforementioned results are extended by allowing q(1, 2) = q(2, 1), but mostly by obtaining results for general sequences (p n ) n≥1 while [EV16] deals only with p n = an -θ for positive constants a and θ. It should be noted that, in order to perfectly mimic the algorithm of the aforementioned article, one should consider the chain

x n = γ n n k=1 1 {i k =1} -1 {i k =2} , which evolves in [-1, 1].
The behavior of this sequence being completely similar to the one we are studying, we chose to work with (4.2) for the sake of consistence.

However, the reader should notice that the invariant measure of the process generated by (4.3) is a Gaussian distribution with variance Σ (p,Υ) 1,1 . In the particular case where p = 0 and θ 1 = θ 2 , this variance writes

Σ (0,Υ) 1,1 = 1 2(1 + Υ) ,
which is, at first glance, different from the variance provided in [START_REF] Engländer | Turning a coin over instead of tossing it[END_REF], which is (under our notation)

σ 2 = 1 a 2 (1 + Υ)
.

The factor a 2 comes from the fact that [START_REF] Engländer | Turning a coin over instead of tossing it[END_REF] studies the behavior of a -1 y n . The factor 2 comes from the choice of normalization mentioned earlier, since x n ∈ [0, 1] and

x n ∈ [-1, 1]. ♦ Whenever D = 2
, it is easier to visualize the dynamics of (X, I) (see Figure 4.2), and we can improve the results of Proposition 3.1 concerning the speed of convergence of the exponential zig-zag process to its stationary measure π. Proposition 4.5 (Ergodicity when D = 2). The Markov process (X t , I t ) t≥0 generated by L Z in (4.4), with values in [0, 1] × {1, 2}, admits a unique stationary distribution Since the inter-jump times of the exponential zig-zag process are spread-out, it is also possible to show convergence in total variation with a method similar to [BMP + 15, Proposition 2.5]. Note that, following Proposition 4.1, the limit distribution of (X t ) t≥0 is the first margin of π, namely β(aθ 1 , aθ 2 ).

π = θ 1 θ 1 + θ 2 β(aθ 1 + 1, aθ 2 ) ⊗ δ 1 + θ 2 θ 1 + θ 2 β(aθ 1 , aθ 2 + 1) ⊗ δ 2 . Moreover, let v = a(θ 1 ∨ θ 2 ), then W ((X t , I t ), π) ≤      2 + 2v |1-v| e -(1∧v)t if v = 1 (2 + t) e -t if v = 1 W ((X 0 , I 0 ), π) e -t if L (I 0 ) = θ1 θ1+θ2 δ 1 + θ2 θ1+θ2 δ 2 . t 1 X 0 It = 2 T 1 It = 1 T 2 It = 2 T 3 X t
Proof of Proposition 4.5: Without loss of generality, let us assume that θ 1 ≥ θ 2 , that is v = aθ 1 . Using Proposition 4.1, it is clear that π is the limit distribution of (X, I). Let us turn to the quantification of the ergodicity of the process. Since the flow is exponentially contracting at rate 1, one can expect the Wasserstein distance of the spatial component X to decrease exponentially. The only issue is to bring I to its stationary measure first. So, consider the Markov coupling (X, I), ( X, I) of L Z on E × E, which evolves independently if I = I, and else follows the same flow with common jumps. We set T 0 = 0 and denote by T n the epoch of its n th jump. If I 0 = I 0 , the first jump is not common a.s., but in any case, since D = 2, I T1 = I T1 a.s. and L (T 1 ) = E (v). Consequently,

E |(X t , I t ) -( X t , I t )| = E |X t -X t | + P(I t = I t ) ≤ t 0 E |X t -X t | T 1 = s v e -vs ds + E |X t -X t | T 1 > t + 1 P(T 1 > t) ≤ 2 e -vt + t 0 E |X s -X s | e -(t-s) v e -vs ds ≤ 2 e -vt +v e -t t 0 e (1-v)s ds ≤ 2 + v 1 -v e -vt - v 1 -v e -t 1 {v =1} + (2 + vt) e -vt 1 {v=1} ≤ 2 + 2v |1 -v| e -(1∧v)t 1 {v =1} + (2 + t) e -t 1 {λ=1} .
Note that if L (I 0 ) = L ( I 0 ), let I 0 = I 0 , so that the coupling (X, I), ( X, I) always has common jumps and

|X t -X t | = |X 0 -X 0 | e -t a.s.
Letting (X 0 , X 0 ) be the optimal Wasserstein coupling entails Wasserstein contraction. The results above hold for any initial conditions ( X 0 , I 0 ). Then, let L ( X 0 , I 0 ) = π to achieve the proof; in particular,

L ( I 0 ) = ν = θ1 θ1+θ2 δ 1 + θ2 θ1+θ2 δ 2 .
In this section, we provide the proofs of the main results of this paper that were stated throughout Section 2.

