Konstantin Ilchev

"COMPENSATION MEASURES IN BULGARIA" INFORMATION REPORT prepared on behalf of EUROPEAN NETWORK ON THE LAW OF WATER AND NATURE 2000

Finally, in May 2011 the national network of protected areas has expanded to reach 34.34 % of Bulgaria, which makes Bulgaria the state-member with the highest percentage of Natura 2000 areas, in proportion to its national territory. Bulgaria is a country whose nature, fauna and flora are particularly varied, its conservation is a national and European issue.

Complex institutional arrangements require restructuring

It accepts the provisions of article 5 of the Law on protected territories ("LPT") July 5, 1999, amended in 2011, OJ No. 19 of 8 March 2011, in Regarding its operating rules are laid down in Articles 8 to 18 of the LPT. The Council shall meet at least twice a year. In case of need or emergency, the Council may be convened by the President or by a third of the members. The Council is headed by the President of the Republic, and in his absence by the President is provided by the Vice President. An agenda is prepared for all meetings, it must be sent to Board members at least five working days before the meeting. By the will of the Minister of Environment or upon proposal of the National Council of biodiversity, third parties may be invited to the deliberations. The Council takes its decisions by a simple majority (50 +1), in case of perfect equality, it is the President who has the last word. A special feature is the requirement for members who voted against a decision of the Council to establish a formal position in writing and notify the President.

The National Biodiversity Council is an administrative structure of public law funded by the budget of the Ministry of Environment and Water. Board members receive remuneration for written opinions and reports to the Council. The amount is set by the Council itself, and depending on the complexity of the problem.

In fact, the National Biodiversity Council is a support structure and support to the Ministry of Environment and Water, which is assigned the functions of some level assessments, programs and investment projects. In this respect, it is plausible that compensation schemes and mitigation come from this administrative structure, but it appears that the Council has not yet been concerned about the issue.

Overview in the field of water A favorable hydrological situation but deficient and even European standards infrastructure out of reach Bulgaria is situated in a poor area of humiditybut it has abundant winter precipitation that feeds enough its basins. Its renewable water resources are close to about 20 km3/year, higher assessed needs 14 km3/year. Because of the uneven distribution of rainfall in the territory and sometimes dilapidated infrastructure, some basins are frequently flooded.

The water supply networks serving almost 99 % of the population but most facilities are outdated. Approximately 20 % of water consumers are experiencing supply difficulties due to high technical losses (66 % on average) and the insufficient number of tanks. Despite the ratherinefficient use, water consumption in Bulgaria (99 l / cap. / Day) is below the European average (150 l / inhab. / Day). With regard to sanitation, the equipment is problematic under: only 70 % of cities (2% of villages) have a network of collection and treatment of wastewater. 75 % of Bulgarians are connected to gathering systems but only 56% of them see their treated wastewater treatment plant. Moreover, industrial effluents have doubled in volume in recent years with the development and progressive transition from industry to sanitation.

The World Bank estimates that more than EUR 12 billion modernization investments necessary in Bulgaria in the water sector to meet European standards.

The application of European standards is therefore out of reach: in the accession negotiations, Bulgaria obtained transitional periods for the implementation of Directive 91/271/EEC, particularly restrictive in sanitation. As a result there is obligation until 2014, to establish a network of sewage collection and processing unit in any community from 2,000 to 10,000.

Bulgaria also needs to implement the Water Framework Directive of 23 October 2000 (2000/60/EC) which requires, in 2015, a high level of quality of groundwater and surface water and the development of each basin committee a management plan, the use of pricing principle by applying the polluter pays principle with the help of development plans and water management guidelines. In late 2011, the development of master plans (expected at the end of the first half of 2013) was awarded to three companies, including the French and Egis Seureca.

After years of preparation, a profound reform of the sector should be launched, probably before the end of 2013. However, it requires the adoption of a new Law on water that has delayed (proposed end of 2012, passed first reading in early 2013, since blocked due to opposition from local officials and the resignation of Borisov government). This text provides for the establishment of a decentralized management governed grouped within departmental association, transfer these associations infrastructure management, the possibility of delegated services between associations and operators (public or private), the introduction of an aggregation scheme for farmers by the regulator as well as the transfer of a small portion of the taxes in investment activity.

Moreover, to support the reform, Bulgaria called in early 2012, technical assistance from the World Bank and the EBRD for the definition of programs in the water sector.

Two areas of business opportunities: generous financing under the European funding (yet hardly used) and public-private partnerships (whose beginning is difficult).

Complex institutional arrangements require restructuring

The state is the sole owner of the national water resources. Five public entities are now competent in the area:

-The Ministry of Environment and Water, which manages the national resource and issues permits for collection and discharge of water . He is the manager of the European program In connection with this, under Article 31 of the aforementioned law, any type of plan, program or investment project likely to have a significant impact on the environment is subject to a compatibility study (CS). This preliminary compatibility study is a kind of pre -study on compliance plan, program or project with the purpose of preserving the ecological state of the environment.

Then the procedure is declining, depending on the case into two procedures: first, the Environmental Assessment (EA) in the case of a plan or program by an organization under public law or with public service missions and, on the other hand, the environmental impact assessment (EIA) in the case of a private investment project.

Article 31 of the Biodiversity Act determines the cases in which a preliminary compatibility study is needed. In factall protected reserve areas, National Park, Natural Monument, maintained Reserve, Parkland, protected area, who are affected by a compatibility study. In addition, there are lists of protected areas approved by order of the Ministry of Environment and Water.

Application conditions and rules of procedure are determined by decree of 31 August 2007, last amended 30 November 2012 on the conditions and rules for achieving compatibility study plans, programs, projects and investments with the purpose and objectives for protected areas.

Article 20 of the Decree provides that in the event of imminent risk of harm to the environment as part of a plan, program or project investment, the competent authority investigates a compatibility study procedure to objectives of environmental protection. When such a decision is made, it includes information on alternative proposals, such as compensatory or mitigation requirements.

At the assessment of plans, programs and projects, the criteria set out in Article 22 of the Decree of 2007. Of the 11 criteria, we will meet the criterion number 8 on mitigation and / or measures restoration measures, and the test number 11, devoted to compensatory measures.

It is further clarified that the initiator plan, program or project investment must take charge, in accordance with Article 23 of the Decree of 2007, the development of documentation including mitigation proposals in order to avoid or reduce the impact of the initiative on the environment, and in addition, an analysis of alternatives must be provided. Article 24 of Decree 2007 states that the bidder must produce further solutions to eliminate or reduce the negative effects of the plan ,program or project , and featuring an elaborate vocabulary for a new term appears , that of remedial measures .

-. Lack of clear distinction between mitigation and compensation measures

Under Bulgarian law, no statutory or regulatory provisions specifically address the issue of the distinction between mitigation and compensation measures. The requirement to provide such action is necessary to study compatibility.

The scope of mitigation and compensation measures is expanded. This applies to all investment plans, programs and projects within the scope of Article 2 of the Decree of 2007, which establishes the circumstances in which a study of accounting is required.

The legal vocabulary is not constant, and it does not allow a clear distinction between the two concepts.

However, under Article 22 of the Decree of 2007, the assessment of environmental impact, as part of the compatibility study, is done according to eleven criteria, including mitigation and / or restoration measures (criterion 8) and Countervailing Measures (criterion 11). In addition, alternative measures should be considered in accordance with criterion 9.

-Case Study "Bansko Ski "

The case study will be devoted to an emblematic case in Bulgaria. This is the ski resort of Bansko which the largest in the Balkans. In fact, it comes to the question of the articulation between tourism development and environmental preservation.

It will also show that in reality the plans of environmental compensation numbers do not exist.

By letter dated 13.08.2012 the Ministry of Environment and Water has decided on the investment project of the company "Ulen" AD which is the concessionaire of the ski area. The project involved the supply of water to the ski slopes in the area ofBunderitsa.

The investment project was to build a water catchment station and ancillary buildings required for this activity. The location of the project, although there has four variants, is in the Pirin National Park but also in protected BG0000209 (habitats) and BG0000209 (birds) areas. In addition, the administrative body in charge has considered that the investment project was not an area that is part of the concession and thereby the waters were high at the time of the decision exclusively owned by the state.

Therefore, the Department refused to permit for pumping based on Article 13 of the Decree of 2007 stating that the investment project is not compatible with the activities permitted in these protected areas by the plan management.

And this is how the problem revolves around the concept of the management plan. Indeed, it prohibited the construction of building, logging and sampling of resource, which makes any investment project incompatible with the objectives of preserving the area. Therefore, in many ways, the question of possible mitigation and compensation did not arise. This is an illustration of the lack of actual environmental compensation measures. This Decision No OVOS-1148/13.08.2012 Department resulted in an application to the State Council of 28.09.2012. The Council of State has referred the case to the Administrative Court of Sofia for the right to be done.

This case is just one example among others of the complexity of the regulatory environment of the operating field. Indeed, very often an investment project is struck by the existence of a management plan, the average length is 10 years, excluding any economic activity. Therefore, we can consider that the preservation of zoned protect as will be optimal in the extent of investment projects cannot take place.

In contrast arises the question of financing the preservation of these areas.

-Financial Instruments

To date, the 26.10.2013, specific plans of financial compensation to investment and / or water management or environmental projects have not been developed. This is why the gaze is fixed on EU funding mechanisms that act as financial instruments for those who are constrained protected areas.

Under the agricultural fund for rural development, the Ministry of Economy in charge of the multiannual program provides a measure number 56 to provide compensatory payments to owners and / or users of land and forests lying in the scope of protected Natura 2000 areas. Are eligible for all areas of natural habitats within the Natura 2000 and all areas supporting bird's network.

Another measure -number 60 -is designed to offset shortfalls that a territory is seen hit by restrictions arising from the preservation requirements of Natura 2000 areas.

In addition, under the Cohesion Fund, the measure number 109 plans to support investment in the application of provisional measures and maintenance of protected areas.

-Case Law

The case law as regards the compensatory and mitigation regimeis not abundant due to its recent introduction in the Bulgarian legal framework since the membership of Bulgaria in the EU.

Neverthelesssince the transposition of "Birds" and "Habitats" Directives, as well as Directive 2001/42/EC on the assessment of the effects of certain plans and programmes on the environment and Directive 2011/92/EC on the assessment of the effects of certain public and private projects on the environment several cases have been ruled outlining and determining the contents of these regimes. However the current case law is not directly linked with the compensation regimes but it rather includes them on the occasion of related issues such as reports on assessment of the impact on the environment.

It is in the framework of procedure of approval of investment project that the Supreme Administrative Court (the "SAC") has ruled (Decision No. 16230 from 18 December 2012) on the compensation and mitigation regime. In accordance with the Environmental Protection Act it is the Supreme Environmental Expert Council (the "SEEC") which issues a positive or negative opinion on the assessment of the impact on the environment. The Ministry of Environment and Water follows the conclusions of the SEEC and thus approves or rejects the report on assessment of the impact on the environment. As being administrative body the decisions issued by the SEEC have to be motivated. In this specific case the SEEC has issued a negative opinion on the report without assessing the findings of the report on assessment of the impact on the environment including the compensation measures proposed. In other words the SEEC has the obligation to consider the compensation measures as condition for the approval of the report.

