Yoann Verger Sraffa

Piero Sraa

Yoann Verger

Sraa and Manara: the mystery of the last article of

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Piero Sraa is a very peculiar kind of scholar. On the one hand, he is one of the few modern economist where we can say his most important publications are almost all of his publications. As an example, one can read Sraa's entry in [START_REF] Beaud | Economic Thought Since Keynes, A History and Dictionary of Major Economists[END_REF], where they list for him 15 mains publications, whereas I can only count in total for him only 18 scientic publications (and, within it, the authorship of only one book -see the appendix for the list of his publications). On the other hand, his inuence on history of economic thought is immense, as it is acknowledge by one of his most famous opponents, Paul A.

Samuelson:

Did any scholar have so great an impact on economic science as Piero Sraa did in so few writings? One doubts it. And there cannot be many scholars in any eld whose greatest works were published exclusively in their second half century of life. Piero Sraa was much respected and much loved. With each passing year, economists perceive new grounds for admiring his genius [START_REF] Samuelson | Sraan economics[END_REF].

Although his publications were not numerous, Sraa was writing a lot: his unpublished papers gathers thousands of notes, drafts, comments and citations that are currently made available online (the online eort began in 2016).

1 In this collection of documents, two les are headed as unpublished draft articles: a draft from 1923, on the economics of fascism (D3/3), and a draft written around 1969, titled : "Produzione congiunta di merci a mezzo di merci. Riposta a C F Manara" (D3/14). The latter is the one that I want to focus on here. I will rst discuss Manara's article, explaining why an answer by Sraa seems important, as this article shakens the most interesting part of Sraa's analysis (section 2). Then I will present Sraa's draft (section 3), and explain why, in my opinion, he nally decided not to publish it (section 4). Finally I will present [START_REF] Dupertuis | Existence of the standard system in the multiple production case: a solution to the manara problem[END_REF] solution to Manara's problem, and I will show how it would have helped Sraa to answer his problematic. 2 Manara's article Manara's article was rst published in Italian in 1968, and then published in English in 1980 (in [START_REF] Pasinetti | Essays on the theory of joint production[END_REF]. This article was widely read and cited (we can nd in Google Scholar 29 citations for the 1980 article and 45 citations for the initial one), and he is cited in two inuential reviews of the Sraan literature, in [START_REF] Pasinetti | Piero sraa and the future of economics[END_REF] article headed Piero Sraa and the future of economics and in [START_REF] Samuelson | Sraan economics[END_REF] article on Sraan economics published in The New Palgrave Dictionary of Economics. So what is its content, and what is at stake concerning its results?

Content of Manara's article

Let us introduced Manara's article by one of the most competent scholar in the eld of Sraan economics, and especially concerning the questions raised by [START_REF] Manara | Ii modello di sraa per la produzione congiunta di merci a mezzo di merci[END_REF], Bertram Schefold:

While the rst part of Sraa's `Production of Commodities by Means of Commodities' received quite considerable attention when the book was rst published, the second passed almost unnoticed for some time. The third part on `Switching of Techniques' stood at the centre of the most interesting controversy on economic theory of the 1960s, but the discussion was again focused on single-product industries. The rst exception to this pattern was Manara's article on `Il modello di Piero Sraa per la produzione congiunta di merci a mezzo di merci' in the Italian journal L'industria, which appeared in 1968. The author, apart from giving an elegant mathematical formulation for the distinction between basics and non-basics in the case of joint production, 2 pointed to two of the diculties of joint- production systems: namely the likelihood of negative prices being associated with some labour vectors at some rates of prot in almost any joint-production system and the nonexistence of the maximum rate of prot and the standard commodity in certain cases. It was not said whether these were to be considered normal or exceptional.

Manara, a mathematician, left the tasks of analysis and interpretation of the `anomalies' to the economists. Manara failed to note that Sraa had simply and explicitly assumed that positive prices and a real standard commodity exist (PCMC, sec. 50)but then Sraa had failed to indicate under what conditions these assumptions would be fullled (Schefold, 1989, p. 44).

Manara [1968] says that these assumptions would be fullled under the condition that, for any system of joint production of commodities by interdependent industries (i.e. with only basic commodities) where A is the matrix of inputs per industry and B the matrix of outputs per industry (with commodities in columns and industries in rows), a real solution to the following equation can be found for the scalar R:

det [B -(1 + R) A] = 0 (1)
If a real solution for R cannot be found, then it means that, from this system of joint production, no Standard system can be constructed. In his article, Manara provides two numerical examples, and he shows that, for the second example, a real solution can not be found for R; but, and this is important for the following, for his rst example also, a real solution cannot be found for R. Hence, for both examples, a Standard system cannot be constructed.

Consequences of Manara's ndings

Sraa (1960, Chapter IV) constructs his imaginary Standard system by applying to the initial system of interdependent industries a set of multipliers x, so that the commodities used as inputs are in the same proportions as the commodities used as outputs, bringing out a rate of physical residue R. In the Standard system, the following equations hold:

2 In single production (i.e. each industry only produces one commodity, and each commod- ity is only produced by one industry), basic commodities are the commodities that are directly or indirectly needed for the production of all commodities of the system. In joint production (one commodity can be produced by dierent industries, and industries can produce several commodities), basic and non-basic commodities cannot be separated so easily. For instance, a commodity that enters all processes of production can be considered non basic in some cases. Sraa (1960, 60) gives a formal (mathematical) denition of the distinction between basic and non-basic commodities that applies for both single and joint production, but he also indicates that, even in the latter case, an economic sense can be given to the distinction: if a tax is imposed on a commodity, and that all prices are impacted by this change, its means that the commodity is basic (Sraa, 1960, 65). Then a system with only basic commodities can be said to be a system of interdependent industries: each industry needs all the other ones to ensure its production.

(

1 + R)xA = xB (2)
Sraa shows that the rate of physical residue R is also the maximum uniform rate of prots of the initial system. Hence, the construction of an imaginary

Standard system helps Sraa to show that, in a system of interdependent industries, the maximum rate of prots is independent from the prices (it is a physical property of the system).

Furthermore, by dening a Standard commodity as a mixed commodity whose commodities are in the same proportion as the commodities in the surplus of the Standard system, and by using this Standard commodity as the numeraire, Sraa shows that the distribution also appears to be independent from the prices (i.e. it can be known prior to prices, once, for instance, the rate of prots is given). This means that the neoclassical theory of distribution, which is to calculate distribution as optimal prices determined by productivity of factors Sraa. In this letter Pasinetti tells Sraa that he shall not use the term incompatibilità (which we can nd in the handwritten notes, for instance in D3/14/28), but rather the terms assoluta inferiorità, which convey no mathematical signication while having an economic meaning. In the typewritten draft, we nd these terms, and no mention of the term incompatibilità. (2) Edizione Einaudi, 1960; ristampa 1969 con la stessa paginatura.

(

Comments

Sraa, then, only considers the second numerical argument of Manara, and dismisses it because it has no economic meaning: in empirical reality, this system could not appear because the second industry is absolutely inferior to the rst one, and would be then ruled-out by competition.

Besides Pasinetti, this draft may have been also presented to Schefold, who was in Cambridge at that time.

4 In his PhD thesis, Schefold uses exactly the same argument to explain that, in a case of imbalance of one industry, as in the second numerical example of Manara, no standard system can be constructed:

Consider (it makes sense for n = 2) systems with A > O.

g j i = b j i a j i (i, j = 1, 2)
is the rate of reproduction of good j in process i.

We dene: a system involving two goods and two processes is imbalanced if one process is better than the other for both goods in that The example proposed by Professor Manara is inecient in this sense

g 1 i , g 2 i > g 1 j , g 2 j for (i, j) = (1, 2) or (2, 1)

Theorem

If the ineciency above (imbalance) is ruled out, a positive standard ratio R and a positive q and p (R) exist [START_REF] Schefold | Mr Sraa on joint production and other essays[END_REF], p.

146).

But Schefold also goes a bit further, by dening the condition where, in joint production systems with more than two interdependent industries, a standard system can be constructed (Schefold, 1989, p. 96-99). This condition is that, for all a j i and b j i , we should observe (possibly after having made the right sub- stitutions of rows and columns in matrices A and B):

0 ≤ b j i a j i < 1 + R < b j j a j j ≤ ∞
Schefold, by assuming the above condition, can then also avoid cases such as the one presented in the rst numerical example of Manara. Unfortunately for Sraa's theory, this above condition is rather strong (slightly more stringent, as it is acknowledge by Schefold 1989, p. 97): for instance an industry where two (or more) of the joint outputs are not also used as inputs by the same industry should be ruled-out, and so independently of the input-output coecients of the other industries. Hence, this condition seems to convey no real economic signicance.

He concludes his development on this above condition by saying:

The main insight obtained in this section is the one derived from the von Neumann model: systems without a semi-positive standard commodity and semi-positive p t (R) suer from specic ineciencies revealed at r = R, much in the same way as systems that are almost all-productive are susceptible to suer from ineciencies revealed at r = 0, but covered at r = r > 0. Both these ineciencies occur in systems where the possibility of substitution is inherent: they are not all-engaging and yet basic (Schefold, 1989, p. 98).

