César Ducruet

Hidekazu Itoh

Olivier Joly
email: olivier.joly@univ-lehavre.fr

Port-region linkages in a global perspective

Keywords: commodity flows, maritime transport, ports, production network, value chain

The recent literature on ports and regional development mostly documents the weakening spatial fix of flows and the negative externalities of ports on their surrounding territories. Except for port impact studies that remain difficultly comparable across locations, such observations are mostly based on simplified measures of port activity, such as total port throughputs, and on developed countries. Arguing that port-region linkages have maintained in more subtle interdependencies, this paper provides a comparative approach including both mature and emerging economies comprising about 1,500 ports situated in more than 300 regions and 40 countries. The statistical analysis of combined commodity traffic data and regional socio-economic data confirms the affinity of certain port traffics with specific local characteristics. A global typology of port regions is proposed while pointing at noticeable influences between traffic volumes, types, and local economic structures. This research thus brings new insights about the territorial embedding of value chains and production networks.

Introduction

Linkages between ports and their host territories are often analyzed from the unidirectional perspective of the local economic and social impacts of port activities. The influence of local socio-economic conditions on port traffics remains much neglected in the literature as well as the more complex interdependencies at stake between port activities and regional development. Although it is recognized that ports continue to play an active role in urban development [START_REF] Hesse | Cities, material flows and the geography of spatial interaction: Urban places in the system of chains[END_REF], many scholars have focused on the erosion of local industry linkages from various angles such as the globalization of manufacturing, trade reorientation, uneven port performance and competitiveness, growing continental connectivity and importance of global carriers' decisions [START_REF] Hoare | British ports and their export hinterlands: A rapidly changing geography[END_REF][START_REF] Slack | Pawns in the game: Ports in a global transportation system[END_REF][START_REF] Hoyle | Ports, Cities and Coastal Zones: Competition and Change in a Multimodal Environment[END_REF][START_REF] Overman | The port geography of UK international trade[END_REF][START_REF] Lee | Trade reorientation and its effects on regional port systems: The Korea-China link along the Yellow Sea rim[END_REF]. The weakening spatial fix of ports was verified empirically through studying the location of maritime Advanced Producer Services on a world level [START_REF] Jacobs | The location and global network structure of maritime advanced producer services[END_REF], while large port terminals appeared to have limited and/or negative externalities of upon their local territory in Canada and the United States [START_REF] Mccalla | Intermodal freight terminals: Locality and industrial linkages[END_REF][START_REF] De Langen | The Economic Performance of Seaport Regions[END_REF][START_REF] Grobar | The economic status of areas surrounding major U.S. container ports: Evidence and policy issues[END_REF][START_REF] Hall | Container ports, local benefits and transportation worker earnings[END_REF][START_REF] Jaffee | Labor and the geographic reorganization of container shipping in the U[END_REF]. Yet, no study has provided a comprehensive and systematic analysis of ports' local linkages taking into account the diversity of port and other economic activities.

In this paper, it is argued that a global and comparative approach to port-region linkages can reveal more complex realities and their various combinations, following earlier works solely based on container traffics of port regions in advanced economies [START_REF] Ducruet | Port Regions and Globalization[END_REF], which underlined the influence of regional economic specialization on traffic growth. Other recent works have also underlined the permanency of strong local linkages of port activities despite the aforementioned constraints and changes [START_REF] Dunford | Regional Development: Port-Industrial Complexes[END_REF][START_REF] Hall | Why are maritime ports (still) urban, and why should policy makers care?[END_REF]. Such a global approach can be seen as a contribution to a better understanding of how freight flows, and to some extent the wider organizational entities they reflect (i.e. global production networks and commodity chains), remain anchored in local territories and what is the nature, level, geographic diversity of such embedding (see [START_REF] Hesse | Global production networks and the role of logistics and transportation[END_REF] for a useful discussion). Does port specialization relates with regional specialization? What is the relevant geographic scale for analysis? Are some port-region linkages more successful than others socially and economically? How do such trends vary according to the situation of the port region within its host country? Answering such questions must bring together a wide set of approaches from various angles, such as transport geography and economics as well as regional science and its recent developments within the so-called New Economic Geography.

Empirically, the study rests on the elaboration of a global database of eighteen port and regional indicators covering more than 1,500 ports located in 300 port regions (40 countries) of the world.

The rest of the paper is organized as follows. The next section (Section 2) introduces the theoretical framework as well as the global database and some preliminary results on the correlations among selected indicators. The core of the paper lies in Section 3 where it is provided a multivariate analysis of port-region linkages as well as a typology of port regions. Section 4 complements the former analyses by a multiple regression simply verifying the mutual influence of port traffic performance and regional economic performance. The conclusion identifies key contributions of this study and discusses them in light of their potential contribution to policy advice and further research pathways.

A global analysis of port regions

Theoretical framework

Most contributions about the mutual influence between transport flows and territorial development remain very theoretical while few of them deal directly with the case of ports and port regions. This paper thus can be seen as an attempt verifying a number of ideas promoted by various schools on the subject of transport and regional development.

Throughout the New Economic Geography, there is a recognition that core and peripheral regions have different roles in attracting firms and functions, with more emphasis on overall transport costs than on the precise nature and volume of transport flows (see [START_REF] Lafourcade | New Economic Geography: The Role of Transport Costs[END_REF] for a useful synthesis). Yet, [START_REF] Tabuchi | Regional Specialization and Transport Costs[END_REF] argue that agglomeration forces are commodity-specific and therefore depend on a certain degree of regional specialization. Notably, heavy industries and industries producing goods with high transport costs are more agglomerated than light industries and industries with lower transport costs. Behrens et al. (2006) as well as [START_REF] Fujita | The role of ports in the making of major cities: Self-agglomeration and hub-effect[END_REF] in the case of ports also considered crucial the situation of the gateway with regard to inland corridors and cities. One major outcome of the New Economic Geography is the belief that increasing the efficiency of transport infrastructure (of which ports) might exacerbate regional disparities and local economic downturn rather than increasing regional benefits [START_REF] Fujita | The Spatial Economy: Cities, Regions and International Trade[END_REF]. Port regions provide a fertile ground to test the hypothetic link between specific commodity flows and specific economic activities taking place within localities, while pointing at different types of port regions according to their logistical and socio-economic profile. Their situation within the national territory and continental context should also be given paramount importance because port regions can be either cores or peripheries.

Other literatures point at the more managerial and strategic dimension of flows and territories, such as those on global commodity chains (GCC) and global production networks (GPN), which also remain very much conceptual or focussed upon one particular place or firm. Recent works have insisted on the need to further integrate such approaches with the one of world city networks (WCN) that are at the contrary more empirical in their study of the location of specialized services in so-called global cities (see [START_REF] Jacobs | Integrating world cities into production networks: The case of port cities[END_REF] for a useful synthesis). Successive stages along the production and distribution chain produce create needs in terms of related activities thus making locations more or less dependent on endogenous and exogenous services. As en effect of logistical constraints in densely urbanized places, the top European cities by their number of maritime Advanced Producer Services (APS) do not always handle higher valued-goods, such as London, nor do they always possess a seaport, such as Paris, Moscow, and Madrid, but certain port cities stand out by their exceptional score mainly due to the presence of a large industrial and logistics cluster, such as Rotterdam and Antwerp [START_REF] Jacobs | Integrating world cities into production networks: The case of port cities[END_REF]. The main difficulty generalizing such linkages lies in the functional and regional diversity of configurations. Thus, this research seeks both transversal and regional trends in port-region linkages: although same economic sectors and commodity types do not develop and combine identically across the world due to historical legacies, socio-cultural differences, and economic imperatives, certain trends might be common to all continents.

Data presentation and methodology

The resulting sample consists in 353 port regions (of which 327 have the complete set of variables) located in 40 countries (Table 1). Ports located in those regions accounted for about 87.5% of total world port tonnage and 37.3% of total world population in 2009, the reference year of the study. One main difficulty of this research has been to access internationally comparable data at infra-national level. Although many other large countries do exert port activities, information collection remained impossible due to various constraints, such as the absence of socio-economic data at infra-national level despite the existence of good port traffic data (e.g. Argentina, Tunisia, Algeria, Singapore, Hong Kong, Cyprus) and vice-versa (e.g. Vietnam, Philippines, Russia, Indonesia). Except from Eurostat and OECD Territorial Database, data were collected from national organizations such as statistical offices, ministries, and associations2 . The tonnage of individual ports was summed at administrative region level to allow a comparison with socio-economic aspects. Analyzing port-region linkages at administrative level remains hindered by two main limitations:

• their mismatch with the contours of port regions and port hinterlands: the proportion of port clients located inside the administrative region was not possible to calculate and even to compare with the actual distribution and weight of clients all over the country and even beyond national boundaries in certain cases. Port-related flows, businesses, and/or employment that can be directly, indirectly related to (or induced by) port activities are often not counted at local level by official statistics. Is was thus impossible to analyze port-region linkages based on harmonized definitions of functional port regions as in urban studies where the functional urban region is preferred to the administrative city that often misses a large part of overall urban activity;

• their heterogeneity in terms of spatial coverage and decentralization levels: despite many efforts from international organizations to harmonize different administrative levels across countries, discrepancies remain in size and decision power. In terms of size, regions in Australia, Canada, but also Brazil and China may appear too large compared with European counterparts whereas the lower administrative level may be spatially too small (e.g. US counties) and often offering limited socio-economic data. In terms of power, even countries with comparable spatial coverage of their regions may function very differently, such as the decentralized French Regions and the centralized British Regions all classified into NUTS-2 administrative level by the European Commission and Eurostat.

The selection of indicators is justified by the limited availability of regional indicators at world level (see Appendices 1 and 2 for an overview of data sources and the full list of variables). Port throughput indicators encompass the most widely accepted categories of commodities such as solid or dry bulks, liquid bulks, and general cargoes, as well as two complementary aspects, import and international. Bulk commodities comprise mostly raw materials, such as cereals, minerals, ores, and coal for solid bulks, liquefied gas, chemicals, oil and oil products for liquid bulks. General cargoes cover a wide range of products that can be either containerized, such as manufactured goods and food products, or in break-bulk form as intermediate inputs or parts (e.g. metals, machinery, etc.). Although those three categories thus remain much aggregated, their respective weight implies certain specialization in terms of cargo handling facilities in the port itself as well as different industrial and logistical linkages with the local and regional economy. However, the precise geographical distribution of such flows is unknown, such as the proportion of transit or transshipment flows. For such reasons it was decided to complement main categories with the volume of import and international throughputs. Import and inbound traffic were considered as one same category although they only partially overlap. While import throughput is always inbound-oriented for inland distribution and export throughput concerns overseas distribution, inbound throughput may in fact include exports (goods shipped from the hinterland to the port) and outbound may include imports (goods shipped from the port to the hinterland). International throughput basically corresponds to non-domestic origins and destinations; it thus excludes coastal shipping from total port activity. Some regions in certain countries remained lacking some of the proposed indicators, such as the unemployment rate of most Brazilian states 3 , the Gross Domestic Product (GDP) of Cyprus districts, and most socio-economic data for a number of Indian states 4 . In several cases, the absence of accurate data obliged using other indicators, such as regional GDP per economic sector instead of employment (Mexico, India), and Gross Value Added (Brazil, South Africa) or average current income per household (Taiwan, Iceland) instead of regional GDP. The 12 original indicators allowed calculating various ratios such as GDP per capita, population density, as well as location quotients at country level (e.g. share of primary sector at region X divided by share of primary sector of the country). Such indices attenuate the influence of structural socio-economic differences and allow international comparison as some countries provide sample surveys only (e.g. India) or use different units (i.e. local currency, constant or current prices for national accounts). Other indices include the population, GDP, and traffic shares (%) in national totals. The share of regional GDP in national GDP, regardless of differences in units, made possible harmonizing the amount of regional GDP in US dollars based on World Bank figures on national level for 2009 5 . Port traffic indices were calculated at world level due to the presence of single-port countries in the sample, such as Malta and Lithuania. Last but not least, the presence of inland capital cities inside certain port regions, such as the Capital Region in South Korea (Incheon, Seoul, and Gyeonggi Province), motivated the study to harmonize such delineation in the case where such capitals where spatially adjacent to (but not included in) the port region, thus resulting in the merger of eight coastal regions with eight inland capital city regions6 . Wherever the inland capital city was not directly adjacent to the port region, such as Berlin being separated from Mecklenburg-Vorpommern (Rostock) by other regions, it was not considered as a single functional gateway, with reference to a classification proposed by [START_REF] Merk | The Competitiveness of Global Port-Cities : The Case of the Seine Axis[END_REF] on freight corridors and inland core economic regions.