Proof of Theorem 2.4: Under Assumption 2.1, let us first assume that p > 0. The matrix (Id +q) is irreducible, and so is (Id +pq). Moreover, ν is also the invariant measure of pq, and Perron-Frobenius Theorem entails that there exist C > 0 and ρ ∈ (0, 1) such that for every n ≥ 1 and i ∈ {1, . . . , D},

d TV δ i (Id +pq) n , ν ≤ Cρ n .
Now, let us prove that (i n ) n≥1 is an asymptotic pseudotrajectory of the dynamical system induced by Id +pq. The limit set of such a system being contained in every global attractor (see [Ben99, Theorems 6.9 and 6.10]), we have

d TV (δ in (Id +pq), i n+1 ) = d TV (δ in (Id +pq), δ in (Id +q n )) ≤ |p n -p| + j =in |r n (i n , j)| ≤ |p n -p| + D i,j=1
|r n (i, j)|, (5.1) and the right-hand side of (5.1) converges to 0, which ends the proof.

The case p = 0 is a mere application of [BBC16, Proposition 3.13].

Asymptotic pseudotrajectories in the non-standard setting

In this section, we prove Theorem 2.9 using results from [START_REF] Benaïm | Ergodicity of inhomogeneous Markov chains through asymptotic pseudotrajectories[END_REF], based on the theory of asymptotic pseudotrajectories for inhomogeneous-time Markov chains. Indeed, with the convention

0 k=1 = 0, let τ n = n k=1 γ k , m(t) = sup{k ≥ 0 : τ k ≤ t}, (5.2)
and define the piecewise-constant processes

X t = ∞ n=1 x n 1 τn≤t<τn+1 , I t = ∞ n=1
i n 1 τn≤t<τn+1 .

(5.3)

We shall show that, as t → +∞, the process (X t , I t ) t≥0 converges in a way (see Figure 5.1) to the exponential zig-zag process (X t , I t ) t≥0 solution of (3.1), that we already studied in Section 3.1. To that end, let (P t ) t≥0 be the Markov semigroup of (X, I), N 1 = (2, . . . , 2, 0) and

F =    f ∈ D(L Z ) ∩ C N1 b : L Z f ∈ D(L Z ), L Z f ∞ + L Z L Z f ∞ + N1 j=0 f (j) ∞ ≤ 1    .
(5.4)

Note that convergence with respect to d F implies convergence in distribution (see [START_REF] Benaïm | Ergodicity of inhomogeneous Markov chains through asymptotic pseudotrajectories[END_REF]Lemma 5.1]).

Lemma 5.1 (Asymptotic pseudotrajectory for non-standard fluctuations). Under the assumptions of Theorem 2.9, the sequence of probability distributions (µ t ) t≥0 is an asymptotic pseudotrajectory of (P t f ) t≥0 with respect to d F .

Moreover, if there exist positive constants A ≥ 1, θ ≤ 1 such that

max j =i (|r n (i, j)|) ≤ A n θ , Figure 5
.1: Sample path of the process (X t ) t≥0 in the setting of Section 4.2 for a = 1, q(1, 2) = 1 3 and q(2, 1) = 2 3 .

then, for any v < aρ(1 + aρ) -1 , there exists a positive constant C such that

d F ∩G (L (X t , I t ), π) ≤ C e -vt .
(5.5)

Moreover, the sequence of processes ((X n+t , I n+t ) t≥0 ) n≥1 converges in distribution, as t → +∞, toward (X π t , I π t ) t≥0 in the Skorokhod space, where (X π , I π ) is a process generated by L Z with initial condition π.