Hence, as Sraa, he explains the fact that in some cases of joint-production, a standard system can not be constructed, by the fact that some industries are less ecient than others. But the condition for the existence of a standard system when the number of industries are more than two is more a mathematical one than an economic requirement.

4 Why Sraa did not publish it?

In this section I try to formulate a plausible conjecture explaining why Sraa decided not to publish the above draft. But rst, several guesses can be attempted to explain the non-publication.

Guesses

A rst guess is that he did not nd the subject important enough to respond in a public way (but he may have had a private explanation with Manara).

Three supports can be advanced for this guess. A rst support is the fact that in opening sentences of the above draft, he says that he was not sure that an answer was necessary (Mi era sembrato che la cosa dovesse essere evidente ad ogni economista così da rendere superua una rettica). But I understand this more as a sarcastic way to introduce the subject.

A second support is a remark of Schefold, that, once, Sraa told him that there have been too much discussion on joint production:

When he {Sraa} was confronted with the diculties of joint production systems, which had surfaced in the 1960s and early 1970s, he told me that, if he had known, he should have written much less about general joint production systems and more about xed capital and land, and that he would have liked to treat only part of the problematic of joint production and, nally, that it might have been better to begin the exposition with the analysis of xed capital, followed only by some thoughts on joint production in general [START_REF] Schefold | Joint production: triumph of economic over mathematical logic?[END_REF].

The last support for this guest comes from Pasinetti himself, responding to an email where I asked him why Sraa did not published the above draft:

Dear Mr. Verger, Thank you for contacting me.

After almost 60 years, I do not recall why Piero Sraa decided not to publish a reply to Manara's 1968 article.

My presumption is that in the end he became convinced that it was not necessary.

My best wishes for your research work.

Sincerely yours, Luigi Pasinetti (personal communication, October 5th, 2017).

From these two points of view, the guess that Sraa thought this discussion on joint production not important is indeed convincing. 5

But, besides my argument in subsection 2.2, we can also nd in the drafts some arguments that do not support this guess. First more than hundred pages of notes in the D3/14 le show that Sraa spend lot of time on the subject, 5 And this is also the point of view of Samuelson:

As Carlo Manara (1980, pp. 9-11) has shown, there may exist no real characteristic vectors to serve as a standard commodity for admissible singletechnique joint product systems.

A catastrophe ? No, no catastrophe. There was little of value (to me, to Ricardo, to Sraa) to be lost and no tragedy in the Manara nding that some b -a matrixes lack real-number characteristic Sraan vectors. [START_REF] Samuelson | Sraa's hits and misses, chapter 3[END_REF].

trying to express his answer in the best possible way. Second, we nd in the le some notes saying that if the article were right, it would shaken the whole classical tradition starting with Adam Smith, for example in the following note:

Si può concludere che la generalità della costruzione di un prodotto Or again in this note: Se l'esempio fosse valido con la mia poverite costruzione crollenbbero anche i pilaobri della economia tradizionali (Sraa's unpublished papers, D3/14: 63).

A second guess for the non-publication is that Sraa did not nd it important to answer himself, as Bertram Schefold was going to publish his thesis on this subject. We can advance two support for this guess: rst we know that Sraa and Schefold were close, as we can see it in a draft of a letter from Sraa to Schefold, dated 10 October 1974, which is also in the unpublished papers of Sraa:

[10 Oct 74 not sent found his of 10 Sept 74!] Dear Schefold, Why on earth did you hesitate? I am delighted to hear from you, hope to hear more, + above all hope to see you! As a rst instalment I shall be glad to see Dr Pennavja. 10 Oct. 74

The above was written when I received your letter of 24.6.74.

Then both got buried in the chaos of my papers and came to the surface only this day. Cristina Pennavaria came on July 5 and I was delighted to meet her. (Sraa's unpublished papers, C279_5).

A second support can be nd in the fact that Sraa was used to send young colleagues in theoretical controversies (as Keynes did with him when he needed some help against Hayek), especially when the controversy went on mathematical grounds. For instance see this report from Sinha on the beginning of the To which Sraa responded: Neither can I.

So Luigi Pasinetti was asked to do the job, and the rest is history.

(Ajit Sinha, Inet Blog, consulted the 04/01/2017, https://www.ineteconomics.org/perspectives/blog/s revolution-in-economic-theory) I can not really provide any fact against this guess; the better way to prove or dismiss it would be to ask Bertram Schefold about it. The only thing that makes me think that it is not the right option is that the third guess is plausible.

Here is the third guess: it does not nd his response convincing enough. This is an obvious guess considering that large parts of the nal version which are crossed. And this would not be actually surprising knowing that Sraa is a meticulous writer, only giving talks or publishing writings when he is absolutely sure of the quality of their content. So one would be not surprised if, in the drafts, we found a logical mistake that would have, ultimately, prevented Sraa to publish the article.

Conjecture: a logical mistake?

We have a rst hint that he was not satised by his answer in some notes in the le D3/14, for instance here: Sraa thinks that his explanation does not prove in a general way that Manara is wrong, and that if an example were to be found where no industry shows assoluta inferiorità and no standard system can be constructed, then it would be a bad hit for his theory.

But this is actually precisely the case of the rst numerical example of Manara. It is then possible that Sraa, discovering this fact, nally decided not to publish his draft; the part where he exposes that it is because of the assoluta inferiorità that Manara's example does not work is totally crossed in the draft above, which can be explained if he found that the rst numerical example of Manara shows no assoluta inferiorità while also being a case where no standard system can be constructed.

We saw in subsection 3.2 that Schefold has also seen that, if there is more than two interdependent industries, the argument of imbalance does not work anymore. But he did not nd a real economic reason to dismiss cases such as the rst numerical example of Sraa, and he only found a mathematical condition to avoid such cases. I believe that Sraa could have been happy with such case, as he always wanted to link theory with empirical facts that one could observe (on the objective approach of Sraa, see Sinha, 2012). 5 Dupertuis and Sinha's solution to Manara's problem: interdependent industries and self-reproducing system [START_REF] Dupertuis | Existence of the standard system in the multiple production case: a solution to the manara problem[END_REF] show that is possible to dismiss both numerical examples of Manara while giving an economic meaning to this dismissal. The key notion is interdependence, or rather mutual dependence:

It is desirable at this stage to explain why the ratios which satisfy the conditions of production have been called 'values' or 'prices' rather than, as might be thought more appropriate, 'costs of production'.

The latter description would be adequate so far as non-basic products were concerned, since, as it follows from what we have seen in the preceding section, their exchange ratio is merely a reection of what must be paid for means of production, labour and prots in order to produce themthere is no mutual dependence.

But for a basic product there is another aspect to be considered.

Its exchange-ratio depends as much on the use that is made of it in the production of other basic commodities as on the extent to which those commodities enter its own production (Sraa, 1960, 7).

We have to remember that Sraa considers only systems of productions where at least one basic commodity is existing, i.e. where at least a minimum level of interdependence between industries exists. This is why the title of his book is Production of commodities by means of commodities: he explicitly rejects systems with no interdependence. This is why also the possibility that an ultimate commodity is produced only by labour is explicitly rejected by Sraa:

The This formulation presupposes the system to be in a self-replacing state; but every system of the type under consideration is capable of being brought to such a state merely by changing the proportion in which the individual equations enter it, (Sraa, 1960, 3, footnote).

However, inside a self-reproducing system, all processes must be non-self-reproducing. This is because prices are dened in the theory of Sraa as the exchange ratios that will allow the system to reproduce itself (and to distribute the surplus according to a given social rule):

There is a unique set of exchange-values which if adopted by the market restores the original distribution of the products and makes it possible for the process to be repeated (Sraa, 1960, 1).

If a process is self-reproducing, there is no need for this process to make exchanges with other processes in order to reproduce itself. The price of the commodity produced by this process and its rate of prots are then determined by the conditions of production of this process only: the interdependence is broken.

Now comes the subject of joint production. If a non-self-reproducing process produces two outputs, the right method to nd the Sraan prices of each single output is to add an other non-self-reproducing process producing the same two outputs, as Sraa itself explains it [START_REF] Sraa | Production of Commodities by Means of Commodities[END_REF], 50, footnote 2). But, if a selfreproducing process jointly produces two outputs, it is not possible to dene the Sraan price of each single output. It is of course possible to calculate prices, if we add to this self-reproducing process an other process of production (either self-reproducing or not). But these prices cannot be the Sraan prices of the two joint products, as no exchanges are needed between both processes. These prices are not the exchange-values which if adopted by the market restores the original distribution of the products and makes it possible for the process to be repeated.

Hence, to the previous restriction that "we shall assume throughout that any system contains at least one basic product" (Sraa, 1960, 6), we must add that, in case of joint production, the system of production must only contain one self-replacing subsystem, as the notion of "mutual dependence" associated with basic commodity in single production is lost if there is more than one self-replacing subsystem in joint production (see [START_REF] Dupertuis | Existence of the standard system in the multiple production case: a solution to the manara problem[END_REF].