[Insert Table 1 about here] 2.2 The specific role of port regions in territorial development Before the analysis of trends and differences among port regions, interesting evidence can be obtained from comparing the relative characteristics of port and non-port regions. As was observed in Table 1, port regions' share in national GDP varies greatly from one country to the other due to multiple factors such as physical geography, political and administrative borders, industrial geography, etc. This approach serves a better understanding of the specificity of port regions, if any, compared with regions having no port activities. For each variable retained (except port throughputs) we calculated the average score of port and nonport regions per continental area, and the ratio between them (Figure 1). Such a broad scale of analysis is justified by the uneven number of port regions per country as seen in Table 1 as well as by the need to verify previous research on the influence of macro-regional backgrounds on logistical patterns [START_REF] Lee | A tale of Asia's world ports: The spatial evolution in global hub port cities[END_REF][START_REF] Rodrigue | Comparative North American and European gateway logistics: The regionalism of freight distribution[END_REF]). Yet, the number of non-port regions is similar to the one of port regions globally, thus making this analysis worthwhile. On a world level, port regions always score higher than non-port regions but this is particularly true for three variables: population share, GDP share, and population density, followed by GDP per capita, GDP volume, and unemployment. This underlines the capacity of port regions to be more productive on average than non-port regions but this goes together with rising spatial, environmental, and social problems already well addressed elsewhere [START_REF] Vallega | Cityports, Coastal Zones and Sustainable Development[END_REF]. No specific economic sector seems preferential on such level of aggregation.

Certain regions keep their specificities, however. Asia stands out by the higher population and GDP share of port regions as well as by the higher importance of the primary (1.22) and secondary (1.14) sectors plus unemployment (1.08). Of course, this Asia-Pacific area is, at least economically, very heterogeneous but this profile might correspond to the importance of extraction and heavy production activities around ports (cf. Australia). Indeed in terms of traffic specialization, Asia has the highest score for solid bulk traffic (1.08)7 . Many Asian port cities face difficult social situations and industrial reconversion, such as Incheon in South Korea (Ducruet et al., 2012), while Shanghai remains dominantly industrial despite its efforts to deploy a global city, service-oriented profile [START_REF] Wang | New port development and global city making: Emergence of the Shanghai-Yangshan multilayered gateway hub[END_REF]. Africa (here only based on Morocco and South Africa) has the highest divergence in terms of population density, unemployment, tertiary and secondary sectors, while its port regions score significantly lower than non-port regions in terms of GDP share, primary sector, and GDP volume. This situation can be explained by the inland location of main decision centers such as Johannesburg and Rabat, the respective capital cities, and of main mining sites creating the bulk of national added value. Port regions thus mostly serve as gateways between inland and overseas locations while suffering from higher social costs and lower economic benefits, sometimes fostering the need for balancing regional development through port-related projects as seen in the case of the Tangier-Med project [START_REF] Ducruet | Maghreb port cities in transition: The case of Tangier[END_REF]. The Americas have the highest positive differential for GDP per capita and the primary sector (together with Asia); many American regions indeed function as energy production centers (e.g. Alaska, Campeche, Amazonas) creating among the highest indices of GDP per capita, while many important cities concentrating economic wealth belong to port regions, such as New York, Los Angeles, and Rio de Janeiro, to name but a few examples. Finally, Europe has in common with Africa a lower score for primary sector and a higher score for population density, its main port cities being often counted at the level of small administrative regions unlike in the United States for instance where the density of largest cities is diluted on the level of states. It also has a significant positive divergence for GDP share, population share, tertiary sector, GDP per capita, and GDP volume.

[Insert Figure 1 about here]

Global trends in port-region linkages

Correlations among main variables

Comparing the correlation between demographic size and port throughput volumes at national and regional level provides some first evidences about the linkages (Figure 2). At country level, the correlation is highly significant since about 82% of port throughput is explained by demographic size. The amount of cargoes thus nicely reflects the size of the national market that imports and exports such goods, notwithstanding local specificities 8 . China, Japan, South Korea, Australia, the Netherlands, and the United States have the largest negative residuals to the model (i.e. higher throughput volume than expected). The international trade of those countries is dominantly using sea transport (e.g. 99% for South Korea) due to physical factors (islands) but also political (borders). Such factors are reinforced when strong emphasis is put on hub functions, such as in South Korea and the Netherlands, where about 40% of total container throughput in Busan and Rotterdam are occupied by transshipment flows, or on massive raw materials imports or exports, such as China and Australia. Conversely India, Brazil, Mexico, Morocco, and Turkey have a lower throughput volume than expected by their demographic size, partly due to logistic chains deficiencies such as cumbersome customs procedures.

[Insert Figure 2 about here]

At sub-national level, the correlation is expectedly lower than at country level (0.39) due to the lower probability for administrative divisions to overlap port hinterlands and market areas. Yet, the fact that almost 40% of port throughput distribution can be explained by local population size is in itself a very fruitful result given the influence of many disturbing factors. Most regions having higher throughput volumes than expected locate in China (e.g. Jiangsu, Zhejiang, Guangdong), South Korea (Gyeongnam, Jeolla), Australia (Western Australia, Queensland), the Netherlands (Zuid Holland), the United States (Louisiana, New York, Texas, California), Japan (Chiba, Kanagawa, Aichi), and Belgium (Antwerp province). This is similar to results obtained at national level that were caused by a concentration of transit flows and/or bulk flows at certain locations. The opposite trend (lower throughput than predicted) mostly applies to Indian states (Maharashtra, West Bengal, Andra Pradesh, Karnataka, Tamil Nadu) and other Chinese provinces (Sichuan, Heilongjiang, Jiangxi, Hunan, Chongqing). Reasons differ from one case to the other since such Chinese provinces only use inland or canal river shipping that is less accessible to larger vessels, and the Indian states face a lack of suitable port infrastructures. While such trends are very much similar to those obtained when plotting total throughput over GDP at both levels, more interesting is the correlation between the respective national shares of port throughput, population, and GDP. About 48-50% of port throughput share thus could be explained by the share of either population or GDP in national totals.

Due to the existence of numerous extreme values, the choice has been made to analyze ranked values instead of initial values. Some large regions such as in China, India, and the United States concentrate very large populations and traffics. Other ones had abnormal index levels in certain categories due to local uniqueness, such as Campeche (Mexico) having the world's highest index of GDP per capita relative to national average as an effect of its offshore oil extraction activity. Correlations between traffic and socio-economic variables on the level of the whole sample remain, however, rather limited (Figure 3). General cargo traffic appears positively correlated with population density and the share (%) of GDP in national total, thus suggesting the importance of urbanization in the concentration of higher valued (and more international) goods. Liquid bulk and import traffics are positively correlated with the amount of GDP per capita, probably because most countries in the selected sample are large importers of energy products (e.g. oil and gas) while regions specialized in energy extraction and transformation often reach high per capita productivity although this is based upon a few industries and/or plants. The negative correlation between total port throughput and the primary sector suggest that such local economies often handle smaller traffic volumes than other regions, but the two other sectors do not correlate significantly with any port traffic variable. All other significant correlations occur within each group (either traffic or socioeconomic variables) except from the absolute figures and the national shares as seen in previous Figure 2.

[Insert Figure 3 about here]

Factor analysis

The factor analysis applied to rank values instead of initial values and provided six relevant factors concentrating about 71% of total variance (Table 2). Main trends first reveal significant evidences on port-region linkages since traffic and regional variables are often grouped together inside factors.

[Insert Table 2 about here]

The most significant trend is an opposition between the weight of port regions (absolute and relative) and their specialization in both solid bulks and the primary sector. On the one hand, this confirms that regions concentrating population and production also tend to concentrate traffics within their country, thereby implying that traffic distribution is not randomly distributed. On the other hand, the significant association between solid bulk traffics and specialization in the primary sector is by no means logical. Commodities such as agricultural products and mine products tend to concentrate in regions specialized in agricultural and mining activities. This opposition also implies that the latter regions are less likely to occupy a central role in their country and in the global hierarchy. Indeed, solid bulk and primary sector specializations are isolated from all other variables in the first factor. On the opposite side, other variables have more or less significance outside the most relevant ones about the national shares, which give more clue about the nature of those "core regions", which tend to attract more weighty and international traffics. Thus, there appears a certain association between absolute traffic size, national importance, and traffic geographic coverage of regions, while those specialized in solid bulks and the primary sector tend to have more modest roles in the port-region hierarchy. This evidence suggests rather logical linkages between economic specialization and throughput specialization: economic activities involving production and transformation handle more raw materials than tertiary activities (of which transport), while the latter have closer affinity with higher valued goods (general cargo) and need higher volumes of energy resources for consumption. As mentioned by [START_REF] Tabuchi | Regional Specialization and Transport Costs[END_REF], higher valued activities are often less spatially concentrated than heavy industries and activities producing goods with high transport costs. It is also the case of many urban economies to be more service-oriented and more densely populated thus increasing the need for consumer goods and energy resources such as oil and gas. As a matter of fact, most largest economic, demographic, and logistical concentrations stand out on this first factor, such as the Seine Axis, Istanbul, Seoul/Incheon, Kuala-Lumpur/Port Klang, Athens, London, Tokyo, New York etc. in radical opposition with more remotely located regions such as Cornwall (United Kingdom), Jeju (South Korea), Northwest Territories (Canada), and Friesland (Netherlands) to name but a few.

The second most significant trend (factor 2) groups together the relative weight of the port region nationally (traffic, population, GDP), international traffic, and primary sector on the one hand, and GDP (total, per capita), import traffic, tertiary sector, density, and population on the other. One can deduce from this trend that more tertiary regions tend to be more wealthy and populated, which in turn fosters imports via local consumption. Examples include many port regions of the United States, Japan, and Europe, but also many Chinese regions due to the demographic and logistical "size effect". The opposite profile is thus better explained by the export of raw materials (international traffics) taking place at less populated and wealthy regions, but which occupy an important role at national level (higher share of population, GDP and traffics). The latter regions include numerous port regions from small and/or single-port European countries (e.g. Malta, Estonia, Romania) that come out mostly due to their national rather than global importance. Yet, no specific commodity type defines such profiles.