Hence the SAC invalidated the administrative deed as non-grounded and sent back the case to be re-examined.

In other case (Decision No. 11209 of 13 September 2012) concerning the authorization of investment project in the extraction industry of gold the SEEC has provided its positive opinion on the report of assessment on the effects on the environment which has been followed by the Ministry of Environment and the Water. This decision has been brought in action by a coalition of NGOs with scope of activity in the field of the environment. The latter have pretended that the decision approving the report has not been grounded and as a result has to be declared null and void. As regards of the compensation measures proposed and described in the report the SAC has found out that they are proportionate vis-à-vis the potential risks for the environment.

Nevertheless in its analysis the SAC underlined that the administrative body in charge has operational independence when adopting the report and it has the discretionary power as regards to its considerations. This suggests certain legal uncertaintyfor investors and applicants when the administrative body assesses the compensation measures. This would not be such an issue but at present there are no official guidelines issued by the government on how the compensation measures should be applied and whether they meet the environmental criteria.

On a related basis the European Commission has recently confirmed (17 October 2013) that it will take Bulgaria to CJEU in the framework of proceedings tofulfill an obligation to protect unique habitats and important species. In the region of Kaliakra (Black Sea shore in the North-East part of Bulgaria) which represents a major migratory route for endangered species of birds was constructed a wind power generation field, as well as other projects without adequate assessments of their environmental effects A reasoned opinion on this matter was sent in June 2012 to Bulgaria. The Governmenthas undertaken significant efforts to restrict the damage and prevent further developments but however failed to comply with a key requirement of the EU Habitats Directive, which obliges Member States to take appropriate measures to avoid the deterioration of habitats and disturbance of species for which the Nature 2000 sites have been designated, and compensate for any damage that occurs. The main critics of the Commission were that the Bulgarian government has permitted the constructions of the above wind turbines without approving any mitigation and/or compensation measures.

The case law of mitigation and compensation arises in disputes between the Ministry of Environment and Water, the regional inspectorates of environment and water on the one hand , and private investors to other.

Introduction

  Overview in the area of Natura 2000 December 31, 2006 was the final date for Bulgaria to make its inventory and the list of protected areas in the European Commission. In March 2007, the Council of Ministers has submitted to the Commission documentation for Natura 2000 protected areas. At that time, in addition to the opinions of associations, 18.2 % of the country had been proposed for the Natura 2000 network. Accordingly, Européan Topic Centre on Biological Diversity held inadequate proposal. Following this observation, the Council of Ministers approved in 2007, 33.8 % of the national territory as Natura 2000 areas.

  Bulgaria there are six categories of protected areasand water basins are also part of the protected areas. The urbanized areas, cities and other land affected by an administrative act of regulation (local development plan, development plan ) are excluded as protected areas. In all categories of protected areas the competent administrative body for their administration and preservation isthe national Biodiversity Council. The National Biodiversity Council gives opinions and proposals to the Minister of Environment and Water issues and issues concerning biodiversity. Its missions are nine in number: -Conduct of national policy and strategy; -Setting priorities for development , and management of the network of protected areas, the development and management of the national Natura 2000 network , the development of management plans for protected areas and protected areas , the development of action plan for biological diversity, the sustainable use of biological resources; -Application of normative rules; -Integration of the conservation of biodiversity into sectorial policies; -Appreciation lists of protected areas and amending list of protected areas; -Evaluation of action plans for biodiversity; -Assessment of applications for release or importation or breeding of non-regional species; -Assessing applications for health interventions in the Reserves; -Other issues and problems on the conservation of biological diversity. The powers of structural body of the Council are divided according to the status of members.

"

  Environment" 2007 -2013 nearly half of the funds are dedicated to projects drinking water and wastewater, -Ministry of Regional Development and Urban Development, responsible for national sector strategy and modernization of facilities, -64 plant operators and water suppliers ( water boards ) that are owned and managed by the state and / or local governments except for the governance of the city , granted in 2000 for a period 25 years to the English company United Utilities and which will was taken back in 2010, Veolia ( the dealer also has charge of the facilities for wastewater treatment) -Local authorities, owners of sanitation, managing programs of environmental protection, construction and maintenance of water treatment facilities, -The regulator that oversees the quality of services and supports the price of services operators (grid connection, water supply, sanitation, treatment etc.). Other public bodies are also involved in this sector: -Executive Agency of the environment; -Directions of water basins; -The Ministry of Agriculture (owner of irrigation infrastructure); -The national electricity company (which has 24 dams and waterfalls) -The Ministry of Economy and Energy (responsible since February 2012 the management and technical control of all dams which 2700 belong to local authorities); -Ministry of Health (responsible for health checks). This administrative yarrow is detrimental to the development of a coherent and sustainable public policy in the field of water. 1. -General Legal Framework From the outset, Compensation Arrangements under Bulgarian law does not exist. Scattered provisions deal with the subject by taking a few items to say compensatory measures to mitigate and reduce the negative effects on the environment. These provisions apply primarily when a plan, program or project investment is protected in the light of national law territory. In this respect, Article 3 of the Law on Biodiversity states that national ecological network includes: -The protected part of the European Natura 2000 network; -The areas protected under national law; -Areas of special protection. Level management plans provided for in Article 27 of the Biodiversity Act, measures must be taken to ensure the preservation of the living conditions of habitats and species protection. These measures are mainly: -Prohibition of certain activities, -Proactive measures against detrimental situations to the preservation of habitat conditions and species protection, -Support measures, coordination and control, -Measures of habitat restoration and reintroduction of species -Conducting scientific research, training and monitoring, This article establishes the beginnings of the future regulation of compensatory and mitigation measures.

As an example here are a few things:

The first case concerns an appeal filed by "Ingenering OOD" against Decision No. 291 / 22.03.2010 of the Ministry of Environment and Water, based on Article 160 of the Water Act .

Indeed, this decision approves the management plan southwest basin of water and the measures adopted by the Environmental Assessment No. 3-2/2009 which prohibits any economic activity in the basin of water between the city of Kyustendil and Greece.

In fact, he refers to the fact that the management plan, prohibiting economic activity, it is detrimental to the measures which it completely stops its investment in two hydro " MVEC Struma " project " MVEC Skrino " for which the environmental assessment took place.

The investment project falls within the protected Natura 2000 BG 0001013, 0002108 BG BGBG 0002100 and 0002107 areas. Mitigation measures under the Environmental Assessment are reducing ballasts, building fish passage and before the physicochemical water balance.

The SAC rejected the applicant's request on the grounds that the mitigation measures were not enough to make the investment project with the requirements of the management plan of southwest water basin.

The second case, No. 2270 of 12.02.2012, concerns an appeal filed by "Hidro OOD " on the rejection of the administrative authorization management of water in the region Danube basin.

With decision number 00056 PV3 -27.05.2009, the administrative authority rejects an approval of Burza river water pumping. Indeed, by letter number 328 of 4.02.2011 , the director of the Regional Inspectorate of Environment and Water certify that the investment project falls within the scope of Natura 2000 protected areas , particularly area " Balkan Zapaden " having to BG0001040 identification number (habitats) and BG0002002 ( birds).

The study of accounting with the object and purpose of nature conservation has resulted in the finding that the application of mitigation and restoration are likely to significantly reduce the negative effects so that the project investment will not produce a significant negative impact.

However, Article 46, paragraph 1 of the Water Act establishes a licensing system for water extraction. To grant such authorization, the competent authority shall take into consideration the availability of the resource needs of the applicant, the ecological status of water, and the goals of preserving the environment and the measures adopted to achieve the good ecological status determined by the management plan of water basins.

Therefore, the Director of the Directorate basin had taken a decision to refuse an investment project based on the management plan for the water basin that imposes restrictions and prohibitions with the aim of preserving habitats natural habitats and species found there. The provisions of Article 118 , j, point 4 of the Water Act provides for a ban on pumping water when the river is within the scope of Article 119a, paragraph 1 , item 5 defining the complex authorization procedures when it comes to preserving habitats or species. In this respect, the facts of the case show that the investment project involved a very Proteus area.

Moreover, in the framework of the management plan was anticipated that mitigation measures will be in effect a ban on construction of new hydraulic power.

The eyes of these elements, the valid decision of the Director of the State Council 29.07.2011 basin and dismisses the applicant's request.

This case illustrates the effect of juxtaposition of administrative procedures and some opacity of authorization schemes and exemption, and this is detrimental to legal certainty for operators.

-. Lack of complete legal system for alternatives measures

Due to the absence of consolidated compensation schemes there is no discussion of alternative measures , although paragraph three of the Annex to Decree 2007 provides in paragraph 7 that " alternative measures are the means of achieve the plan, program or project investment without producing a negative impact on the protected areas. Alternative measures may include: choosing another location, differentiation of geographical boundaries, change the master plan and implementation schedule or simply the use of alternative technologies. "

When the deciding authority that the investment project is likely to produce a significant impact on the environment, the latter decided to investigate a compatibility study in which alternative measures should be considered.

Then the assessment of the effect of a plan, program or project is assessed according to the criteria set out in Article 22 of the Decree of 2007, including the presence or absence of alternatives (item 9).

It is plausible that an alternative measure could go to open an investment project, but in reality the absence of specific plans as well as clear definitions do not allow for the efficient functioning of alternatives.

-Conclusion

In conclusion, it is imperative to standardize mitigation and compensation measures, and alternatives as legal uncertainty and sovereign appreciation of the managing authorities is detrimental to the Bulgarian economy.

The majority of protected areas are governed by rigid plans and non-scalable management, and therefore no investment project can not intervene.

Moreover, the absence of clear and precise legal definitions of what constitutes a compensation mitigation measure or allow private operators in the permanent doubt validation of their proposal.

Least, financial compensation damage to the environment do not exist, for example in the ski resort of Bansko could very well produced a double effect: that of restoring an ecological status for a degraded zone and the consolidation of tourism.

Finally, the absence of the Environmental Code in Bulgaria is detrimental to the building-up and the predictability of the whole legal systems in the field of environment.


