Éric N Kéré

Romuald S Kinda
email: skinda@univ-ouaga2.bf

Eric N Kéré
email: kereneric@yahoo.fr

Catherine Araujo Bonjean

Keywords: Food security, Climate change, Spatial spillovers, Spatial econometrics, Developing countries. JEL Codes C23, Q17, Q18, Q54 𝐹𝑆

This article analyzes the role of spatial spillovers in the relationship between climate change and food security in developing countries over the period of 1971-2010. Using a Samuelson's spatial price equilibrium model (theoretically) and Spatial Durbin Model (empirically), results show a strategic substitutability between the levels of food availability in the countries suggesting that an increase of food availability in a given country decreases the food availability of neighboring countries. Second climate change (water balance variability, droughts, floods and extreme temperatures) reduces food availability both in the affected countries and its main food trading partners. Third, food demand factors in a country may have the opposite (asymmetric) effect on its major trading partners. Fourth, supply factors have symmetric impact on food availability.

Introduction

Over the last two decades, the number of people affected by extreme poverty has decreased from 1.9 billion in 1990 to 836 million in 2015 and the percentage of undernourished people has dropped by almost half from 23.3 per cent to 12.9 per cent.

In 2015, the United Nations launched the Sustainable Development Goals (SDGs) initiative that aims to end poverty and hunger by 2030. Addressing issues related to food security is included in year fifteen of the development agenda. According to The State of Food Insecurity in the World report (SOFI, 2015), 795 million people were undernourished in 2015. Although hunger has been reduced in developing countries, progress has been hindered by several factors including political instability, less inclusive economic growth and climate events [START_REF] St | The opening of Pandora's Box: climate change impacts on soil fertility and crop nutrition in developing countries[END_REF].

The economic literature on the impact of climate change on food security and production can be divided into two strands. In one body of literature, several authors develop theoretical arguments or prospective studies which indicate that climatic variability has a negative impact on agricultural production and decreases food availability. For instance, [START_REF] Christensen | Climate change 2007: The physical science basis[END_REF] show that food production is highly vulnerable to the influence of adverse weather and [START_REF] Ringler | Climate Change Impacts on Food Security in Sub-Saharan Africa[END_REF] and St. Clair and Lynch (2010) conclude that climatic variability is a factor of childhood malnutrition in Sub-Saharan Africa. The second body of literature has found mitigated effects of climate change on food production. For example, [START_REF] Eckersten | Modelling C, N, water and heat dynamics in winter wheat under climate change in southern Sweden[END_REF] conclude that in areas suffering from water stress, increased rainfall can actually increase agricultural production. However, they also find that excessive rainfall can contribute to soil degradation and reduce both soil oxygen (nitrogen leaching, runoff and soil erosion areas) and agricultural productivity. In the United States, several studies show that climate change has a significant and negative impact on agriculture [START_REF] Adams | Global climate change and agriculture: an economic perspective[END_REF][START_REF] Schlenker | Will US agriculture really benefit from global warming? Accounting for irrigation in the hedonic approach[END_REF], among others).

However, using panel data, [START_REF] Deschênes | The economic impacts of climate change: evidence from agricultural output and random fluctuations in weather[END_REF] have demonstrated that climate change has an insignificant or slightly positive effect. Using panel data for Asian countries from 1998 to 2007, [START_REF] Lee | Impact of Climate Change on Agricultural Production in Asian Countries: Evidence from Panel Study[END_REF] show that high temperatures and increased precipitation in summer increase agricultural production. In the case of Ethiopia, von Braun (1991) concludes that a 10% decrease in the amount of rainfall below the long term average leads to a 4.4% reduction in food production. Finally, [START_REF] Badolo | Climatic Variability and Food Security in Developing Countries[END_REF] show that the negative effects of climatic variability on food security are exacerbated in the presence of civil conflicts and are high for countries that are vulnerable to food price shocks. These countries are highly dependent on food imports and have an agricultural sector that is sensitive to climatic events.

Another body of literature focuses on the integration of agricultural markets as a contributing factor of food security in developing countries [START_REF] Barrett | Distinguishing Between Equilibrium and Integration in Spatial Price Analysis[END_REF][START_REF] Baulch | Transfer Costs, Spatial Arbitrage, and Testing for Food Market Integration[END_REF][START_REF] Goodwin | Spatial market integration in the Presence of threshold Effects[END_REF][START_REF] Ravallion | Testing Market Integration[END_REF]. These studies have shown that many food markets are not well integrated. During the global food crisis of 2006-2008, agricultural commodities and food prices dramatically increased. One of the factors of this spike in prices was climatic shocks in the form of droughts and floods.

Rainfall instability and extreme temperatures negatively affect agricultural crops and lead to lower food supplies in international markets, which contributes to a rise in agricultural prices. Increases in food prices can have a strong negative impact on households in developing countries where agricultural products form the basis for food consumption. The 2006[START_REF]Crop Prospects and Food Situation[END_REF] crises illustrated a link between the international food market and spillover effects. Indeed, these crises can be explained by the increase in demand due to a richer diet in India and China, the increase in world population and competition with biofuels on the one hand, and on the other, the decline in production due to droughts (Romania, Lesotho, Somalia, Ghana), floods (Ecuador, Bolivia, Sri Lanka) and a particularly hard winter (southern China and Argentina) in 2007 (FAO, 2008). These spatially localized phenomena have spread around the world and led to food riots in developing countries (Senegal, Ivory Coast, Egypt, Haiti, Indonesia, Philippines, Cameroon, and others) that originally were not affected by these phenomena. In addition, ignoring the spatial interactions in regression models (OLS, panel data, etc.) can not only bias the standard deviations but can impact the value of the estimates. Therefore, it is important to explicitly address the spatial patterns underlying the determination of food availability that may exist in market mechanisms because many developing countries, especially the poorest countries, depend on international food markets. This paper investigates the importance of spatial spillovers in the relationship between climate change and food security in 53 developing countries over the period of 1971-2010. To the best of our knowledge, no study has focused on the spatial dependencies of food availability and cross-border effects generated by climate change. We model theoretically the spatial interdependence of countries' food availability levels using a Samuelson's spatial price equilibrium model. This spatial interdependence of trade in goods between a country and its trading partners is empirically modeled using a trade connectivity matrix (imports of food commodities) and a Spatial Durbin Model with fixed effects. Results are as follows: First, we find strategic substitutability between the levels of food availability in the countries suggesting that the increase of food availability in a given country decreases the food availability of neighboring countries. Second climate change (water balance variability, droughts, floods and extreme temperatures) reduces food availability both in the affected countries and its main food trading partners. The adverse effect of climate change on food availability in developing countries is more explained by spillover effects than direct effects.

Third, demand factors (income per capita, population density, population growth, dependence ratio) in a country may have the opposite (asymmetric) effect on its major trading partners. Fourth, supply factors (water balance variability, drought, flood, extreme temperature, water balance, arable and cereal lands,) have symmetric impact on food availability.

The remainder of the paper is as follows. Section 2 defines the concepts of climate change and food security and presents the theoretical model. Sections 3 and 4 describe the empirical strategy and discuss the results. Concluding remarks are offered in last section.

Conceptual Framework

In this section we will define concepts and theoretically analyze the nature of spillover effects in food security.

Definition and measures of food security and climate change

According to the 1996 World Food Summit "Food security [is] a situation that exists when all people, at all times, have physical, social and economic access to sufficient, safe and nutritious food that meets their dietary needs and food preferences for an active and healthy life." Food availability in a country depends on its imports and exports to other countries. A country's trade relations can lead to strategic behaviors, spatial spillovers and interdependence on the amount of food available between countries. For example the total food production is distributed among countries through imports and exports, the quantities exported by a country are no longer available for domestic consumption: it is a zero sum game. By contrast, food security indicators of social and economic access to and use of available food depend largely on individual and household conditions so it is unlikely that there is a spatial interdependence of these food security indicators across countries. Therefore, in this study we will focus only on analyzing the effects of spillovers between the food availability of each country/on the national level.

According to the United Nations Intergovernmental Panel on Climate Change (IPCC, 2007;p30), "Climate change refers to a change in the state of the climate that can be identified (e.g., using statistical tests) by changes in the mean and/or the variability of its properties, and that persists for an extended period, typically decades or longer." In this study we will focus particularly on climate variability, which measures the change in climate from its mean state (standard deviations, the occurrence of extremes). Climate change is measured by 1) the standard deviation of the growth rate of water balance which is the difference between rainfall and evaporation and 2) extreme events (droughts, floods and extreme temperature).

Sources of spatial spillovers: Samuelson's spatial price equilibrium model

To determine the source and nature of the spatial interactions between countries, we use Samuelson's (1952) simple spatial price equilibrium model. This model allows for computing the competitive equilibrium of prices and quantities in several spatially separated markets given the domestic demand and supply functions and transport costs. In the presence of trade, it allows us to analyze the interdependence between food quantities available in each country at the equilibrium and the transmission of supply and demand shocks from one country to another through market mechanisms.

In this paper, we consider two countries indexed by i and j which produce and consume a homogenous food commodity. We assume that demand and supply are linear functions that can be written as:

𝐹𝐷 𝑖 = 𝛾 𝑖 -𝑃 𝑖 𝐹𝑆 𝑖 = 𝛿 𝑖 + 𝑃 𝑖 (1) 𝐹𝐷 𝑗 = 𝛾 𝑗 -𝑃 𝑗 𝐹𝑆 𝑗 = 𝛿 𝑗 + 𝑃 𝑗 (2)
where 𝐹𝐷 𝑖 and 𝐹𝐷 𝑗 are the demand for food in country i and j, 𝐹𝑆 𝑖 and 𝐹𝑆 𝑗 their supply of food and 𝑃 𝑖 and 𝑃 𝑗 the price of food. We further assume that 𝛿 𝑖 > 𝛿 𝑗 and 𝛾 𝑖 > 𝛾 𝑗 , therefore 𝛾 𝑖 + 𝛿 𝑖 > 𝛾 𝑗 + 𝛿 𝑗 . In others words, food supply and demand are higher in country i than in country j.

In autarky these two markets are independent and the equilibrium price in each country is given by: 𝑃 𝑖 * = 𝛾 𝑖 -𝛿 𝑖 2 ; 𝐹𝐷 𝑖 * = 𝐹𝑆 𝑖 * = 𝛾 𝑖 + 𝛿 𝑖 2 ;

(3)

𝑃 𝑗 * = 𝛾 𝑗 -𝛿 𝑗 2 ; 𝐹𝐷 𝑗 * = 𝐹𝑆 𝑗 * = 𝛾 𝑗 + 𝛿 𝑗 2 • (4)
Suppose 𝛾 𝑖 -𝛿 𝑖 > 𝛾 𝑗 -𝛿 𝑗 , the equilibrium price and quantity will be higher in country i than in country j. The equilibrium quantity is equivalent to the food availability of each country. Now, assume that trade takes place between these two countries and that a climate shock (𝐶𝑆 𝑗 ) affects country j. The supply function of j can be rewritten as follows:

𝐹𝑆 𝑗 = (𝛿 𝑗 -𝐶𝑆 𝑗 ) + 𝑃 𝑗 . Suppose transportation costs are zero; therefore, we have a unique equilibrium price for both countries. Note that this assumption only affects the magnitude of trade flows and not their direction. We also assume that for food sovereignty and survival reasons, supply and demand for food commodities (mainly basic goods) are always greater than zero. For example, a country will not import all of the food it consumes--it will produce a part. The equilibrium can be found by equating the total food shipments and the total receipts:

(𝐹𝐷 𝑖 -𝐹𝑆 𝑖 ) + (𝐹𝐷 𝑗 -𝐹𝑆 𝑗 ) = 0

(5)

Solving the equation ( 5), the equilibrium price and quantities traded are given by 𝑃 𝑇 * = 𝛾 𝑖 -𝛿 𝑖 + 𝛾 𝑗 -𝛿 𝑗 + 𝐶𝑆 𝑗 4

(6)

Proposition 1. The amount of food commodities available in countries i (j) and j (i) are strategic substitutes.