Turning to the third trend (factor 3), the type of commodities handled takes it whole meaning in relation to the socio-economic features of regions. On the one hand, solid bulk is positively related with the overall size of the region (traffic, GDP, population) and to the secondary sector as well as to the primary sector to a lesser extent. This profile typically indicates the possibility for certain regions to be large in size as well as specialized in more traditional activities. This specialization is directly apparent in the preference to raw materials traffic. Examples include many inland port regions in the United States (Michigan, Minnesota, Ohio, Indiana, West Virginia), Canada (Ontario), China (Anhui, Sichuan, Heilongjiang) but also coastal regions exporting large quantities of minerals and agricultural goods such as in Australia (Western Australia, Queensland), Mexico (Baia California), and India (Orissa, Andrah Pradesh). The opposite profile clearly groups the tertiary sector with general cargo, liquid bulk, import, and international traffics, as well as with GDP per capita (index) and unemployment. This profile comprises many urbanized regions playing a central role in the national and global economy (e.g. Oslo, Reykjavik, Bremen, Stockholm, Hamburg, Dusseldorf, Rome, Genoa, London, Athens, Lisbon, Casablanca) but also a number of peripheral ones being remotely located from core economic regions (e.g. Northwest Territories, Arica y Parinacota, Nunavut, Alaska, Mellersta Norrland) and/or islands and enclaves (e.g. Corsica, Hawaii, Ceuta, Melilla, Madeira, Malta, Canary Islands, Baleares, Azores). In fact, the tertiary sector encompasses a wide range of activities: urban regions are more likely to specialize in private services (e.g. finance, transport, retail) while peripheral regions that lack of a productive economy tend to rely on the public sector. They have in common to exhibit higher unemployment levels than national average. Lastly, attention should be paid to the fourth trend (factor 4) that depicts a profile of urban region specialized in solid bulks but with higher unemployment levels, opposed to a profile of industrial region specialized in liquid bulks but with higher GDP per capita than national average. One typical example of the former profile is Campeche state in Mexico, which concentrate most of the country's offshore oil extraction and has the world's highest index of GDP per capita relative to national average. The former profile often corresponds to peripheral regions facing difficult reconversion due to over-specialization in traditional activities, such as Gangwon in South Korea and Eastern province in Taiwan, which locate away from core economic regions.

Although further understanding the geographic logic of each factor would necessitate their cartographical representation, this article opted for mapping only the typology of port regions based on the same factor analysis.

A global typology of port regions

Eight types of port regions have been retained from the non-hierarchical clustering analysis applied to the six relevant factors (see Appendix 2 for a more detailed description of clusters). Each cluster is defined by a positive influence of certain variables appearing by order of importance, as well as by those having negative influence. The first two clusters (1 and 2) have in common certain issues such as higher unemployment levels, specialization in liquid bulk traffic, smaller share in total world port throughputs, and lack of industrial activities. In addition, "deprived port regions" have poor economic performance and higher specialization in primary activities, while "peripheral port regions" tend to be richer and rely dominantly on imports and services suggesting a dominance of local consumption. The two next clusters (3 and 4) also share certain features, such as being important demographic, economic, and port concentrations in their host country and handling more international traffics than world average, while concentrating noticeable shares of world port throughputs and liquid bulks. Their territorial profile differs, however, as "metropolitan port regions" are richer, more densely populated, and more service-oriented with lesser production activities but handling more general cargo, and "industrial port regions" typically resemble maritime industrial development areas where production and transformation activities dominate as well as liquid bulks. The next clusters (5 and 6) have in common to be large regions with high GDP with lower international and general cargo traffics than world average. "Productive port regions" are richer and more industrialized than average while handling more imports and liquid bulks, and "bulky port regions" are better defined by a large share of world port throughputs (50%), specialization in solid bulk traffics, and population density. Finally, the two last clusters (7 and 8) share a smaller size and lower liquid bulk traffics than average. "Transit port regions" specialize in higher valued goods and industrial activities while "traditional port regions" specialize in solid bulk traffics and the primary sector. The last cluster can be seen as a typical class of agricultural and mining regions having lesser population density and higher exports than average, while transit port regions have lesser services.

The spatial distribution of clusters (Figure 4) shows interesting patterns in terms of recurrences in the location of certain types within countries and continents. Certain types are concentrated at specific areas, such as "bulky port regions" comprising many Chinese, Indian, Australian, North American regions as well as Kwazulu-Natal (South Africa) and southern Brazil, but no such type appears in Europe. Conversely, "metropolitan port regions" are few outside Europe: they underline the importance of some large cities such as Casablanca, Western Cape (Cape Town), Gujarat, Maharashtra (Mumbai), Kerala, Hyogo (Kobe), Osaka, and Kanagawa (Yokohama), others being national capital regions, such as Valparaiso-Santiago, New York, Selangor-Kuala Lumpur, Taipei, Seoul-Incheon, Auckland, and Tokyo. It is surprising that despite their metropolitan dimension, Shanghai and Beijing-Tianjin do not belong to this category, perhaps due to the still "developing" character of China defined by the importance of industrial sector and bulk traffics, but the same also applies to New South Wales (Sydney) in a totally different context. Inside Europe, not only national capitals but also other cities belong to this category: Zuid and North Holland (Rotterdam and Amsterdam), Hamburg, Provence (Marseilles), Liguria (Genoa), Hampshire (Southampton), Pomorskie (Gdansk), Skane (Malmo), and Izmir. Their location echoes the continental coreperiphery pattern of economic and logistics activities [START_REF] Lee | A tale of Asia's world ports: The spatial evolution in global hub port cities[END_REF] with few cases away from the European megalopolis: Lisbon, Attiki (Athens-Piraeus), Istanbul, Stockholm, Helsinki, and Oslo. By contrast, many remotely located regions belong to the first two clusters and confirm the socio-economic difficulties of many UK and southern European regions. Elsewhere, deprived regions also locate away from core economic centers of their country (e.g. Hainan, Sabah, Hokkaido, Nova Scotia, Newfoundland), with the notable exception of California and Florida being often seen as rather dynamic port regions. Peripheral port regions underline the insular effect as many of them are islands or territories separated from mainland, such as Hawaii, Alaska, Okinawa, Baleares, Corsica, Ceuta and Melilla (Spanish enclaves in Morocco), Ionia Nisia (Greece), but also some states in the US such as Maine, Rhode Island, and Delaware. Industrial port regions also concentrate very much in Europe while often including large regional cities (e.g. Barcelona, Bilbao, Nantes, Antwerp, Bremen, Goteborg, Bergen) just like outside Europe with the examples of Texas (Houston), Veracruz, Antofagasta, Western Australia (Perth), Adelaide, Taichung, and many Korean regions of which Gyeongnam province (Busan). Productive port regions concentrate in certain parts only of the most globalized areas, such as Northern Italy in Europe and a number of regions in the United States and Japan. Those regions levy high rents from transformation activities (e.g. Louisiana) after importing raw materials for domestic redistribution (e.g. steelworks and petrochemical industries). Transit port regions often locate near metropolitan regions due to their gateway function (e.g. Valencia for Madrid, East Anglia/Felixstowe and Kent for London, Tangier in Morocco, Johor in Malaysia), while traditional port regions appear as small primary centers of economic and port activity dedicated to local resources such as in Mexico, Chile, Morocco, and Turkey.

[Insert Figure 4 about here]

Mutual influences between port and regional activities

A multiple regression investigates which variables most influence total regional port throughput on the one hand and total regional GDP on the other as dependent variables. The stepwise method was used to best select the variables having most significance on the model's results [START_REF] Goldberg | Note on stepwise least-squares[END_REF][START_REF] Gujarati | Basic Econometrics[END_REF]. Four models are compared in their ability explaining each dependent variables, with detailed scores provided in Appendix 4. The first model (A1 and B1) includes only the original traffic and socio-economic variables used in previous sections. The other models contain additional variables, such as the dummy distinguishing amongst coastal and inland (river) port regions (models A2 and B2), the level of logistical performance of the belonged country as measured by the World Bank Logistics Performance Index [START_REF] Arvis | Connecting to Compete 2012[END_REF] as well as the DHL Global Connectedness Index [START_REF] Ghemawhat | DHL Global Connected Index 2012. Analyzing Global Flows and their Power to Increase Prosperity[END_REF] (models A3 and B3) 9 , and dummies for each continent where regions locate (models A4 and B4). Those newly added variables bring in contextual information about the environment in which port regions operate and develop, although the integration of national-level variables should in practice make use of multilevel regressions rather than classic multiple regressions, but this aspect has been kept for further research on these issues.

On the level of the whole sample, main results show that export-oriented traffics will affect positively total throughput and this effect is even more pronounced at richer countries. However, imports affect positively regional GDP as this relates to the ability of regions to consume flows locally and/or redistribute them towards the hinterland: the influence of imports is indeed higher when Europe and North America dummies are included, i.e. mostly import regions with important inland connectivity [START_REF] Rodrigue | Comparative North American and European gateway logistics: The regionalism of freight distribution[END_REF]. The coastal dummy has positive and negative influence on throughput and GDP respectively since seaports favor traffic concentration and economies of scale while inland regions tend to be more productive than coastal regions (see also [START_REF] Lever | Regional economic growth and port activities in European cities[END_REF]. The shares (%) of throughput and GDP have both negative scores, thereby implying that port regions may grow economically and logistically without being highly concentrated on a national level. This can be explained by the fact that higher shares often correspond to smaller countries where one or two regions concentrate a majority of GDP and/or throughput (e.g. Iceland, Malta, Estonia) but without equivalent importance in terms of absolute production/traffic volumes globally. The secondary sector has a strong positive effect on regional GDP mostly in developing 9 The LPI is a composite measure including 6 sub-categories (efficiency of customs and border management clearance, quality of trade and transport infrastructure, ease of arranging competitively priced shipments, competence and quality of logistics services, ability to track and trace consignments, frequency with which shipments reach consignees, and within scheduled or expected delivery times). The DHL index is also composed of various sub-scores covering trade, capital, information, and people flows analyzed both in terms of total volumes (breadth) and importance in the national economy (depth). countries, thus suggesting that transformation activities still generate more profits in relation with port activities, even in advanced countries. In comparison, the tertiary sector will have rather negative effects on GDP but mostly in less developed countries (model A2). While the DHL index does not play a significant role in either throughput or GDP, the LPI index that is much more defined by logistical quality and performance has a strong positive influence on both GDP and throughputs, for the latter mostly in poorer countries. Unsurprisingly, Asia and Oceania have positive influence on throughputs while North America is negative, thus confirming the hierarchy observed in previous Figure 4 and the importance of the continental context in which port regions develop and operate.

Conclusion

Despite all the methodological constraints inherent to a global and comparative analysis of local economies, this research has provided a snapshot of port-region linkages across more than 1,500 ports located in more than 300 regions and 40 countries. The central hypothesis of the research was the existence of functional and spatial affinities between types of traffics and types of regions. Several statistical analyses brought a number of answers as well as new questions about the mutual influence of port activities and regional development.