The implication of the lack of such conditions is the apparent impossibility to build a standard system. However this diculty is only apparent: following the method displayed in Dupertuis and Sinha [2009], in order to construct the standard system, we will have to limit the system to the self-reproducing sub-system. If there is several self-reproducing sub-systems, several standard systems can be constructed. But, as we already said, it is true that, if in the self-reproducing sub-system the number of commodities produced is higher that the number of processes, some prices will stay indeterminate.

Conclusion

Focusing on self-reproducing systems with interdependent industries, we can always construct the standard system, and thus we can always express the rate of physical residue of such systems. Thus an economic reason to dismiss Manara's numerical argument is found. We can see clearly that the rst process does not need the second to reproduce itself: it is self-reproducing (while the second process is not, as only 0.99 of the second commodity is produced whereas one unit is required as input). We can then limit the analysis at the rst process, the second being a non-basic process.

And here is the second numerical example of Manara (processes are again in column):

A =   1 2 1 2 1 3 1 2 2   B =  
2.9 1.2 1.9 1.2 2.9 3.9 0.1 1.2 3.9

 

Here, it is the third process which is self-reproducing. Thus there is no need to take into account the two others to search for rate of physical residue of this system.

What would have been history if Sraa had nd this way to dismiss Manara's argument? Surely it would have been much harder for neoclassical theorists such as Samuelson to argue that Sraa's theory is just a special case of the neoclassical theory.

Of course, it may be that my conjecture is wrong, and that Sraa, after some attempts to respond, became convinced that it was not necessary. But I do not think so. Looking at the notes in the D3/14 le, one can see the urgent

Appendix

Here is the list of all Sraa's scientic publications, dened as publications in scientic journals, academic publications and book publication and edition:

1. Monetary ination in Italy during and after the war.

  So it seems important to me to present below the integral transcription of this presumed nal version. I use [ ] to signal footnotes and handwritten insertions in the margin or above or below the line, and I use {} to indicate my additions: PRODUZIONE CONG[U]INTA DE MERCI A MEZZO DI MERCI. RIPOSTA A C.F. MANARA 1 In un articolo pubblicato su questa rivista (1) Carlo Felice Manara ha presentato un'analisi in termini matematici di un capitolo del mio libro Produzione di merci a mezzo di merci (2) e di questo gli sono grato. [a capo] {the next sentences are crossed} Fin dall'inizio del suo scritto il Manara ha voluto esimersi da ogni responsabilità per il contenuto economico delle ipotesi o esempi che egli propone (3). Questa sarebbe stata una felice ispirazione se forma e contenuto potessero così facilmente essere separati. Ma subito diventa chiaro che la critica che egli fa è appunto basata su di un suo sbaglio di natura economica. [a capo] Mi era sembrato che la cosa dovesse essere evidente ad ogni economista così da rendere superua una rettica. Viceversa, ho constatato recentemente che l'articolo in questione ha messo fuori strada alcuni giovani studiosi; e di qui è sorta la necessità di una riposta. {end of the crossed section} 2 Prima di venire alla questione essenziale c'è un punto da chiarire. Il capita[o]le[o] al quale il Manara dedica la sua analisi è intitolato: Il sistema tipo con prodotti congiunti.(4) [(1) Il modello di Piero Sraa per la produzione congiunta di merci a mezzo di merci, L'industria, 1968, Pag. 318.

  Samuelson at the MIT, Cambridge, Massachusetts, had set his doctoral student David Levhari the task of disproving a proposition of Sraa regarding re-switching of techniques. Levhari published his refutation of Sraa's proposition in the Quarterly Journal of Economics in 1965. Georey Harcourt once recounted to me that he was perhaps the rst person in Cambridge, UK, to have come across this paper by Levhari at the Applied Economics Library. He went straight to Sraa and told him that there is a chap at MIT Cambridge who claims that your re-switching proposition is false. Sraa responded: No, he is wrong, and you show it to him! Harcourt responded: Me? I can't do matrix algebra.

  avvede? nota spiritosa cito di nuvo scuza commenti generalità negata: non vero. Ma esempio serabbe interessante Mentre non è provato (no ha mai detto che sia scupre ma solo in generale) M. fallisce: il suo errore non prova niente, sua raorza la così che la generalità novi è scossa: caso mai la presunzione è raorzata Il fallimento di M. rinforze la presuzione che nei casi economicamente possiblii non c'è è generale Ma sarebbe interessanta se qualcuno che maggio pazienza e fortuna del M. lo trovarse Si noti che L'esempio del M., non solo era stato esclusivo dall condiz. di soluzioni reali che avevo fatti explicitamente ; ma sarebbe stato esclusi dalla condizione prelim [economica] elementare che il sistema [della convenienza]. che si tratti di un sistema capace vitale capace di esistere o di montenersi in vita. Sarebbe interessante se il manara o qualcuno [più paziente] meno meno sfortunato di lui trovarse un esempio di un sistema di prodotti base [in sè] economicamente accettabile [possibile], che non avesse soluzione raz. per il sistema la merce tipo. (Sraa's unpublished papers, D3/14: 7274).

  Remember the second numerical example (the processes are in column) :

  need of Sraa to try to save his life work. And one can only wonder how broken was his heart when he nally crossed out the last version of what would have been his last article.

1

  The online eort is directed by Giancarlo de Vivo and Murray Mil- gate with the collaboration of the Wren archivist Jonathan Smith.

													It is ex-
	pected	to	be	completed	in	2017	and	can	be	found	at	the	following	url:
	https://janus.lib.cam.ac.uk/db/node.xsp?id=EAD%2FGBR%2F0016%2FSRAFFA

  Se nell'esempio del Manara cambiano l'ultimo membro da 0.99d a 1.09d il caso diventa possibile: e il sistema si trove già formato in sistema tipo, con il saggio massimo del protto che può essere calcolato in termin della une o dell'altro merce, = 2.234-2.1 . . . %. (Sraa's unpublished papers, D3/14: 61).

	tipo, e con non e diversa da quella che [non è soggesta a condizioni
	più restrittive di quella] di qualunque[siasi] altra teoria generale di
	un sistema di industrie interdipendenti[.] la capace di mantenersi
	Se questo fosse vero il M. non solo avrebbe demolito [schiacciato]
	la mia povera [modesta] capanne ma, nuovo Sansone, avrebbe fatto
	crollare tutto i édicio colonnato dell' economia moderna. che Adam
	Smith in
	2.1	= 0.133 2.1
	o 6, 6

  1920, doctorate thesis. 2. The bank crisis in Italy. 1922, Economic Journal 32, June, pp. 17897. 3. Obituary of Mafeo Pantaloni. 1924, Economic Journal 34, December, pp. 648653.


The more journal peer review (JPR) became a scientific topic, the more it has been the subject of criticisms and controversies. Repeated fake reports, confirmed reviewers' biases, lack of reproducibility, and a recurrent inability to detect fraud and misconduct have apparently condemned JPR in its supposedly traditional form. In fact, just like previous historical reforms and inventions, JPR has again been the object of many proposals to "fix it" since the beginning of the 21 st century . Though these proposals are very diverse, two main directions 1 have been identified: open peer review on one side, post-publication peer review (PPPR) on the other .

2

Even if its actual forms depend on 22 different definitions , the first one relies on transparen 3 cy and accountability to clean JPR from suspicion and criticism. These values are infused in pre-publication peer review by the mutual knowledge of reviewers and authors, the publication of referees' reports, the immediate availability of manuscripts or the organisation of a real discussion between authors, referees, editor-in-chief and sometimes whoever is interested in the manuscript. It has been implemented successfully in many new electronic-only journals and in working papers/preprint servers that are finally finding popularity, long after their success based on the ArXiv model has been announced . The second one is also very diverse, in 4 cluding old practices such as citation counting in most natural and life sciences and book reviews in humanities and social sciences. It is also driven by innovations, such as the "publish first, review second" model developed by F1000 , the implementation of automated instru 5 ments by the Altmetrics movement or the systematic labelling of problematic published ar 6 ticles through an "expression of concern", a "correction" or a "retraction" by journals . Would 7 it be through modelisation , common sense or experience, PPPR supporters strongly believe 8 in the wisdom of the crowd and consequently empower readers as good judges to valuate and evaluate the published scientific literature .

These two "fixes" share a common device, the open commenting of published articles, which is both an open peer review practice as it is visible to all readers and PPPR as it comes after the publication and often the certification of articles. At their intersection, it should thus thrive and indeed many journals have proposed this feature. As an experiment in 2006, Nature offered the "open comment" possibility to authors: only 5% of them agreed on the procedure, and among the 71 articles, less than half of them got comments and these were mostly non substantive . New publishers also embraced it: BioMedCentral opened comments after it shifted 10 to an open access journal model in 2002, so did PlOS One, with very little success . Even the 11 PubMedCentral platform built them, and less than 1 paper out of 1000 were ever commented before the PubMedCommons was recently shut down . These failures question the many rea 12 sons why scientists don't seem eager to become commentators in the digital age, whereas the "Letter to the Editor" format has thrived for ages. For Angela Cochran, it is ultimately the lack of community, which leads to this absence of engagement, contrary to other modes of scholarly communication such as workshops, conferences or formal journal activities .