The characteristics of country j may also have an effect on food availability in country i. First, a reduction of production in country j due to a shock will lead to an increase in the equilibrium price ( Second, a change of the demand level will have an asymmetric effect. Indeed, an increase in demand in country j will lead to higher prices in this country. This will result in additional imports from i to j until the new equilibrium price is reached. All things being equal, this will increase the quantity of food in country j at the equilibrium ( 𝜕𝐹𝐷 𝑇𝑗 * 𝜕𝛾 𝑗 > 0) and reduce the available food of country i (

𝜕𝐹𝐷 𝑇𝑖 * 𝜕𝛾 𝑗 < 0).
Proposition 3. Increase (decrease) in the demand of food in country j (i) increases (decrease) the food availability in country j (i) but reduces (increases) the food availability of country i (j). The effect of a demand shock is asymmetric between i and j.

We are interested in understanding the main channels of spatial spillover effects between food availability in a country and its trading partners, and particularly in the contagion effect of climatic shocks on food availability. The following section will empirically analyze these spatial spillover effects.

Empirical strategy

To investigate the importance of spatial dependences on the effect of climate change on food security in developing countries, this section presents our data sources and the econometric setting.

Data and sources

This study is based on yearly panel data. It covers a period from 1971 to 2010 for 53 developing countries. The choice of countries and the period of study are related to the fact that spatial econometric models require a balanced panel (without missing data). The data on food security (food availability) come from the Food and Agriculture Organization of the United Nations (2015). Food availability is the sum of production, stocks and the trade balance (importsexports) of each of the 53 countries. In this study we consider the main cereals (maize, rice, sorghum, millet and wheat), soybeans and sugar because they represent the main source of food for direct human consumption as well as the main input of meat production in several developing countries. The food availability indicator obtained is a simple average of food availability of these commodities expressed in g/person/year. As a robustness check we use the total availability of food in the country. Figure 1 shows the spatial distribution of food availability indicators for the countries in our sample. indicator measures the number of events that occurred in a country in a given year as these three natural disasters do not have the same effects on food security. For example, droughts should have a greater negative effect on food security than other natural disasters. Two reasons justify the choice of the number of occurrence of each natural disaster to measure the impact of climate shocks instead of the monetary damage caused by natural disasters or the number of affected people. Indeed, data on monetary damage, number of people affected and the number of deaths are likely to be subject to measurement errors and endogenously determined by the level of development (Skidmore and Toya, 2002). According to Elhorst (2014) it is difficult, if not impossible, to derive Maximum Likelihood (ML) or Bayesian estimators of models with spatial dependence and additional endogenous explanatory variables (Elhorst, 2014). Source: Own calculations using data from CRED (2015).

Econometric setting

In this study we are primarily interested in the comovement of food availability across countries according to the intensity of their trade relations. This corresponds to a global 2 spatial spillover scenario because a change in supply or demand for food in a given country will lead to a sequence of adjustments between supply and demand on the international market until a new long term steady state equilibrium is reached.

Following Anselin (2001) and [START_REF] Lesage | What regional scientists need to know about spatial econometrics[END_REF] and 𝛽 2 are the parameters to be estimated. 𝜖 𝑖𝑡 is the error term in the model and we allow this error term to be serially correlated. According to the first law of geography:

"[E]verything is related to everything else, but near things are more related than distant things" (Tobler, 1970: 236). This law explains why geographical distance has been widely used in spatial econometrics to measure connectivity. However, in a context of market integration, the economic distance matrices are more relevant in certain circumstances. In this study, W is a trade connectivity matrix. It is based on the value of agri-food trade (imports) flows taken from the UN Comtrade database for the period of 1971-2010. Each country is linked to all its trading partners, but the intensity of connectivity is stronger when the share of imports from a country is high.

This trade connectivity matrix is more suited to our case study than an inverse distance matrix. Indeed, the availability of food in a country depends more on the volume of imports from other countries than its geographical proximity with these countries. For instance, Burkina Faso is geographically closer to Benin that India, However, a shock on India's rice production (world rice production) will have a greater effect on Burkina Faso than a shock in Benin. We normalize the matrix W by dividing each element (i, j) by the sum of the line.

According to [START_REF] Pace | Spatial Dependence in Regressors[END_REF], the use of Instrumental Variables and Generalized Method of Moments to estimate a SDM is less effective than ordinary least squares unless the number of observations is greater than 500,000 because the second order spatially lagged explanatory variables in a SDM are weak instruments and they do not properly identify the spatial autocorrelation coefficient. In this study we use the maximum likelihood method estimator developed by Elhorst (2010a) and [START_REF] Lee | Estimation of spatial autoregressive panel data models with fixed effects[END_REF] and implemented in Stata by [START_REF] Hughes | A Command to Estimate Spatial Panel Models in Stata[END_REF] to estimate our model.

Results

Specification tests

We test the appropriateness of the SDM for analyzing the spatial interdependence of food safety against the Spatial Autoregressive (SAR) model and the Spatial Error Model (SEM). Following Elhorst (2014), we test the joint nullity of all coefficients of spatially lagged explanatory variables(𝛼 2 = 𝛽 2 = 0). This test, significant at the 1% level, allows us to reject the specification SAR. The equality (𝛼 2 = -𝜌𝛼 1 ) and

(𝛽 2 = -𝜌𝛽 1 ) is rejected by the 𝜒 2 at a level of significance of 1% which allows us to reject the SEM. Finally, the Hausman test is carried out on SDM and allows us to reject the hypothesis of independence between the random effects and explanatory variables. Therefore, the fixed effects model is preferred in this study. All the specification tests are presented at the bottom of Table B.1 in the Appendix.

General pattern of spatial spillovers

The presence of strategic substitutability in countries' food availability levels in our theoretical model is confirmed by a significant value of the spatial autocorrelation coefficient (𝜌) (table B.1). Put differently, the food availability of a given country tends to decrease along with that of its neighbours. These spillover effects allow for the sharing of world food production between the world's leading food producers and countries with low food production. However, in the case of a global crisis strategic substitutability may also exacerbate food availability and food insecurity by contributing to the leakage to rich countries.

The existence of spatial interactions has important policy implications. Indeed, any change of an explanation variable will generate a direct and an indirect/spillover effects (Lesage and Pace, 2009). The direct effect measures the impact of a change in the value of each explanatory variable in country i on food availability in country j.

The indirect effect measures the impact of a change in the value of each explanatory variable in country i on food availability in all other countries (j). Following [START_REF] Lesage | What regional scientists need to know about spatial econometrics[END_REF], indirect effects capture spillover effects because they arise in all countries but indirect effects decrease with the intensity of trade between countries.

Effect of climate change on food security

The main contribution of this study is the identification of spatial spillover effects of all of the explanatory variables, in general, and more particularly of the impact of climate change. Indeed, we show that the variability of water balance, droughts, floods and extreme temperatures not only negatively affect food availability of the treated/affected country, but also that of its major food trading partners. Table 1 shows that an increase of one point in the variability of the water balance will lead to a of 0.0131g/day/per capita reduction in the country's food availability and to a 0.0205g /day/per capita reduction in the economic neighborhood for a total effect of 0.0336g/day/per capita. These results are consistent with the predictions of our theoretical model. Previous papers have shown that climatic events have negative effects in the country's food production [START_REF] Christensen | Climate change 2007: The physical science basis[END_REF][START_REF] Lee | Impact of Climate Change on Agricultural Production in Asian Countries: Evidence from Panel Study[END_REF], food availability and security [START_REF] Badolo | Climatic Variability and Food Security in Developing Countries[END_REF]. However our results are innovative and important because they show that climate events can also reduce food availability in neighbouring countries. Consequently, the omission of the spillover effects in the literature would have led to the underestimation of the climate change effect on food security. In addition, these results allow anticipating the impact of climate indicators that come from partner countries and identifying countries that will be most affected in the future.

Figure 3 shows the average total impact of climate events (water balance variability, drought, flood, extreme temperatures) by country from 1971 to 2010. It shows that America has suffered more adverse effects of droughts, temperature extremes and variability of water balance while Asia and Africa were more affected by droughts, extreme temperatures and floods. In addition, it also appears that drought events are the most harmful to food availability. Indeed, over the period studied they generated an average decline in food availability of between 0.13 and 0.38g/day/per capita. In addition, figure 4 shows the importance of direct and indirect of climate indicators on per capita food availability in developing countries. The total negative of climate change indicators on food availability in developing countries is more related to indirect effect (spillover effects) than direct effect. In the literature, climate change can be defined as a change in the trend of the occurrence of climatic phenomena. To measure this change in trend, we compute the difference between the average values of our climate change indicators in the first ten years (1970)(1971)(1972)(1973)(1974)(1975)(1976)(1977)(1978)(1979)(1980) of our study and their average values during the last ten years (2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010). The results are presented in Table A.3. If the difference is positive then there is an accentuation of the climate phenomenon between the two periods these countries. The higher the value, the higher the impact of climate change on food security has increased between the two periods. These countries are the losers of climate change. In contrast, if the difference is negative then there has been a reduction in the occurrence of climatic phenomenon between the two sub-periods.

The countries that are in this situation have benefited from a reduction of the negative effects of climate change between the two periods.

For

Spillover effects and explanatory variables

The percentage of arable land and the area of land devoted to cereal production in a given country increase the food availability both of this country and of its major trading partners. The water balance and percentage of people of working age (dependency ratio) have a direct positive effect on food availability and a nonsignificant indirect effect. These results validate proposition 2 of the theoretical model. GDP per capita has a positive direct effect and negative indirect effect on food availability. In other words, economic resources improve the capacity of a given country to increase food availability. However an increase of GDP per capita from its major trading partner countries can reduce this food availability through food imports.

This expected effect is consistent with the predictions of proposition 3. Contrariwise, density of population and inflation have a significant positive direct effect and a nonsignificant indirect effect. The sign of the direct effect is in line with our expectations.

Indeed, an increase in demand (population density) or prices (inflation) will boost food supply and lead to an increase in food availability through imports. The population growth rate has both a direct and an indirect negative and significant effect on food availability. This result seems intuitive because population growth increases pressure on world agricultural resources. Finally, democracy has a counter-intuitive direct effect. It has a negative direct impact on food security in the country while the indirect effect has the expected positive impact.

Robustness checks

In this section, we test whether our results are robust to the inclusion of all food commodities in the calculation of food availability and to a change of the econometric specification used.

First, in the case of climatic events affecting cereal production, countries may substitute cereals with other foods. In this case, the reduction of cereal availability will have no effect on total food availability. As a robustness check, we repeat the analysis using the total food availability (expressed in kilocalories/person/year) as the dependent variable instead of cereal availability. The results, reported in Table B.2 and B.3 in the Appendix, confirm strategic substitutability between food availability levels and the negative effect of climate change on food security. Indeed, the direct and indirect effects associated with climate change variables are negative and significantly different from zero, except for the direct effect of the variability of the water balance. Furthermore, the spatial autocorrelation coefficient is negative and significantly different from zero at the 1% level, although it is relatively lower.