Generally speaking, the specialization and spatial distribution of port traffics is far from being randomly distributed over space. Port regions vary in function and importance according to the continental context. Although the same traffics such as liquid bulks may be associated with different types of local economies (e.g. poorer or richer), their concentration reveals various combinations. Deprived and metropolitan port regions both handle liquid bulks but this occurs in totally different socio-economic contexts and for different reasons, from transit and transformation to consumption. Traffic specialization is thus highly related with local skills and does not have the same meaning everywhere. Metropolitan port regions handle lesser traffic volumes but more valued goods than bulky port regions. Thus, the types of commodities passing through the port(s) of a given region not only reflect but also shape local economic structure and wealth. Export traffics foster total throughputs while import traffics foster economic growth. Such trends are largely due to the heterogeneity of the sample composed of both emerging and advanced economies. Yet, core and peripheral regions can be identified on all continents based on their common features. This territorial embedding raises important policy issues. First, the path-dependency of traffic specialization suggests that port development can be a driver of regional development only under certain conditions. Reviving local economies that face important socio-economic difficulties through attracting new cargo flows might not always be a relevant option if policy-makers do not take into account the link with local industries. For instance, the emphasis put on containerization by numerous industrial port regions can result in weak local impacts due to the absence of specific skills and markets, while remaining a rather artificial implant bound to transit flows. Removing port activities based on environmental and aesthetic arguments notably through waterfront redevelopment in coastal cities may also have underestimated negative impacts if their traffics still contribute to some extent to industrial and logistical development.

Further research shall insist more on the dynamics of traffic and regional specialization, perhaps by taking inspiration from existing empirical works on regional development with an evolutionary perspective (see for instance [START_REF] Neffke | How do regions diversify over time? Industry relatedness and the development of new growth paths in regions[END_REF]. Such an approach would better elucidate the co-evolution between the growth and diversification/specialization of both traffics and regions. Attracting certain traffic types in certain localities might have positive or negative effects on further port and regional performance. Other research pathways include the need to verify the influence of neighborhood on port regions' profiles notably via spatial autocorrelation tests as well as through applying multilevel rather than classic multiple regressions in order to better understand the role of the national context. Enlarging the current sample of port regions will face difficult challenges of data availability, but recent works on global maritime flows suggest the possibility complementing traditional port throughput data by vessel movement data while lowering problems of comparability across countries (see [START_REF] Van Hamme | Territorial Impact of Globalization on Europe and its Regions (TIGER), Draft Final Report, The European Spatial Planning Observatory Network (ESPON) 2013 Programme[END_REF].

Figure 3 :Figure 4 :

 34 Figure 3: Spearman correlations among port-region variables

Table 1 : Study sample

 1

	Country	No. port regions	No. ports	Avg. area (000s sq. km)	Total port throughput (million metric tons)	Share (%) in national GDP	Unit
	Australia	7	39	1098.5	880.9	97.9	State	TL2
	Belgium	3	5	3.0	202.0	40.5	Province NUTS-2/TL2
	Brazil	20	80	332.6	732.9	83.0	State	-
	Bulgaria	2	2	17.1	21.9	22.8		NUTS-2
	Canada	10	90	940.4	464.0	80.3		TL2
	Chile	11	19	58.8	105.9	43.6		TL2
	China	18	89	166.9	8031.7	76.8	Province	-
	Croatia	1	1	24.7	19.7	32.0		NUTS-2
	Denmark	5	34	8.6	81.8	100.0	NUTS-2/TL2
	Estonia	3	6	6.3	34.4	76.6		NUTS-3
	Finland	3	22	77.6	89.5	90.1	NUTS-2/TL2
	France	16	20	23.6	310.3	36.4	Région NUTS-2/TL2
	Germany	7	39	10.8	259.7	20.6	NUTS-2/TL2
	Greece	11	25	10.8	110.9	96.2	NUTS-2/TL2
	Iceland	2	40	51.5	6.2	100.0	Region	-
	India	12	187	116.4	821.1	60.9	State	-
	Ireland	2	10	34.2	40.8	100.0	NUTS-2/TL2
	Italy	13	40	16.0	455.7	66.2	NUTS-2/TL2
	Japan	39	172	8.0	1966.4	87.1	Prefecture	TL3
	Latvia	2	3	7.0	58.9	63.7		NUTS-3
	Lithuania	1	2	5.2	34.3	12.4		NUTS-3
	Malaysia	10	17	29.3	380.8	55.4	State	-
	Malta	1	2	0.2	3.4	94.4		NUTS-3
	Mexico	13	33	65.9	102.6	34.6		TL2
	Morocco	9	12	67.9	40.0	52.9	Province	-
	Netherlands	6	14	4.0	482.1	64.7	NUTS-2/TL2
	New Zealand	2	15	132.5	42.7	100.0	Island	TL2
	Norway	6	26	42.4	145.7	93.9	NUTS-2/TL2
	Poland	2	5	20.6	45.0	9.6	NUTS-2/TL2
	Portugal	6	8	14.5	60.0	95.7	NUTS-2/TL2
	Romania	1	3	35.8	35.2	10.5		NUTS-2
	Slovenia	1	1	8.0	13.3	56.3		NUTS-2
	South Africa	3	8	130.9	275.2	37.6	Province	-
	South Korea	7	26	14.2	1032.1	100.0		TL2
	Spain	12	28	18.1	360.7	65.8	NUTS-2/TL2
	Sweden	8	30	51.3	140.4	100.0	NUTS-2/TL2
	Taiwan	4	13	9.0	235.7	100.0	Region	-
	Turkey	14	51	24.2	291.1	73.4		NUTS-2
	United Kingdom	24	52	7.9	489.6	65.8	NUTS-2/TL2
	United States	36	287	165.9	2052.0	86.8	State	TL2
	All countries	353	1,556	115.9	20963.3	72.5
	Europe	148	484	18.9	3787.3	53.1
	Asia-Pacific	99	358	132.3	13391.4	82.7
	Americas	90	419	261.5	3457.5	83.1
	Africa	12	20	83.7	315.2	41.3

Table 2 : Description of most significant factors Figure 1: Socio-economic convergence or divergence between port and non-port regions Figure 2: Correlations among throughput, population, and GDP at region and country levels

 2

	Population share	GDP share	Primary sector	GDP volume	Population size	Secondary sector	GDP per capita	Unemployment	Tertiary sector	Population density
	2.92	2.67
			1.22	1.10	0.99	1.14	0.93	1.08	0.86
										0.51
										ASIA
	1.03	1.02	1.22	1.06	1.01	1.04	1.18	1.07	1.01	0.75
										AMERICAS
	1.46	1.50	1.01	1.06	1.01	1.03	1.07	1.04	1.03	1.21
										WORLD
										2.39
	1.62	1.75
			0.93	1.04	1.01	0.99	1.07	1.01	1.09
										EUROPE
										2.50
	1.02	0.89		0.94	0.98	1.18	1.00	1.35	1.28
			0.44
										AFRICA

All data used in this study were freely accessible online or by request towards relevant organizations.

The unemployment rate is only available for 6 metropolitan cities so it was used as a proxy for state unemployment.

India's Union Territories of Lakshadweep and Andaman and Nicobar Islands, as well as Goa and Puducherry.

http://data.worldbank.org/

The cases include Haute Normandie and Ile-de-France (Paris, France), Valparaiso and Santiago (Chile), New South Wales and Canberra (Australia), Selangor, Kuala Lumpur, and Putrajaya (Malaysia), Maryland and Washington DC (USA), Tianjin and Beijing (China), Inner and Outer London (UK).

This corresponds to the average index among port regions

Weighting total port throughputs using the method of[START_REF] Charlier | Sur le concept de tonnages pondérés en économie portuaire. L'exemple du Northern Range[END_REF] to reduce discrepancies between tonnage units and value units did not provide a better determination coefficient (0.80). Some noticeable effects are the better visibility of Malaysia's transshipment functions for containers and the more balanced profile of the United States. At region level, the same method also produced a worse fit (0.32).


Ahmad Hably

Rogelio Lozano

Mazen Alamir

Jonathan Dumon

Observer-based control of a tethered wing wind power system: indoor real-time experiment

High altitude wind energy (HAWE) is a new field of renewable energy that has received an increasing attention during the last decade. Many solutions were proposed to harvest this energy including the usage of kites and wings. In this paper, a novel wind power system based on a tethered wing is presented. An observer-based control strategy that avoids the need for precise knowledge of the dynamic model is then applied. Numerical simulations show the performance of the proposed control scheme which is validated in real time on an indoor experimental set-up of Gipsa-lab control system department.

I. INTRODUCTION

Conventional wind turbines convert wind's kinetic energy into electrical energy through a rotor coupled with an alternator. As the wind is stronger and more steady at high altitudes, the wind turbines manufacturers try to build their wind turbines as big as possible. The cost of building these wind turbines is mainly due to the cost of steel and concrete used in the construction without forgetting the complexity due to their size. The research to overcome these drawbacks has led to the birth of many airborne wind energy conversion systems (AWEC). There are 3 main types of wind energy production systems such as balloons, airborne wind turbines and finally tethered wings and kites. The principle of the balloon developed by Magenn Power Inc. and called Magenn's Air Rotor System (MARS) is based on the rotation of a helium-filled balloon stationary at an altitude between 200m to 350m around a horizontal axis connected to an electrical machine. The produced energy is then transmitted to the ground by a conductive cable. The second solution studied by several companies such as Sky WindPower, Joby energy [8] or Makani Power [9], is composed of one or several airborne wind turbines to harvest energy directly in high-altitude winds and send it to the ground through conductive tethers. A third solution is to use power kites as renewable energy generators [START_REF] Ahmed | High altitude wind power systems: A survey on power kites[END_REF]. One can cite the "Kite Wind Generator" of Politecnico di Torino [START_REF] Canale | Kitegen: A revolution in wind energy generation[END_REF], [START_REF] Canale | High altitude wind energy generation using controlled power kites[END_REF] and the "Laddermill" of the Delft University of Technology [START_REF] Williams | Optimal cross-wind towing and power generation with tethered kites[END_REF]. These constructions have the same operation principle, which is to make the kite fly following a periodic trajectory (circular or figure-eight shape) that produces a positive amount of energy. As the kite moves upwards pulled by the wind, the dynamo generates power. When the kite is moving upwards, the wing's angle of attack is kept at a great value in order to produce a high lift force. During the phase when with the CNRS-University of Grenoble, Control Systems Dept. Rue de la Houille Blanche, Domaine Universitaire, Saint Martin d'Heres, France ahmad.hably,(rogelio.lozano,mazen.alamir,jonathan.dumon)@gipsalab.grenoble-inp.fr the kite is moving downwards the angle of attack is chosen in a way such that almost no energy is consumed bringing back the kite to its lowest altitude. As a consequence, there will be a positive amount of energy generated at the end of each cycle. Developing control strategies that enable the kite-based wind power systems to operate autonomously is a crucial issue. Indeed a kite or a wing subject to wind and attached to the ground with a rope may produce oscillations which might lead to crash. Nonlinear model predictive control (NMPC) has been applied in ( [START_REF] Ilzhofer | Nonlinear MPC of kites under varying wind conditions for a new class of large-scale wind power generators[END_REF], [START_REF] Canale | High altitude wind energy generation using controlled power kites[END_REF], [START_REF] Ahmed | Power maximization of a closedorbit kite generator system[END_REF]. Trajectory tracking methods including neural network control [START_REF] Furey | Evolution of neural networks for active control of tethered airfoils[END_REF], robust control [START_REF] Podgaets | Robust control of laddermill wind energy system[END_REF], direct-inverse control [START_REF] Novara | Direct data-driven inverse control of a power kite for high altitude wind energy conversion[END_REF], and nonlinear adaptive tracking control [START_REF] Baayen | Automatic trajectory tracking control of kites[END_REF] has been also proposed. In this paper, we propose a novel wind power system similar to the third solution mentioned above but instead of using kites, a tethered light yet robust wing is used to turn a motor with the rope. An observer-based control strategy is employed. The observer is used to estimate badly known aerodynamic parameters that need to be reconstructed. This control strategy allows us to overcome system's modeling errors and the need of expensive sensors. Our objective is to have a complete solution starting from the theory (modelling, observers, control and simulations) and ending with practice (validation and experimental results) 1 . This paper is organized as follows. In section II, the aerodynamical model of the wing-based wind power system will be presented. The control strategy will be detailed in section III followed by validation results in simulation (section IV) and in experiment (section V). Finally, after a discussion on the power produced in section VI, the paper will end with conclusions and perspectives in section VII.