13 Nevertheless, there is an exception to the disappointment with open commentary in PPPR, which is the empirical case for this presentation: PubPeer, where commentators come in herds and comments flourish . The only explanation given for this peculiar success is the possibili 14 ty, largely used, to publish anoymized comments on the platform. So, how can you embrace the openness of discussion and, at the same time, enable anonymous commentators? What kind of PPPR practices is it connected with? Does it inform our views on traditional peer review and how? To answer these questions, we will first describe how the platform has been built and works, then to what kind of dynamics it leads as far as anonymity is concerned, then typify the arguments used for and against anonymity in PPPR, discuss its effects on published papers, before concluding on the way debates could be organized in PPPR. These first results are based on the systematic qualitative analysis of both threads on PubPeer, articles on specialized websites on PubPeer and anonymity (Scholarly Kitchen, RetractionWatch…) and on editorials from scientific journals that have commented on anonymity in PPPR .

was mostly to educate young physicians, discuss applicability of published papers in clinical and health settings and develop critical thinking about the literature . This last objective was 17 at the heart of PubPeer creators, as they thought that raising specific questions about a given paper was insightful for the scientific debate. Moreover, the online translation of journal clubs would bring two new components to the debate: firstly, authors of the papers could actually answer the questions addressed to them; secondly, these discussions would be shared and cumulative across departments . Strongly opposed to the evaluation of scientists and papers 18 based on the journals in which they publish, they envisioned a device that is very similar to the initial project of Garfield with ISI in the 1950s: to have a unique place for the discussion and community surrounding any paper . To do so, they would create a thread for each paper 19 (using DOI as identification) and, once a comment has been posted, generate an automatic email to the authors of the paper in order to them to answer and start a conversation.

Anonymity was not included by design: yet, the three founders were anonymous, but comments had initially been signed, until founders were repeatedly contacted by young scientists being afraid to comment in the open view of their senior peers. So they implemented this possibility by March 2013 and rapidly saw the number of comments rising on the platform . An 20 ticipating trolling and other problems pervasive within internet anonymous forums, they add a moderation step for these anonymous users, aimed at verifying comments are "based on publicly verifiable information" and are "not ad hominem" . Though the moderation didn't pre 21 vent the expansion of the website, the massive publication of such comments led to heated discussions about anonymity (see section 2 and 3). In 2015, PubPeer became a Californian foundation, thus legally "forcing" one of its founders to become the public figure of the website, Brandon Stell, an American biologist working for the French institution CNRS.

From then, Stell has been invited to many conferences on misconduct, scientific integrity or peer review and PubPeer has still grown, embracing a major public role in PPPR at least in biology . They have developed additional features, notably a browser plugin to show existing 22 PubPeer comments when users visit a given article page, and recently, some paying dashboards for journals to manage all existing comments on the articles they have published. But their dream of organized debates on the published literature has faced notable critics and unexpected twists, because of their use of anonymous comments.

Discussing papers and anonymity

The PubPeer website contains a list of threads which are searchable with standard published literature metadata (DOI, authors, title, keywords, journal…). Yet, there are "general threads" on journal management or JPR, such as the accountability of COPE member journals or "the sociology of rejection" (sic!) on the treatment of manuscripts, a few PubPeer Blog entries, but the quasi-totality of threads are discussions about a specific published paper. Despite the moderation processes, anonymity often becomes an obstacle for at least some stakeholders in the debate.

In a typical example, a recent thread about a paper concerning the treatment for certain patients starts with an very strong anonymous comment, which is automatically given a name from the "tree of life", Ehrendorferia chrysantha . As the corresponding author has subse 23 quently received an email, she answers and starts by pointing out anonymity as a problem: "However, I am eager to know what makes you to be ashamed of your real name, of your profession, your specialty and your affiliation? Why do you want to hide your real name under the plant name of Ehrendorferia chrysantha?", then showing her faith in another form of PPPR as she states that her article has already received a high number of views. In his reply, the first commentator excludes personal questions and tries to come back to the content of the paper : "I am not ashamed of my name or profession but I don't see how that makes a difference so I am using anonymous login and the names is automatically generated by Pubpeer. I admit that I also made a comment about the author's identity when I pointed out that the study has been done by single author in a private clinic but I think it's a noteworthy in this case." Then, the editor of the journal answers, renewing the critique of anonymity: "First, in my opinion it is a disrespect to the author to contact her anonymously. If you pretend that you are a scientist, you should not use an anonymous way to debate scientific issues". Finally, the first commentator comes back, thanking for answers, but considering they were not addressing the weaknesses of the paper, he concludes: "I hope people who read this paper can be critical and nobody makes any decision on treatment solely based on this paper".

In disciplines where the value of statements is strongly attached to their author, the organisation of the debate with anonymous contributors often derived on a debate about anonymity, with conflicting views between those who wish to have content-based comments and others who consider valid contributions only from known persons. As such, it is the very definition of "peer" in PubPeer which is at stake.

The pros and cons of anonymity

If many threads never lead to any kind of real debate, whether because they shift to anonymity discussions or just by the lack of answers from authors or other commentators, the "success" of PubPeer, both in number of comments and in some spectacular cases (see section 4), has led to very constructed opposing views on anonymity in PPPR. We present in Table 1 a first synthesis, notably based on these exchanges .

24

Table 1 Main arguments on anonymity in PPPR

The first two lines of Table 1 derive directly from the conception of PubPeer. Its founders strongly state that their one of its kind success is the proof that anonymity has to be part of commenting in PPPR; otherwise the omerta of the academic milieu would prevent public criticism to flourish, as it is shown by the failures of other commenting platforms. It is the anonymity that allows commentators to raise issues on content itself, especially very technical questions on methods. Conversely, opponents and critics underline the non-constructive engagement produced by anonymous commentators, so whatever number they might be, it is useless for science. More specifically, for them, commentators seem to be obsessed to find image manipulation in seemingly every figure of any published paper: so, rather than really "commenting", they just put an image on the website with arrows or squares pointing out supposed problems .

25

The last two lines of Table 1 point out to the organization of debates in science. For PubPeer founders and supporters, the protection of commentators is an absolute goal, for debate to happen, especially because they observed careers being crushed simply by the voicing out of young scientists. And for them, open debate, in the most classical scholasticism fashion, should be focused on arguments and counter-arguments on content, forbidding judgments about persons in favor of discussion on facts and writings. Conversely, its opponents would care about the protection of authors, especially in a context where fraud and misconduct has become a common issue in life sciences. They would defend a "civil" science in which participants in a debate would be known to one another and avoid personal accusations, in fact "be critical but fair" . Indeed, supporters and opponents of anonymity as organized in PubPeer agree on what would be an ideal world, in which scientific debate would not need anonymity, where raising questions about a paper would not be seen as an accusation . They also agree on problems in 27 science and share an interest in PPPR, but only diverge on the means to organise PPPR commenting . There is a striking similarity with the 1970s historical debates on anonymity in 28 JPR, which predated the "single blind" and "double-blind" labels to designate competing practices . But there is also a major difference: if scandals and misconduct cases were alrea 29 dy there in the 1970s, especially in the biomedical sciences in the US , they were not directly 30 raised by peer review devices but rather by independent inquiries.

PubPeer outcomes

As a matter of fact, PubPeer and PPPR have been taken into a highly critical configuration, in which science and JPR have been considered in full crisis. As soon as anonymous commenting was available, the founders were deeply surprised by the number of comments on fabrication and manipulation of data as they were expecting something completely different: "when we created PubPeer, we expected to facilitate public, on-the-record discussions about the finer points of experimental design and interpretation, similar to the conversations we all have in our journal clubs" . In the context of the "reproducibility crisis" and of massive discoveries 31 of fraud, the PubPeer platform has become a key actor not only in the commenting of affairs and scandals, but also in their genesis and development.

Consequently, the desired outcomes, which initially revolved around scientific debate have been gradually transformed to "correcting science", then "fraud detection". PubPeer has gradually become a home not for whistleblowers but for alarm raisers as they would only point out potential problems rather than directly expressing a judgment about the authors actions .

32

For example, a commentator underlines about a problem on a PLOS paper, which, in his opinion, should have led to an expression of concern after his previous comment on the PLOS platform had no answer for more than a year ; by doing so, he shows the supposed alarming lack of response from journals . On a larger scene, the questioning of the "scientific esta 33 blishment" by the "anonymous cowards" about a series of Nature journals articles has become directly staged by the PubPeer founders in a blog entry .

34

In fact, the debunking of big affairs/scandals through the use of PubPeer. has become another possible outcome. One of the most preeminent cases has been the Voinnet affair, a French plant biologist who has been accused of using fabricated data in part of his papers. One paper after the other, his whole production seems to have been dissected and, more particularly, the figures in it . Moreover, it works the other way around: when expressions of concern and re 35 tractions happen, the role played by PubPeer and its anynomized commentators is underlined in the reports, sustaining a never-ending dynamic of suspicion on certain authors/labs . As 36 more comments appeared on Pubpeer for some papers, the more concerned journals and institutions were pressured to launch investigations about the papers they published .