Second, methodologically, our use of the SDM is the most suitable model to analyze the spatial interdependence of the level of food availability between countries; however, the error terms of this model can be spatially correlated. This correlation can be explained by the omission of spatially correlated explanatory variables such as climatic, hydrological and soil fertility factors. In addition, neighboring countries are more likely to use similar agricultural practices, which could also lead to a spatial correlation of the error term. This spatial correlation is more related to geographical proximity than to commercial relations between countries. According to [START_REF] Lesage | What regional scientists need to know about spatial econometrics[END_REF], who equates these spatial interactions to local spillovers, a neighborhood matrix is more suitable for this analysis. Thus, we consider a general nesting spatial model with a spatially lagged dependent variable (economic matrix W), spatially lagged explanatory variables and spatially correlated error term (geographical neighborhood matrix𝑊 2 ). The use of a different neighborhood matrix between the dependent variable and the error term facilitated the identification of this model (Elhorst, 2014). The estimated model is written as follows:

𝐹𝑆 𝑖𝑡 = 𝜌 𝑊 𝐹𝑆 𝐽𝑡 + 𝛼 1 𝐶𝐶 𝑖𝑡 + 𝛼 2 𝑊𝐶𝐶 𝑖𝑡 + 𝛽 1 𝑋 𝑖𝑡 + 𝛽 2 𝑊𝑋 𝑖𝑡 + 𝜇 𝑖 + 𝜆𝑊 2 𝑢 𝑗𝑡 + 𝜖 𝑖𝑡 (12) With 𝑊 2 the matrix of 5 nearest neighbors, 𝜆 the spatial autocorrelation coefficient of error terms and 𝑊 2 𝑢 𝑗𝑡 is referred to as the spatially-lagged error.

The results presented in Table B.4 in the Appendix show that the spatial autocorrelation coefficient of the error terms is not significant. This reinforces both our choice of the SDM in this study and the nature of spatial interactions that we have highlighted.

Concluding remarks

Global warming poses considerable uncertainty on agricultural production and food availability and, thus, on the livelihoods of the most vulnerable populations.

Therefore rigorous analyses based on improved methodologies are needed to inform public decisions.

In this article we theoretically and empirically investigate the existence of spatial interactions between the levels of food availability of countries and analyse the spatial spillover effects of climatic change (water balance variability, droughts, floods and extreme temperatures) on food availability. Our contribution to theory is the modeling of the impact of shocks and the spillover effects that result using a Samuelson's spatial price equilibrium model. We empirically use a Spatial Durbin Model in which the spatial interdependence of food availability levels between the countries is explicitly modeled using a trade connectivity matrix (imports of food commodities).

Results are as follows: firstly we show that the level of food availability in a given country is a strategic substitute of those of its main trading partners. In other words, an increase of food availability in a given country reduces the food availability of its main trading partners. Second, climatic events (water balance variability, droughts, floods and extreme temperatures) have spillover effects. Indeed they reduce food availability in both the country and in its economic neighborhood. The adverse effect of climate change on food availability in developing countries is more explained by spillover effects than direct effects. Third, results establish that demand factors (income per capita, population density, population growth, dependence ratio) in a country may have the opposite (asymmetric) effect on its major trading partners.

Final, the effect of supply factors (water balance variability, drought, flood, extreme temperature, water balance, arable and cereal lands,) on food availability is symmetric.

Our results have clear policy implications. A better knowledge about the nature of the spatial effects of climate change can improve the design and implementation of climate policies aimed at fighting against food insecurity in developing countries. Indeed, instead of the emergency response strategy against food insecurity, our results suggest to establish mechanisms that identify countries that most are affected and vulnerable to climate change. In addition, mitigation and adaptation policies could be implemented to countries by taking into account the spatial spillover effects of climate change. In other words, because the vulnerability depends on the type of climate events that developing countries are exposed to, they could identify and implement policies that serve multiple objectives (development, adaptation and mitigation). These strategies may contribute to circumscribe the impact of climate change on food security, but also to anticipate, plan and optimize food safety policies.

Figure 1 :

 1 Figure1: Map of food availability per capita in selected developing countries

  Figure 2 shows the spatial distribution of the climate change indicators of our sample. The control variables are the main determinants of food security. They include the level of economic development (proxied by the natural logarithm of real GDP per capita), demographic characteristics of the population (logarithm of population density, logarithm of the dependency ratio 1 and logarithm of the population growth rate), agricultural factors (cereal production land and logarithm of arable land), inflation (consumer prices) and the quality of institutions (polity IV). They are taken from the World Development Indicators (World Bank) and Polity IV project.Descriptive statistics of all of the variables used in the paper are presented in Appendix A.1 The proportion of people between the ages of 15 and 65 years.

Figure 2 :

 2 Figure 2: Map of climate change indicators in selected developing countries.

  each country, we can compute the direct and indirect effects of the change in the trend of indicators of climate change on food security by multiplying the marginal effects (direct and indirect) by this trend. The impact of the change in trend of climate change indicators on food availability by country is presented in Figure 5. This graphshows different pictures. In South America and Africa, the impact of water balance variability on food availability has increased between the two sub-periods. In addition Africa, India, Nepal, and Southeast Asia are those that experienced the largest increases in the impact of droughts. In contrast the impact of flooding has increased especially in Asia and Africa. Finally, India, China, South Africa, Central America, Brazil and Bolivia are those who have known a sharp rise of the impact of extreme temperatures on food availability. Final, figure6concludes that the marginal contribution of spillover effects of the trend of climate change indicators is higher than the direct effect.

Figure 3 :

 3 Figure 3: Map of the average decline in per capita food availability caused by climate change indicators (1971-2010).

Figure 4 :

 4 Figure 4: Average marginal impact (direct and indirect) of climate change indicators on per capita food availability.

Figure 5 :

 5 Figure 5: Map of the impact of climate change (change in trend) on food availability per capita.

Figure 6 :

 6 Figure 6: Average marginal impact (direct and indirect) of the trend of climate change indicators on per capita food availability.

Flood

  *** (0.0180) 0.6764*** (0.0180) 0.6725*** (0.0181) 0.6752*** (0.0181) Polity2 -0.0103*** (0.0011) -0.0102*** (0.0011) -0.0103*** (0.0011) -0.0102*** (0.0011) LDependency_ratio 0.0722*** (0.0206) 0.0705*** (0.0223) 0.0704*** (0.0224) 0.0707*** (0.0224) Inflation 0.0031*** (0.0007) 0.0029*** (0.0006) 0.0029*** (0.0006) 0.0029*** (0*** (0.0130) 0.0363*** (0.0135) 0.0360*** (0.0133) 0.0360*** (0.0134) Cereal_land 0.5347*** (0.0894) 0.5181*** (0.0646) 0.5178*** (0.0648) 0.5168*

  𝐹𝑆 𝑖𝑡 = 𝜌 𝑊 𝐹𝑆 𝐽𝑡 + 𝛼 1 𝐶𝐶 𝑖𝑡 + 𝛼 2 𝑊𝐶𝐶 𝑖𝑡 + 𝛽 1 𝑋 𝑖𝑡 + 𝛽 2 𝑊𝑋 𝑖𝑡 + 𝜇 𝑖 + 𝜖 𝑖𝑡 𝐹𝑆 𝑖𝑡 is food security, W is the spatial weight matrix, 𝑋 𝑖𝑡 the other explanatory variables (country characteristics : GDP per capita, population density, cereal yields, arable lands, etc.) in country i at period t and 𝜌 is the spatial autocorrelation coefficient. 𝐶𝐶 𝑖𝑡 is the climatic change variable and our interest variable. 𝛼 1 , 𝛼 2 , 𝛽 1

	(11)
	where,
	advantage of producing unbiased estimates even if the underlying data generator
	process is a Spatial Autoregressive Model (Elhorst 2010b). Finally, we separately
	estimate the impact of climate change indicators as they are strongly correlated. The
	model to be estimated is written as follows:

we therefore estimate a Spatial Durbin Model (SDM) with fixed effect panel data. 3 This model includes spatially lagged endogenous and exogenous variables and takes into account temporal dimensions and observed and unobserved heterogeneity of countries. In addition, the SDM has the 2 LeSage (2014) distinguishes between local and global spillovers. Global spillovers arise if spatial spillovers concern the neighbors of country i, but also the neighbors of its neighbors, and so on. Local spillovers imply only interactions between direct neighbours.

3

Initially, we also try to estimate a Dynamic Spatial Durbin Model with fixed effects but unfortunately the model is non-concave.

Table 1 : Climate change and food availability: marginal effects

 1

		Food_day_g	Food_day_g	Food_day_g	Food_day_g
		Coef.	Std. Err.	Coef.	Std. Err.	Coef.	Std. Err.	Coef.	Std. Err.
	Direct
	Water balance variability -0.0131*	(0.0077)
	Drought			-0.0002*** (0.0001)
	Flood					0.0000	(0.0001)
	Extreme temperature							-0.0001** (0.0001)
	Water balance	0.0078**	(0.0032) 0.0080*** (0.0029) 0.0082*** (0.0032) 0.0081*** (0.0030)
	Lgdp	0.0370*** (0.0040) 0.0368*** (0.0038) 0.0367*** (0.0041) 0.0367*** (0.0040)
	Lpop_dens	0.0755*** (0.0154) 0.0756*** (0.0150) 0.0754*** (0.0162) 0.0754*** (0.0154)
	Pop_growth	-0.0026*** (0.0005) -0.0026*** (0.0005) -0.0026*** (0.0005) -0.0026*** (0.0005)
	Larable	0.0524*** (0.0042) 0.0524*** (0.0043) 0.0524*** (0.0044) 0.0523*** (0.0043)
	Cereal_land	0.6711*** (0.0176) 0.6726*** (0.0186) 0.6704*** (0.0177) 0.6715*** (0.0181)
	Polity2	-0.0104*** (0.0012) -0.0103*** (0.0012) -0.0103*** (0.0012) -0.0103*** (0.0012)
	LDependency_ratio	0.0738*** (0.0186) 0.0718*** (0.0212) 0.0727*** (0.0220) 0.0719*** (0.0224)
	Inflation	0.0031*** (0.0008) 0.0028*** (0.0008) 0.0029*** (0.0009) 0.0028*** (0.0008)
	Indirect
	Water balance variability -0.0205*	(0.0121)
	Drought			-0.0004*** (0.0001)
	Flood					-0.0002*** (0.0000)
	Extreme temperature							-0.0002*** (0.0001)
	Water balance	-0.0065	(0.0039) -0.0046	(0.0050) -0.0043	(0.0054) -0.0043	(0.0051)
	Lgdp	-0.0708*** (0.0061) -0.0712*** (0.0067) -0.0713*** (0.0072) -0.0714*** (0.0070)
	Lpop_dens	0.0110	(0.0304) 0.0096	(0.0306) 0.0100	(0.0321) 0.0098	(0.0307)
	Pop_growth	-0.0080*	(0.0046) -0.0071	(0.0048) -0.0075	(0.0053) -0.0072	(0.0051)
	Larable	0.0150**	(0.0072) 0.0158**	(0.0078) 0.0157**	(0.0080) 0.0156**	(0.0077)
	Cereal_land	0.2595*** (0.0486) 0.2456*** (0.0366) 0.2494*** (0.0337) 0.2455*** (0.0347)
	Polity2	0.0012	(0.0014) 0.0020	(0.0018) 0.0017	(0.0018) 0.0018	(0.0018)
	LDependency_ratio	-0.0484	(0.0988) -0.0458	(0.0992) -0.0460	(0.1056) -0.0443	(0.1018)
	Inflation	0.0020	(0.0015) 0.0006	(0.0014) 0.0009	(0.0014) 0.0008	(0.0014)
	Total
	Water balance variability -0.0336*** (0.0116)
	Drought			-0.0006*** (0.0001)
	Flood					-0.0001*** (0.0000)
	Extreme temperature							-0.0004*** (0.0001)
	Water balance	0.0014	(0.0012) 0.0033	(0.0023) 0.0039	(0.0025) 0.0038	(0.0023)
	Lgdp	-0.0338*** (0.0024) -0.0345*** (0.0031) -0.0347*** (0.0034) -0.0347*** (0.0031)
	Lpop_dens	0.0865*** (0.0157) 0.0853*** (0.0163) 0.0853*** (0.0166) 0.0852*** (0.0159)
	Pop_growth	-0.0106**	(0.0050) -0.0097*	(0.0052) -0.0101*	(0.0057) -0.0098*	(0.0056)
	Larable	0.0674*** (0.0036) 0.0682*** (0.0041) 0.0681*** (0.0042) 0.0679*** (0.0039)
	Cereal_land	0.9306*** (0.0480) 0.9183*** (0.0313) 0.9198*** (0.0288) 0.9170*** (0.0302)
	Polity2	-0.0092*** (0.0023) -0.0083*** (0.0027) -0.0086*** (0.0028) -0.0085*** (0.0027)
	LDependency_ratio	0.0253	(0.0827) 0.0260	(0.0806) 0.0267	(0.0860) 0.0276	(0.0816)
	Inflation	0.0051*** (0.0020) 0.0034**	(0.0016) 0.0038**	(0.0017) 0.0036**	(0.0017)
	Standard errors in parentheses; * p<0.10, * p<0.05, * p<0.01.