The proof of Lemma 5.1 consists in checking [BBC16, Assumptions 2.1, 2.2, 2.3 and 2.7.ii)] and relies on three ingredients:

• Convergence of a kind of discrete infinitesimal generator L n , which characterizes the dynamics of (X, I), to L Z defined in (1.3).

• Smoothness of the limit semigroup (P t ) t≥0 and control of its derivatives with respect to the initial condition of the process.

• Uniform boundedness of the moments of (x n , i n ) n≥1 up to some order, which is trivially satisfied here since E is compact.

Proof of Lemma 5.1: In what follows, the notation O (as n → +∞) is uniform over x, i, f . We define

L n f (x, i) = γ -1 n+1 E[f (x n+1 , i n+1 |x n = x, i n = i],
and we study the convergence of L n to L Z in the sense of [START_REF] Benaïm | Ergodicity of inhomogeneous Markov chains through asymptotic pseudotrajectories[END_REF]

. Let (x, i) ∈ E and χ N1 (x, i) = D k=1 x 2 k . We recall that q n (i, i) = 1 + O(p n ) as n → +∞. With n = 1 n ∨ max j =i (|r n (i, j)|),
we have

L n f (x, i) = j =i q n+1 (i, j) γ n+1 f γ n+1 γ n x + γ n+1 e j , j -f (x, i) + 1 -j =i q n+1 (i, j) γ n+1 f γ n+1 γ n x + γ n+1 e i , i -f (x, i) = j =i p n+1 γ n+1 (q(i, j) + r n+1 (i, j)) f (x, j) -f (x, i) + χ (1,...,1,0) (x, i) f (1,...,1,0) ∞ O (γ n+1 ) + 1 + O(p n+1 ) γ n+1 γ n+1 γ n -1 + γ n+1 e i • ∇ x f (x) + χ N1 (x, i) f (N1) ∞ O γ 2 n+1 = L Z f (x, i) + χ N1 (x, i) f (N1) ∞ O ( n+1 ) .
We turn to the study of the regularity of the limit semigroup, following [Kun84]. Let t > 0 and note that P t f ∞ ≤ f ∞ . Moreover, the process (X, I) is solution of the following SDE (we emphasize below the dependence on the initial condition):

(X x,i t , I x,i t ) = (x, i) + t 0 A(X x,i s -, I x,i s -) + e I x,i s - ds + D j=1 t 0 B I x,i s -,j (X x,i s -, I x,i s -)N I x,i s -,j (ds), (5.6)
where N i,j is a Poisson process of intensity aq(i, j)1 {i =j} and the matrices A and B i,j are defined in (3.2).

Then, if we denote by η

x,i,k,h t = h -1 (X x+he k ,i t , I x+he k ,i t ) -(X x,i t , I x,i t )
, we recover from (5.6) that the process η x,i,k,h satisfies the ODE

η x,i,k,h t = (e k , 0) + t 0 Aη x,i,k,h s - ds, so that η x,i,k,h t
= (e -t e k , 0). Thus, η x,i,k,h admits a continuous modification (notably at h = 0) and ∂ k (X x,i , I x,i ) = (e -t e k , 0) is continuous. Using similar arguments, ∂ k ∂ l (X x,i , I x,i ) = 0. Gathering those expressions, and since f N is bounded for every multi-index N ≤ N 1 , it is clear that P t f ∈ C N1 , with, for any j, k ≤ D,

∂ j (P t f )(x, i) = E ∂ j f (X x,i t , I x,i t ) e -t , ∂ j ∂ k (P t f )(x, i) = E ∂ j ∂ k f (X x,i t , I x,i t ) e -2t .
Hence, for any j ≤ N 1 ,

(P t f ) (j) ∞ ≤ f (j) ∞ . Finally, for any n ≥ 1, |x n | ≤ 1, so that sup n≥1 χ N1 (x n , i n ) = sup n≥1 2 k=0 |x n | k ≤ 3.
Hence, we can apply [BBC16, Theorems 2.6 and 2.8.ii)] with

N 1 = N 2 = d 1 = d = (2, . . . , 2, 0), C T = 1, M 2 = 3
, to obtain the existence and the announced properties of π as well as

lim n→+∞ d F (L (x n , i n ), π) = 0. Moreover, following [BBC16, Remark 2.5], λ(γ, ) = -lim sup n→+∞ log(γ ∨ n ) n k=1 γ k ≤ θ A .
Finally, using Proposition 3.1 together with [BBC16, Theorem 2.8.ii)] entails (5.5). Recall the compactness of E, then we can apply [BBC16, Theorem 2.12] and achieve the proof.