II. AERODYNAMICAL MODEL

The tethered wing wind power system proposed in this paper is composed of a ground-based motor that will apply a traction force to a tether connected to a light-weight wing. Length of the tether r is controlled with pitch angle of the wing α u and angle of flight θ is stabilized with the control of spoilers (see figure 1). The wing's roll angle is mechanically stabilized. Its dynamical model can be given by :

θ = 1 r -2 θ ṙ + F T M (1) r = 1 M + M IM r θ2 M + F R -T (2)
Ṫ = β T u T -T (3) αu = β a u a -α u ( 4
)
where r is the tether's length from the wing to the motor on the ground, θ is the angle that the tether makes with respect to the horizon, M is the mass of the wing, M IM = I R 2 with I the inertia of the rotor and R the radius of the drum, T is the traction on the rope, α u is the pitch angle of the wing with respect to the ground. Equations 3 and 4 represent the first order dynamic response of actuators used in the experimental plateform to control respectively the traction in the tether and the pitch angle of the wing. Aerodynamical forces F R and F T are respectively the radial and tangential forces acting on the wing due to lift force L and drag force D and are expressed as:

F R = L sin(θ -α w ) + D cos(θ -α w ) -P sin θ (5) F T = L cos(θ -α w ) -D sin(θ -α w ) -P cos θ (6)
with α w is the wind angle defined later in equation ( 12) and P is the weight of the wing. Lift and drag forces can be expressed by:

L = 0.5ρSv 2 r C L , D = 0.5ρSv 2 r C D ( 7
)
where ρ is the air density, S is the wing surface, v r as we will see after is the norm of the apparent wind velocity vector (equation 13). Lift coefficient C L and drag coefficient C D are given in the following equations:

C L = ∂C L ∂α (α w + α u ) + C L0 (8)
C D = C 2 L πeλ + C D0 + β θ u θ (9)
with C L0 , C D0 , β θ are positive coefficients, e is the Oswald efficiency factor, and λ is the aspect ratio. u θ is the control of the spoilers that modifies the drag of the wing. Note that angle of attack α is equal to the sum of pitch angle α u and angle of apparent wind α w due to the movement of the wing. Horizontal relative air speed v h and vertical airspeed v v depend on the movements of the wing. When the wing is in a stationary position, the variations on θ and on r are equal to 0 (i.e θ = ṙ = 0). One has:

v h = V + r θ sin θ -ṙ cos θ (10) v v = r θ cos θ + ṙ sin θ ( 11
)
with V is the airspeed with respect to the ground. Using these equations, one obtains angle of wing's apparent wind velocity vector with respect to the horizon α w and its norm v r .

α w = -arctan r θ cos θ + ṙ sin θ V + r θ sin θ -ṙ cos θ (12) v r = (r θ cos θ + ṙ sin θ) 2 + (V + r θ sin θ -ṙ cos θ) 2 (13)
In the following sections, the control strategy of each subsystem composing the tethered wing wind power system will be presented.

III. THE CONTROL STRATEGY

The control strategy adopted here is to apply an observerbased feedback to the wing-based system in order to force some variables (namely r and the traction force T ) to track a certain periodic profile. The control strategy adopted here is to apply an observer-based feedback to the wing-based system in order to force some variables (namely r and the traction force T ) to track a periodic profile related to the mean power produced. The use of observers is essential since in such a system, model uncertainties must be handled and several variables that cannot be measured must be reconstructed.

A. The control of the angular velocity

We will first present the control strategy for angular displacement θ which will be stabilized using spoilers' control u θ . We have experimentally noticed that without spoilers, it is difficult to stabilize angular position θ because there exist severe oscillations at high angles of attack.Such oscillations have been damped by using spoilers on the wing. Note that the equation (1) can be rewritten in the following condensed form:

θ = χ + Qu θ ( 14
)
where χ represents the relatively badly known term

Q = - ρSv 2 r β θ 2M r sin(θ -α w ) χ = - 2 ṙ θ r + L cos(θ -α w ) -P cos θ rM - ρSv 2 r ( C 2 L πeλ + C D0 ) sin(θ -α w ) 2rM
This suggests that χ can be estimated using the following Luenberger estimator based on the sole measurement of θ:

χ = 0 0 1 X1 (15) Ẋ1 = (A 1 -L 1 C 1 ) X1 + B 1 u θ + L 1 θ (16)
where the observer's state is X 1 = [θ θ χ] and

A 1 :=   0 1 0 0 0 1 0 0 0   ; B 1 :=   0 Q 0   ; C 1 := 1 0 0 (17)
while L 1 is the observer gain obtained for instance using LQE design. Q is a slow time-varying parameter. This gives the control law given by ( 15)-( 16) together with

u θ = - 1 Q χ + λ 1 θ ( θ -ω ref ) + λ 2 θ (θ -θ ref ) (18)
where θ ref and ω ref are respectively some desired angle and angular velocity while λ 1 θ and λ 2 θ are some design parameters. Note that feedback law (18) only needs the estimation of θ, θ and χ.

B. The control of traction force T

The periodic traction reference trajectory is given by:

T ref (t) := T 0 + ∆ T sin( 2πt τ ) (19)
with the condition that T 0 -∆ T ≥ T min and τ is the period. This together with (3) leads to the following feedback law:

u T = T + 1 β T Ṫref -λ 2 (T -T ref ) (20)
with λ 2 a positive control parameter.

C. The control of radius r

Consider the following reference trajectory on state variable r:

r ref = r 0 -∆ r • Ṫref (21)
The reason behind this choice is that in case of perfect tracking, produced power expressed by P := ṙT would be positive. This suggests the following reference trajectory on F R involved in (2):

F ref R = (u T -r θ2 M ) + (M + M IM )h ( 22
)
where h is given by:

h = rref -λ r1 (r -r ref ) -λ r2 ( ṙ -ṙref ) (23)
with λ r1 and λ r2 are some positive control parameters. Here again based on (2), an estimation FR of F R can be obtained based on a Luenberger observer given by:

Ẋ2 = (A 2 -L 2 C 2 ) X2 + u T M + M IM + L 2 r (24) FR = 0 0 1 X2 (25)
where X 2 = [r ṙ F R ] and matrices A 2 , C 2 are given by:

A 2 :=    0 1 0 M (ω ref ) 2 M +M IM 0 1 M + M IM 0 0 0    C 2 := 1 0 0 (26)
and gain matrix L 2 is obtained using LQE design.

Having estimation FR and desired value F ref R , the following feedback can be defined:

u a = u amin + u a max -u amin 2 (27)
+ u a max -u amin 2 • tanh γ F ref R -FR
with γ is a design variable, u amin and u a max are respectively the lower limit and the upper limit of u a . Equation ( 27) together with ( 18) and (20) completely define the proposed feedback. The complete system (observers, feedback control, and the tethered wing power system) is shown on figure 2.

O 1 O 2 K 1 K 2 K 3 u a u θ u T r, ṙ θ, θ T x x x x x FR χ V V r ref , ṙref θ ref , θref T ref , Ṫref

IV. SIMULATION RESULTS

The proposed control strategy is first validated in simulation using values of coefficients of the experimental set-up given in the following table. Our objective is to stabilize θ around θ ref = 60 o with a null angular velocity. In addition, the traction force T must follow a periodic reference (equation 19) of a period τ = 30 sec. Several combinations of the control parameters (λ r1 , λ r2 , λ 1 θ , λ 2 θ , and γ) can be used. In fact, this depends whether one wants to focus on the stabilization of the dynamics of θ, r, or on both. In figure 3, the evolution of tether's angle θ and the control of angle of the spoilers u θ is depicted. One can clearly see the stabilization of angle θ at the desired value with an angular velocity equals to zero. On figure 4, the variations of tether's length r with respect to the reference trajectory and control u a are plotted. By changing u a , the wing arrives to follow the imposed reference on r. One can also clearly see the periodic evolution of the traction force T and its control u T on figure 5.

V. EXPERIMENTAL VALIDATION

In order to validate the proposed control strategy, an experimental set-up has been built in the Gipsa-lab laboratory for testing the proposed wind power system presented in the previous sections. An indoor wind tunnel has been built to test our prototype independently of the weather conditions (see figure 6). The air flow is produced at speeds up to 12 m/s. The wind tunnel is composed of 9 brushless motors of 800W each distributed on a surface of 1.8 m 2 . The dynamomotor system is a Maxon 2260L DC 97W driven by a 4 quadrants amplifier Maxon ADS 50/10. A wing has been built using foam and carbon fiber tubes to increase the rigidity (figure 7). We have chosen a wing profile NACA 0018. The yaw angular position ψ of the wing is controlled using a rudder (figure 7), which is controlled by PID gyroscopic control system FUTABA GY 5020 that stabilizes orientation ψ and angular velocity ψ . The wing has ailerons driven by servomechanisms which can be deployed to form spoilers. The spoilers are used to vary the drag to introduce damping in the wing displacement. For simplicity, the pitch angle α u is controlled using three strings attached to servomechanisms on the ground. Two tethers are attached to the back tips of the wing and one is attached to the front of the wing. In a future work we will consider controlling the angle of attack with ailerons which is a more realistic. The observer-based feedback controls tested in the simulation part are implemented on the experimental set-up using the xPC target real-time toolbox of Matlab (figure 8). Experimental results are similar to those obtained in the previous

VI. DISCUSSION ON THE POWER PRODUCED

As mentioned in the introduction, the system presented in this paper is a kite-based wind power system. The power produced using the control developed in section IV and in section V is shown respectively on figure 12 and figure 13. The mean power is positive although it is not big because of two reasons:

• Our main concern is to control the trajectory tracking by developing control laws robust with respect to modeling errors without the need of expensive sensors to measure aerodynamical forces and variables badly known.

• The small size of the experimental set-up. Simulations show that the wing's surface, the wind speed, the limits on r, and the reference trajectory play an important role in modifying the produced power. Moreover, the small size makes difficult to have crosswind trajectories that lead to better performances in terms of produced power.

VII. CONCLUSIONS AND PERSPECTIVES

A. Conclusion

In this paper we have studied a wind power generation system mainly composed of a wing attached to a dynamo with a tether. The wing subject to airflow generates a lift force which drives the dynamo on the ground to generate power. We have focused on such a configuration because it is less expensive than current wind power systems and could be used in rural areas to go to high altitudes without infrastructure. We have presented an aerodynamical model of the system. We have also proposed an observer-based control strategy to stabilize the wing's flight angle and to track a periodic reference on the tether's length. An indoor experimental set-up has been built including a wind tunnel, a wing with spoilers and a control system for the pitch angle. Both simulation and experimental results show promising results on the generated power.