Table A

 A Standard errors in parentheses; * p<0.10, * p<0.05, * p<0.01. Table B.4: Climate change and food availability: General nesting stpatial model results

	Standard errors in parentheses; * p<0.10, * p<0.05, * p<0.01.
	.2. List of countries Table B.2: Climate change and total food availability: Estimation results Table B.3: Climate change and total food availability: Marginal effects
	Algeria Benin Bolivia Brazil Water balance variability -0.0345 Lfood_day_g Gabon Lfood_day_g Food_day_g Ghana Guatemala Honduras Coef. Std. Err. Coef. Lfood_day_g Lfood_day_g Food_day_g Std. Err. Panama Lfood_day_g Lfood_day_g Food_day_g Paraguay Peru Coef. Std. Err. Main (0.0395) Coef. Std. Err. Coef. Std. Err. Coef. Std. Err. Coef. Std. Err. Coef. Std. Err. Coef. Std. Err. Direct Water balance variability -0.0338 Main (0.0335) Drought -0.0011*** (0.0003) Water balance variability -0.0134 (0.0089) Philippines Burkina Faso Hungary Rwanda Burundi India Flood -0.0001 (0.0001) Water balance 0.0184*** (0.0068) 0.0199*** (0.0062) 0.0206*** (0.0065) Senegal Drought Extreme temperature -0.0011*** (0.0003) Flood -0.0001 (0.0001) Drought -0.0002*** (0.0001)	Lfood_day_g Lfood_day_g Food_day_g Coef. Std. Err. Coef. Std. Err. Coef. Std. Err. 0.0208*** (0.0065) 0.0010*** (0.0003)
	Cameroon Extreme temperature Lgdp	Indonesia 0.3445*** (0.0133) 0.3431*** (0.0139) Sierra Leone 0.3431*** (0.0138)	0.0010*** (0.0003) 0.3430*** (0.0128)
	Water balance Lpop_dens Pop_growth	Central African Republic 0.4881*** (0.0370) Korea, Rep. 0.4875*** (0.0416) South Africa 0.4872*** (0.0401) 0.0178*** (0.0063) 0.0194*** (0.0059) 0.0201*** (0.0059) 0.0203*** (0.0058) 0.4868*** (0.0365) -0.0075 (0.0052) -0.0072 (0.0054) -0.0072 (0.0055) -0.0074 (0.0052)
	Lgdp Larable	Chad	Liberia 0.3420*** (0.0133) 0.3408*** (0.0127) 0.3408*** (0.0127) 0.3409*** (0.0126) Sri Lanka 0.1211*** (0.0256) 0.1208*** (0.0272) 0.1203*** (0.0264) 0.1207*** (0.0251)
	Lpop_dens Cereal_land	Chile	Madagascar 0.4855*** (0.0417) 0.4842*** (0.0413) 0.4842*** (0.0415) 0.4848*** (0.0412) Sudan 2.8417*** (0.1449) 2.8387*** (0.1290) 2.8285*** (0.1266) 2.8174*** (0.1158)
	China Colombia LDependency_ratio Pop_growth Polity2 Congo, Rep. -0.0093* -0.0325*** (0.0042) Malaysia Mauritania (0.0056) -0.0091 -0.0315*** (0.0042) Syrian Arab Republic (0.0056) -0.0091 (0.0056) -0.0092 -0.0319*** (0.0041) -0.0320*** (0.0042) (0.0056) 0.4929*** (0.0974) 0.4933*** (0.0840) 0.4918*** (0.0958) 0.4912*** (0.0961) Togo Mexico Cereal_land 2.8415*** (0.1587) 2.8390*** (0.1335) 2.8275*** (0.1306) 2.8172*** (0.1278) Trinidad and Tobago Larable 0.1221*** (0.0282) 0.1221*** (0.0276) 0.1217*** (0.0276) 0.1212*** (0.0275) Inflation 0.0070*** (0.0025) 0.0057** (0.0025) 0.0060** (0.0025) 0.0063*** (0.0023)
	Costa Rica Cote d'Ivoire Dominican Republic -0.0320*** (0.0042) -0.0311*** (0.0044) -0.0314*** (0.0043) -0.0316*** (0.0043) Morocco Tunisia Nepal Turkey Nicaragua Uruguay LDependency_ratio Polity2 Total Water balance variability -0.2615*** (0.0405) 0.5010*** (0.0971) 0.4936*** (0.0964) 0.4941*** (0.0963) 0.4941*** (0.0968) Drought -0.0015*** (0.0005)
	Ecuador Egypt, Arab Rep. 0.0065*** (0.0025) 0.0053** Niger Extreme temperature Inflation Flood Nigeria Water balance -0.0391** (0.0152) -0.0193 Wx Lgdp -0.2870*** (0.0280) -0.2961*** (0.0358) (0.0024) 0.0056** -0.0004*** (0.0001) (0.0025) 0.0060** Venezuela -0.0002 (0.0135) -0.0178 (0.0130) -0.0156 -0.2973*** (0.0347) -0.2980*** (0.0314) (0.0024) (0.0002) (0.0120)
	Water balance variability -0.2802*** (0.0428) Lpop_dens -0.3553*** (0.0736)	-0.3609*** (0.0873)	-0.3621*** (0.0825)	-0.3597*** (0.0761)
	Drought Pop_growth		-0.2203*** (0.0125)	-0.0017*** (0.0004) -0.2198*** (0.0139)	-0.2208*** (0.0142)	-0.2229*** (0.0143)
	Flood Larable Cereal_land		0.3119*** (0.0345) 1.6168*** (0.3453)	0.3154*** (0.0476) 1.4737*** (0.2934)	0.3150*** (0.0432) -0.0004*** (0.0001) 1.4778*** (0.2853)	0.3114*** (0.0388) 1.4616*** (0.2574)
	Extreme temperature Polity2	0.0068	(0.0062)	0.0105			(0.0084)	0.0094		(0.0070)	-0.0001 0.0077	(0.0003) (0.0072)
	Water balance LDependency_ratio	-0.0394** (0.0167) -0.0177 -0.0974 (0.3685) -0.1080		(0.0130) -0.0161 (0.3843) -0.0939	(0.0128) -0.0140 (0.3905) -0.1014	(0.0126) (0.3730)
	Lgdp Inflation		-0.2793*** (0.0318) -0.2887*** (0.0365) -0.2896*** (0.0363) -0.2913*** (0.0360) -0.0046 (0.0084) -0.0138* (0.0083) -0.0125 (0.0084) -0.0108 (0.0086)
	Lpop_dens Indirect		-0.3387*** (0.0842) -0.3465*** (0.0898) -0.3475*** (0.0896) -0.3475*** (0.0890)
	Pop_growth Water balance variability -0.2953*** (0.0265) -0.2406*** (0.0178) -0.2377*** (0.0190) -0.2384*** (0.0190) -0.2406*** (0.0190)
	Larable Drought Flood		-0.0026*** (0.0006) 0.3453*** (0.0465) 0.3422*** (0.0509) 0.3408*** (0.0497) 0.3374*** (0.0497) -0.0005** (0.0002)
	Cereal_land Extreme temperature	1.9777*** (0.5006) 1.7693*** (0.3485) 1.7664*** (0.3428) 1.7512*** (0.3378) 0.0008** (0.0004)
	Polity2 Water balance		0.0035 -0.0207	(0.0062) 0.0079 (0.0151) 0.0006		(0.0081) 0.0067 (0.0115) 0.0028	(0.0078) 0.0050 (0.0111) 0.0052	(0.0083) (0.0103)
	LDependency_ratio Lgdp Lpop_dens	-0.0893 0.0574*** (0.0187) (0.4326) -0.1100 0.0470* 0.1328*** (0.0426) 0.1266**	(0.4348) -0.1040 (0.0252) 0.0459* (0.0507) 0.1251*** (0.0483) (0.4305) -0.1052 (0.0241) 0.0450** 0.1271*** (0.0457) (0.0225) (0.4294)
	Inflation Pop_growth		-0.0036 -0.2279*** (0.0145) (0.0082) -0.0138 -0.2270*** (0.0163) (0.0086) -0.0124 -0.2280*** (0.0166) (0.0082) -0.0104 -0.2303*** (0.0165) (0.0084)
	Larable		0.4330*** (0.0233)	0.4362*** (0.0301)	0.4354*** (0.0281)	0.4321*** (0.0271)
	Rho Cereal_land Polity2		4.4586*** (0.4707) -0.0867*** (0.0334) -0.0739** (0.0294) -0.0720** (0.0295) -0.0705** (0.0295) 4.3124*** (0.3996) 4.3064*** (0.3893) 4.2789*** (0.3480) -0.0257*** (0.0043) -0.0210*** (0.0069) -0.0225*** (0.0058) -0.0243*** (0.0056)
	Sigma2_e LDependency_ratio	0.0201*** (0.0026) 0.0201*** (0.0026) 0.0202*** (0.0026) 0.0201*** (0.0026) 0.3956 (0.3323) 0.3852 (0.3554) 0.3979 (0.3480) 0.3898 (0.3273)
	Observations Inflation		0.0024	2040 (0.0083)	-0.0081	2040	(0.0080)	-0.0065	2040	(0.0082)	-0.0045	2040	(0.0084)
	Log-pseudolikelihood		-1087		-1087		-1088		-1091
	Standard errors in parentheses; * p<0.10, * p<0.05, * p<0.01.

Acknowledgements

This work has benefited from the support of the ECOWAS staff exchange program.