37

This role in uncovering misconduct has also led PubPeer in front of courts, as defamation was raised by a biologist who was fired after an investigation in his university had been triggered by PubPeer comments. In particular, he wanted judges to force PubPeer to reveal the identity of anonymous commentators . While PubPeer even received support from ACLU in the US, 38 the reaction of the platform show once again the importance of anonymity for them, as they first warned their users about the case, then urged them to use Tor to be completely anonymous , and finally developed a system in which they would have not even computer traces of 39 users.

Conclusion : debating published literature

As this last example show, for PubPeer founders, anonymity is at the heart of its PPPR system: though they regret it, it seems the only way for them to have massive insightful comments about papers, targeting problematic issues . To have open information, they insist they 40 need anonymity for their commentators and at least some part of their "efficiency" is now judged based not only on the debates they host, but rather on the debunked misconduct they informed, mostly as an alarm raiser platform.

Then, the contrasted opinions on this kind of PPPR may be summed up into the couple "Vigilante Science/ Vigilants for Science", the former one being uttered by the editor-in-chief of Plant Physiology about the anonymous crowd of commentators, the latter one by PubPeer 41 founders, pointing out that the real problem was "flood of low-quality, overinterpreted and ultimately unreliable research" . They both show that, thanks to the PubPeer platform, public 42 commenting of published literature has found its niche, far from initial aims: PPPR as a postpublication problem resolution, where fraud and misconduct, incorrectly treated by JPR, would be, if not solved, inquired and dealt with.

While Altmetrics, the Open Citation movements and others have supported a view of PPPR as "usage measurements", where the content of the papers themselves is untouched and JPR (and especially "sound science JPR") reinforced, anonymous PPPR commenting undermines journals as certification devices and appear as a competitor/corrector . Whether they are angry, 43 seek for revenge or idealistic, the wisdom of the anonymous crowd questions more profoundly the weight and qualities attributed to JPR.

  Pubpeer: vigilante science, journal club or alarm raiser?The controversies over anonymity in post-publication peer review.Didier Torny (Mines ParisTech, PSL Research University, CSI-i3 UMR CNRS 9217)

1. PubPeer: a success story?PubPeer has been conceived by 3 biologists in 2012 as an online version of "journal clubs", a common practice at least in many medical departments for more than a century . Their aim

Pros Cons

Efficiency


Brahim Boufoussi
email: boufoussi@ucam.ac.ma

Marco Dozzi
email: dozzi@iecn.u-nancy.fr

Raby Guerbaz
email: r.guerbaz@ucam.ac.ma

Sample path properties of the local time of multifractional Brownian motion

Keywords: Multifractional Brownian motion, Local asymptotic self similarity, Chungtype law of iterated logarithm, Local times

We establish estimates for the local and uniform moduli of continuity of the local time of multifractional Brownian motion, B H = (B H(t) (t), t ∈ R + ). An analog of the Chung's law of the iterated logarithm is studied for B H and used to obtain the pointwise Hölder exponent of the local time. A kind of local asymptotic self similarity is proved to be satisfied by the local time of B H .

Introduction

The multifractional Brownian motion (mBm), B H = (B H(t) (t), t ∈ R + ), is a Gaussian process which extends the fractional Brownian motion (fBm) by allowing the Hurst function to vary with time. This provides a tool to model systems whose regularity evolves in time, such as internet traffic or images. Recently, [START_REF] Boufoussi | On the local time of the multifractional Brownian motion[END_REF] have investigated, under mild regularity conditions on H(.), the existence of jointly continuous local times of mBm. Their objective was to explore the link between the pointwise regularity of the local time and the irregularity of the underling process. This effect of inverse regularity was observed by [START_REF] Berman | Local nondeterminism and local times of Gaussian processes[END_REF] for the case of uniform regularity.

The first aim of this paper is to establish estimates for the local and uniform moduli of continuity for the local time of mBm. Upper bounds for the moduli of continuity of local times have been obtained by [START_REF] Kôno | Hölder conditions for the local times of certain Gaussian processes with stationary increments[END_REF] for Gaussian processes with stationary increments, and more recently [START_REF] Csörgö | On moduli of continuity for local times of Gaussian processes[END_REF] have proved upper bounds for the moduli of continuity of the maximum of local times for stationary Gaussian processes and Gaussian processes with stationary increments. However, contrary to classical results, the moduli of continuity obtained in this paper depend on the point at which the regularity is studied. This is natural, since mBm has a regularity evolving in time. The approach of the present paper seems to be useful for extending the results of [START_REF] Kôno | Hölder conditions for the local times of certain Gaussian processes with stationary increments[END_REF] and [START_REF] Csörgö | On moduli of continuity for local times of Gaussian processes[END_REF] to Gaussian processes without stationary increments.

Chung's form of the law of iterated logarithm for mBm is established in section 4. This result is used to prove that the pointwise Hölder exponent of its local time with respect to time is equal to 1 -H(t) uniformly in x.

The second main objective of the paper concerns the question whether the sample path properties of mBm can be transferred to its local time. If H(.) is not constant, the mBm is no longer self-similar. However, it is proved by Lévy Véhel and Peltier (1995, Proposition 5) that if H is β-Hölder continuous and sup t∈R + H(t) < β, a property called local asymptotic self-similarity remains, and is defined as follows:

lim ρ-→0 + law B(t + ρu) -B(t) ρ H(t) , u ∈ R = law{B H(t) (u), u ∈ R}, (1)
where B H(t) is fBm with Hurst parameter H(t). It is the purpose of section 5 to prove that the local time of mBm has a kind of local asymptotic self similarity. Through this result, we obtain some local limit theorems corresponding to the mBm. We will use C, C 1 , ... to denote unspecified positive finite constants which may not necessarily be the same at each occurrence.

Preliminaries

In this section, we present some notation and collect facts about multifractional Brownian motion and its local times.

Multifractional Brownian motion

The mBm was introduced independently by Lévy [START_REF] Véhel | Multifractional Brownian motion: definition and preliminary results[END_REF] and [START_REF] Benassi | Elliptic Gaussian random processes[END_REF]. The definition due to Lévy-Véhel and Peltier (1995) is based on the moving average representation of fBm, where the constant Hurst parameter H is replaced by a function H(t) as follows :

B H(t) (t) = 1 Γ(H(t) + 1/2) 0 -∞ [(t -u) H(t)-1/2 -(-u) H(t)-1/2 ] W (du) + t 0 (t -u) H(t)-1/2 W (du) ; t ≥ 0, (2)
where

H(t) : [0, ∞) -→ [µ, ν] ⊂ (0,
B H(t) (t) = R e itξ -1 |ξ| H(t)+1/2 d W (dξ), (3)
where W (ξ) is the Fourier transform of the series representation of white noise with respect to an orthonormal basis of L 2 (R); we refer to [START_REF] Cohen | From self-similarity to local self-similarity : the estimation problem[END_REF] for the precise definition and for the fact that B H is real-valued.

Various properties of mBm have already been investigated in the literature, related, for instance, to its pointwise and uniform Hölder regularity as well as the local Hausdorff dimension of its sample paths. More precisely, it is known from Lévy Véhel and Peltier (1995, Proposition 8) that with probability one, for each t 0 ≥ 0 the Hölder exponent at t 0 of mBm is H(t 0 ). Recall that the pointwise Hölder exponent of a stochastic process X at t 0 is defined by

α X (t 0 , ω) = sup α > 0, lim ρ→0 X(t 0 + ρ, ω) -X(t 0 , ω) ρ α = 0 . (4)
In addition, according to the same authors, the local Hausdorff dimension of the graph is

2 -min t∈[a,b] H(t), almost surely for each interval [a, b] ⊂ R + .
According to the previous results, the regularity of mBm depends on the regularity of H. Furthermore, by using Lemma 3.1 in [START_REF] Boufoussi | On the local time of the multifractional Brownian motion[END_REF] we can prove easily that the irregularity of H implies the irregularity of mBm. More precisely, the points of discontinuities of H are also discontinuities of B (see Ayache (2001) Proposition 1).

Local times

We end this section by briefly recalling some aspects of the theory of local times. For a comprehensive survey on local times of both random and non random vector fields, we refer to [START_REF] Geman | Occupation densities[END_REF], [START_REF] Dozzi | Occupation density and sample path properties of N-parameter processes[END_REF] and [START_REF] Xiao | Strong local nondeterminism and sample path properties of Gaussian random fields[END_REF].