B. Perspectives

For a future work, an observer of the wind speed (supposed to be known in the actual paper) will be designed. Another option is to design a nonlinear observer based on the measurement of θ and u θ to estimate the Q term of equation ( 14) to overpass the need of the exact value of wind speed. Further study will be done to obtain three dimensional optimal cycles that maximizes the generated power and the possibility of using the observer-based control approach for tracking these optimal trajectories. The choice of the cycle's period and the slope of the reference signal plays an important role in this direction. The slope in the production phase (when the wing pulls up the tether) must be less than the slope in the consumption phase (when the wing is pulled by the motor) in order to increase the total result.

VIII. ACKNOWLEDGMENT

Fig. 1 .

 1 Fig. 1. The whole wind power system (wing+tether+motor) in the airflow.

Fig. 2 .

 2 Fig. 2. The complete system: O 1 and O 2 are respectively the observers of F R and χ. K 1 , K 2 , and K 3 are respectively the state feedback control of ua (equation 27), u θ (equation 18), and u T (equation 20).

Fig. 3 .

 3 Fig.3. Simulation: Evolution of the tether's angle θ and the control u θ .

Fig. 4 .

 4 Fig. 4. Simulation: Evolution of the tether's length r with respect to r ref with the control input ua.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Simulation: Evolution the traction force T and its reference and the control u T .

Fig. 7 .

 7 Fig. 7. The wing with the spoilers in a medium drag position and the rudder used for the ψ angle stabilization.

Fig. 8 .

 8 Fig. 8. Flight test for the wind power system in the GIPSA wind tunnel.

Fig. 10 .

 10 Fig. 10. Real-time experiment: Evolution of the tether's length r with respect to r ref with the control input ua. r ref and T ref are chosen in order to have a positive produced power.

Fig. 11 .

 11 Fig. 11. Real-time experiment: Evolution the traction force T and its reference. The maximum and minimum values are determined by the limits of αu of the experimental set-up.

Fig. 12 .

 12 Fig.[START_REF] Podgaets | Robust control of laddermill wind energy system[END_REF]. The power produced in the simulation. The mean power over one period (between 22 sec and 52 sec) is equal to 0.0041886 W.

Fig. 13 .

 13 Fig.[START_REF] Williams | Optimal cross-wind towing and power generation with tethered kites[END_REF]. The power produced in the real time experiment. The mean power over one period is equal to 0.0042503 W.

The reader is invited to visit the "cerfs-volants" section of website[1] to have more details on the activities of Gipsa-lab in this field.

The authors would like to thank the technical staff of Gipsa-lab who provided an important technical support in the construction of the experimental set-up.


Thierry Hargé

Borel summation of the small time expansion of the heat kernel with a vector potential * †

Keywords: heat kernel, quantum mechanics, vector potential, first order perturbation, semi-classical, Borel summation, asymptotic expansion, Wigner-Kirkwood expansion, Poisson formula, Wiener integral, Feynman integral; A.M.S. subject classification: 35K08, 35C20, 30E15, 81Q30

Let pt be the heat kernel associated to the Laplacian with a vector potential. We prove under rather strong assumptions on this potential that the small time expansion of pt is Borel summable. An explicit formula for pt plays a central role. In the periodic case, a Poisson's formula is introduced.

Introduction

Let ν 1. Let c 0 , . . . , c ν be regular square matrix-valued functions on Ê ν . Let H be the operator defined by

H := -(∂ 2 x1 + • • • + ∂ 2 xν ) + 2 c 1 (x)∂ x1 + • • • + c ν (x)∂ xν + (∂ x1 c 1 + • • • + ∂ xν c ν ) -c 0 (x).
(1.1) Let p t (x, y) be the heat kernel associated to this operator. Let p conj t (x, y) be the conjugate heat kernel given by p t (x, y) = (4πt) -ν 2 exp -(x -y) 2 4t p conj t (x, y).

In a previous work about the scalar potential case (c 1 = 0, . . . , c ν = 0), we found conditions on c 0 providing Borel summation of the small time expansion of the conjugate heat kernel. We also studied the partition function on the torus. In this paper, similar questions are considered in the vector potential case. A general motivation for this work is a question asked by Balian and Bloch in [B-B]. Can quantum quantities be exactly recovered with the help of classical quantities? Indeed, coefficients of the small time expansion are classical quantities and then Borel summation gives a positive answer to this question for the quantum quantity y|e -tH |x = p t (x, y). See [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF] for partial comments about these questions and references therein. Another motivation is the introduction and the use of a so-called deformation formula which gives an "explicit" expression of the conjugate heat kernel (see Proposition 3.2). Note also that we always work in a complex setting (t ∈ , x, y ∈ ν ).

Let us now introduce more precisely our results. The following asymptotic expansion (Minakshisundaram-Pleijel expansion) is well known p conj t (x, y) =a 0 (x, y) + a 1 (x, y)t + • • • + a r-1 (x, y)t r-1 + t r O t→0 + (1).

We shall prove, under rather strong assumptions on c 0 , . . . , c ν , that this expansion is Borel summable and that its Borel sum is equal to p conj t (x, y). Assume now that c 0 , . . . , c ν are defined on the torus (Ê/ ) ν with values in a space of d × d matrices and that c 0 is Hermitian whereas c 1 , . . . , c ν are anti-Hermitian. Let

λ 1 λ 2 • • • λ n • • • , λ n →
+∞ be the eigenvalues of the operator H acting on periodic d -valued functions. We shall prove the Poisson formula: for small t ∈ , Ret > 0,

+∞ n=1 e -λnt = (4πt) -ν 2 q∈ ν e -q 2 4t u q (t), (1.2)
where, for q

∈ ν u q (t) = a 0,q + a 1,q t + • • • + a r-1,q t r-1 + • • • (1.3)
In (1.3), each expansion is Borel summable and u q denotes the Borel sum of such an expansion. Let us compare the scalar and the vector potential cases. We consider in [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF]

perturbations of -(∂ 2 x1 + • • • + ∂ 2 xν ) + α(x 2 1 + • • • + x 2 ν ) with α ∈ Ê.
For the vector potential case, we prefer to consider only perturbations of

-(∂ 2 x1 + • • • + ∂ 2 xν )
, and then focus on the perturbation term. This allows us to work easily with the Borel transform of the conjugate heat kernel. The proofs are then simplified as already remarked in [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF]. In what follows, we prove that this Borel transform is analytic on the complex plane and is exponentially dominated by the Borel variable on parabolic domains which are symmetric with respect to the positive real axis. One can expect that this result can be improved (in the scalar case, the Borel transform is exponentially dominated by the square root of Borel variable on the same domains).

Our choice of notation in (1.1) is imposed by our deformation formula without any reference to a Hermitian product. Now, let σ 1 , . . . , σ ν be Dirac matrices

(σ j σ k + σ k σ j = δ jk ). Then a Halmitonian like σ • ∂ x -ia(x)
2 is covered by our results both in an Abelian setting (electromagnetic field) and a non-Abelian setting (Yang-Mills field).

In the scalar potential case, the deformation formula is considered by E. Onifri [On] in a heuristic way and can be obtained with the help of Wiener's (or Feynman's) formula and Wick's theorem [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF]Appendix]. In the vectorial potential case, this connection also exists but it seems more difficult to exhibit it. In [F-H-S-S], an algorithm, using Wick's theorem, is proposed for computing, in a covariant invariant way, the coefficients of the small time expansion of p t (x, x).

Notation and main results

For

z = |z|e iθ ∈ , θ ∈] -π, π], let z 1/2 := |z| 1/2 e iθ/2 . Let T > 0. Let + := {z ∈ |Re(z) > 0}, D T := {z ∈ ||z| < T }, D + T := D T ∩ + and DT := z ∈ |Re( 1 z ) > 1 T .
DT is the open disk of center T 2 and radius T 2 . Let κ > 0. Let Sκ := {z ∈ |d(z, [0, +∞[) < κ} and

S κ := {z ∈ ||Imz 1/2 | 2 < κ} = {z ∈ |Rez > 1 4κ Im 2 z -κ}.
S κ is the interior of a parabola which contains Sκ (see figures 2.1 and 2.2 in [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF]).

We work with finite dimensional spaces of square matrices. We always consider multiplicative norms on these spaces (|AB| |A B|, for A and B square matrices) and we assume that |½| = 1. For A = (a i,j ) 1 i,j d with a i,j ∈ , we set

A * = (ā j,i ) 1 i,j d . For λ, µ ∈ ν , we denote λ • µ := λ 1 µ 1 + • • • + λ ν µ ν , λ := ( λ1 , . . . , λν ), Imλ := (Imλ 1 , . . . , Imλ ν ), λ 2 := λ • λ, |λ| := (λ • λ) 1/2 (if λ ∈ Ê ν , |λ| = √ λ 2
). Later, we shall extend these notations to operators and measures.

Let Ω be an open domain in m and let F be a complex finite dimensional space. We denote by A(Ω) the space of F -valued analytic functions on Ω, if there is no ambiguity on F .

Let B denote the collection of all Borel sets on Ê m . An F -valued measure µ on Ê m is an F -valued function on B satisfying the classical countable additivity property (cf. [Ru]). Let | • | be a norm on F . We denote by |µ| the positive measure defined by

|µ|(E) = sup ∞ j=1 |µ(E j )|(E ∈ B),
the supremum being taken over all partitions {E j } of E. In particular, |µ|(Ê m ) < ∞. Note that dµ = hd|µ| where h is some F -valued function satisfying |h| = 1 |µ|-a.e. In the sequel, we shall consider vector spaces F ν and F ν+1 with the following norms (depending on the norm of E wdθ) for every E ∈ B. Equivalently, there exists a vector-valued function h such that dµ = hdθ and |h| w θ-a.e. For instance, let u 1 , . . . ., u ν be measurable -valued functions and λ 1 , . . . , λ ν be F -valued measures. Then |u•dλ| |u|d|λ| (where u•dλ

F ). For f = (f 0 , . . . , f ν ) ∈ F ν+1 , we set f := (f 1 , . . . , f ν ) and |f | ⋆ := max(|f |, |f 0 |). If f = (f 0 , . . . , f ν ) is an F ν+1 -valued function on Ê m and θ is a positive measure such that Ê m |f 0 |dθ < ∞, . . . , Ê m |f ν |dθ < ∞,
:= u 1 dλ 1 +• • •+u ν dλ ν ).
We refer to [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF] for a rigorous definition of Borel and Laplace transform. Roughly speaking, assuming that f (respectively f ) is a function of a complex variable t (respectively τ ), f is the Laplace transform of f if

f (t) = +∞ 0 f (τ )e -τ t dτ t (2.1) whereas f is the Borel transform of f = a 0 + a 1 t + • • • + a n t n + • • • if f (τ ) = ∞ r=0 a r r! τ r .
With suitable assumptions, these two transforms are inverse each to other. In the whole paper, sums indexed by an empty set are, by convention, equal to zero.