This work was supported by the LABEX IDGM+ (ANR-10-LABX-14-01) within the program "Investissements d'Avenir" operated by the French National Research Agency (ANR). + 𝛾 𝑗 -𝛿 𝑗 + 𝐶𝑆 𝑗 4

(7)

𝐹𝐷 𝑇𝑖 * = 3𝛾 𝑖 + 𝛿 𝑖 -𝛾 𝑗 + 𝛿 𝑗 -𝐶𝑆 𝑗 4

(8)

𝐹𝑆 𝑇𝑗 * = 𝛾 𝑖 -𝛿 𝑖 + 𝛾 𝑗 + 3𝛿 𝑗 -3𝐶𝑆 𝑗 4

(

Appendix

A. Description, data sources and list of countries 0.170 0.106 -0.064 0.000 0.000 0.000 0.000 0.500 0.500 0.000 0.000 0.000 South Africa 0.081 0.109 0.028 0.000 2.600 2.600 0.000 3.300 3.300 0.000 2.600 2.600 Sri Lanka 0.262 0.186 -0.076 0.100 0.300 0.200 0.100 0.400 0.300 0.000 0.000 0.000 Sudan 0.032 0.048 0.017 0.100 0.900 0.800 0.100 1.100 1.000 0.000 0.000 0.000 Syrian Arab Republic 0.103 0.086 -0.017 0.300 0.000 -0.300 0.300 0.000 -0.300 0.000 0.000 0.000 Togo 0.114 0.144 0.030 0.000 0.400 0.400 0.000 0.400 0.400 0.000 0.000 0.000 Trinidad and Tobago 0.143 0.219 0.075 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Tunisia 0.074 0.070 -0.003 0.100 0.000 -0.100 0.200 0.000 -0.200 0.000 0.000 0.000 Turkey 0.076 0.065 -0.011 0.000 0.000 0.000 0.400 0.300 -0.100 0.200 0.100 -0.100 Uruguay 0.172 0.255 0.083 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Venezuela 0.096 0.269 0.174 1.200 0.000 -1.200 2.000 0.000 -2.000 0.000 0.000 0.000

B. Empirical Results


Bertrand Cloez
email: bertrand.cloez@supagro-inra.fr

Marie - Noémie

Marie-Noémie Thai

FLEMING-VIOT PROCESSES : TWO EXPLICIT EXAMPLES

research documents, whether they are published or not. The documents may come

INTRODUCTION

In discrete space, the Fleming-Viot particle system has been studied by many authors [START_REF] Asselah | Quasistationary distributions and Fleming-Viot processes in finite spaces[END_REF][START_REF] Asselah | Fleming-Viot selects the minimal quasi-stationary distribution: The Galton-Watson case[END_REF][START_REF] Asselah | A note on the rightmost particle in a Fleming-Viot process[END_REF][START_REF] Ferrari | Quasi stationary distributions and Fleming-Viot processes in countable spaces[END_REF][START_REF] Marić | Fleming-Viot particle system driven by a random walk on $\mathbb{N}[END_REF]. Such a system is a mean field particle system described in the following way: we consider N copies of an absorbed Markov chain and, instead of being absorbed, one chain jumps randomly on the state of another one. It is well known that, when the number of copies tends to infinity, the empirical measure converges to the law of the initial chain conditioned not to be absorbed, see for instance [START_REF] Cloez | Quantitative results for the Fleming-Viot particle system and quasi-stationary distributions in discrete space[END_REF][START_REF] Del Moral | A Moran particle system approximation of Feynman-Kac formulae[END_REF][START_REF] Groisman | Simulation of quasi-stationary distributions on countable spaces[END_REF][START_REF] Villemonais | General approximation method for the distribution of Markov processes conditioned not to be killed[END_REF]. Convergence to equilibrium as time goes to infinity is less known. In [START_REF] Rousset | On the control of an interacting particle estimation of schrödinger ground states[END_REF][START_REF] Villemonais | Interacting particle systems and Yaglom limit approximation of diffusions with unbounded drift[END_REF], this question is addressed for some models. Nevertheless, to our knowledge, there are few results on the expression of the invariant distribution or on the explicit rates of convergence. This paper is concerned with studying two specific models for which the invariant distribution is explicit and extend the investigation started by Cloez and Thai [START_REF] Cloez | Quantitative results for the Fleming-Viot particle system and quasi-stationary distributions in discrete space[END_REF]. Let Q = (Q i,j ; i, j ∈ F * ∪ {0}) be the transition rates matrix of an irreducible and positive recurrent continuous time Markov process (X t ) t≥0 on a countable state space F = F * ∪ {0}. We think of 0 as an absorbing state. Let µ be the initial law of (X t ) t≥0 and let µT t be its law at time t conditioned on non absorption up to time t. That is defined, for all non-negative function f on F * , by µT t f = µP t f µP t 1 {0} c = y∈F * P t f (y)µ(y)

y∈F * P t 1 {0} c (y)µ(y)
, where (P t ) t≥0 is the semigroup associated with the transition matrix Q and we use the convention f (0) = 0. For every x ∈ F * , k ∈ F * and non-negative function f on F * , we also set

T t f (x) = δ x T t f and µT t (k) = µT t 1 {k} , ∀t ≥ 0.
A quasi-stationary distribution (QSD) for Q is a probability measure ν qs on F * satisfying, for every t ≥ 0, ν qs T t = ν qs .

The particle system we are focusing on was initially introduced in [START_REF] Del Moral | On the stability of measure valued processes with applications to filtering[END_REF][START_REF] Del Moral | A Moran particle system approximation of Feynman-Kac formulae[END_REF] for approximating the conditioned semigroup (T t ) t≥0 and the QSD ν qs . It is convenient to think of particles as being indistinguishable, and to consider the occupation number η with, for k ∈ F * , η(k) = η (N ) (k) representing the number of particles at site k. The configuration (η t ) t≥0 is a Markov process with state space E = E (N ) defined by

E = η : F * → N | i∈F * η(i) = N .
Date: February 25, 2016.

Applying its generator to a bounded function f gives

Lf (η) = L (N ) f (η) = i∈F * η(i)   j∈F * (f (T i→j η) -f (η)) Q i,j + Q i,0 η(j) N -1   , (1)
for every η ∈ E, where, if η(i) = 0, the configuration T i→j η is defined by T i→j η(i) = η(i) -1, T i→j η(j) = η(j) + 1, and T i→j η(k) = η(k) k / ∈ {i, j}.

The present paper is a continuation of [START_REF] Cloez | Quantitative results for the Fleming-Viot particle system and quasi-stationary distributions in discrete space[END_REF] in which the following limits are studied and quantified:

µ N t t→∞ N →∞ μT t t→∞ ! ! µ N ∞ N →∞ } } ν qs
where µ N is the associated empirical distribution of the particle system defined, for η ∈ E, by

µ N t = 1 N k∈F * η(k)δ {k} .
For countable space F , the ergodicity of the Fleming-Viot process is not guaranteed. In [START_REF] Cloez | Quantitative results for the Fleming-Viot particle system and quasi-stationary distributions in discrete space[END_REF],

Cloez and Thai show that under some conditions, the particle system converges exponentially fast to equilibrium for a suitable Wasserstein coupling distance. Let us recall the different distances given by the autors. For η, η ∈ E, let d be the distance defined by

d(η, η ) = 1 2 j∈F |η(j) -η (j)|, (2)
and for any two probability measures µ and µ on E, let W d (µ, µ ) be the Wasserstein coupling distance between these two laws defined by

W d (µ, µ ) = inf X∼µ X∼µ E d(X, X) , (3)
where the infimum runs over all the couples of random variables with marginal laws µ and µ .

Theorem 1.1 (Theorem 1.1 of [8]). Let λ = inf i,i ∈F *   Q i,i + Q i ,i + j =i,i Q i,j ∧ Q i ,j   and for i ∈ F * , p 0 (i) = Q i,0 . If ρ = λ -(sup(p 0 ) -inf(p 0 )
) then for any processes (η t ) t>0 and (η t ) t>0 generated by (1), and for any t ≥ 0, we have

W d (Law(η t ), Law(η t )) ≤ e -ρt W d (Law(η 0 ), Law(η 0 )).
In particular, if ρ > 0 then there exists a unique invariant distribution ν N satisfying for every

t ≥ 0, W d (Law(η t ), ν N ) ≤ e -ρt W d (Law(η 0 ), ν N ).
This theorem gives the existence and uniqueness of the invariant distribution, but this one is not explicit. Actually, there are few models for which an explicit formula of the invariant distribution is given. This observation is one of the motivations of the present paper. So, it is interesting to consider and study a model satisfying the previous point : the example of random walk on a complete graph. An interesting point of the complete graph approach is that it permits to reduce the difficulties of the Fleming-Viot to the interaction. Due to its simple geometry, several explicit formulas are obtained such as the invariant distribution, the correlations and the spectral gap. It seems to be new in the context of Fleming-Viot particle systems. A second model for which the invariant distribution is explicit is the two point case, the study of the particle system is then reduced to a birth and death process with quadratic rates. The bound obtained in Theorem 1.1 is not optimal. Nevertheless, the coupling introduced in [START_REF] Cloez | Quantitative results for the Fleming-Viot particle system and quasi-stationary distributions in discrete space[END_REF] in order to prove Theorem 1.1, permits us to obtain the spectral gap as rate of convergence. Moreover, we show that the spectral gap of the Fleming-Viot process is always bounded from below by a positive constant not depending on the number of particles.

The remainder of the paper is as follows. Section 2 is dedicated to the study of random walk on the complete graph and Section 3 to that of the two point case.

COMPLETE GRAPH DYNAMICS

In all this section, we study the example of a random walk on the complete graph. Let us fix K ∈ N * , p > 0 and N ∈ N * , the dynamics of this example is as follows: we consider a model with N particles and K +1 vertices 0, 1, . . . , K. The N particles move on the K vertices 1, . . . , K uniformly at random and jump to 0 with rate p. When a particle reaches the node 0, it jumps instantaneously over another particle chosen uniformly at random. This particle system corresponds to the model previously cited with parameters

Q i,j = 1 K , ∀i, j ∈ F * = {1, . . . , K}, i = j and Q i,0 = p, ∀i ∈ F * .
The generator of the associated Fleming-Viot process is then given by

Lf (η) = K i=1 η(i)   K j=1 (f (T i→j η) -f (η)) 1 K + p η(j) N -1   , (4)
for every function f and η ∈ E.

A process generated by ( 4) is an instance of inclusion processes studied in [START_REF] Giardinà | Correlation inequalities for interacting particle systems with duality[END_REF][START_REF] Grosskinsky | Condensation in the inclusion process and related models[END_REF][START_REF] Grosskinsky | Dynamics of condensation in the symmetric inclusion process[END_REF]. It is then related to models of heat conduction. One main point of [START_REF] Giardinà | Correlation inequalities for interacting particle systems with duality[END_REF][START_REF] Grosskinsky | Condensation in the inclusion process and related models[END_REF] is a criterion ensuring the existence and reversibility of an invariant distribution for the inclusion processes. In particular, they give an explicit formula of the invariant distribution of a process generated by (4) and we give this expression in Subsection 2.3. They also study different scaling limits which seem to be irrelevant for our problems. Another application of this example comes from population genetics. Indeed, this model can also be referred as neutral evolution, see for instance [START_REF] Etheridge | Some mathematical models from population genetics[END_REF][START_REF] Watterson | Reversibility and the age of an allele. I. Moran's infinitely many neutral alleles model[END_REF]. More precisely, consider N individuals possessing one type in F * = {1, . . . , K} at time t. Each pair of individuals interacts at rate p. Upon an interacting event, one individual dies and the other one reproduces. In addition, every individual changes its type (mutates) at rate 1 and chooses uniformly at random a new type in F * . The measure µ N t gives the proportions of types. The kind of mutation we consider here is often referred as parent-independent or the house-of-cards model. In all this section, for any probability measure µ on E, we set in a classical manner

E µ [•] = F * E x [•]µ(dx) and P µ = E µ [1 • ];
similarly Cov µ and Var µ are defined with respect to E µ .