Christophe Profeta

Kouji Yano

Yuko Yano

Local time penalizations with various clocks for one-dimensional diffusions

HAL is

Introduction

The systematic studies of penalizations started in 2003 with the works of Roynette, Vallois and Yor, essentially on Brownian motion; see for instance [START_REF] Roynette | Limiting laws associated with Brownian motion perturbed by normalized exponential weights[END_REF], [START_REF] Roynette | Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time[END_REF], or [START_REF] Roynette | Penalising Brownian paths[END_REF] for a monograph on this subject. Since then, many authors have generalized their results to other processes. When dealing with weights involving local times (L t , t ≥ 0), we may refer in particular to Debs [START_REF] Debs | Penalisation of the standard random walk by a function of the one-sided maximum, of the local time, or of the duration of the excursions[END_REF] for random walks, Najnudel, Roynette and Yor [START_REF] Najnudel | A global view of Brownian penalisations[END_REF] for Markov chains and Bessel processes, Yano, Yano and Yor [START_REF] Yano | Penalising symmetric stable Lévy paths[END_REF] for stable processes, or Salminen and Vallois [START_REF] Salminen | On subexponentiality of the Lévy measure of the diffusion inverse local time; with applications to penalizations[END_REF] and Profeta [START_REF] Profeta | Penalizing null recurrent diffusions[END_REF] for linear diffusions. In most of these papers, the authors focus on penalizations with a natural clock, letting the time t go to infinity in quantities such as P[F s f (L t )] P[f (L t )] where f is a positive integrable function and (F s ) is a bounded adapted process. This in turn requires some assumptions on the considered processes, see for instance Salminen and Vallois [START_REF] Salminen | On subexponentiality of the Lévy measure of the diffusion inverse local time; with applications to penalizations[END_REF], where the authors introduce a large family of diffusions for which local time penalization results apply.

In this paper, we shall rather study local time penalizations with different clocks, i.e.

we shall study the limit of quantities such as

P[F s f (L τ )] P[f (L τ )]
as τ tends to infinity in a certain sense along a parametrized family of random times. Examples of such results already appear in the literature, essentially when dealing with processes conditioned to avoid 0, i.e. with the function f (u) = 1 {u=0} . We refer to Knight [START_REF] Knight | Brownian local times and taboo processes[END_REF] for Brownian motions, Chaumont and Doney [START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF] and Doney [START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF] for Lévy processes, and Yano and Yano [START_REF] Yano | On h-transforms of one-dimensional diffusions stopped upon hitting zero[END_REF] for diffusions. Note in particular that, in general, different choices of τ lead to different limits, hence different penalized processes.

(1) Université Evry Val d'Essonne; (2) Kyoto University; (3) Kyoto Sangyo University. (1)(2) These authors were supported by JSPS-MAEDI Sakura program. (2) This author was supported by KAKENHI 26800058 and partially by KAKENHI 24540390 and KAKENHI 15H03624. (3) This author was supported by KAKENHI 23740073.

We consider here a generalized one-dimensional diffusion X (in the sense of Watanabe [START_REF] Watanabe | On time inversion of one-dimensional diffusion processes[END_REF]) defined on an interval I whose left boundary is 0, with scale function s(x) = x and speed measure m(dx). We assume that the function m : [0, +∞) → [0, +∞] is non-decreasing, right-continuous, null at 0 and such that

m is     
strictly increasing on [0, ℓ ′ ), flat and finite on [ℓ ′ , ℓ), infinite on [ℓ, +∞) (1.1) where 0 < ℓ ′ ≤ ℓ ≤ +∞. The choice of the right boundary point of I will depend on m; see [START_REF] Yano | On h-transforms of one-dimensional diffusions stopped upon hitting zero[END_REF]Section 2] for the boundary classification; see also Section 7. As for the left boundary point, we assume that 0 is regular-reflecting for X. This implies in particular that X admits a local time at 0, which we shall denote (L t , t ≥ 0), normalized so that P x ∞ 0 e -qt dL t = r q (x, 0) r q (0, 0) ,

where r q (x, y) denotes the resolvent density of X with respect to m(dy). Let φ q and ψ q be the two classical eigenfunctions associated to X via the integral equations, for x ∈ [0, ℓ):

φ q (x) =1 + q x 0 dy (0,y] φ q (z)m(dz), (1.3)
ψ q (x) =x + q x 0 dy (0,y] φ q (z)m(dz). (1.4)
Set

H(q) = lim x↑ℓ ψ q (x) φ q (x) . (1.5)
Denoting by m(∞) the limit lim x→+∞ m(x), we have lim q↓0 H(q) = ℓ and lim

q↓0 qH(q) = 1 m(∞) =: π 0 . (1.6)
With these notations, the resolvent density of X is given by

r q (x, y) = r q (y, x) = H(q)φ q (x) φ q (y) - ψ q (y) H(q) , 0 ≤ x ≤ y, x, y ∈ I ′ , (1.7)
where I ′ is defined in [13, Section 2] (see also Section 7), We finally define, following [START_REF] Yano | On h-transforms of one-dimensional diffusions stopped upon hitting zero[END_REF],

h q (x) =r q (0, 0)r q (0, x), (1.8)

h 0 (x) = lim q↓0 h q (x) = x -π 0 x 0 m(y)dy, (1.9)
and we call h 0 the normalized zero resolvent.

Let us outline the main results of the paper. For simplicity, we assume ℓ ′ = ℓ = ∞ and we take up the following three cases:

the boundary ∞ m(∞) (1,∞) xm(dx) (i) type-1-natural = ∞ = ∞ (ii) type-2-natural < ∞ = ∞ (iii) entrance < ∞ < ∞
The Brownian motion reflected at 0 is an example of case (i), where π 0 = 0 and h 0 (x) = x.

Some other examples will be given in Section 7. Let us now present our main results of the local time penalizations with various clocks.

1 • ) Let e q denote an exponential random variable with parameter q which is independent of the diffusion considered. We may adopt {e q : q > 0} as a clock since e q → ∞ in law as q ↓ 0.

Theorem 1.1. Let f ∈ L 1 + and x ≥ 0. For any bounded stopping time T and any bounded adapted process (F t ),

H(q)P x F T f (L eq ); T < e q -→ q↓0 P x [F T N h 0 ,f T ] and H(q)P x F T f (L eq ) -→ q↓0 P x [F T M h 0 ,f T ] (1.10)
where the P x -supermartingale N h 0 ,f and the P x -martingale M h 0 ,f are defined by

N h 0 ,f t = h 0 (X t )f (L t ) + +∞ 0 f (L t + u)du, t ≥ 0 (1.11)
and

M h 0 ,f t = N h 0 ,f t + π 0 t 0 f (L u )du, t ≥ 0.
(1.12) 2 • ) For a ∈ I, let T a = inf{t ≥ 0 : X t = a} denote the first hitting time of a by X. We may adopt {T a : a ≥ 0} as a clock since T a → ∞ a.s. as a → ∞.

Theorem 1.2. Assume that ∞ is natural. Let f ∈ L 1 + and x ≥ 0. For any bounded stopping time T and any bounded adapted process (F t ),

aP x [F T f (L Ta ); T < T a ] -→ a↑+∞ P x [F T M s,f T ] and aP x [F T f (L Ta )] -→ a↑+∞ P x [F T M s,f T ] (1.13)
where M s,f is the P x -martingale defined by

M s,f t = X t f (L t ) + +∞ 0 f (L t + u)du, t ≥ 0. (1.14) 3
• ) For a ≥ 0, let (L a t , t ≥ 0) denote the local time of X at level a, and define its right-continuous inverse:

η a u = inf{t ≥ 0, L a t > u}. (1.15)
We may adopt as a clock {η a u : a ≥ 0} for a fixed u > 0, since η a u → ∞ in law as a → ∞.

Theorem 1.3. Assume that ∞ is either type-1-natural or type-2-natural. Let f ∈ L 1 + , x ≥ 0 and u > 0. For any bounded stopping time T and any bounded adapted process (F t ),

aP x F T f (L η a u ); T < η a u -→ a↑+∞ P x [F T M s,f T ] and aP x F T f (L η a u ) -→ a↑+∞ P x [F T M s,f T ] (1.16)
where M s,f is the P x -martingale defined above.

We may also adopt as a clock {η a u : u ≥ 0} for a fixed a > 0, since η a u → ∞ a.s. as u → ∞.

P x [F T M β,α T ] (1.17)
where M β,α is the P x -martingale defined by

M β,α t = 1 + β(X t ∧ a) 1 + βa exp -βL t + β 1 + βa L a t , t ≥ 0. (1.18)
This paper is organized as follows. The local time penalizations are studied with an independent exponential clock in Section 2, then with a hitting time clock in Section 3 and finally with inverse local time clocks in Section 4. In Section 5, we study universal σ-finite measures. In Section 6, we characterize the limit measure for an exponential weight. The final section, Section 7, is an appendix on our boundary classification.

Local time penalization with an exponential clock

Let L 1 + denote the set of non-negative functions f on [0, ∞) such that ∞ 0 f (u)du < ∞. Lemma 2.1. Let f ∈ L 1 + , q > 0 and x ∈ I. Then P x [f (L eq )] = 1 H(q) h q (x)f (0) + r q (x, 0) r q (0, 0) ∞ 0 e -u/H(q) f (u)du . (2.1)
Proof. Using the excursion theory, we have

P 0 ∞ 0 f (L t )qe -qt dt =P 0 u ηu η u- f (u)qe -qt dt (2.2) =P 0 u f (u)e -qη u- T 0 (p(u)) 0 qe -qt dt (2.3) =P 0 ∞ 0 f (u)e -qηu du n 1 -e -qT 0 (2.4) = ∞ 0 f (u)e -u/H(q) du • 1 H(q) , (2.5)
where we write p for the excursion point process and set η u = s≤u T 0 (p(s)), the inverse local time at 0. We now obtain

P x [f (L e q )] =P x ∞ 0 f (L t )qe -qt dt (2.6) =P x T 0 0 f (L t )qe -qt dt + P x e -qT 0 P 0 ∞ 0 f (L t )qe -qt dt (2.7) =f (0) 1 - r q (x, 0) r q (0, 0) + r q (x, 0) r q (0, 0) • ∞ 0 f (u)e -u/H(q) du • 1 H(q) . (2.8)
This yields (2.1).

We have the following remarkable formulae.

Theorem 2.2. Let f ∈ L 1 + and x ∈ I. Then the following assertions hold:

(i) If ℓ < ∞, i.e., 0 is transient, then

P x [f (L ∞ )] = 1 ℓ xf (0) + 1 - x ℓ ∞ 0 e -u/ℓ f (u)du . (2.9)
(ii) If π 0 > 0, i.e., 0 is positive recurrent, then

P x ∞ 0 f (L t )dt = 1 π 0 h 0 (x)f (0) + ∞ 0 f (u)du . (2.10)
Proof. (i) Suppose f is bounded. Denote g = sup{t : X t = 0}. Then, at every sample point, we have f (L (1/q)e 1 ) = f (L g ) = f (L ∞ ) for q > 0 small enough. Hence we obtain P x [f (L eq )] = P x [f (L (1/q)e 1 )] → P x [f (L ∞ )] as q ↓ 0 by the dominated convergence theorem. It is obvious that the right hand side of (2.1) converges to that of (2.9). Hence we obtain (2.9).