ODE and SDE methods in the standard setting

In the present section, we successively provide proofs for Theorems 2.11 and 2.12. We shall prove the former with a method involving an asymptotic pseudotrajectory for some interpolated process, similarly to Section 5.1 and [START_REF] Benaïm | A stochastic approximation approach to quasi-stationary distributions on finite spaces[END_REF]. On the contrary, the fluctuations obtained for (x n ) n≥1 in Theorem 2.12 are obtained through a more classic result for stochastic algorithms, namely the SDE method developed in [START_REF] Duflo | Algorithmes stochastiques[END_REF] (see also [START_REF] Kushner | Stochastic approximation and recursive algorithms and applications[END_REF]).

Proof of Theorem 2.11: In the following, we mimic the proof of [BC15, Lemma 2.4] (see also [START_REF] Métivier | Théorèmes de convergence presque sure pour une classe d'algorithmes stochastiques à pas décroissant[END_REF][START_REF] Benaïm | Vertex-reinforced random walks and a conjecture of Pemantle[END_REF]). Indeed, for any n ≥ 1, (2.4) writes

x n+1 = x n + γ n+1 (ν -x n ) + γ n+1 (e in+1 -ν).
Let the sequence (τ n ) n≥0 and the function m be as in (5.2), and define the interpolated process

X τn+s 0 = x n + s x n+1 -x n τ n+1 -τ n ,
for all s ∈ [0, γ n+1 ) and n ≥ 0. We will show that X is an asymptotic pseudotrajectory (and a -pseudotrajectory) for the flow Φ(x, t) = ν + e -t (x -ν) associated to the ODE ∂ t Φ(x, t) = ν -Φ(x, t).

From [Ben99, Proposition 4.1] it suffices to show that, for all T > 0,

lim t→+∞ ∆(t, T ) = 0, with ∆(t, T ) = sup 0≤h≤T m(t+h) k=m(t) γ k (e i k+1 -ν) , (5.7)
and

lim t→+∞ log(∆(t, T )) t ≤ -. (5.8)
Consider h defined in (2.6). Then,

γ n+1 e in+1 -ν = γ n+1 D j=1 q(i n+1 , j) h in+1 -h j = γ n+1 p n D j=1 (p n q(i n , j) -q n (i n , j)) h in+1 + γ n+1 p n h in+1 -E h in+1 |i n + γ n+1 p n D j=1 (q n (i n , j) -p n q(i n , j)) h j +   γ n+1 D j=1 q(i n , j)h j -γ n D j=1 q(i n , j)h j   +   γ n D j=1 q(i n , j)h j -γ n+1 D j=1 q(i n+1 , j)h j   .
(5.9)

We shall bound each term of the sum (5.9) separately. We easily have

γ n+1 D j=1 (q n (i n , j) -p n q(i n , j)) h j p n = γ n+1 D j=1 r n (i n , j)h j ≤ h 1 R n γ n+1 and γ n+1 D j=1 (p n q(i n , j) -q n (i n , j)) h in+1 = γ n+1 h in+1 D j=1 r n (i n+1 , j) ≤ h 1 R n γ n+1 ,
where h 1 = sup j i |h i,j | and R n = sup i j |r n (i, j)|. Also, for some constant C > 0,

γ n+1 D j=1
q(i n , j)h j -γ n D j=1 q(i n , j)h j ≤ C(γ n -γ n+1 ).

Note that (γ n D j=1 q(i n , j)h j -γ n+1 D j=1 q(i n+1 , j)h j ) is the main term of a telescoping series. It remains to bound the norm of the sum of γ Moreover, as Then, bounding all the other terms of (5.9), we find lim t→+∞ log(∆(t, T ) t ≤with = min λ(γ, γ/p), λ(γ, γ), λ(γ, R) = λ(γ, γ/p) ∧ λ(γ, R).

E   D j=1 q k (i k , j)(h l,j -E[h l,i k+1 |i k ])(h c,j -E[h c,i k+1 |i k ])   =E   D j=1 q k (i k , j)(h l,j h c,j -E[h l,i k+1 |i k ]E[h c,i k+1 |i k ])   =p k E   j =i k q(i k , j)h l,j h c,j   + E     1 -p k j =i k q(i k , j)   h l,i k h c,i k
Since the flow Φ converges to ν exponentially fast at rate 1, use [Ben99, Theorem 6.9 and Lemma 8.7] to achieve the proof.