2.1. The associated killed process. We define the process (X t ) t≥0 by setting

X t = Z t if t < τ 0 if t ≥ τ,
where τ is an exponential variable with mean 1/p and (Z t ) t≥0 is the classical complete graph random walk (i.e. without extinction) on {1, . . . , K}. We have, for any bounded function f ,

T t f (x) = E [f (X t ) | X 0 = x, X t = 0] , t ≥ 0, x ∈ F * .
The conditional distribution of X t is simply given by the distribution of Z t :

P(X t = i | X t = 0) = P(Z t = i).
The study of (Z t ) t≥0 is trivial. Indeed, it converges exponentially fast to the uniform distribution π K on {1, . . . , K}. We deduce that for all t ≥ 0 and all initial distribution µ,

d TV (µT t , π K ) = K i=1 |P µ (X t = i | τ > t) -π K (i)| ≤ e -t .
Thus in this case, the conditional distribution of X converges exponentially fast to the Yaglom limit π K .

2.2.

Correlations at fixed time. The special form of L, defined at (4), makes the calculation of the two-particle correlations at fixed time easy.

Theorem 2.1 (Two-particle correlations). For all k, l ∈ {1, . . . , K}, k = l and any probability measure µ on E, we have for all t ≥ 0

Cov µ (η t (k), η t (l)) = E µ [η 0 (k)η 0 (l)] e - 2K(N -1+p) K(N -1) t + -N + 1 + 2pN K(N -1 + 2p) (E µ [η 0 (k)] + E µ [η 0 (l)])e -t -E µ [η 0 (k)] E µ [η 0 (l)] e -2t + -N 2 (p + 1) + N K 2 (N -1 + p) .
Remark 2.2 (Limit t → +∞). By the previous theorem, we find for any probability measure µ

lim t→+∞ Cov µ (η t (k), η t (l)) = -N 2 (p + 1) + N K 2 (N -1 + p) = Cov(η(k), η(l)),
where η is distributed according to the invariant distribution; it exists since the state space is finite, see the next section.

Remark 2.3 (Limit N → +∞). If Cov µ (η 0 (k), η 0 (l)) = 0 then for all k, l ∈ {1, . . . , K}, k = l and any probability measure µ, we have

Cov µ η t (k) N , η t (l) N ∼ N e -2t Cov µ η 0 (k) N , η 0 (l) N ,
where

u N ∼ N v N iff lim N →+∞ u N v N = 1.
Proof of Theorem 2.1. For k, l ∈ {1, .., K}, let ψ k,l be the function η → η(k)η(l). Applying the generator (4) to ψ k,l we obtain

Lψ k,l (η) = - 2K(N -1 + p) K(N -1) η(k)η(l) + N -1 K (η(k) + η(l)).
So, for all t ≥ 0,

Lψ k,l (η t ) = - 2K(N -1 + p) K(N -1) η t (k)η t (l) + N -1 K (η t (k) + η t (l)).
Using Kolmogorov's equation, we have

∂ t E µ (η t (k)η t (l)) = - 2K(N -1 + p) K(N -1) E µ (η t (k)η t (l)) + N -1 K (E µ (η t (k)) + E µ (η t (l))). (5) Now if ϕ k (η) = η(k) then Lϕ k (η) = N K -η(k).
We deduce that, for every t ≥ 0,

∂ t E µ (η t (k)) = N K -E µ (η t (k)) and E µ (η t (k)) = E µ (η 0 (k))e -t + N K .
Solving equation ( 5) ends the proof.

2.3. Properties of the invariant measure. As (η t ) t≥0 is an irreducible Markov chain on a finite state space, it is straightforward that it admits a unique invariant measure. In fact, this invariant distribution is reversible and we know its expression.

Theorem 2.4 (Invariant distribution). The process (η t ) t≥0 admits a unique invariant and reversible measure ν N , which is defined, for every η ∈ E, by

ν N ({η}) = Z -1 K i=1 η(i)-1 j=0 N -1 + Kpj j + 1 ,
where Z is a normalizing constant.

This result is a slight generalisation of [13, Section 4] and [15, Theorem 2.1].

Proof. A measure ν is reversible if and only if it satisfies the following balance equation

ν({η})C(η, ξ) = ν({ξ})C(ξ, η) (6)
where

ξ = T i→j η and C(η, ξ) = L1 ξ (η) = η(i)(K -1 + pη(j)(N -1) -1 ).
Due to the geometry of the complete graph, it is natural to consider that ν has the following form

ν({η}) = 1 Z K i=1 l(η(i)),
where l : {0, . . . , N } → [0, 1] is a function and Z is a normalizing constant. From (6), we have

l(η(i))l(η(j))η(i)(N -1 + Kpη(j)) = l(η(i) -1)l(η(j) + 1)(η(j) + 1)(N -1 + Kp(η(i) -1)),
for all η ∈ E and i, j ∈ {1, . . . K}. Hence,

l(n) l(n -1) n N -1 + Kp(n -1) = l(m) l(m -1) m N -1 + Kp(m -1) = u,
for every m, n ∈ {1, . . . , N } and some u ∈ R. Finally,

ν({η}) = K i=1   u η(i) η(i)-1 j=0 N -1 + Kpi i + 1 l(0)   = l(0) K u N K i=1 η(i)-1 j=0 N -1 + Kpj j + 1 , and Z = 1/(l(0) K u N ).
In particular, we have directly Corollary 2.5 (Invariant distribution when p = 1/K). If p = 1/K then the process (η t ) t≥0 admits a unique invariant and reversible measure ν N , which is defined, for every η ∈ E, by

ν N ({η}) = Z -1 K i=1 N -2 + η(i) N -2 ,
where Z is a normalizing constant given by

Z = (K + 1)N -K -1 KN -K -1 .
Corollary 2.6 (Marginal laws when p = 1/K). If p = 1/K then for all i ∈ {1, . . . , K} we have

P ν N (η(i) = x) = 1 Z N -2 + x N -2 KN -K -x (K -1)N -K ,
Proof. Firstly let us recall the Vandermonde binomial convolution type formula: let n, n 1 , . . . , n p be some non-negative integers satisfying

p i=1 n i = n, we have r -1 n -1 = r 1 +•••+rp=r p j=1 r j -1 n j -1 .
The proof is based on the power series decomposition of z

→ (z/(1 -z)) n = p i=1 (z/(1 -z)) n i .
Using this formula, we find

P ν N (η(i) = x) = x∈E 1 P ν N (η = (x 1 , . . . , x i-1 , x, x i+1 . . . , x K )) = 1 Z N -2 + x N -2 x∈E 1 i-1 l=1 K l=i+1 N -2 + x l N -2 = 1 Z N -2 + x N -2 (K -1)(N -1) + N -x -1 (K -1)(N -1) -1 ,
where

E 1 = {x = (x 1 , . . . , x i-1 , x i+1 . . . , x K )|x 1 + • • • + x i-1 + x i+1 • • • + x K = N -x}.
We are now able to express the particle correlations under this invariant measure.

Theorem 2.7 (Correlation estimates). For all i = j ∈ {1, . . . , K}, we have

|Cov ν N (η(i)/N, η(j)/N )| ∼ N p + 1 K 2 N ,
Proof. Let η be a random variable with law ν N . As η(1), . . . , η(K) are identically distributed and

K i=1 η(i) = N we have Cov ν N (η(i)/N, η(j)/N ) = - Var ν N (η(i)/N ) K -1 .
Using the results of Section 2.4, we have

L(η(i) 2 ) = η(i) 2 -2 - 2p N -1 + η(i) 2N K + 2pN N -1 + K -2 K + N K .
Using the fact that L(η(i) 2 )dν N = 0 and η(i)dν N = N K , we deduce that

η(i) 2 dν N = N [(2N + K -2)(N -1) + 2KN p + K(N -1)] 2K 2 (N -1 + p) .
Finally,

Var ν N (η(i)) = η(i) 2 dν N - η(i)dν N 2 = N (K -1)(N p + N -1) K 2 (N -1 + p) ,
and thus, for i = j,

|Cov ν N (η(i)/N, η(j)/N )| ∼ N p + 1 K 2 N .
Remark 2.8 (Proof through coalescence methods). Maybe we can use properties of Kingman's coalescent type process (which is a dual process) to recover some of our results (as for instance the previous correlation estimates). Indeed, after an interacting event, all individuals evolve independtly and it is enough to look when the first mutation happens (backwards in time) on one of the genealogical tree branches. Nevertheless, we prefer to use another approach based on Markovian techniques.

Remark 2.9 (Number of sites). Theorem 2.7 gives the rate of the decay of correlations with respect to the number of particles, but we also have a rate with respect to the number of sites K.

For instance when p = 1/K and if η is distributed under the invariant measure, then

|Cov ν N (η(i)/N, η(j)/N )| ∼ K 1 K(K -1)N .
The previous theorem shows that the occupation numbers of two distinct sites become noncorrelated when the number of particles increases. In fact, Theorem 2.7 leads to a propagation of chaos:

Corollary 2.10 (Convergence to the QSD). We have

E ν N d TV (µ N , π K ) ≤ K(p + 1) N ,
where π K is the uniform measure on {1, . . . , K}.

Proof. By the Cauchy-Schwarz inequality, we have

E ν N η(k) N - 1 K ≤ E ν N η(k) N - 1 K 2 1 2 = Var ν N η(k) N 1/2 ≤ (K -1)(p + 1) K 2 N .
Summing over {1, . . . , K} ends the proof.

Result [8, Theorem 1.2] and its corollaries states that there exist C, θ > 0 such that

E ν N d TV (µ N , π K ) ≤ C N θ .
All constants are explicit and we have θ < 1/2. The last corollary then gives a better bound. To our knowledge, it is the first time that this rate of convergence is obtained for the Fleming-Viot process in discrete space. With spectral arguments, this type of result was obtained for diffusion processes in [START_REF] Rousset | On the control of an interacting particle estimation of schrödinger ground states[END_REF]. This bound is achieved because of the absence of bias term. Indeed,

∀k ∈ F * , E ν N [µ N (k)] = 1 K = π K (k).
The bad term in [8, Theorem 1.2] comes from, with the notations of its proof, the estimation of |u k (t) -v k (t)| and Gronwall Lemma.

Remark 2.11 (Parameters depending on N ). A nice application of explicit rates of convergence is to consider parameters depending on N . For instance, we can now consider that p = p N depends on N , this does not change neither the conditioned semi-goup nor the QSD but this changes the dynamics of our interacting-particle system. The last corollary gives that if lim

N →∞
p N /N = 0 then the empirical measure converges to the uniform measure.

2.4.

Long time behavior and spectral analysis of the generator. In this subsection, we point out the optimality of Theorem 1.1 in this special case. Conditions in Theorem 1.1, which seems to be a bit strong, are tight in the complete graph dynamics. In that case, λ = ρ = 1 and the bound obtained is optimal in terms of contraction. Moreover, the obtained rate is exactly the spectral gap.