Theorem 2.1 Let ε > 0. Let λ 0 , . . . , λ ν be measures on Ê ν with values in a complex finite dimensional space of square matrices verifying for q ∈ {0, . . . , ν}

Ê ν exp(εξ 2 )d|λ q |(ξ) < ∞.
(2.2)

Let c q (x) := exp(ix • ξ)dλ q (ξ). Let c(x) := c 1 (x), . . . , c ν (x) and ∂ x • c := ∂ x1 c 1 + • • • + ∂ xν c ν . Let u 1 be the solution of    ∂ t u= ∂ 2 x u -2c(x) • ∂ x u + c 0 (x) -∂ x • c u u| t=0 + = δ x=y ½ . (2.3)
Let v be defined by u=(4πt) -ν/2 e -(x-y) 2 4t v. Then v admits a Borel transform v (with respect to t) which is analytic on 1+2ν . Let κ, R > 0. Let C be defined by

C := Ê ν exp ε 2 ξ 2 + (1 + R)|ξ| + 2κ ε d|λ| ⋆ (ξ).
Then, for every

(τ, x, y) ∈ S κ × 2ν such that |Imx| < R and |Imy| < R, |v(τ, x, y)| e C|x-y| e C(|τ |+2|τ | 1/2 ) .
(2.4)

Remark 2.2 By (2.4) and (2.1), the solution u of (2.3) through v is meaningful for t ∈ , Re 1 t > 1 C . Since S κ contains Sκ , (2.4) implies that the small time expansion of v is Borel summable and that v is equal to the Borel sum of this expansion (cf. [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF]).

Remark 2.3 One can expect that (2.4) can be improved. Let us assume that ν = 1, that the measure λ 1 takes its values in and satisfies (2.2) where ε is replaced by 2ε. Let

c(x) := exp(ix • ξ)d λ(ξ)
where λ is a -valued measure on Ê satisfying (2.2). Let us choose c 0 := c 2 1 + c. Then c 0 is the Fourier transform of a measure satisfying (2.2) and the system (2.3) is equivalent to

   ∂ t u= ∂ x -c 1 (x) 2 u + c(x)u u| t=0 + = δ x=y .
Performing the substitution u = exp where

C 1 := Ê e R|ξ| d|λ 1 |(ξ), C := 2 Ê ν exp 2κ ε + ε 2 ξ 2 + R|ξ| d| λ|(ξ) 1/2 .
In particular, unlike in the general case (see Remark 2.2), the solution u of (2.3) through v is meaningful for t ∈ + , which is more natural. Actually the proof of Theorem 2.1 uses a deformation formula which does no take into account the fondamental notion of magnetic field (or curvature). The substitution (2.5) is efficient because the magnetic field vanishes in the one-dimensional case.

Corollary 2.4 Let ε > 0 and d ∈ AE * . For every (j, q) ∈ {0, . . . , ν} × ν , let c j,q be square matrices acting on d such that c 0,-q = c * 0,q and c j,-q = -c * j,q for every j ∈ {1, . . . , ν}. Assume that

q∈ ν e 4π 2 εq 2 |c j,q | < ∞
for every j ∈ {0, . . . , ν}. Let c j (x) = q∈ ν c j,q e 2iπq•x . Let Λ 1 Λ 2

• • • be the eigenvalues of the operator

H := -∂ 2 x + 2c(x) • ∂ x + ∂ x • c -c 0 (x)
acting on d -valued functions defined on the torus (Ê/ ) ν . For each q ∈ ν , there exists a function ŵ(q, .) analytic on satisfying

• For every κ > 0, there exists T, K > 0 such that, for every q ∈ ν and τ ∈ S κ ,

| ŵ(q, τ )| Ke |q| T e |τ |
T .

• There exists T such that for every t ∈ D

T +∞ n=1 e -Λnt = (4πt) -ν/2 q∈ ν e -q 2 4t +∞ 0 e -τ t ŵ(q, τ ) dτ t .
(2.6)

Remark 2.5 Let us consider the usual Hermitian product on L 2 (Ê/ ) ν , d . By our assumptions on the coefficients c j,q , H can be viewed as a self-adjoint operator with compact resolvent. The proof of Corollary 2.4 is similar to the proof of Corollary 3.3 in [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF] and is therefore omitted.

Remark 2.6

• Formula (2.6) can be viewed as a Poisson formula (see also [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF]): for each q ∈ ν , there are numbers a 0,q , a 1,q , . . . ∈ and functions R 0,q , R 1,q , . . . ∈ A( D T ) such that, for every r 0 and t ∈ D

T +∞ n=1 e -Λnt = (4πt) -ν/2
q∈ ν e -q 2 4t a 0,q + a 1,q t + • • • + a r-1,q t r-1 + R r,q (t) ,

(2.7) and for κ small enough, there exist M > 0 such that

|R r,q (t)| M r! κ r |t| r ,
for every r 0, q ∈ ν , t ∈ D T .

• One can easily check, using the deformation formula (3.3), that

a 0,q = [0,1] ν Tr Texp 1 0 q • c(x + sq)ds dx.
with the following definition for the ordered exponential

½ + 1 0 q • c(x + s 1 q)ds 1 + 0<s1<s2<1 q • c(x + s 2 q)q • c(x + s 1 q)ds 1 ds 2 + • • •
3 Proof of the theorems Our proofs use the following deformation formula which gives a representation of the heat kernel. First, we need Notation 3.1 Let λ 0 , . . . , λ ν be measures on Ê ν with values in a complex finite dimensional space of square matrices. Let us assume that for every R > 0 and q ∈ {0, . . . , ν} Ê ν exp(R|ξ|)d|λ q |(ξ) < +∞.

(3.1)

Let c and c 0 as in Theorem 2.1. For t,x 1 , . . . , x n , y 1 , . . . y n ∈ 2n+1 and 0 < s

1 < • • • < s n < 1, let V n (t, x 1 , y 1 , . . .) := (x n -y n ) • c y n + s n (x n -y n ) + tc 0 y n + s n (x n -y n ) • • • (x 1 -y 1 ) • c y 1 + s 1 (x 1 -y 1 ) + tc 0 y 1 + s 1 (x 1 -y 1 )
.

Let P n be the operator acting on A( 2νn ) defined by

P n := n j,k=1 ∂ x j∧k • ∂ y j∨k ,
where j ∧ k := min(j, k) and j ∨ k := max(j, k). Finally, for ξ = (ξ 1 , . . . , ξ n ) ∈ Ê νn , let

|ξ| 1 := |ξ 1 | + • • • + |ξ n |,
and let λ be the measure on Ê νn defined by

d νn ||λ||(ξ) = d|λ| ⋆ (ξ n ) • • • d|λ| ⋆ (ξ 1 ).
Proposition 3.2 Let λ 0 , . . . , λ ν be as in Notation 3.1. Let v be defined by

v = ½ + n 1 v n (3.2)
where

v n (t, x, y) := 0<s1<•••<sn<1 exp(tP n )V n (t, x 1 , y 1 , . . .) x1 = x, y1 = y . . . xn = x, yn = y d n s. (3.3) Let R > 0 and let Υ R := {(x, y) ∈ 2ν ||Imx| < R, |Imy| < R}. Let T R := Ê ν exp (1 + R)|ξ| d|λ| ⋆ (ξ) -1 . Then v ∈ A(D + TR × Υ R ). The function u := (4πt) -ν/2 e -(x-y) 2 4t
v is a solution of (2.3).

Let us give another useful expression of the function v. Let z = (z 1 , . . . , zn ) ∈ νn and ξ = (ξ 1 , . . . , ξ n ) ∈ Ê νn . Let µ z be the measure defined on Ê νn by

d νn µ z (ξ) = zn • dλ(ξ n ) + tdλ 0 (ξ n ) • • • z1 • dλ(ξ 1 ) + tdλ 0 (ξ 1 ) . Let s = (s 1 , . . . , s n ) such that 0 < s 1 < • • • < s n < 1. Let Pn be the operator defined by Pn := i n j,k=1 (1 -s j∨k )ξ j∨k • ∂ zj∧k -s j∧k ξ j∧k • ∂ zj∨k -2 1 j<k n ∂ zj • ∂ zk . Let s(1 -s) • n ξ ⊗ ξ := n j,k=1 s j∧k (1 -s j∨k )ξ j • ξ k , y + s(x -y) • ξ := y + s 1 (x -y) • ξ 1 + • • • + y + s n (x -y) • ξ n .
Remark 3.3 The following identity holds

v n = 0<s1<•••<sn<1 Ê νn e i y+s(x-y) •ξ × e -ts(1-s)•nξ⊗ξ e t Pn d νn µ z (ξ) z1 = x -y . . . zn = x -y d n s.
(3.4) Remark 3.3 and Proposition 3.2 will be proved together; we shall need the following remarks and lemma. Remark 3.4 will partially explain the shape of the operator P n in (3.3). Remark 3.5 (cf. Lemma 4.1 in [START_REF] Hargé | Borel summation of the small time expansion of the heat kernel[END_REF]) is crucial for the proof of the convergence of the integrals in (3.4) and for Theorem 2.1.

Remark 3.4

∂ t + n j=1 x j -y j t • ∂ xj , tP n = n j=1 ∂ xj • n j=1 ∂ xj . Remark 3.5 Let ξ ∈ Ê νn and 0 < s 1 < • • • < s n < 1. Then 0 s(1 -s) • n ξ ⊗ ξ n(ξ 2 1 + • • • + ξ 2 n ). (3.5)
Lemma 3.6 Let λ 0 , . . . , λ ν be as in Notation 3.1. Then, for m, n ∈ AE, m n, there exist measures µ n,m defined on Ê νn which are polynomial with respect to (x, y) and which do not depend on t such that

e t Pn d νn µ z (ξ) z1 = x -y . . . zn = x -y = m n t m d νn µ n,m (ξ) (3.6)
where

|d νn µ n,m (ξ)| n!e |ξ|1 × p + q = m 2p + q n p, q 0 1 p! |x -y| n-2p-q (n -2p -q)! d νn λ||(ξ). (3.7) Proof Let b γ (ξ) := -i 1 j γ s j ξ j + i γ j n (1 -s j )ξ j . Then Pn = n γ=1 b γ (ξ) • ∂ zγ -2 1 α<β n ∂ zα • ∂ zβ .
For α, β, γ = 1, . . . , n, α < β, let a γ := b γ (ξ) • ∂ zγ and A α,β := ∂ zα • ∂ zβ be the operators acting on linear combinations of monomials such as j∈J zj where J ⊂ {1, . . . , n}. Since

a 2 γ = A 2 α,β = 0, a γ A α,β = 0 if γ ∈ {α, β} and A α,β A α ′ ,β ′ = 0 if {α, β} ∩ {α ′ , β ′ } = ∅, one gets 1 r! n γ=1 a γ -2 1 α<β n A α,β r = I1,...,Ip,J (-2) p γ∈J a γ p k=1 A min(I k ),max(I k ) . (3.8)
Here, p 0 and the sum runs over all collections of pairwise disjoint subsets I 1 , . . . , I p , J of {1, . . . , n}, such that

|I 1 | = • • • = |I p | = 2 and |J| + p = r, without ordering on I 1 , . . . , I p .
Expanding e t Pn using (3.8) implies that the left hand side of (3.6) is equal to I1,...,Ip,J,K (-2t) p t |J|+|K| P {I},J,K (3.9) where P {I},J,K := {α,β},γ,δ,ε

dλ(ξ α ) • dλ(ξ β ) b γ (ξ) • dλ(ξ γ ) (x -y) • dλ(ξ δ ) dλ 0 (ξ ε ) > . (3.10)
Here we use the following convention: First, p ∈ AE, I 1 , . . . , I p , J, K are subsets of {1, . . . , n} such that

|I 1 | = • • • = |I p | = 2 and I 1 , . . . , I p , J, K are pairwise disjoint (hence 2p + |J| + |K| n).
The sum (3.9) runs over all such subsets without ordering as far as I 1 , . . . , I p are concerned.