For the general case, we have the formula (2.9) for f ∧ n and from this, letting n → ∞, we obtain (2.9) for f . (ii) We may rewrite (2.1) as

P x ∞ 0 f (L t )e -qt dt = 1 qH(q) h q (x)f (0) + r q (x, 0) r q (0, 0) ∞ 0 e -u/H(q) f (u)du . (2.11)
Letting q ↓ 0, we obtain

P x ∞ 0 f (L t )e -qt dt → P x ∞ 0 f (L t )
dt by the monotone convergence theorem, and hence we obtain (2.10).

Let F 0 t = σ(X s : s ≤ t) and F t = F 0 t+ . Note that e q is independent of F ∞ := σ( t F t ). Lemma 2.3. Let f ∈ L 1 + and x ∈ I. For q > 0, set N q t = H(q)P x f (L eq )1 {t<eq} |F t , M q t = H(q)P x f (L eq )|F t (2.

12)

and

N h 0 ,f t =h 0 (X t )f (L t ) + 1 - X t ℓ ∞ 0 e -u/ℓ f (L t + u)du, (2.13)
M h 0 ,f t =N h 0 ,f t + A h 0 ,f t , (2.14)
A h 0 ,f t =π 0 t 0 f (L u )du. ( 2

.15)

Then the following assertions hold:

(i) N q t → N h 0 ,f t and M q t → M h 0 ,f t , P x -a.s. as q ↓ 0; (ii) (N h 0 ,f t ) is a P x -supermartingale.
Proof. In what follows in this section we sometimes write N t , M t and A t simply for

N h 0 ,f t , M h 0 ,f t and A h 0 ,f t , respectively. (i) Since f (a + •) ∈ L 1
+ , we have, applying the Markov property, N q t =H(q)e -qt P Xt [f (a + L eq )] a=Lt (2.16)

=e -qt h q (X t )f (L t ) + r q (X t , 0) r q (0, 0) ∞ 0 e -u/H(q) f (L t + u)du . (2.17) It is now clear that N q t → N t , P x -a.s. Since A q t :=M q t -N q t (2.18) =H(q)P x [f (L eq )1 {eq≤t} |F t ] (2.19) =qH(q) t 0 f (L u )e -qu du, (2.20)
we obtain A q t → A t and M q t → M t , P x -a.s. (ii) Since for s ≤ t we have 1 {t<eq} ≤ 1 {s<eq} , we easily see that (N q t ) is a P xsupermartingale. For s ≤ t, we apply Fatou's lemma to obtain

P x [N t |F s ] ≤ lim inf q↓0 P x [N q t |F s ] ≤ lim inf q↓0 N q s = N s , (2.21)
which shows that (N t ) is a P x -supermartingale.

Theorem 2.4. Let f ∈ L 1 + and x ∈ I. Then, for any finite stopping time T , it holds that

N q T -→ q↓0 N h 0 ,f T in L 1 (P x ). (2.22)
Consequently, for any bounded adapted process (F t ), it holds that

lim q↓0 H(q)P x [F T f (L e q ); T < e q ] = P x [F T N h 0 ,f T ]. (2.23)
Proof. Observe first by Fatou's lemma that

P x [N T ] ≤ lim inf n→∞ P x [N T ∧n ] ≤ P x [N 0 ] < ∞. (2.24)
Let us compute N q T . We have

N q T =e -qT h q (X T )f (L t ) + e -qT r q (X T , 0) r q (0, 0) ∞ 0 e -u/H(q) f (L T + u)du (2.25) =(I) q + (II) q .
(2.26)

We write similarly

N T =h 0 (X T )f (L T ) + 1 - X T ℓ ∞ 0 e -u/ℓ f (L T + u)du (2.27) =(I) + (II).
(2.28) Since (II) q ≤ ∞ 0 f (u)du, we may apply the dominated convergence theorem to obtain (II) q → (II) in L 1 (P x ).

If π 0 = 0, then we have (I

) q ≤ X T f (L T ) = h 0 (X T )f (L T ) ≤ N T . If π 0 > 0 and ℓ ′ is regular-reflecting, then we have h 0 (x) ≥ cx with c = h 0 (ℓ ′ )/ℓ ′ > 0, since h 0 (x) is concave. We now have (I) q ≤ X T f (L T ) ≤ c -1 h 0 (X T )f (L T ) ≤ c -1 N T . In both cases, since P x [N T ] ≤ P x [N 0 ] < ∞,
we may apply the dominated convergence theorem to obtain (I) q → (I) in L 1 (P x ).

If π 0 > 0 and ℓ ′ is either entrance or natural, we have (I) q ≤ X T f (L T ). Since we see by (ii) of Lemma 3.2 that

P x [X T f (L T )] ≤ P x [N s,f T ] ≤ P x [N s,f 0 ] = xf (0) + 1 - x ℓ ∞ 0 e -u/ℓ f (u)du < ∞, (2.29)
we may apply the dominated convergence theorem to obtain (I) q → (I) in L 1 (P x ).

Therefore we have obtained the former assertion.

For the latter assertion, we have

H(q)P x [F T f (L eq ); T < e q ] = P x [F T N q T ] -→ q↓0 P x [F T N T ].
(2.30) Theorem 2.5. Let f ∈ L 1 + and x ∈ I. Let T be a finite stopping time such that

P x T 0 f (L u )du < ∞. (2.31)
Then it holds that

M q T → M h 0 ,f T in L 1 (P x ). (2.32)
Consequently, for any bounded adapted process (F t ), it holds that

lim q↓0 H(q)P x [F T f (L e q )] = P x [F T M h 0 ,f T ]. (2.33)
Proof. By (2.31), we have

T 0 f (L u )e -qu du → T 0 f (L u )du in L 1 (P x ). This shows that A q T → A T in L 1 (P x ), which implies M q T → M T in L 1 (P x )
. The latter assertion is obvious.

Theorem 2.6. The condition (2.31) is satisfied whenever T is a bounded stopping time.

For any f ∈ L 1 + , it holds that

M h 0 ,f t = h 0 (X t )f (L t ) + 1 - X t ℓ ∞ 0 e -u/ℓ f (L t + u)du + π 0 t 0 f (L u )du (2.34) is a P x -martingale. Consequently, the identity N h 0 ,f t = M h 0 ,f t -A h 0 ,f
t may be regarded as the Doob-Meyer decomposition of the supermartingale (N h 0 ,f t ).

Proof. Since

q 2 ∞ 0 P x t 0 f (L u )du e -qt dt = P x ∞ 0 f (L u )qe -qu du = P x f (L eq ) < ∞ (2.35)
and since t → P x t 0 f (L u )du is increasing, we see that P x t 0 f (L u )du < ∞ for all t ≥ 0. In other words, the assumption (2.31) is satisfied when T is a bounded stopping time. This shows that (M t ) is a P x -martingale. Remark 2.7. If ℓ ′ is type-1-natural, then the identity (2.34) becomes

M h 0 ,f t = X t f (L t ) + ∞ 0 f (L t + u)du, (2.36)
which is nothing else but the Azema-Yor martingale. In this sense we may regard the identity (2.34) as a generalization of the Azema-Yor martingale. Another generalization will be given in Theorem 3.5.

Remark 2.8. If we take f (u) = 1 {u=0} , we have

M h 0 ,f t = h 0 (X t )1 {T 0 >t} + π 0 (T 0 ∧ t).
(2.37)

In particular, from the identity

P x [M h 0 ,f 0 ] = P x [M h 0 ,f t ], we obtain h 0 (x) = P x [h 0 (X t ); T 0 > t] + π 0 P x [T 0 ∧ t], (2.38)
which verifies the first assertion of Theorem 6.4 of [START_REF] Yano | On h-transforms of one-dimensional diffusions stopped upon hitting zero[END_REF].

Local time penalization with a hitting time clock

In this section we assume that ℓ(= ℓ ′ ) is either entrance or natural. Since any point in [0, ℓ) is accessible but ℓ is not, we have

P x (T a → ∞ as a ↑ ℓ) = 1. (3.1) Lemma 3.1. Let f ∈ L 1 + and x ∈ I.
Then, for any a ∈ I with x < a,

P x [f (L Ta )] = 1 a xf (0) + 1 - x a ∞ 0 e -u/a f (u)du . (3.2)
Proof. Let P a x denote the law of X •∧Ta under P x . Then we have

P x [f (L Ta )] = P a x [f (L ∞ )]. (3.3)
Since {X, P a x } is a diffusion process on [0, a] where a is a regular-absorbing boundary, we may use (i) of Theorem 2.2 and obtain (3.2). Lemma 3.2. Let f ∈ L 1 + and x ∈ I. For any a ∈ I with x < a, set

N a t = aP x f (L Ta )1 {t<Ta} |F t , M a t = aP x [f (L Ta )|F t ] (3.4)
and

M s,f t =X t f (L t ) + 1 - X t ℓ ∞ 0 e -u/ℓ f (L t + u)du. (3.5)
Then the following assertions hold:

(i) N a t → M s,f t and M a t → M s,f t , P x -a.s. as a ↑ ℓ; (ii) (M s,f t ) is a P x -supermartingale and is a local P x -martingale.

Proof. In what follows in this section we sometimes write M t simply for M s,f t .

(i) Since f (b + •) ∈ L 1 + , we have, by Lemma 3.1, N a t =a P Xt [f (b + L Ta )]| b=Lt 1 {t<Ta} (3.6) = X t f (L t ) + 1 - X t a ∞ 0 e -u/a f (L t + u)du 1 {t<Ta} . (3.7) Since T a → ∞ as a ↑ ℓ, we have N a t → M t , P x -a.s. Set A a t = M a t -N a t = af (L Ta )1 {Ta≤t} . (3.8)
Since A a t → 0, P x -a.s., we have M a t → M t , P x -a.s. (ii) In the same way as (ii) of Lemma 2.3, we can see that (M t ) is a P x -supermartingale.

It is obvious that (M a t ) is a P x -martingale. Let {a n } be a sequence of I such that a n ↑ ℓ. If we take σ n = inf{t : X t > a n }, we have A a σn∧t = af (L Ta )1 {Ta≤σn∧t} = 0 for any a > a n , so that we have M a σn∧t → M σn∧t in L 1 (P x ) as a ↑ ℓ. This shows that (M t ) is a local P x -martingale. Theorem 3.3. Let f ∈ L 1 + and x ∈ I. Then, for any finite stopping time T , it holds that

N a T -→ a↑ℓ M s,f T in L 1 (P x ). (3.9)
Consequently, for any bounded adapted process (F t ), it holds that

aP x [F T f (L Ta ); T < T a ] -→ a↑ℓ P x [F T M s,f T ]. (3.10)
Proof. We have

N a T =X T f (L T )1 {T <Ta} + 1 - X T a ∞ 0 e -u/a f (L T + u)du1 {T <Ta} , (3.11)
M T =X T f (L T ) + 1 - X T ℓ ∞ 0 e -u/ℓ f (L T + u)du. (3.12)
Since N a T ≤ M T and since

P x [M T ] ≤ lim inf n→∞ P x [M T ∧n ] ≤ P x [M 0 ] < ∞, (3.13)
we may apply the dominated convergence theorem to obtain (3.9). The remaining assertion is obvious.