Proof of Theorem 2.12: We have y n+1 = y n + y n α n+1 α n (1 -γ n+1 ) -1 + γ n+1 α n+1 (e in+1 -ν).

Recall (5.9), so that q(i n , j)h j -D j=1 q(i n+1 , j)h j   .

γ
Then, by (5.10) and similar computations,

E[ n |F n ] = 0 E[ n n |F n ] = Σ (p,Υ) + o(p n ), sup n≥1 E [| n | q ] < +∞, q > 2,
where Σ (p,Υ) is defined in (1.2). Classically, we should prove that lim n→+∞ r n = 0, in order to work in the framework of [Duf96, Hypothèse H4-4], which is quite difficult. Nevertheless, rather than checking that lim n→+∞ r n = 0 it is sufficient2 to prove that for any T > 0, where m(t) is defined in (5.2). Then, let r

(1)

n+1 = y n 1 γ n+1 α n+1 α n (1 -γ n+1 ) -1 + γ n+1 1 + Υ 2 + α n+1 γ n+1 γ n+1 p n D j=1
r n (i n , j)(h in+1 -h j ), r

n+1 = α n+1 γ n+1 γ n+1   D j=1 (2)
q(i n , j)h j -D j=1 q(i n+1 , j)h j   .

The sequence ( r

n ) n≥1 goes to 0 a.s. and in L 1 straightforwardly under our assumptions. Furthermore

γ n+1 r (2) n+1 = α n+1 γ n+1 D j=1
q(i n , j)h j -α n+2 γ n+2 D j=1 q(i n+1 , j)h j + (α n+2 γ n+2 -α n+1 γ n+1 ) D j=1 q(i n+1 , j)h j .

(5.12)

  2 and Proposition 3.4 below by replacing the Lebesgue measure L on R D by the Lebesgue measure on the linear subspace spanned by the recurrent class of Id +q. ♦ Proof of Lemma 3.2: The proof is mainly based on Hörmander-type conditions for switching dynamical systems obtained in [BH12, BLBMZ15]. Using the notation of [BLBMZ15], let

Proposition 3. 6 (

 6 Ergodicity of the Ornstein-Uhlenbeck process). The Markov process (Y t ) t≥0 generated by L O in (1.1), with values in R, admits a unique stationary distribution N 0, Σ (p,Υ) .

Figure 4 . 1 :

 41 Figure 4.1: Probability density functions of π 1 = D(2, 2, 5), π 2 = D(1, 3, 5), π 3 = D(1, 2, 6), for θ 1 = 1, θ 2 = 2, θ 3 = 5.

Figure 4 . 2 :

 42 Figure 4.2: Typical path of the exponential zig-zag process when D = 2.

  n+1 p -1 n h in+1 -E[h in+1 |i n+1 ] . For all m, n ≥ 1 and l = 1, ..., D, set M m,n (l) = n k=m γ k+1 p k h l,i k+1 -E h l,i k+1 |i k+1 .The sequence (M m,n (l)) m≥n is a martingale andE [M m,n (l)M m,n (c)] = i k , j)(h l,j -E[h l,i k+1 |i k ])(h c,j -E[h c,i k+1 |i k ])   .