Let X = (X(t), t ∈ R + ) be a real-valued separable random process with Borel sample functions. For any Borel set B ⊂ R + , the occupation measure of X on B is defined as follows

µ B (A) = λ{s ∈ B : X(s) ∈ A} for all A ∈ B(R),
where λ is the one-dimensional Lebesgue measure on R + . If µ B is absolutely continuous with respect to Lebesgue measure on R, we say that X has a local time on B and define its local time, L(B, .), to be the Radon-Nikodym derivative of µ B . Here x is the so-called space variable, and B is the time variable. By standard monotone class arguments, one can deduce that the local times have a measurable modification that satisfies the following occupation density formula: for every Borel set B ⊂ R + , and for every measurable function

f : R → R + , B f (X(t))dt = R f (x)L(B, x)dx.
Recently, [START_REF] Boufoussi | On the local time of the multifractional Brownian motion[END_REF] have proved that if the Hurst function is Hölder continuous with exponent β and if sup t≥0 H(t) < β, the mBm has a jointly continuous local time, i.e. the mapping (t, x) → L(t, x) is continuous. In addition, this local time has the following Hölder continuities : It satisfies for any compact U ⊂ R (i)

sup x∈U |L(t + h, x) -L(t, x)| |h| γ < +∞ a.s., (5)
where γ < 1 -H(t) and |h| < η, η being a small random variable almost surely positive and finite, (ii) for any I ⊂ [0, T ] with small length, sup

x,y∈U,x =y

|L(I, x) -L(I, y)| |x -y| α < +∞ a.s., (6)
where α <

  1 2 sup I H(t) - 1 2   ∧ 1.
These results have been used to obtain the local and pointwise Hausdorff dimension of the level sets of mBm. We refer to [START_REF] Boufoussi | On the local time of the multifractional Brownian motion[END_REF] for definition and results.

Moduli of continuity of the local time

Throughout this section, the Hurst function H : R + → [µ, ν] ⊂ (0, 1) is assumed to be measurable. The notation B = (B(t), t ≥ 0) means that either representation, (2) or (3) can be chosen. Moreover, we say that H satisfies the condition (H β ) if :

H is β-Hölder continuous with sup s≥0 H(s) < β.

We give the following improvement of Theorem 3.1 of [START_REF] Boufoussi | On the local time of the multifractional Brownian motion[END_REF]. Note that the first part reproduces this theorem without assuming (H β ). Theorem 3.1. Consider a measurable function H(.) : R + → [µ, ν] ⊂ (0, 1). The mBm with Hurst function H(.) given by (2) admits on any interval [a, b] ⊂ [0, ∞) a square integrable local time. Moreover for both representations (2) and (3), if (H β ) holds, the existence of square integrable local times implies that H(t) < 1 for almost all t.

Proof. Let us write for simplicity

B H(t) (t) = t -∞ K H(t) (t, u)dW (u),
where

K H(t) (t, u) = 1 Γ(H(t) + 1/2) (t -u) H(t)-1/2 + -(-u) H(t)-1/2 + .
For any t > s, taking the variance of B H(t) (t) -B H(s) (s), we get

V ar(B H(t) (t) -B H(s) (s)) ≥ V ar(B H(t) (t) -B H(s) (s)/W (u), u ≤ s) = V ar(B H(t) (t)/W (u), u ≤ s), (7)
where the last equality follows from the fact that B H(s) (s) is measurable with respect to σ(W (u), u ≤ s). Moreover we can write

B H(t) (t) = t -∞ K H(t) (t, u)dW (u) = s -∞ K H(t) (t, u)dW (u) + t s K H(t) (t, u)dW (u).
Hence, by using the measurability of

s -∞ K H(t) (t, u)dW (u) with respect to σ(W (u)/ u ≤ s), we obtain V ar(B H(t) (t)/W (u), u ≤ s) = V ar t s K H(t) (t, u)dW (u)/W (u), u ≤ s = V ar t s K H(t) (t, u)dW (u) , (8)
where, to obtain the last equality, we have used the fact that

t s K H(t) (t, u)dW (u) is independent of σ(W (u), u ≤ s)
(by the independence of the increments of the Brownian motion). Combining ( 7) and ( 8) and denote

C = sup u∈[µ,ν] Γ(1/2 + u), we obtain V ar(B H(t) (t) -B H(s) (s)) ≥ 1 Γ(1/2 + H(t)) 2 t s (t -r) 2H(t)-1 dr ≥ 1 2νC 2 (t -s) 2H(t) . Therefore [a,b] [a,b] E[B H(t) (t) -B H(s) (s)] 2 -1/2 dsdt ≤ √ 2νC [a,b] [a,b] |t -s| -sup r≥0 H(r) dsdt. (9)
The last integral is finite because sup r≥0 H(r) < 1.

By
E(B(t) -B(s)) 2 ≤ C µ,ν |t -s| 2H(t) , for s, t ∈ [a, b]. (10)
[E(B(t) -B(s)) 2 ] 1/2 dsdt.
Then, [START_REF] Dozzi | Occupation density and sample path properties of N-parameter processes[END_REF]

implies that b a b a 1 (t -s) H(t) dsdt < ∞. Consequently, H(t) < 1 for almost all t ∈ [a, b]. Since R + is a countable union of small intervals, hence the result is proved.
In what follows, we are interested in the local and pointwise oscillations (at each t) of the local time of the mBm. Theorem 3.2. Let {B H(t) (t), t ≥ 0} be the mBm given in [START_REF] Ayache | Du mouvement Brownien fractionnaire au mouvement Brownien multifractionnaire[END_REF] with Hurst function H(.) satisfying the assumption (H β ). Then, for every t ∈ R + and any x ∈ R, there exist positive and finite constants C 1 and C 2 such that

lim sup δ→0 L(t + δ, B H(t) (t)) -L(t, B H(t) (t)) δ 1-H(t) (log log(δ -1 )) H(t) ≤ C 1 a.s. ( 11
)
lim sup δ→0 L(t + δ, x) -L(t, x) δ 1-H(t) (log log(δ -1 )) H(t) ≤ C 2 a.s. (12)
Proof. Let t ≥ 0 be a fixed point and 0 < h < 1 and denote H t,t+h = sup s∈[t,t+h] H(s).

According to the Fourier analytic approach of Berman (1969) (see also [START_REF] Davies | Local Hölder conditions for the local times of certain stationary Gaussian processes[END_REF], expression ( 27)), we have

E L(t + h, B H(t) (t)) -L(t, B H(t) (t)) m = 1 (2π) m [t,t+h] m R m E e i P m j=1 u j (B H(s j ) (s j )-B H(t) (t)) m j=1 du j m j=1 ds j . Let B(s) = B H(s) (s) -B H(t) (t), s ≥ 0 and denote by R(s 1 , ..., s m ) the covariance matrix of ( B(s 1 ), ..., B(s m )) for distinct s 1 , ..., s m . Let U = (u 1 , ..., u m ) ∈ R m and U T denote the transpose of U . According to (16) below we have, det R(s 1 , ..., s m ) > 0. Hence, the change of variable V = R 1/2 U , implies R m E e i P m j=1 u j e B(s j ) du 1 ...du m = (2π) m/2 (det R(s 1 , ..., s m )) 1/2 . Therefore E L(t + h, B H(t) (t)) -L(t, B H(t) (t)) m = 1 (2π) m/2 [t,t+h] m 1 (det R(s 1 , ...., s m )) 1/2 ds 1 ...ds m = m! (2π) m/2 t<s 1 <...<sm<t+h 1 (det R(s 1 , ...., s m )) 1/2 ds 1 ...ds m . (13)
In addition

det R(s 1 , ..., s m ) = V ar( B(s 1 ))V ar( B(s 2 )/ B(s 1 ))....V ar( B(s m )/ B(s 1 ), ..., B(s m-1 )). ( 14
)
Using similar arguments as in the proof of Theorem 3.1, we obtain for any r, s

∈ [t, t+h], such that r < s, V ar B(s)/ B(u), u ∈ A, u ≤ r ≥ V ar B(s) -B(r)/ W (u), u ≤ r ≥ V ar B H(s) (s)/ W (u), u ≤ r ≥ 1 2νC 2 (s -r) 2H(s) , (15)
where C = sup x∈[µ,ν] Γ(x + 1/2). Combining ( 14) and ( 15) we obtain

det R(s 1 , ..., s m ) ≥ 1 (2νC 2 ) m m j=1 (s j -s j-1 ) 2H(s j ) ≥ 1 (2νC 2 ) m m j=1 (s j -s j-1 ) 2H t,t+h , (16)
where s 0 = 0 and (s j -s j-1 ) H(s j ) ≥ (s j -s j-1 ) H t,t+h , since (s j -s j-1 ) < 1.

According to [START_REF] Jeganathan | Convergence of functionals of sums of fractionally integrated processes[END_REF] we have

E L(t + h, B H(t) (t)) -L(t, B H(t) (t)) m ≤ νC 2 π m/2 m! t<s 1 <...<sm<t+h m j=1 1 (s j -s j-1 ) H t,t+h ds 1 ...ds m .
Now, by an elementary calculation (cf. [START_REF] Ehm | Sample function properties of multi-parameter stable processes[END_REF]), for all m ≥ 1, h > 0 and b j < 1,

t<s 1 <...<sm<t+h m j=1 (s j -s j-1 ) -b j ds 1 ...ds m = h m- P m j=1 b j m j=1 Γ(1 -b j ) Γ(1 + m -m j=1 b j )
.