Lemma 3.4. Suppose that ℓ is natural. Then

aP x (T a ≤ t) -→ a↑ℓ 0 for all t ≥ 0. (3.14)
Proof. If ℓ < ∞, i.e., ℓ is type-3-natural, then (3.14) is obvious.

Suppose ℓ = ∞. Then we have

aP x (T a ≤ t) ≤ ae t P x [e -Ta ] = e t φ 1 (x) • a φ 1 (a) . (3.15) Since ℓ = ∞ is natural, we have φ 1 (a) = 1 + a 0 dx (0,x] φ 1 (y)dm(y) ≥ a 0 dx (0,x] dm(y) -→ a↑ℓ ∞ (3.16)
and, for a > 1,

φ ′ 1 (a) ≥ (0,a] dm(x) x 0 φ ′ 1 (y)dy ≥ φ ′ 1 (1) (1,a] dm(x) x 1 dy -→ a↑ℓ ∞.
(3.17) Thus, by the l'Hôpital rule, we obtain a/φ 1 (a) → 0 as a ↑ ℓ = ∞. Therefore we obtain (3.14).

Theorem 3.5. Suppose that ℓ is natural. Then, for any f ∈ L 1 + and for any bounded stopping time T , it holds that

M a T -→ a↑ℓ M s,f T in L 1 (P x ). (3.18)
Consequently, for any bounded adapted process (F t ), it holds that

aP x [F T f (L Ta )] -→ a↑ℓ P x F T M s,f T . (3.19)
It also holds that

M s,f t = X t f (L t ) + 1 - X t ℓ ∞ 0 e -u/ℓ f (L t + u)du (3.20) is a P x -martingale.
Proof. Suppose that f ∈ L 1 + is bounded. Since A a T → 0, P x -a.s. and since

P x [A a T ] ≤ a f ∞ P x (T a ≤ T ) -→ a↑ℓ 0, (3.21)
we see that A a T → 0 in L 1 (P x ). Hence we obtain (3.18) and (3.19) in this special case. We now see that

P x [M s,f t ] = P x [M s,f 0 ], i.e., P x X t f (L t ) + 1 - X t ℓ ∞ 0 e -u/ℓ f (L t + u)du =xf (0) + 1 - x ℓ ∞ 0 e -u/ℓ f (u)du (3.22)
holds for all t ≥ 0 and all bounded f ∈ L 1 + . By considering f ∧ n, taking n → ∞ and applying the monotone convergence theorem, we can drop the boundedness assumption and obtain (3.22) for all t ≥ 0 and all f ∈ L 1 + . By (ii) of Lemma 3.2, we see, for any f ∈ L 1 + , that (M s,f t ) is a P x -supermartingale with constant expectation, which turns out to be a P x -martingale.

Let f ∈ L 1 + . Since (M s,f t ) is a P x -martingale, we may apply the optional stopping theorem to see that

P x [A a T ] =P x [X Ta f (L Ta ); T a ≤ T ] (3.23) ≤P x [M Ta ; T a ≤ T ] (3.24) =P x [M T ; T a ≤ T ] -→ a↑ℓ 0. ( 3

.25)

Since A a T → 0, P x -a.s., we see that A a T → 0 in L 1 (P x ). Hence we obtain (3.18) and (3.19) in a general case. Remark 3.6. Suppose ℓ is entrance. We claim that (M t ) is not a true P x -martingale if f (0) > 0. Suppose (M t ) were a P x -martingale. On one hand we would have

P x [M t∧T 0 ] = M 0 = xf (0) + ∞ 0 f (u)du. (3.26)
On the other hand, we see that P x [M t∧T 0 ] is equal to

P x [M t ; t < T 0 ] + P x [M T 0 ; t ≥ T 0 ] = P x [X t ; t < T 0 ]f (0) + ∞ 0 f (u)du. (3.27)
Hence we would have P x [X t ; t < T 0 ] = x, which would contradict the fact that the scale function s(x) = x is not P x -invariant (see Theorem 6.5 of [START_REF] Yano | On h-transforms of one-dimensional diffusions stopped upon hitting zero[END_REF]).

Local time penalization with inverse local time clocks

4.1 Limit as a tends to infinity with u being fixed Suppose ℓ ′ (= ℓ = ∞) is either entrance, type-1-natural or type-2-natural. We thus have, for any x ∈ I and any u > 0,

P x (η a u < ∞) = 1 and η a u -→ a→∞ ∞ in law under P x ; (4.1)
in fact, we have

P x [e -qη a u ] → 1 as q ↓ 0, 0 as a → ∞, (4.2)
since we have r q (a, a) -→ q↓0 ∞, φ q (a) -→ a→∞ ∞ and

P x [e -qη a u ] = P x [e -qTa ]P a [e -qη a u ] = φ q (x) φ q (a) exp - u r q (a, a) . (4.3)
For ν ≥ 0, we denote by I ν (x) the modified Bessel function of the first kind, which may be represented as a series expansion formula (see e.g. [START_REF] Lebedev | Special functions and their applications[END_REF], eq. (5.7.1) on page 108) by

I ν (x) = ∞ n=0 (x/2) ν+2n n!Γ(ν + n + 1) , x > 0. (4.4)
We recall the asymptotic formulae (see e.g. [START_REF] Lebedev | Special functions and their applications[END_REF], Section 5.16): For any u > 0 and f ∈ L 1 + ,

I ν (x) ∼ x↓0 (x/2) ν Γ(1 + ν) , I ν (x) ∼ x→∞ e x √ 2πx . ( 4
P a [f (L η a u )] =e -u/a f (0) + ∞ 0 f (y)ρ a u (y)dy, (4.7)
where

ρ a u (y) = e -(u+y)/a u/y a I 1 2 √ uy a . (4.8)
Proof. Let p a (v) denote the point process of excursions away from a and n a its excursion measure. Since L increases only on the intervals (η a v-, η a v ), we have

L η a u = v≤u: p a (v)∈{T 0 <∞} (L η a v -L η a v-) = v≤u: p a (v)∈{T 0 <∞} L Ta (p a (v)). (4.9)
Since n a (T 0 < T a ) = 1/a < ∞, the sum of (4.9) is a finite sum, and so we see that {(L η a u ) u≥0 , P a } is a compound Poisson process with Lévy measure n a (L Ta ∈ ds; T 0 < T a ). Let λ 0 a = inf{v : p 0 (v) ∈ {T a < ∞}}. Then we have P 0 (L Ta > s) = P 0 (T a > η 0 s ) = P 0 (λ 0 a > s) = e -sn 0 (Ta<∞) = e -s/a . (4.12)

Thus we obtain (4.6).

Let {S n } be a process with i.i.d. increments P(S n -S n-1 > s) = e -s/a such that S 0 = 0 and let N be a Poisson variable with mean u/a which is independent of {S n }. Then we have L η a u law = S N , and hence

P a [f (L η a u )] =P(N = 0)f (0) + ∞ n=1 P(N = n)P[f (S n )] (4.13) =e -u/a f (0) + ∞ n=1 e -u/a (u/a) n n! ∞ 0 f (y) (y/a) n-1 (n -1)! e -y/a dy a . (4.14)
Thus, using (4.4), we obtain (4.7).

Lemma 4.2. For u > 0, x, a ∈ I and f ∈ L 1 + , it holds that Proof. When a ≤ x, we have

P x [f (L η a u )] = x ∧ a a P a [f (L η a u )] + 1 - x a + P a [f (e 1/a + L η a u )] (4.15) = x ∧ a a P a [f (L η a u )] + 1 a 1 - x a + ∞ 0 f (y) ρ a u (y)dy, ( 4
P x [f (L η a u )] = P x [f (L Ta + L η a u • θ Ta )] = P a [f (L η a u )]
, which proves identity (4.15). Suppose x < a. Using Lemma 3.1, we have

P x [f (L η a u )] =P x f L Ta + L η a u • θ Ta (4.18) = x a P a [f (L η a u )] + 1 a 1 - x a P a ∞ 0 e -v/a f (v + L η a u )dv , (4.19)
which coincides with (4.15). Using the same notation as that of the proof of Lemma 4.1, we obtain By (4.5), there exists a constant C such that

P a [f (e 1/a + L η a u )] = ∞ n=0 P(N = n)P[f (S n+1 )] ( 4
I ν (x) ≤ Cx ν for 0 < x ≤ 1, I ν (x) ≤ Ce x for x ≥ 1. (4.22)
Lemma 4.3. For any u > 0, a > 0 and y > 0, it holds that

ρ a u (y) ≤ 2Cu a 2 , ρ a u (y) ≤ C. ( 4

.23)

For any fixed u > 0 and y > 0, it holds that

ρ a u (y) -→ a→∞ 1. ( 4
N a,u t =aP x f (L η a u )1 {t<η a u } | F t , (4.28) M a,u t =aP x f (L η a u ) | F t . ( 4

.29)

Then it holds that N a,u t → M s,f t and M a,u t → M s,f t in probability with respect to P x as a → ∞, where M s,f t has been defined in (3.5).

Proof.

In what follows in this section we sometimes write M t simply for M s,f t . (i) By the strong Markov property and by Lemma 4.2, we have, for a > X t ,

N a,u t = a P Xt [f (b + L η a u-c )] b=Lt c=L a t 1 {t<η a u } = (I) a + (II) a , (4.30)
where

(I) a =X t e -u-c a f (b) + ∞ 0 f (b + y)ρ a u-c (y)dy b=Lt c=L a t 1 {t<η a u } , (4.31)
(II) a = 1 - X t a ∞ 0 f (b + y) ρ a u-c (y)dy b=Lt c=L a t 1 {t<η a u } . (4.32)
Letting a → ∞, we deduce from Lemma 4.3 that in probability with respect to P x

(I) a -→ a→∞ X t f (L t ), (4.33)
(II) a -→ a→∞ ∞ 0 f (L t + y)dy. (4.34)
We thus obtain N a,u t → M s,f t in probability with respect to P x . Set

A a,u t = M a,u t -N a,u t = af (L η a u )1 {η a u ≤t} . (4.35)
Since A a,u t → 0, we obtain M a,u t → M s,f t in probability with respect to P x .

Theorem 4.5. Let f ∈ L 1 + , x ∈ I and u > 0. Then, for any finite stopping time T , it holds that

N a,u T -→ a→∞ M s,f T in L 1 (P x ).
(4.36)

Consequently, for any bounded adapted process (F t ), it holds that

aP x [F T f (L η a u ); T < η a u ] -→ a→∞ P x [F T M s,f T ]. (4.37)
Proof. By the proof of Lemma 4.4 and by Lemma 4.3, we obtain, for a > 1,

N a,u t ≤X t f (L t ) + 2Cu a + C ∞ 0 f (L t + y)dy (4.38) ≤M s,f t +(2Cu + C) ∞ 0 f (y)dy, (4.39)
where the last quantity is integrable with respect to P x . Thus we obtain the desired result by the dominated convergence theorem.