Second, if I 1 , . . . , I p , J, K satisfy the previous assumptions, the product in (3.10) runs over all {α, β} = I 1 , . . . , I p , γ ∈ J, ε ∈ K and δ lying in the complementary of

I 1 ∪ • • • ∪ I p ∪ J ∪ K.
Third, the symbol {•} > means that the terms of the product are ordered. For instance, if n = 6, p = 1, I 1 = {1, 5}, J = {3} and K = {2, 4},

P {I},J,K = ν ℓ=1 (x -y) • dλ(ξ 6 )dλ ℓ (ξ 5 )dλ 0 (ξ 4 )b 3 (ξ) • dλ(ξ 3 )dλ 0 (ξ 2 )dλ ℓ (ξ 1 ).
Then (3.6) holds with

d νn µ n,m (ξ) := I1, . . . , Ip, J, K p + |J| + |K| = m (-2) p P {I},J,K .
Let p, j, k ∈ AE such that 2p + j + k n. There are n! 2 p p!j!k!(n-2p-j-k)! subsets I 1 , . . . , I p , J, K such that |J| = j and |K| = k in (3.9). Moreover

|b γ (ξ) • dλ(ξ γ )| |ξ| 1 d|λ|(ξ γ ).
Then

|d νn µ n,m (ξ)| p + j + k = m 2p + j + k n n! p!j!k!(n -2p -j -k)! |ξ| j 1 |x-y| n-2p-j-k d νn ||λ||(ξ).
Since, for q 0, j+k=q

1 k! |ξ| j 1 j! e |ξ|1 , (3.11)
one gets (3.7).

We can now prove Proposition 3.2.

Proof We claim that the right hand side of (3.4) is well defined and analytic on + × 2ν . Let |λ| as in Lemma 3.6. For n 1 and (t, x, y) ∈ + × 2ν , let (3.12)

Let R > 0 and suppose now that (x, y) ∈ Υ R . By (3.5) and Lemma 3.6

|d νn W n (t, x, y)| n!e (R+1)|ξ|1 2p + q n p, q 0 1 p! |x -y| n-2p-q (n -2p -q)! |t| p+q d νn λ (ξ).
Let

A := Ê ν exp (1 + R)|ξ| d|λ| ⋆ (ξ). Then 0<s1<•••<sn<1 Ê νn |d νn W n (t, x, y)| A n 2p+q n 1 p! |x -y| n-2p-q (n -2p -q)! |t| p+q .
Therefore

Q := 1 + n 1 0<s1<•••<sn<1 Ê νn |d νn W n (t, x, y)|
e A|x-y| p,q 0

A 2p+q |t| p+q p! since, for p, q ∈ AE, n 2p+q

A n-2p-q |x -y| n-2p-q (n -2p -q)! = e A|x-y| .

Hence Q < +∞ if A|t| < 1. Set T R := 1 A .
Then ṽn and hence ṽ := 1 + n 1 ṽn are well defined on D + TR × Υ R . By dominated convergence theorem, one can also check that ṽ is analytic on D + TR × Υ R . Let us now prove that ṽn = v n . One has

V n (t, x 1 , y 1 , . . .) = Ê νn exp i y 1 +s 1 (x 1 -y 1 ) •ξ 1 +• • •+i(y n +s n (x n -y n ) •ξ n × (x n -y n ) • dλ(ξ n ) + tdλ 0 (ξ n ) • • • (x n -y n ) • dλ(ξ n ) + tdλ 0 (ξ n ) . Then ∂ xα • ∂ y β V n (t, x 1 , y 1 , . . .) = Ê νn exp i y 1 + s 1 (x 1 -y 1 ) • ξ 1 + • • • × (I + II + III)d νn µ z (ξ) z1 = x -y . . . zn = x -y .
where

I := -s α (1 -s β )ξ α • ξ β , II := -∂ zα • ∂ zβ , III := i (1 -s β )ξ β • ∂ zα -s α ξ α • ∂ zβ .
This proves that ṽn = v n and therefore Remark 3.3. Then v = ṽ and v ∈ A(D + TR × Υ R ). We claim that the function

u = (4πt) -ν/2 e -(x-y) 2 4t v, (3.13)
where v is given by (3.2), is a solution of (2.3). Indeed, any solution u of (2.3) is obtained, using (3.13), from a solution v of the conjugate equation

   ∂ t + x-y t • ∂ x v = ∂ 2 x v + (x-y)•c(x) t v + c 0 (x) -∂ x • c(x) v -2c(x) • ∂ x v v| t=0 + ,x=y = ½ . Let v 0 = ½. It is then sufficient to verify that v n satisfies    ∂ t + x-y t • ∂ x v n = ∂ 2 x v n + (x-y)•c t v n-1 + (c 0 -∂ x • c)v n-1 -2c • ∂ x v n-1 v n | t=0 + ,x=y = 0 (3.14) for (t, x, y) ∈ D + TR × Υ R , n 1. It suffices to check (3.14) for (t, x, y) ∈ ]0, T R [×Υ R . By (3.3) v n (t, x, y) = 0<s1<•••<sn<t exp(tP n )V ♮ n (t, x 1 , y 1 , . . .) x1 = x, y1 = y . . . xn = x, yn = y d n s (3.15) where V ♮ n (t, x 1 , y 1 , . . .) := x n -y n t • c(y n + s n x n -y n t ) + c 0 (y n + s n x n -y n t ) • • • x 1 -y 1 t • c(y 1 + s 1 x 1 -y 1 t ) + c 0 (y 1 + s 1 x 1 -y 1 t ) .
One has ∂ t + x-y t • ∂ x v n = I + II where I is obtained by differentiating the domain of the integral in (3.15) whereas II is obtained by differentiating the integrand. Then

I = 0<s1<•••<sn-1<t exp(tP n )V ♮ n (t, x 1 , y 1 , . . .) sn = t x1 = x, y1 = y . . . xn = x, yn = y d n-1 s. Note that V ♮ n (t, x 1 , y 1 , . . .)| sn=t = x n -y n t • c(x n ) + c 0 (x n ) V ♮ n-1 (t, x 1 , y 1 , . .

.).

Since

P n = ∂ xn • ∂ yn + 2( n-1 j=1 ∂ xj ) • ∂ yn + P n-1 , one gets I = (x -y) • c(x) t + c 0 (x) v n-1 -∂ x • c(x)v n-1 -2c(x) • ∂ x v n-1 . (3.16)
Let us evaluate II. By Remark 3.4

∂ t + n j=1 x j -y j t •∂ xj e tPn = n j=1 ∂ xj 2 e tPn +e tPn ∂ t + n j=1 x j -y j t •∂ xj . (3.17)
Let j ∈ {1, . . . , n} and α ∈ {1, . . . , ν}. Denoting by x j,α the α-coordinate of x j ∈ ν , one has

∂ t + x j -y j t • ∂ xj x j,α -y j,α t = 0. (3.18) Then ∂ t + n j=1 x j -y j t • ∂ xj V ♮ n (t, x 1 , y 1 , . . .) = 0. (3.19)
Hence, by (3.17) and (3.19) |τ | p+q |x -y| n-2p-q p!(p + q)!(n -2p -q)! × exp 2κ 1/2 s(1 -s) • n ξ ⊗ ξ d νn λ (ξ).

∂ t + x -y t • ∂ x -∂ 2 x exp(tP n )V ♮ n (t, x 1 , y 1 , . . .) x1 = x, y1 = y . . . xn = x, yn = y = 0. Then II = ∂ 2 x v n . ( 3
By (3.5)

2κ 1/2 s(1 -s) • n ξ ⊗ ξ 2κ 1/2 n(ξ 2 1 + • • • + ξ 2 n ) 2 × 2κn ε 1/2 × ε 2 (ξ 2 1 + • • • + ξ 2 n ) 2κn ε + ε 2 (ξ 2 1 + • • • + ξ 2 n ).
Hence

|d νn Fn | exp (1 + R)|ξ| 1 + ε 2 (ξ 2 1 + • • • + ξ 2 n )ε + 2κn ε × n! 2p+q n
|τ | p+q |x -y| n-2p-q p!(p + q)!(n -2p -q)! d νn λ (ξ). |τ | p+q |x -y| n-2p-q p!(p + q)!(n -2p -q)! .

Then C n-2p-q |x -y| n-2p-q (n -2p -q)! = e C|x-y| .

Q := 1 + n 1 0<s1<
Hence Q < ∞. This proves that vn and v are well defined on 1+2ν since κ and R are arbitrary. By dominated convergence theorem, one can also check that vn , hence v, are analytic on 1+2ν . Since p!(p + q)! p!p!q! (2p)!q! 2 2p

Q e C|x-y| e 2C |τ | 1/2 e C|τ | .

Then v satisfies (2.4). By (3.12), (3.23) and the definition of K m (Lemma 3.7), vn is the Laplace transform of v n = ṽn . Then v defined by (3.2) is the Laplace transform of v. Hence Proposition 3.2 implies Theorem 2.1.

  one can define an F ν -valued (respectively F ν+1 -valued) measure λ (respectively λ) by setting dλ = f dθ (respectively dλ = f dθ). Then d|λ| = |f |dθ and d|λ| ⋆ = |f | ⋆ dθ. Let θ be a positive measure, w be a positive function and µ be a normed vector space valued measure. The notation |dµ| wdθ means that |µ(E)| E wdθ (or |µ|(E)

  in the previous system allows one to use the scalar case Borel summation result [Ha4, Theorem 3.1]. Therefore (2.4) can be improved as follows: |v(τ, x, y)| e C1|x-y| e C|τ | 1/2

d

  νn W n (t, x, y) := e i y+s(x-y) •ξ e -ts(1-s)•nξ⊗ξ e t Pn d νn µ z z1 = x -y . . . zn = x -y and ṽn (t, x, y) := 0<s1<•••<sn<1 Ê νn d νn W n (t, x, y).

  .20) Hence, by (3.16) and (3.20), we have checked (3.14) (the second line is trivial). For the proof of Theorem 2.1, we need [Ha4, Lemma 4.9]: Lemma 3.7 Let m 1. For B ∈ , let τ -→ K m (B, τ ) be the Borel transform of the function t -→ t m exp(-Bt). Then, for τ ∈ and B 0 |K m (B, Proof First, let us define v. Remark 3.3 and Lemma 3.6 suggest the following construction. Let d νn Fn , vn , v be defined by d νn Fn := m n exp i y + s(x -y) • ξ K m s(1 -s) • n ξ ⊗ ξ, τ d νn µ n,m (ξ), (3.22) vn (τ, x, y) := 0<s1<•••<sn<1 Ê νn d νn Fn d n s, that vn and v are well defined. Let κ, R > 0. Let (x, y) ∈ Υ R and τ ∈ S κ . By (3.7) and (3.21) |d νn Fn | e (1+R)|ξ|1 n! 2p+q n

LetC

  := Ê ν exp 2κ ε + ε 2 ξ 2 + (1 + R)|ξ| d|λ| ⋆ (ξ). Then 0<s1<•••<sn<1 Ê νn |d νn Fn |d n s C n 2p+q n

  ••<sn<1 Ê νn |d νn Fn |d n s

	e C|x-y|	p,q 0	|τ | p+q p!(p + q)!	C 2p+q

since, for p, q ∈ AE, n 2p+q

Since c 0 , . . . , cν are analytic and bounded on Ê ν , (2.3) admits a unique analytic solution on Ê + × Ê

2ν .