Therefore E L(t + h, B H(t) (t)) -L(t, B H(t) (t)) m ≤ m! νC 2 π m/2 h m(1-H t,t+h ) (Γ(1 -H t,t+h )) m Γ(1 + m(1 -H t,t+h ))
According to Stirling's formula, we have

(m!/Γ(1 + m(1 -H t,t+h ))) ≤ M m m! H t,t+h , m ≥ 2,
for a suitable finite number M . Therefore

E L(t + h, B H(t) (t)) -L(t, B H(t) (t)) h 1-H t,t+h m ≤ C m m! H t,t+h . (17)
We shall now prove that for any K > 0 there exists a positive and finite constant A > 0, depending on t, such that for u small enough,

P L(t + h, B H(t) (t)) -L(t, B H(t) (t)) ≥ Ah 1-H t,t+h u H t,t+h ≤ exp(-K/u). (18)
Consider u first of the form u = 1/m. Combining Tchebychev's inequality and ( 17) we obtain

P L(t + h, B H(t) (t)) -L(t, B H(t) (t)) h 1-H t,t+h ≥ Am H t,t+h ≤ E L(t + h, B H(t) (t)) -L(t, B H(t) (t)) Ah (1-H t,t+h ) m H t,t+h m ≤ C m A m ( 1 m ) mH t,t+h (m!) H t,t+h .
Using again Stirling's formula, the last expression is at most

C m A m (2πm) H t,t+h
/2 e -H t,t+h m . This can be written

exp m [log(C/A) -H t,t+h ] + H t,t+h 2 [log(m) + log(2π)] , (19)
it suffices to take A > C and m 0 large such that for any m ≥ m 0 , ( 19) is dominated by e -2Km . Moreover for u sufficiently small, there exists m ≥ m 0 such that u m+1 < u < u m and since m ≥ 1,

m m + 1 ≥ 1 2
. This proves [START_REF] Lim | Fractional Brownian motion and multifractional Brownian motion of Riemann-Liouville type[END_REF].

In addition, if we take u(h) = 1/ log log(1/h) and consider first h m of the form 2 -m , then [START_REF] Lim | Fractional Brownian motion and multifractional Brownian motion of Riemann-Liouville type[END_REF] implies

P L(t + h m , B H(t) (t)) -L(t, B H(t) (t)) ≥ Ah 1-H t,t+hm m (log log(1/h m )) H t,t+hm ≤ m -2 ,
for large m. Consequently, by using the Borel-Cantelli lemma and monotonicity arguments, we obtain

L(t + h, B H(t) (t)) -L(t, B H(t) (t)) h 1-H t,t+h ≤ A (log log(1/h)) H t,t+h a.s.
In addition the Hölder regularity of H implies that

log log(1/h) H t,t+h log log(1/h) H(t) = e Kh β log log log 1/h → 1 as h → 0 + Therefore (log log(1/h)) H t,t+h ≤ K(log log(1/h)) H(t)
for h small enough. Moreover, using the same arguments (See Remark 3.7 in Boufoussi et al. ( 2006)), we obtain

h 1-H t,t+h ≤ Kh 1-H(t) . (20)
This completes the proof of [START_REF] Ehm | Sample function properties of multi-parameter stable processes[END_REF]. Since [START_REF] Geman | Occupation densities[END_REF] is proved in a same manner, the proof is omitted here.

Remark 3.3. The result of the previous theorem, with sup s∈[t,t+h] H(s) instead of H(t), may be proved even when (H β ) is not satisfied.

We shall also prove the following uniform result.

Theorem 3.4. Let {B H(t) (t), t ∈ [0, 1]} be the mBm in [START_REF] Ayache | Du mouvement Brownien fractionnaire au mouvement Brownien multifractionnaire[END_REF] with arbitrary Hurst function and denote H 0,1 = sup s∈[0,1] H(s). Then for all x ∈ R, there exists C 3 > 0 such that lim sup

|t-s| 0 + t,s∈[0,1] |L(t, x) -L(s, x)| |t -s| 1-H 0,1 (log(1/|t -s|)) H 0,1 ≤ C 3 a.s. (21)
Proof. By using the same arguments as above, without using the Hölder continuity of H, we show that with probability one, there exists m 0 (ω) such that for all m ≥ m 0 (ω) and k = 1, ....., 2 m , we have

L(k2 -m , x) -L((k -1)2 -m , x) ≤ A2 -(1-H 0,1 )m (m log(2)) H 0,1 . a.s.
Then by proceding essentially as in [START_REF] Kôno | Hölder conditions for the local times of certain Gaussian processes with stationary increments[END_REF] we prove [START_REF] Xiao | Strong local nondeterminism and sample path properties of Gaussian random fields[END_REF].

Remark 3.5. The results in this section are based on the following property of the mBm:

V ar B H(t+h) (t + h)/B H(s) (s) : 0 ≤ s ≤ t ≥ Ch 2H t,t+h , (22)
X(t) = 1 Γ(H(t) + 1/2) t 0 (t -u) H(t)-1/2 dW u .
So our results extend those of [START_REF] Kôno | Hölder conditions for the local times of certain Gaussian processes with stationary increments[END_REF] to Gaussian processes without stationary increments property. Remark 3.6. A closer examination of the proof of Theorem 2.1 in Csörgö et al. (1995) shows that the property of stationarity of increments is just used to prove two main ingredients presented in their Lemmas 3.1 and 3.3 and which can in our case be replaced easily by the following properties 1. Let A n be the covariance matrix of a Gaussian vector {ζ i ; 1 ≤ i ≤ n}. Then, it is known that the conditional variance can be written as

V ar(ζ i /ζ l , l = i, 1 ≤ l ≤ n) = |A n | |A (i) n | , where A (i)
n is the submatrix of A n obtained by deleting the ith row and column.

T →∞ sup 0≤t≤b T L(t + a T , x) -L(t, x) a 1-H T (log(b T /a T ) + log log(a T + 1/a T )) H < C < ∞ a.s. ( 23
)
where a T and b T are nonnegative functions in T ≥ 0 such that

1 + b T a T -→ +∞ as T → +∞.

Chung's law of the iterated logarithm

The following theorem establishes an analogue of Chung's law of the iterated logarithm for mBm. The result for fBm has been obtained by [START_REF] Monrad | Small values of Gaussian processes and functional laws of the iterated logarithm[END_REF].

Theorem 4.1. Let B be a multifractional Brownian motion, assume that H satisfies the condition (H β ). Then the following Chung type law of iterated logarithm holds:

lim inf δ→0 sup s∈[t 0 ,t 0 +δ] |B(s) -B(t 0 )| (δ/ log | log(δ)|) H(t 0 ) = C H(t 0 ) , a.s. ( 24
)
where the constant C H(t 0 ) is the one appearing in Chung's law for the fBm of Hurst parameter H(t 0 ).

Proof. We first introduce the process { B(t) = B(t) -B H(t 0 ) (t), t ≥ 0}, where B H(t 0 ) denotes a fBm with Hurst parameter H(t 0 ). According to Theorem 3.3 of Monrad and Rootzén (1995), the fBm with Hurst exponent H(t 0 ) satisfies (24). Furthermore,

B(t) -B(t 0 ) = (B(t) -B H(t 0 ) (t)) + (B H(t 0 ) (t) -B H(t 0 ) (t 0 )), = B(t) + (B H(t 0 ) (t) -B H(t 0 ) (t 0 )).
Then (24) will be proved if we show that

lim δ→0 sup t∈[t 0 ,t 0 +δ] | B(t)| (δ/ log | log(δ)|) H(t 0 ) = 0. a.s.
On the other hand, Lemma 3.1 in [START_REF] Boufoussi | On the local time of the multifractional Brownian motion[END_REF], implies that there exists a positive constant K such that sup

t∈[t 0 ,t 0 +δ] E( B(t)) 2 ≤ Kδ 2β .
Hence, according to Theorem 2.1 in Adler (1990, p. 43) and a symmetry argument, we have

P sup t∈[t 0 ,t 0 +δ] | B(t)| ≥ u ≤ 2P sup t∈[t 0 ,t 0 +δ] B(t) ≥ u ≤ 4 exp    - u -E sup t∈[t 0 ,t 0 +δ] B(t) 2 Kδ 2β    . ( 25
)
Denote for simplicity Λ = sup t∈[t 0 ,t 0 +δ] B(t). By (25) we obtain

E (Λ) ≤ +∞ 0 P sup t∈[t 0 ,t 0 +δ] | B(t)| > x dx ≤ 4 +∞ 0 exp - [x -EΛ] 2 Kδ 2β dx = 4 √ Kδ β √ 2 ∞ - √ 2/K EΛ δ β e -y 2 /2 dy ≤ 4 √ Kπδ β
It follows that