Theorem 4.6. Suppose that ℓ ′ (= ℓ = ∞) is either type-1-natural or type-2-natural. Let f ∈ L 1 + , x ∈ I and u > 0. Then, for any bounded stopping time T , it holds that

M a,u T -→ a→∞ M s,f T in L 1 (P x ). (4.40)
Consequently, for any bounded adapted process (F t ), it holds that

aP x F T f (L η a u ) -→ a→∞ P x F T M s,f T . (4.41)
Proof. Since (M s,f t ) is a P x -martingale, we may apply the optional stopping theorem to see that

P x [A a,u T ] =P x X η a u f (L η a u ); η a u ≤ T (4.42) ≤P x M η a u ; η a u ≤ T (4.43) =P x [M T ; η a u ≤ T ] -→ a→∞ 0. (4.44)
Since A a,u T → 0, P x -a.s., we see that A a,u T → 0 in L 1 (P x ). Hence we obtain (4.40) and (4.41).

Limit as u tends to infinity with a being fixed

Suppose ℓ ′ (= ℓ = ∞) is either entrance, type-1-natural or type-2-natural. We thus have, for any x, a ∈ I,

P x (η a u < ∞) = 1 and η a u -→ u→∞ ∞ P x -a.s. (4.45)
In fact, η a u increases to a limit η a ∞ which must be infinite P x -a.s. by (4.3). For the clock τ = η a u in u > 0, we only consider the weights f (L η a u ) for f (u) = e -βu and f (u) = 1 {u=0} .

Lemma 4.7. Let x, a ∈ I, β > 0 and t > 0. For u > 0, set It also holds that (M β,a t ) is a P x -martingale.

N u,β,a t = e βu 1+βa P x e -βL η a u 1 {t<η a u } F t , M u,
Proof. Let us first prove that P x [e cL a t ] < ∞ for all c > 0 and t > 0. Following the same argument as in the proof of Lemma 2.1, we obtain P a exp cL a eq = 1 r q (a, a)

∞ 0 e cu e -u/rq(a,a) du.

(4.54)

Since r q (a, a) → 0 as q → ∞, we may take q > 0 large enough so that r q (a, a) < 1/c. This shows that P a exp cL a eq < ∞. By the monotonicity, we see that P x [e cL a t ] < ∞ for all t > 0.

The fact that L a t admits exponential moments implies that M β,a t ∈ L 1 (P x ) for all t > 0. Thus, by the dominated convergence theorem, we see that N u,β,a t -→ u→∞ M β,a t in L 1 (P x ) for all t > 0.

We second note that, for q > 0, P x (η a u ≤ t) ≤ e qt P x [e -qη a u ] ≤ e qt P a [e -qη a u ] = e qt e -u/rq(a,a) . (4.55)

We may take q > 0 large enough so that r q (a, a) < (1 + βa)/β. Then we obtain

P x [A u,β,a t ] ≤ e βu 1+βa P x (η a u ≤ t) ≤ e qt exp - 1 r q (a, a) - β 1 + βa u -→ u→∞ 0. (4.56)
Thus we obtain A u,β,a t -→ u→∞ 0 in L 1 (P x ) for all t > 0, which implies M u,β,a t -→ u→∞ M β,a t in L 1 (P x ) for all t > 0.

Theorem 4.9. Let x, a ∈ I. For u > 0 and t > 0, set

N u,∞,a t =e u/a P x (t < η a u < T 0 | F t ) (4.57) M u,∞,a t =e u/a P x (η a u < T 0 | F t ) (4.58) M ∞,a t = X t ∧ a a e L a t /a 1 {t<T 0 } . (4.59)
Then it holds that

N u,∞,a t -→ u→∞ M ∞,a t and M u,∞,a t -→ u→∞ M ∞,
:= M u,∞,a t -N u,∞,a t = e u/a 1 {η a u <T 0 } 1 {η a u ≤t} . ( 4

.63)

The remainder of the proof is the same as that of Theorem 4.8.

Universal σ-finite measures

In this section we find universal σ-finite measures for the local time penalizations.

for all non-negative predictable processes (F u ). We also have the last exit decomposition formula:

P x [F t ; T 0 ≤ t] = t 0 P x [dL u ] P (u) x,0 • n [t-u] [F t ] (5.9)
for all non-negative F t -measurable functionals (F t ), where we denote

n [t] (•) = n(• ∩ {t < T 0 }). (5.10)
For h = h 0 or h = s, let P h x denote the law of h-transform:

P h x (A; t < ζ) = 1 h(x) P 0 x [1 A h(X t )] (x > 0), (5.11)
P h 0 (A; t < ζ) =n[1 A h(X t )]
(5.12)

for A ∈ F t . Note that, when h = h 0 or h = s, the coordinate process under P h x never hits zero; see [START_REF] Yano | On h-transforms of one-dimensional diffusions stopped upon hitting zero[END_REF]Theorems 7.6 and 7.3]. We now define

P h x = ∞ 0 P x [dL u ] P (u)
x,0 • P h 0 + h(x)P h x .

(5.13) Theorem 5.2. Suppose 0 is recurrent. Let f ∈ L 1 + and x ∈ I. Let t be a constant time and let F t be a bounded F t -measurable functional. Then

lim q↓0 H(q)P x [F t f (L eq ); t < e q ] = P h 0 x [F t f (L ζ ); t < ζ]. (5.14)
Proof. By Theorem 3.3, it suffices to show

P h 0 x [F t f (L ζ ); t < ζ] = P x [F t N h 0 ,f t ].
(5.15) Denote g = sup{t < ζ : X t = 0}, where sup ∅ = 0. On the set {0 = g ≤ t < ζ}, we have

P h 0 x [F t f (L ζ ); 0 = g ≤ t < ζ] =h 0 (x)P h 0 x [F t f (L t ); t < ζ] (5.16) =P x [F t f (L t )h 0 (X t ); t < T 0 ].
(5.17)

On the set {0 < g ≤ t < ζ}, we have

P h 0 x [F t f (L ζ ); 0 < g ≤ t < ζ] = t 0 P x [dL u ] P (u) x,0 • P h 0 [F t f (L t ); t < ζ] (5.18) = t 0 P x [dL u ] P (u) x,0 • n [F t f (L t )h 0 (X t )] (5.19) =P x [F t f (L t )h 0 (X t ); T 0 ≤ t].
(5.20)

On the set {t < g < ζ}, we have The proof is parallel to that of Theorem 5.2, where we use Theorem 3.5 instead of Theorem 3.3. So we omit it.

P h 0 x [F t f (L ζ ); T 0 ≤ t < g] = ∞ t P x [dL u ]P (u) x,0 [F t f (L u )] (5.21) =P x F t ∞ t f (L u )dL u (5.22) =P x F t ∞ Lt f (u)du . ( 5

Exponential weights

Let us investigate the example where we take f (x) = e -cx , c > 0.

(6.1)

For h = h 0 or s, the supermartingale N t = N h,f t is given as

N t = h c (X t )e -cLt (6.2)
where

h c (x) = h(x) + 1 -x ℓ c + 1 ℓ . (6.3)
Since (N t ) is a supermartingale, we may define the subprobability measure Q h,c x by

Q h,c x (A; t < ζ) = P x h c (X t ) h c (x) e -cLt ; A for A ∈ F t and t ≥ 0. (6.4)
Then the process {X, (Q h,c x ) x∈I } is a diffusion on I. The corresponding speed measure and the scale function are given as

m h,c (x) = (0,x] h c (y) 2 dm(y), s h,c (x) = x 0 dy h c (y) 2 .
(6.5) Denote φ h,c q = h c (0) • φ q + cψ q h c , ρ h,c q = h c (0) • ρ q h c . (6.6)

Then we see that ϕ = φ h,c q (resp. ρ h,c q ) is a positive increasing (resp. decreasing) solution to the differential equation

D m h,c D s h,c - π 0 h c ϕ = qϕ (6.7)
which satisfies the boundary condition f (0) = 1 and D s h,c φ h,c q (0) = 0, (

where we have used

h c (0) = 1 c + 1 ℓ , (h c ) ′ (0) = c c + 1 ℓ .
(6.9) Theorem 6.1. The resolvent operator for the diffusion {X, (Q h,c x ) x∈I } is given as

Q h,c
x ∞ 0 e -qt f (X t )dt = I r h,c q (x, y)f (y)dm h,c (y), q > 0, (6.10) where r h,c q (x, y) = r h,c q (y, x) = H(q) h c (0) 2 (cH(q) + 1) φ h,c q (x)ρ h,c q (y), x, y ∈ I, x ≤ y. (6.11)

Consequently, 0 for {X, (Q h,c x ) x∈I } is regular-reflecting.

Proof. Let ϕ c (x) = ϕ(x)h c (x). Then we have • R q ϕ c (0) H(q) . (6.16)

Since P x [e -qT 0 ]R q ϕ c (0) = R q ϕ c (x) -R 0 q ϕ c (x), we obtain

Q h,c
x ∞ 0 e -qt ϕ(X t )dt = 1 h c (x) 1 cH(q) + 1 R q ϕ c (x) + cH(q) cH(q) + 1 R 0 q ϕ c (x) . (6.17)

From this we obtain (6.11).

Remark 6.2. The boundary classification at ℓ ′ is the same as that for the h-transform of the stopped process; see Theorems 7.3 and 7.6 of [START_REF] Yano | On h-transforms of one-dimensional diffusions stopped upon hitting zero[END_REF].

(ii) Let us study the case b(x) = cνx ν-1 with c > 0 and ν > 0, i.e.,

Lf = 1 2 f ′′ -cνx ν-1 f ′ on C c ((0, ∞)), (7.2)
which we may call the power drift. If ν = 1, then it is a Brownian motion with constant negative drift. If ν = 2, then it is an Ornstein-Uhlenbeck process. In this case s ′ = e cx ν , s =

. 5 ) 4 . 1 . 0 ( 1

 54101 Lemma Let a ∈ (0, ∞). Then the process {(L η a u ) u≥0 , P a } is a compound Poisson process with Laplace transformP a e -βL η a u = exp -u ∞ e -βs )1 a 2 e -s/a ds = e -uβ 1+βa . (4.6)

(4. 10 )

 10 By the strong Markov property of n a , we have n a (L Ta > s; T 0 < T a ) = n a (T 0 < ∞)P 0 (L Ta > s) = 1 a P 0 (L Ta > s).(4.11)

  (4.4), we obtain (4.16).