EE

  k , j)(h l,j -h l,i k )(h c,j -h c,i k ) ) k , j)(h l,i k -h l,j ) k , j)(h c,j -h c,i k ) i k , j)(h l,j -E[h l,i k+1 |i k ])(h c,j -E[h c,i k+1 |i k ]) j)(h l,j -h l,i )(h c,j -h c,i) ) ν l -1 i=l ) (ν c -1 i=c ) + o(p k ).(5.10) As a consequence of (5.10), there exists some constant C > 0 such thatD l=1 E M m,n (l) 2 ≤ C n k=m γ 2 k+1 p k .By Doob's inequality and Assumption 2.8, it follows that, for every k ≥ 0,E sup 0≤h≤T |M m(kT ),m(kT +h) | ≤ 2 D l=1 |M m(kT ),m((k+1)T ) (l)| 2 ≤ 2C m((k+1)T ) lim k→+∞ sup 0≤h≤T |M m(kT ),m(kT +h) | = 0 and then lim k→+∞ ∆(kT, T ) = 0 in probability. By the triangle inequality and [Ben99, Proposition 4.1], (5.7) holds. |M m(kT ),m((k+1)T ) (l)| 2 ≤ 2C k≥m(T ) which implies lim k→+∞ sup 0≤h≤T |M m(kT ),m(kT +h) | = 0 a.s. Then, lim k→+∞ ∆(kT, T ) = 0 a.s. and lim t→+∞ ∆(t, T ) = 0 since ∆(t, T ) ≤ 2∆( t/T T, T ) + ∆(( t/T + 1)T, T ).In order to obtain a -pseudotrajectory, use Markov's and Doob's inequalities so thatP sup 0≤h≤T |M m(kT ),m(kT +h) | ≥ e -kT α ≤ e kT α 2C m((k+1)T ) j=m(kT ) γ j+1 γ j+1 p j ≤ 2C(T + 1)e kT α sup j≥m(kT ) γ j+1 p j .Now, for all ε > 0 and k large enough, sup j≥m(kT ) γ j+1 p j ≤ exp -(λ(γ, γ/p)) -ε)τ m(kT ) ≤ exp (-(λ(γ, γ/p)) -ε)kT ) , where λ(γ, γ/p) is defined in (2.5). Hence, P sup 0≤h≤T |M m(kT ),m(kT +h) | ≥ e -kT α ≤ 2C(T + 1) exp (kT (α -λ(γ, γ/p) + ε)) , and by the Borel-Cantelli lemma, we have lim sup k→+∞ 1 k sup 0≤h≤T |M m(kT ),m(kT +h) | ≤ -λ(γ, γ/p) a.s.

  n+1 (e in+1 -ν) = γ n+1 p n (h in+1 -E[h in+1 |i n ]) + b n ,with a remainder term b n converging to 0. Now, we want to use [Duf96, Théorème 4.II.4]. In our setting, its notation readsy n+1 = y n + γ n (h(y n ) + r n+1 ) + γ n n+1 ,

	with
	n+1 =	1 + Υ 2	-1/2 1 √ p n	(h in+1 -E[h in+1 |i n ]), γ n+1 =	1 + Υ 2	γ 2 n+1 α 2 n+1 p n	, h : z → -z,
	and
			r n+1 = y n	1 γ n+1	α n+1 α n	(1 -γ n+1 ) -1 + γ n+1	1 + Υ 2
			+	α n+1 γ n+1	γ n+1 p n	D j=1	r n (i n , j)(h in+1 -h j )
			+	α n+1 γ n+1	γ n+1	 	D j=1

The algebric term indecomposable also exists for matrices, and is sometimes mistaken for irreducibility. Throughout this paper, a Markov chain (or its associated transistion matrix) is said indecomposable if it admits a unique recurrent class.

This assertion can be easily checked at the end of[START_REF] Duflo | Algorithmes stochastiques[END_REF] p.156], whose proof is based on usual arguments on diffusion approximation, such as[START_REF] Ethier | Markov processes[END_REF]. The decomposition (5.11) is often assumed in more recent generalizations, see for instance[START_REF] Fort | Central limit theorems for stochastic approximation with controlled Markov chain dynamics[END_REF]. Note that we cannot use directly[START_REF] Fort | Central limit theorems for stochastic approximation with controlled Markov chain dynamics[END_REF], which besides does not provide functional convergence.

Acknowledgements: Both authors acknowledge financial support from the ANR PIECE (ANR-12-JS01-0006-01) and the Chaire Modélisation Mathématique et Biodiversité.

The first line of (5.12) is a telescoping series and is bounded by α n γ n+1 which goes to 0. The second line of (5.12) is bounded by,

(5.13) for some C > 0. Since (5.12) is a telescoping series as well, and goes to 0, we established the announced decomposition (5.11). As a conclusion, the diffusive limit (Y t ) t≥0 is the solution of (3.10), which trivially admits V : z → z as a Lyapunov function, as required in [Duf96, Hypothèse H4-3]. The only use of an assumption on the eigenelements of Σ (p,Υ) would be to guaranty the existence, uniqueness of and convergence to an invariant distribution for Y, which was already proved in Proposition 3.6.