Corollary 2.12 (Wasserstein contraction). For any processes (η t ) t>0 and (η t ) t>0 generated by (4), and for any t ≥ 0, we have

W d (Law(η t ), Law(η t )) ≤ e -t W d (Law(η 0 ), Law(η 0 )).
In particular, when (η 0 ) follows the invariant distribution ν N associated to (4), we get for every

t ≥ 0 W d (Law(η t ), ν N ) ≤ e -t W d (Law(η 0 ), ν N ).
In particular, if λ 1 is the smallest positive eigenvalue of -L, defined at (4), then we have

1 = ρ ≤ λ 1 .
Indeed, on the one hand, let us recall that, as the invariant measure is reversible, λ 1 is the largest constant such that lim

t→+∞ e 2λt R t f -ν N (f ) 2 L 2 (ν N ) = 0, (7)
for every λ < λ 1 and f ∈ L 2 (ν N ), where (R t ) t≥0 is the semi-group generated by L. See for instance [START_REF] Bakry | L'hypercontractivité et son utilisation en théorie des semigroupes[END_REF][START_REF] Saloff-Coste | Lectures on finite Markov chains[END_REF]. On the other hand, if λ < 1 then, by Theorem 1.1, we have

e 2λt R t f -ν N (f ) 2 L 2 (ν N ) = e 2λt E ((δ η R t )f -(ν N R t )f ) 2 ν N (dη) ≤ 2e 2λt f 2 ∞ E W d (δ η R t , ν N R t ) 2 ν N (dη) ≤ 2e 2(λ-1)t f 2 ∞ E W d (δ η , ν N ) 2 ν N (dη),
and then (7) holds. Now, the constant functions are trivially eigenvectors of L associated with the eigenvalue 0, and if, for k ∈ {1, . . . , K}, l ≥ 1 we set ϕ (l) k : η → η(k) l then the function ϕ

(1) k satisfies Lϕ

(1)

k = N/K -ϕ (1)
k .

In particular ϕ

(1) k -N/K is an eigenvector and 1 is an eigenvalue of -L. This gives λ 1 ≤ 1 and finally λ 1 = 1 is the smallest eigenvalue of -L. By the reversibility, we have a Poincaré (or spectral gap) inequality

∀t ≥ 0, R t f -ν N (f ) 2 L 2 (ν N ) ≤ e -2t f -ν N (f ) 2 L 2 (ν N ) .
Remark 2.13 (Complete graph random walk). If (a i ) 1≤i≤K is a sequence such that

K i=1 a i = 0 then the function K i=1 ϕ (1)
i is an eigenvector of L. However, if L is the generator of the classical complete graph random walk, La = -a and then a is also an eigenvector of L with the same eigenvalue.

Let us finally give the following result on the spectrum of L:

Lemma 2.14 (Spectrum of -L). The spectrum of -L is included in

K i=1 λ l i | l 1 , . . . , l K ∈ {0, . . . , N } ,
where ∀l ∈ {0, . . . , N }, λ l = l + l(l -1)p N -1 .

Proof. For every k ∈ {1, . . . , K} and l ∈ {0, . . . , N }, we have

Lϕ (l) k (η) = -λ l ϕ (l) k (η) + Q l-1 (η)
, where Q l-1 is a polynomial whose degree is less than l -1. A straightforward recurrence shows that whether there exists or not a polynomial function ψ

(l) k , whose degree is l, satisfying Lψ (l) k = -λ l ψ (l) k (namely ψ (l)
k is an eigenvector of L). Indeed, it is possible to have ψ (l) k = 0 since the polynomial functions are not linearly independent (F is finite). More generally, for all l 1 , . . . , l K ∈ {1, . . . , N }, there exists a polynomial Q with K variables, whose degree with respect to the i th variable is strictly less than l i , such that the function φ

: η → K i=1 η(k i ) l i + Q(η) satisfies Lφ = -λφ where λ = K i=1 λ l i .
Again, provided that φ = 0, φ is an eigenvector and λ an eigenvalue of -L. Finally, as the state space is finite, using multivariate Lagrange polynomial, we can prove that every function is polynomial and thus we capture all the eigenvalues.

Remark 2.15 (Cardinal of E). As card(F

* ) = K, we have card(E) = N + K -1 K -1 = (N + K -1)! N !(K -1)! .
In particular, the number of eigenvalues is finite and less than card(E).

Remark 2.16 (Marginals). For each k, the random process (η t (k)) t≥0 , which is a marginal of a process generated by (4), is a Markov process on N N = {0, . . . , N } generated by

Gf (x) = (N -x) 1 K + px N -1 (f (x + 1) -f (x)) + x K -1 K + p(N -x) N -1 (f (x -1) -f (x)),
for every function f on N N and x ∈ N N . We can express the spectrum of this generator. Indeed, let ϕ l : x → x l , for every l ≥ 0. The family (ϕ l ) 0≤l≤N is linearly independent as can be checked with a Vandermonde determinant. This family generates the L 2 -space associated to the invariant measure since this space has a dimension equal to N + 1. Now, similarly to the proof of the previous lemma, we can prove the existence of N + 1 polynomials, which are eigenvectors and linearly independent, whose eigenvalues are λ 0 , λ 1 , . . . , λ N .

THE TWO POINT SPACE

In all this section we denote by p 0 the function i ∈ F * → Q i,0 . We consider a Markov chain defined on the states {0, 1, 2} where 0 is the absorbing state. Its infinitesimal generator G is defined by

G =   0 0 0 p 0 (1) -a -p 0 (1) a p 0 (2) b -b -p 0 (b),  
where a, b > 0, p 0 (1), p 0 (2) ≥ 0 and p 0 (1) + p 0 (2) > 0. The generator of the Fleming-Viot process with N particles applied to bounded functions f :

E → R reads Lf (η) = η(1) a + p 0 (1) η(2) N -1 (f (T 1→2 η) -f (η)) + η(2) b + p 0 (2) η(1) N -1 (f (T 2→1 η) -f (η)). (8)
3.1. The associated killed process. The long time behavior of the conditionned process is related to the eigenvalues and eigenvectors of the matrix:

M = -a -p 0 (1) a b -b -p 0 (2) .
Indeed see [18, section 3.1]. Its eigenvalues are given by

λ + = -(a + b + p 0 (1) + p 0 (2)) + (a -b + p 0 (1) -p 0 (2)) 2 + 4ab 2 , λ -= -(a + b + p 0 (1) + p 0 (2)) -(a -b + p 0 (1) -p 0 (2)) 2 + 4ab 2 ,
and the corresponding eigenvectors are respectively given by

v + = a -A + √ A 2 + 4ab and v -= a -A - √ A 2 + 4ab , where A = a -b + p 0 (1) -p 0 (2). Also set ν = v + /(v + (1) + v + (2)
). From these properties, we deduce that Lemma 3.1 (Convergence to the QSD). There exists a constant C > 0 such that for every initial distribution µ, we have

∀t ≥ 0, d TV (µT t , ν) ≤ Ce -(λ + -λ -)t .
Proof. See [START_REF] Méléard | Quasi-stationary distributions and population processes[END_REF]Theorem 7] and [START_REF] Méléard | Quasi-stationary distributions and population processes[END_REF]Remark 3].

Note that

λ + -λ -= (a + b) 2 + 2(a -b)(p 0 (1) -p 0 (2)) + (p 0 (1) -p 0 (2)) 2
and ρ = a + b -(sup(p 0 ) -inf(p 0 )) defined in Theorem 1.1. We have then λ + -λ -> ρ when sup(p 0 ) > inf(p 0 ). In particular Theorem 1.1 seems not optimal.

3.2. Explicit formula of the invariant distribution. Firstly note that, as

∀η ∈ E, η(1) + η(2) = N,
each marginal of (η t ) t≥0 is a Markov process:

Lemma 3.2 (Markovian marginals). The random process (η t (1)) t≥0 , which is a marginal of a process generated by [START_REF] Cloez | Quantitative results for the Fleming-Viot particle system and quasi-stationary distributions in discrete space[END_REF], is a Markov process generated by G defined by

Gf (n) = b n (f (n + 1) -f (n)) + d n (f (n -1) -f (n)), (9)
for any function f and n ∈ N N = {0, . . . , N }, where

b n = (N -n) b + p 0 (2) n N -1 and d n = n a + p 0 (1) N -n N -1 .
Proof. For every η ∈ E, we have η = (η(1), N -η(1)) thus the Markov property and the generator are easily deducible from the properties of (η t ) t≥0 .

From this result and the already known results on birth and death processes [START_REF] Chafaï | Intertwining and commutation relations for birth-death processes[END_REF][START_REF] Chen | From Markov chains to non-equilibrium particle systems[END_REF], we deduce that (η t (1)) t≥0 admits an invariant and reversible distribution π given by

π(n) = u 0 n k=1 b k-1 d k and u -1 0 = 1 + N k=1 b 0 • • • b k-1 d 1 • • • d k ,
for every n ∈ N N . This gives

π(n) = u 0 N n n k=1 b(N -1) + (k -1)p 0 (2) a(N -1) + (N -k)p 0 (1) , and
u -1 0 = 1 + N k=1 b(N -1) + kp 0 (2) a(N -1) + kp 0 (1) .
Similarly, as η t (2) = N -η t (1), the process (η t (2)) t≥0 is a Markov process whose invariant distribution is also easily calculable. The invariant law of (η t ) t≥0 , is then given by

ν N ((r 1 , r 2 )) = π ({r 1 }) , ∀(r 1 , r 2 ) ∈ E.
Note that if p 0 is not constant then we can not find a basis of orthogonal polynomials in the L 2 space associated to ν N . It is then very difficult to express the spectral gap or the decay rate of the correlations.

3.3. Rate of convergence. Applying Theorem 1.1, in this special case, we find:

Corollary 3.3 (Wasserstein contraction).
For any processes (η t ) t>0 and (η t ) t>0 generated by [START_REF] Cloez | Quantitative results for the Fleming-Viot particle system and quasi-stationary distributions in discrete space[END_REF], and for any t ≥ 0, we have

W d (Law(η t ), Law(η t )) ≤ e -ρt W d (Law(η 0 ), Law(η 0 )),
where ρ = a + b -(sup(p 0 ) -inf(p 0 )). In particular, when (η 0 ) follows the invariant distribution ν N of (8), we get for every t > 0

W d (Law(η t ), ν N ) ≤ e -ρt W d (Law(η 0 ), ν N ).
This result is not optimal. Nevertheless, the error does not come from the coupling choice of [START_REF] Cloez | Quantitative results for the Fleming-Viot particle system and quasi-stationary distributions in discrete space[END_REF] but it comes from how the distance is estimated. Indeed, this coupling induces a coupling between two processes generated by G defined by [START_REF] Del Moral | On the stability of measure valued processes with applications to filtering[END_REF]. More precisely, let L = L Q + L p be the generator of the coupling introduced in the proof of [8, Theorem 1.1] in this special case. We set G = G Q +G p , where for any n, n ∈ N N and f on E × E,

L Q f ((n, N -n), (n , N -n )) = G Q ϕ f (n, n ), L p f ((n, N -n), (n , N -n )) = G p ϕ f (n, n ), and ϕ f (n, n ) = f ((n, N -n), (n , N -n )). It satisfies, for any function f on N N and n > n two elements of N N , G Q f (n, n ) = na f (n -1, n -1) -f (n, n ) + (N -n )b f (n + 1, n + 1) -f (n, n ) + (n -n)b f (n + 1, n ) -f (n, n ) + (n -n)a f (n, n -1) -f (n, n ) ,
and

G p f (n, n ) = p 0 (1) n(N -n ) N -1 f (n -1, n -1) -f (n, n ) + p 0 (2) n(N -n ) N -1 f (n + 1, n + 1) -f (n, n ) + p 0 (1) n(n -n) N -1 f (n -1, n ) -f (n, n ) + p 0 (2) (N -n )(n -n) N -1 f (n, n + 1) -f (n, n ) + p 0 (2) n(n -n) N -1 f (n + 1, n ) -f (n, n ) + p 0 (1) (N -n )(n -n) N -1 f (n, n -1) -f (n, n ) .
Now, for any sequence of positive numbers (u k ) k∈{0,...,N -1} , we introduce the distance δ u defined by

δ u (n, n ) = n -1 k=n u k ,
for every n, n ∈ N N such that n > n. For all n ∈ N N \{N }, we have Gδ u (n, n + 1) ≤ -λ u δ u (n, n + 1) where

λ u = min k∈{0,...,N -1} d k+1 -d k u k-1 u k + b k -b k+1 u k+1 u k ,
and thus, by linearity, Gδ u (n, n ) ≤ -λ u δ u (n, n ), for every n, n ∈ N N . This implies that for any processes (X t ) t≥0 and (X t ) t≥0 generated by G , and for any t ≥ 0, W δu (Law(X t ), Law(X t )) ≤ e -λut W δu (Law(X 0 ), Law(X 0 )).