P x ∞ 0 ee 0 e+ 1 H

 001 -qt ϕ c (X t )e -cLt dt (-qt ϕ c (X t )dt + P x [e -qT 0 ]P 0 ∞ -qt ϕ c (X t )e -cLt dt (6.13)=R 0 q ϕ c (x) + P x [e -qT 0 ]P 0 u e -cu-qη(u-) T 0 (p(u)) 0 e -qt ϕ c (p(u) t )dt (6.14) =R 0 q ϕ c (x) + P x [e -qT 0 ]P 0 ∞ 0 e -cu-qη(u) du n T 0 0 e -qt ϕ c (X t )dt (6.15) =R 0 q ϕ c (x) + P x [e -qT 0 ] • 1 c

4 ) 2 ∞ 1 {s 1 x 1 ee 1 e cy ν dy = x 1 (e cy ν ) ′ y 1 = e cy ν y 1-ν cν x 1 + ν -1 cν x 1 e≥ 1 cν ∞ 1 x 1 -ν dx -c ′ ∞ 1 e≥ 1 cν ∞ 1 y 1 -≤ 1 cν ∞ 1 y 1 -

 42111111111111111 In particular, we have m(∞) < ∞. Note thatJ := 1 (x)s(1)}dm(x) =∞ cy ν dy e -cx ν dx (7-cx ν dx e cy ν dy.(7.6) We shall prove that∞ is type-2-natural if 0 < ν ≤ 2, entrance if 2 < ν < ∞.(7.7)If 1 ≤ ν ≤ 2, thenx cy ν y -ν dy (7.9)≥e cx ν x 1-ν cν c ′ (7.10)for some constant c ′ > 0. Hence we haveJ -cx ν dx = ∞. ν dx ≥e -cy ν y 1-ν cν and J ν dy = ∞. (7.14) If ν > 2, then∞ y e -cx ν dx ≤ e -cy ν y 1-ν cν and J ν dy < ∞.(7.15)

  Suppose ℓ ′ is either entrance, type-1-natural or type-2-natural. Let f ∈ L 1 + and x ∈ I. Let t be a constant time and let F t be a bounded F t -measurable functional. Then lim

	.23)
	Therefore we obtain (5.15).
	Theorem 5.3.

a↑ℓ aP x [F t f (L Ta ); t < T a ] = P s x [F t f (L ζ ); t < ζ].

(5.24)

The transient case

Theorem 5.1. Suppose ℓ < ∞, i.e., 0 is transient. Let f ∈ L 1 + and x ∈ I. Let t be a constant time and let F t be a bounded F t -measurable functional. Then

(5.1)

If, in particular, ℓ is type-3-natural, then

Proof. By Theorems 2.4 and 2.5, we see that (5.1) is equivalent to

where

On the other hand, we use (i) of Theorem 2.2 and obtain

Thus we obtain (5.3).

Using Theorems 3.3 and 3.5 instead of Theorems 2.4 and 2.5, we can obtain (5.2) in the same way as above.

The recurrent case

Let P (u)

x,y denote the law of the bridge with duration u starting from x and ending at y. Following [START_REF] Fitzsimmons | Markovian bridges: construction, Palm interpretation, and splicing[END_REF], this measure can be characterized by

where p u (x, y) denotes the transition density of the process X with respect to m(dy). We write symbolically

We have the conditioning formula:

x,0 [F u ] (5.8)

Appendix: the boundary classification

The following tables explain the boundary classification which we take from [START_REF] Yano | On h-transforms of one-dimensional diffusions stopped upon hitting zero[END_REF] and the recurrence property of the corresponding diffusion to each class:

Let us give some examples. Let X be a diffusion on [0, ∞) where 0 is the reflecting boundary and whose local generator on (0, ∞) is given by


Romain Deltour
email: romain.deltour@inria.fr

Agnès Guerraz
email: agnes.guerraz@inria.fr

Cécile Roisin
email: cecile.roisin@inria.fr

Multimedia Authoring for CoPs

One way of providing technological support for CoPs is to help participants to produce, structure and share information. As this information becomes more and more multimedia in nature, the challenge is to build multimedia authoring and publishing tools that meets CoPs requirements. In this paper we analyze these requirements and propose a multimedia authoring model and a generic platform on which specific CoPs-oriented authoring tools can be realized. The main idea is to provide template-based authoring tools while keeping rich composition capabilities and smooth adaptability. It is based on a component-oriented approach integrating homogeneously logical, time and spatial structures. Templates are defined as constraints on these structures.

Introduction

In order to support the activities of Communities of Practice, the Palette project [START_REF]Palette[END_REF] will provide tools for document production and for document reuse in heterogeneous applications. The objective is to reduce the current limitations caused by the proliferation of data sources deploying a variety of modalities, information models and encoding syntaxes. This will enhance applicability and performances of document technologies within pedagogically consistent scenarios.

The LimSee3 project aims at defining a document model dedicated to adaptive and evolutive multimedia authoring tools, for different categories of authors and applications, to easily generate documents in standard formats (see the authoring process showed in Fig. 1). Our approach is to focus on the logical structure of the document while keeping some semantics of proven technologies such as SMIL [START_REF]W3C SMIL[END_REF]. This provides better modularity, facilitates the definition of document templates, and improves manipulation and reusability of content.

This paper is organized as follows: Sect. 2 presents a scenario example that will be developed throughout the paper and thereby analyzes CoPs requirements for authoring multimedia documents. We then define the main concepts on which multimedia authoring tools are based and we classify existing approaches in the light of these concepts. Section 4 introduces our LimSee3 document model and Sect. 5 shows how it can be used for the development of authoring tools tuned for specific CoPs. Last section presents the current state of our development and our perspectives in the context of the Palette project. The instrumentation of CoPs heavily relies on communication technologies. In this paper we are concerned with communication through sharing and collaborative authoring of information. We are studying scenarios where experience and knowledge are shared by means of multimedia data, such as annotated video or synchronized slideshow. The key point is that in CoPs, content readers are also content creators but usually have no skills in multimedia authoring. We develop below a concrete scenario of how a particular CoP shares information and then we identify the main requirements of multimedia authoring in such situations.

Experience sharing between reps

Studies of experiences at companies such as Xerox [START_REF] Brown | Organizational learning and communities-of-practice: Toward a unified view of working, learning and innovation[END_REF] have demonstrated that CoPs, as the copier repair technicians ("tech reps") CoP, are a very effective way for professionals to share informal or tacit knowledge gained from experience in the field. This sharing of tips, which could not be found in training manuals or classroom settings, was critical to help the tech reps do a better job and was even ultimately fostered by Xerox.

The practice of creating and exchanging stories has two important aspects. First of all, telling stories helps to diagnose the state of a troublesome machine. Reps begin by extracting a history from the users of the machine and with this and the machine as their starting point, they construct their own account. If they cannot tell an adequate story on their own then they seek help from specialists or colleagues (over coffee or lunch).

Brown took example on one service call observed by the ethnograph Orr in [START_REF] Orr | Sharing Knowledge, Celebrating Identity: War Stories and Community Memory in a Service Culture[END_REF]. A rep confronted a machine that produced copious raw information in the form of error codes and obligingly crashed when tested. As the error codes and the nature of the crashes did not correspond, the case immediately fell outside the directive training and documentation provided by the organization. Unfortunately, the problem also fell outside the rep's accumulated, improvised experience ; his technical specialist was equally baffled. Solving the problem in situ required constructing a coherent account of the malfunction out of the incoherence of the data and the documentation. To do this, the rep and the specialist embarked on a long story-telling procedure. They explored the machine or waited for it to crash for collecting data such as logs, screenshots, sound records. The rep and specialist recalled and discussed other occasions on which they had encountered some of the present symptoms via phone calls, webcam records, user feedback... Each story presented an exchangeable account that could be examined and reflected upon to provoke old memories and new insights. Yet more tests and more stories were thereby generated. The story-telling process continued forming a purposeful progression from incoherence to coherence.

Ultimately, these stories generated sufficient interplay among memories, tests, the machine's responses, and the ensuing insights to lead to diagnosis and repair. Through story-telling, these separate experiences converged, leading to a shared diagnosis of previously encountered but unresolved symptoms. Rep and specialist were now in a position to modify previous stories and build a more insightful one. They both increased their own understanding and added to their community's collective knowledge. A story, once in the possession of the community, can then be used -and further modified -in similar diagnostic sessions.

The information units that are exchanged in this particular CoP are multimedia story documents that are composed of sequences of story steps where data elements are heterogeneous and multimedia. The challenges are to enrich information with the synchronization of data elements (for instance a phone call with the corresponding webcam excerpt) and to provide a document structure enabling knowledge sharing and reusability (of experience stories).

Basic requirements

The cooperative platform to be provided to the CoPs must have the two following basic features: (i) authoring tool of stories dedicated to tech reps ; (ii) access tool to read the existing stories on different devices (desktop PC, PDA, mobile phone...). Looking more closely at the ways in which CoPs participants are producing multimedia information, we can identify some requirements for the authoring and presentation platform:

1. Simple and efficient authoring paradigms -because CoPs members are not (always) computer science technicians. 2. Easy and rapid handling of the authoring tool -because new members can join CoPs. 3. Modular and reusable content -because multimedia information results in a co-construction process between members. 4. Evolutive structuring of documents -because of the dynamic nature of CoPs. 5. Use of standard formats -because CoPs need portability, easy publishing process and platform-independence.

Basically, our approach proposes a template mechanism to cope with requirements 1 and 2, a component-based structuring enabling requirements 3 and 4, and relies on proven standard technologies to ensure the last requirement. Before further stating our authoring model, we present in the next section the main concepts and approaches of multimedia authoring on which this work is based.

In traditional text oriented document systems, the communication mode is characterized by the spatial nature of information layout and the eye's ability to actively browse parts of the display. The reader is active while the rendering itself is passive. This active-passive role is reversed in audio-video communications: active information flows to a passive listener or viewer. As multimedia documents combine time, space and interactivity, the reader is both active and passive. Such documents contain different types of elements such as video, audio, still-picture, text, synthesized image, and so on, some of which having intrinsic duration. Time schedule is also defined by a time structure synchronizing these media elements. Interactivity is provided through hypermedia links that can be used to navigate inside the same document and/or between different documents.

Due to this time dimension, building an authoring tool is a challenging task because the WYSIWYG paradigm, used for classical documents, is not relevant anymore: it is not possible to specify a dynamic behavior and to immediately see its result. Within the past years, numerous researches have presented various ways of authoring multimedia scenarios, focusing on the understanding and the expressive power of synchronization between media components: approaches can be classified in absolute-based [START_REF]Adobe Authorware 7 and Director MX[END_REF], constraint-based [START_REF] Buchanan | Automatic temporal layout mechanisms[END_REF], [START_REF] Jourdan | Madeus, an authoring environment for interactive multimedia documents[END_REF], event-based [START_REF] Sénac | Modeling logical and temporal synchronization in hypermedia[END_REF] and hierarchical models [START_REF]W3C SMIL[END_REF], [START_REF] Van Rossum | CMIFed : a presentation environment for portable hypermedia documents[END_REF]. Besides, to cope with the inherent complexity of this kind of authoring, several tools [START_REF]Adobe Authorware 7 and Director MX[END_REF], [START_REF]MS Producer for PowerPoint[END_REF], [START_REF] Hua | LazyCut: Content-aware template based video authoring[END_REF] have proposed limited but quite simple solutions for the same objective. Dedicated authoring, templatebased authoring and reduced synchronization features are the main techniques to provide reasonable editing facilities. But we can notice that these tools generally also provide scripting facilities to enrich the authoring capabilities and therefore loose in some way their easiness.

Beside timelines, script languages and templates, intermediate approaches have been proposed through "direct manipulation" and multi-views interface paradigms. IBM XMT authoring tool [START_REF]Authoring in XMT[END_REF] and SMIL tools such as LimSee2 [3] and Grins [START_REF] Grins | [END_REF] are good examples. In LimSee2, the time structure of SMIL is represented for instance in a hierarchical timeline as shown in of Fig. 2 (4). Time bars can be moved or resized to finely author the timing scenario. This kind of manipulation has proven very useful to manipulate efficiently the complex structures representing time in multimedia XML documents.

However even if XMT and SMIL are well-established languages, the abovementioned tools are too complex for most users because they require a deep understanding of the semantics of the language (e.g. the SMIL timing model). Moreover these models generally put the time structure at the heart of the document whereas it does not always reflect exactly the logical structure in the way it is considered by the author. Our approach instead sets this logical dimension as the master structure of the document, which is a tree of modular components each one specifying its own time and spatial structures. Additionally, the document can be constrained by a dedicated template mechanism.

A template document is a kind of reusable document skeleton that provides a starting point to create document instances. Domain specific template systems are a user-friendly authoring solution but require hardly extensible dedicated transformation process to output the rendering format. We chose on the contrary to tightly integrate the template syntax in the document: the template is itself a document constrained by schema-like syntax. The continuum between both template and document permits to edit templates as any other document, within the same environment, and enables an evolutive authoring of document instances under the control of templates. There is no need to define a dedicated language to adapt to each different use case.

We believe that the combination of document structuring and template definition will considerably help CoPs in (i) reusability of materials, (ii) optimization of the composition and life cycle of documents, (iii) development and transmission of knowledge, (iv) drawing global communities together effectively.

The LimSee3 Authoring Language

Main Features

In the LimSee3 project, we define a structured authoring language independently of any publication language. Elements of the master structure are components that represent semantically significant objects. For instance a story report document is a list of step components. Each step is composed of several media objects and describes a phase of the story (failure description, machine exploration...). Components can be authored independently, integrated in the document structure, extracted for reusability, constrained by templates or referenced by other components.

The different components of a multimedia document are often tightly related one with another: when they are synchronized or aligned in space, when one contains an interactive link to another, and so on. Our approach, which is close to the one proposed in [START_REF] Silva | NCL 2.0: integrating new concepts to XML modular languages[END_REF] is for each component to abstract its dependencies to external components by giving them symbolic names that are used in the timing and layout sections. This abstraction layer facilitates the extraction of a component from its context, thus enhancing modularity and reusability.

Finally, the goal was to rely on proven existing technologies, in both contexts of authoring environments and multimedia representation. The timing and positioning models are wholly taken from SMIL. Using XML provides excellent structuring properties and enables the use of many related technologies. Among them are XPath, used to provide fine-grained access to components, and XSLT, used in templates for structural transformation and content generation.

The authoring language is twofold: it consists in a generic document model for the representation of multimedia documents, and it defines a dedicated syntax to represent templates for these documents.

Document Model

A document is no more than a document element wrapping the root of the object hierarchy and a head element containing metadata. This greatly facilitates the insertion of the content of a document in a tree of objects, or the extraction of a document from a sub-tree of objects.

A compound object is a tree structure composed of nested objects. Each compound object is defined by the object element with the type attribute set to compound. It contains a children element that lists children objects, a timing element that describes its timing scenario and a layout element that describes its spatial layout.

The value of the required localId attribute uniquely identifies the component in the scope of its parent object, thereby also implicitly defining a global identifier id when associated with the localId of the ancestors. In Example 1, the first child of object step1 has the local id copyLog and hence is globally identified as step1.copyLog.

The timing model, and similarly the positioning model, is taken from SMIL 2.1. The timing element defines a SMIL time container. The timing scenario of a component is obtained by composition of the timed inclusions defined by the timeRef elements, whose refId attributes are set to local ids of children.

<document xmlns="http://wam.inrialpes.fr/limsee3/" xmlns:smil="http://www.w3.org/2005/SMIL21/"> <head><!--some metadata --></head> <object localId="step1" type="compound"> <children> <object type="text" localId="copyLog">...</object> <object type="image" localId="screenshot">...</object> <object type="compound" localId="AnnotatedVid">...</object> </children> <timing timeContainer="par"> <timeRef refId="AnnotatedVid" begin="0s"/> <smil:seq begin="0s"> <timeRef refId="screeshot"/> <timeRef refId="copyLog"/></smil:seq></timing> <layout height="100" width="100"> <layoutRef refId="AnnotatedVid" left="0"/>...</layout> </object></document> Example 1. A simple story step LimSee3 document A media object is actually a simple object that wraps a media asset, i.e. an external resource (such as an image, a video, an audio track, a text...) referenced by its URI. It is defined by the object element with the type attribute set to either text, image, audio, video or animation. The URI of the wrapped media asset is the value of the src attribute. Example 2 shows a text media object with local id menuItem1 which wraps the media asset identified by the relative URI ./medias/item1.txt.

Area objects inspired from the SMIL area element can be associated with media objects. They are used for instance to structure the content of a media object or to add a timed link to a media object. An area is defined as an object element with the type attribute set to area. For instance, in Example 2 the media object menuEntry1 has a child area which defines a hyperlink.

Relations of dependency between objects are described independently of their semantics in the document. External dependencies are declared with ref elements grouped inside the related child element of objects. The value of refId of a ref element is the id of the related element and the value of localId is a symbolic name that is used within the object to refer to the related object. For instance, in Example 2, object menuItem1 describes a text that links to the object story.step1, by first declaring the relation in a ref element and then using this external object locally named target to set the value of the href attribute of the link, using attribute and value-of elements taken form XSLT. <object localId="menuItem1" type="txt" src="./medias/item1.txt"> <related><ref localId="target" refId="story.step1"/></related> <children><object type="area" localId="link"/></children> <timing> <attribute name="begin"> <value-of refName="target" select="@id"/>.begin</attribute> <timeRef refId="link"> <attribute name="href"> #<value-of refName="target" select="@id"/></attribute> </timeRef></timing>...</object> Example 2. A LimSee3 object with external dependency relations

Templates

Template nodes aim at guiding and constraining the edition of the document. In order to have better control and easy GUI set up, the language includes two template nodes: media zone and repeatable structure.

A media zone is a template node that defines a reserved place for a media object. It is represented by the zone element, that accepts a type attribute (text, img, audio, video, animation, any, or a list of these types) to define what types of media object can be inserted in this zone. The author can also specify content that will be displayed to invite the user to edit the media zone with the invite element (of any media type). For instance Example 3 shows a media zone for an image, with textual invitation. During the authoring process zone elements are filled with media objects inserted by the user.

A repeatable structure, represented by the objList element, is a template node that defines a homogeneous list of objects. Each item of the list matches a model object declared in the model child of the list. The cardinality of the list can be specified with the minOccurs and maxOccurs attributes. Example 3 shows a story template document based on an objList named step-list, and partially instanciated with three compound objects respecting the step model. Thanks to the use of XSLT-like syntax, the timing scenario can be specified independently of the content of children instances.

It is possible to lock parts of a document with the locked attribute, to prevent the author from editing anything. This permits for instance to guide more strongly inexperienced users by restricting their access to the only parts of the document that make sense to them. <object localId="story" type="compound"> <children> <objList localId="step-list" maxOccurs="20"> <model name="step"> <object type="compound">...</object></model> <object type="compound" localId="step1">...</object> <object type="compound" localId="step2">...</object> <object type="compound" localId="step3">...</object> </objList></children> <timing> <smil:seq begin="1s"> <for-each select="children/objList[@name="step-list"]/object"> <timeRef> <attribute name="refId"> <value-of select="@localId"/> </attribute></timeRef></for-each></smil:seq> </timing>...</object> Example 3. A partially instanciated story template Figure 3 (2) also shows the creation of a template document from an existing document. The main structure of the document, in this case a sequence of story steps, can be constrained by template nodes such as repeatable structures. Additionally, inter-object relations described in Sect. 4 facilitate the extraction of components from their context so that they can be reused in other documents. In the tech reps CoP, a possible workflow is to first create a story report from scratch (1), then to extract a template document from this report (2), along with a dedicated GUI, to ease the creation of further story reports (1'). This is a typical example of participative design leading to the development of a dedicated tool based on the LimSee3 generic platform.

The LimSee3 model leads to the development of authoring tools that fit the requirements of Sect. 2.2. We are defining a generic platform that permits to manipulate all the elements defined in the model (documents, compound objects, timing and layout details, relations...). It provides features based on the proven authoring paradigms described in Sect. 3 such as multi-views, timeline, structure tree an 2D canvas. In the reps CoP example described in Sect. 2, a tech rep could have used the generic GUI to create the story report ex-nihilo, as shown in Fig. 3 (1), incrementally adding story steps by creating and integrating new objects in the document (resulting in the LimSee3 document of Example 1). Once fully authored, the story report can be persistently added to the base of documentation maintained by the company, and published on demand to any output format (provided its semantics is included in the document model).

Another approach is to use a domain-specific template with dedicated GUI, as shown in Fig. 3 (1'). For instance, a template for a story report could consist in a repeatable structure of story steps. These steps could be instanciated from existing template components such as an audio zone for phone calls, a text zone for machine logs, .... The constraints of the template would guide the tech rep in the creation of the document, reflected in the GUI by dedicated buttons or menu items such as "add a story step", "insert a phone call record", or a formbased interface for adding titles or comments to multimedia content. In the underneath manipulated model, the tight integration of template nodes in the document ensures a smooth evolution from the template to the final document.

Conclusion

The model presented in this paper develops a practice-based approach to multimedia authoring dedicated to communities where collaborative and participative design is of high importance. It improves reusability with template definitions and with the homogeneous structuring of documents. This document model is being implemented as cross-platform java software. In the context of Palette, we will use this model to develop dedicated authoring tools for pedagogical CoPs.

Fig. 1 .

 1 Fig. 1. The authoring process in LimSee3

Fig. 2 .

 2 Fig. 2. Multiview authoring in LimSee2

Fig. 3 .

 3 Fig. 3. Authoring with LimSee3