Note that, for every n, n ∈ N N , we have

min(u)d((n, N -n), (n , N -n )) ≤ δ u (n, n ) ≤ max(u)d((n, N -n), (n , N -n )),
and then for any processes (η t ) t≥0 and (η t ) t≥0 generated by [START_REF] Cloez | Quantitative results for the Fleming-Viot particle system and quasi-stationary distributions in discrete space[END_REF], and for any t ≥ 0, we have

W d (Law(η t ), Law(η t )) ≤ max(u) min(u) e -λut W d (Law(η 0 ), Law(η 0 )).
Finally, using [START_REF] Chen | From Markov chains to non-equilibrium particle systems[END_REF]Theorem 9.25], there exists a positive sequence v such that λ v = max u λ u > 0 is the spectral gap of the birth and death process (η t (1)) t≥0 . These parameters depend on N and so we should write the previous inequality as

W d (Law(η t ), Law(η t )) ≤ C(N )e -λ N t W d (Law(η 0 ), Law(η 0 )), (10)
where C(N ) and λ N are two constants depending on N . In conclusion, the coupling introduced in Theorem 1.1 gives the optimal rate of convergence but we are not able to express a precise expression of λ N and C(N ). Nevertheless, in the section that follows, we will prove that, whatever the value of the parameters, the spectral gap is always bounded from below by a positive constant not depending on N .

3.4.

A lower bound for the spectral gap. In this subsection, we study the evolution of (λ N ) N ≥0 . Calculating λ N for small value of N (it is the eigenvalue of a small matrix) and some different parameters show that, in general, this sequence is not monotone and seems to converge to λ + -λ -.

We are not able to prove this, but as it is trivial that for all N ≥ 0, λ N > 0, we can hope that it is bounded from below. The aim of this section is to prove this fact. Firstly, using similar arguments of subsection 2.4, we have λ N ≥ ρ, for every N ≥ 0. This result does not give us information in the case ρ ≤ 0. However, we can use Hardy's inequalities [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques, volume 10 of Panoramas et Synthèses [Panoramas and Syntheses[END_REF]Chapter 6] and mimic some arguments of [START_REF] Miclo | An example of application of discrete hardy's inequalities. Markov Process[END_REF] to obtain: Theorem 3.4 (A lower bound for the spectral gap). If ρ ≤ 0 then there exists c > 0 such that

∀N ≥ 0, λ N > c.
The rest of this subsection aims to prove this result. Hardy's inequalities are mainly based on the estimation of the quantities B N,+ and B N,-defined for every i ∈ N by

B N,+ (i) = max x>i   x y=i+1 1 π(y)d y   π([x, N ]), (11)
and

B N,-(i) = max x<i i-1 y=x 1 π(y)b y π([1, x]).
We recall that π = π N is the invariant distribution defined in Subsection 3.2 and jumps rates b and d also depend on N . More precisely, [START_REF] Miclo | An example of application of discrete hardy's inequalities. Markov Process[END_REF]Proposition 3] shows that if one wants to get a "good" lower bound of the spectral gap, one only needs to guess an "adequate choice" of i and to apply the estimate

λ N ≥ 1 4 max{B N,+ (i), B N,-(i)}
.

So, we have to find an upper bound for these two quantities. Before to give it, let us prove that the invariant distribution π is unimodal. Indeed, it will help us to choose an appropriate i.

Lemma 3.5 (Unimodality of π). The sequence (π(i + 1)/π(i)) i≥0 is decreasing.

Proof of Lemma 3.5. For all i ∈ {1, . . . , N }, we set

g(i) = π(i + 1) π(i) = (N -i)(b(N -1) + ip 0 (2)) (i + 1)((a + p 0 (1))(N -1) -ip 0 (1))
.

It follows that

g(i + 1) -g(i) = Λ N (i) (i + 1)((a + p 0 (1))(N -1) -ip 0 (1))(i + 2)((a + p 0 (1))(N -1) -(i + 1)p 0 (1)) where Λ N (i) = (N -i -1)(b(N -1) + (i + 1)p 0 (2))(i + 1)((a + p 0 (1))(N -1) -ip 0 (1)) -(N -i)(b(N -1) + ip 0 (2))(i + 2)((a + p 0 (1))(N -1) -(i + 1)p 0 (1)) = -[b(N -1) -p 0 (2)] [(N + 1) (a(N -1) -p 0 (1)) + p 0 (1)(N -i)(N -i -1)] -p 0 (2) i 2 + 3i + 2 (a(N -1) -p 0 (1)) ≤ 0.
We deduce the result.

Proof of Theorem 3.4. Without less of generality, we assume that p 0 (1) ≥ p 0 (2) and we recall that ρ ≤ 0. We would like to know where π reaches its maximum i * since it will be a good candidate to estimate B N,+ (i * ) and B N,-(i * ). From the previous lemma, to find it, we look when π(i + 1)/π(i) is close to one. We have, for all i ∈ {1, . . . , N },

π(i + 1) π(i) = b i d i+1 = 1 + (p 0 (1) -p 0 (2))(i -i 1 )(i -i 2 ) (i + 1) ((a + p 0 (1))(N -1) -ip 0 (1)) , (12)
where i 1 and i 2 are the two real numbers given by

i 1 = N (a + b + p 0 (1) -p 0 (2)) -(a + b + 2p 0 (1)) - √ ∆ 2(p 0 (1) -p 0 (2)) and i 2 = N (a + b + p 0 (1) -p 0 (2)) -(a + b + 2p 0 (1)) + √ ∆ 2(p 0 (1) -p 0 (2))
,

where

∆ = [N (a + b + p 0 (1) -p 0 (2)) -(a + b + 2p 0 (1))] 2 -4(N -1)(bN -a -p 0 (1))(p 0 (1) -p 0 (2)).
In particular, 1 ≤ i 1 ≤ N ≤ i 2 . Furthermore, if . denotes the integer part then

π( i 1 + 2) π( i 1 + 1) ≤ 1 ≤ π( i 1 + 1) π( i 1 ) .
Let us define m N = i 1 + 1 and l N = 2( √ N + 1). Using a telescopic product, we have

π(m N + l N ) π(m N ) = π(m N + l N - √ N -1) π(m N ) √ N +1 j=1 π(m N + l N -j + 1) π(m N + l N -j) ,
Using Lemma 3.5 and the previous calculus, we have that the sequences (π(i)) i≥m N and (π(i + 1)/π(i)) i≥0 are decreasing and then

π(m N + l N ) π(m N ) ≤ π(m N + l N - √ N ) π(m N + l N - √ N -1) √ N +1
. Now using [START_REF] Ferrari | Quasi stationary distributions and Fleming-Viot processes in countable spaces[END_REF] and some equivalents, there exists a constant δ 1 > 0 (not depending on N ) such that

π(m N + l N - √ N ) π(m N + l N - √ N -1) ≤ 1 - δ 1 √ N .
Using the fact that 1 -x ≤ e -x for all x ≥ 0, we finally obtain π(m N + l N )/π(m N ) ≤ e -δ 1 . Similar arguments entail the existence of δ 2 > 0 (also not depending on N ) such that π(m Nl N )/π(m N ) ≤ e -δ 2 . In conclusion, using Lemma 3.5, we have shown that for all i ≥ m N and j ≤ m N , the following inequalities holds:

π(i + l N ) ≤ e -δ 1 π(i) and π(j -l N ) ≤ e -δ 2 π(j).

We are now armed to evaluate B N,+ (m N ) defined in [START_REF] Etheridge | Some mathematical models from population genetics[END_REF]. Firstly, using the expressions of the death rate d and m N , there exist γ > 0 (not depending on N ) and N 0 ≥ 0 such that for all N ≥ N 0 and all i ≥ m N + 1, d i ≥ γN . Let us fix x ≥ m N + 1, using that (π(i)) i≥m N is decreasing, we have Using these three estimates, we deduce that, for every N ≥ N 0 ,

B N,+ (m N ) ≤ 1 γN l N 1 -e -δ 1 2 ≤ 1 γN 2( √ N + 1) 1 -e -δ 1 2 ≤ 16 γ(1 -e -δ 1 )
.

The study of B N,-(m N ) is similar. As stated in Lemma 3.1, the rate of convergence λ of the conditioned semi-group to the QSD is explicit. For a fixed N ≥ 1, the Fleming-Viot process is a reversible Markov chain (for this example) on a finite space, it then converges to equilibrium at rate λ N , where λ N is the eigenvalue of its generator closer to 0. Theorem 3.4 shows that the rate of convergence to equilibrium of the Fleming-Viot process is uniformly bounded. Nevertheless, there is some natural questions such as the convergence of λ N to λ or the monotonicity of this sequence. Also, do we have λ N ≤ λ or λ N ≥ λ? When there are few particles, the generator of the Fleming-Viot process is a square matrix of size 2N and one can explicitly calculate its spectrum (with the help of a computer). In figure 1, each graphic represents, with different parameters, the evolution of the spectral gap of the conditioned process λ = λ + -λ -(detailed in Lemma 3.1), of the Fleming Viot particle system λ N and of the upper-bound ρ in Theorem 1.1, with respect to the number of particles N . Graphics (i) and (ii) illustrate that when osc(p 0 ) = p 0 (2) -p 0 (1) remains constant for two parameter choices then the Fleming-Viot dynamics are different although the conditioned semigroups are the same. More p 0 (1) is large more jumps there are; these graphics seem to show that the rate of convergence is dragged down by the interactions. Graphic (iii) shows that λ N is neither increasing nor decreasing and neither upper nor lower than λ. Graphic (iv) shows that when osc(p 0 ) is small then all curves are close.

In any case, it seems that λ N converges to λ but this point remains an open question.

-δ 1 .

 1 N +1≤k-il N ≤x} 1 π(k -il N ) ≤ {i,k|m N +1≤k-il N ≤x} e -δ 1 i π(k) Similarly, we have π([x, N ]) = {k,i|x≤k+il N ≤N } 1 {x+il N ≤N } Π N (k + il N ) ≤ l N π(x)1 -e -δ 1 .

FIGURE 1 . 5 3. 5 .

 155 FIGURE 1. Evolution of the spectral gap with respect to the number of particles. Details are described in Subsection 3.5

 Corollary 3.6 (Correlations). If (η t ) t≥0 is a process generated by [START_REF] Cloez | Quantitative results for the Fleming-Viot particle system and quasi-stationary distributions in discrete space[END_REF] then we have for all t ≥ 0,

If ρ ≤ 0, the right-hand side of the previous inequality explodes as t tends to infinity whereas these correlations are bounded by 1. Nevertheless, using Theorem [8, Theorem 2.6], Remark [8, Remark 2.7] and Inequality [START_REF] Del Moral | A Moran particle system approximation of Feynman-Kac formulae[END_REF], we can prove that there exists two constants C (N ), depending on N , and K, which does not depend on N , such that

where C(N ) is defined in [START_REF] Del Moral | A Moran particle system approximation of Feynman-Kac formulae[END_REF]. Even if Theorem 3.4 gives an estimate of λ N , C(N ) is not (completely) explicit and we do not know if the right-hand side of the previous expression tends to 0 as N tends to infinity. This example shows the difficulty of finding explicit and optimal rates of the convergence towards equilibrium and the decay of correlations.