(u -EΛ) 2 ≥ 1 2 u 2 -(EΛ) 2 ≥ 1 2 u 2 -16Kπδ 2β
Consequently (25) becomes

P sup t∈[t 0 ,t 0 +δ] | B(t)| ≥ u ≤ C exp - u 2 Kδ 2β . ( 26
)
Since H(t 0 ) < β, there exists 0 < ξ < β -H(t 0 ). Consider δ n = n 1/(2(ξ+H(t 0 )-β)) and u n = δ

H(t 0 )+ξ n
. Therefore, according to (26), we have

∞ n=1 P sup t∈[t 0 ,t 0 +δn] | B(t)| ≥ u n ≤ ∞ n=1 exp(- 1 K n) < ∞
It follows from the Borel Cantelli lemma that there exists n 0 = n 0 (ω) such that for all n ≥ n 0 , sup

s∈[t 0 ,t 0 +δn] | B(s)| ≤ δ H(t 0 )+ξ n almost surely. Furthermore, for δ n+1 ≤ δ ≤ δ n , we have almost surely sup s∈[t 0 ,t 0 +δ] | B(s)| ≤ sup s∈[t 0 ,t 0 +δn] | B(s)| ≤ δ H(t 0 )+ξ n ≤ δ H(t 0 )+ξ δ n δ n+1 H(t 0 )+ξ ≤ 2 θ δ H(t 0 )+ξ where θ = H(t 0 ) + ξ 2(β -H(t 0 ) -ξ)
. Hence

lim δ→0 sup t∈[t 0 ,t 0 +δ] | B(t)| (δ/ log | log(δ)|) H(t 0 ) ≤ lim δ→0 δ ξ (log | log(δ)|) H(t 0 ) = 0. a.s.
This completes the proof of the theorem.

Remark 4.2. Observe that by the ideas used to prove the previous theorem many laws of iterated logarithm (LIL) proved for fBm can now be obtained for mBm and with the same constants. For example, we have the following LIL (c.f. [START_REF] Li | Gaussian processes: inequalities, small ball probabilities and applications[END_REF], equation (7.5) for the fBm)

lim sup δ→0 sup s∈[t 0 ,t 0 +δ] |B(s) -B(t 0 )| δ H(t 0 ) (log | log(δ)|) 1/2 = √ 2V H(t 0 ) (B), a.s. ( 27
)
where

V H(t 0 ) ( B) = π H(t 0 )Γ(2H(t 0 )) sin(πH(t 0 )) and V H(t 0 ) (B H ) = 0 -∞ (1 -s) H(t 0 )-1 2 -(-s) H(t 0 )-1 2 2 ds 1/2 + 1 2H(t 0 ) Γ(H(t 0 ) + 1/2) .
Other laws of iterated logarithm and uniform moduli of continuity have been obtained for B by Benassi et al. (1997, Theorem 1.7) via wavelet techniques.

Since the local time is a two parameter process, we precise the meaning of the pointwise Hölder exponent in time, uniformly in space. The pointwise Hölder exponent of sup x∈R L(., x) is defined by

α L (t) = sup α > 0, lim sup δ→0 sup x∈R L(t 0 + δ, x) -L(t 0 , x) δ α = 0 . (28)
Since the fraction above is positive, the lim sup may be replaced by lim or lim inf.

We are now able to prove the following

δ = R L([t 0 , t 0 + δ], x)dx ≤ sup x∈R L([t 0 , t 0 + δ], x) sup s,t∈[t 0 ,t 0 +δ] |B(s) -B(t)| ≤ 2 sup x∈R L([t 0 , t 0 + δ], x) sup s∈[t 0 ,t 0 +δ] |B(s) -B(t 0 )|. (29)
Combining ( 24) and (29) we obtain that there exists a positive constant C such that

lim sup δ→0 sup x∈R L(t 0 + δ, x) -L(t 0 , x) δ 1-H(t 0 ) (log log(δ -1 )) H(t 0 ) ≥ C a.s.
Which with the definition of the pointwise Hölder exponent proves the result.

Asymptotic results

It is well known that techniques for proving limit theorems related to self similar processes use the self similarity of their local times. It is natural to expect the same when dealing with locally asymptotically self similar processes (lass for brevity). Thus it will be of some interest to know if the local times satisfy a kind of lass property.

Lass for local times

The answer to the preceding question is affirmative in the case of the mBm and the result is given by the following Theorem 5.1. Let B be a multifractional Brownian motion, assume that H satisfies the condition (H β ). Then, for any fixed t 0 , the local time of mBm is locally asymptotically self similar with parameter 1 -H(t 0 ), in the sense that for every x ∈ R, the processes

{Y ρ (t, x), t ∈ [0, 1]} ρ>0 defined by Y ρ (t, x) = L t 0 + ρt, ρ H(t 0 ) x + B(t 0 ) -L t 0 , ρ H(t 0 ) x + B(t 0 ) ρ 1-H(t 0 ) ,
converge in law to the local time, { (t, x), t ∈ [0, 1]}, of the fBm B H(t 0 ) with Hurst parameter H(t 0 ), i.e.

lim ρ→0 law{Y ρ (t, x), t ∈ [0, 1]} = law{ (t, x), t ∈ [0, 1]} ( 30
)
where the convergence is in the space of continuous functions endowed with the norm of the uniform convergence.

The motivation to consider the processes Y ρ follows from the occupation density formula and the lass property (1) of mBm. Indeed, according to the occupation density formula, the local time has the following representation

L(t, x) = 1 2π +∞ -∞ e -iux
t 0 e iuB(s) ds du.

Consequently,

Y ρ (t, x) = 1 2πρ 1-H(t 0 ) R e -iy(ρ H(t 0 ) x+B(t 0 )) t 0 +ρt t 0 e iyB(s) dsdy.

Using the change of variables r = s -t 0 ρ and ρ H(t 0 ) y = v, the right hand side of the previous expression becomes

1 2π R e -ivx t 0 exp iv B(t 0 + ρr) -B(t 0 ) ρ H(t 0 ) drdv,
which is the local time of the Gaussian process

B ρ (r) = B(t 0 + ρr) -B(t 0 ) ρ H(t 0 ) , r ∈ [0, 1] .
We need the following lemma for the proof of the finite dimensional convergence. If {X n (t), t ∈ [0, 1]} n≥1 is a family of processes which converges in law in D([0, 1]) to X, then φ I (X n ) converges in law to φ I (X).

Proof. The lemma is a consequence of the continuity of the map X(.) → φ I (X(.)) in J 1 topology on D([0, 1]) at almost all sample points of the process X, which is proved on page 11 in [START_REF] Kesten | A limit theorem related to a new class of self similar Processes[END_REF].

Proof of Theorem 5.1. To prove the convergence in law, we proceed in two steps. First we prove the tightness of the family {Y ρ (t, x), t ∈ [0, 1]} ρ>0 in the space of continuous functions. By using [START_REF] Véhel | Multifractional Brownian motion: definition and preliminary results[END_REF] and [START_REF] Monrad | Small values of Gaussian processes and functional laws of the iterated logarithm[END_REF], for ρ small enough, we obtain

E |Y ρ (t, x) -Y ρ (s, x)| m = E L t 0 + ρt, ρ H(t 0 ) x + B(t 0 ) -L t 0 + ρs, ρ H(t 0 ) x + B(t 0 ) m ρ (1-H(t 0 ))m , ≤ C m |t -s| (1-H(t 0 ))m .
We can take m > 1-H(t 0 ) to get the tightness. Now, we prove the convergence of the finite dimensional distributions of Y ρ , as ρ tends to 0, to those of the local time of the fBm, B H(t 0 ) , with Hurst parameter H(t 0 ). We need to show that for any d ≥ 1, a 1 , ..., a d ∈ R and t 1 , ...., t d ∈ [0, 1], the following convergence holds,

d j=1 a j Y ρ (t j , x) W =⇒ d j=1
a j (t j , x) as ρ → 0.

We will show the convergence of the corresponding characteristic function. More precisely we prove that

E exp iλ d j=1 a j Y ρ (t j , x) -E exp iλ d j=1
a j (t j , x) -→ 0, as ρ → 0.

Let's introduce the following notation :

φ ε,x (X)(t) = 1 ε t 0 1 [x,x+ε] (X(s))ds, I ε,ρ 1 = E exp iλ d j=1 a j Y ρ (t j , x) -E exp iλ d j=1 a j φ ε,x (B ρ )(t j ) , I ε,ρ 2 = E exp iλ d j=1 a j φ ε,x (B H(t 0 ) )(t j ) -E exp iλ d j=1 a j φ ε,x (B ρ )(t j ) , and
I ε 3 = E exp iλ d j=1 a j φ ε,x (B H(t 0 ) )(t j ) -E exp iλ d j=1 a j (t j , x) . Therefore E exp iλ d j=1 a j Y ρ (t j , x) -E exp iλ d j=1 a j (t j , x) ≤ I ε,ρ 1 + I ε,ρ 2 + I ε,ρ 3 . (31)
On the other hand, Y ρ is the local time of B ρ and by using the mean value theorem and the occupation density formula we obtain

I ε,ρ 1 ≤ C max 1≤j≤d E |Y ρ (t j , x) -φ ε,x (B ρ )(t j )| = C max 1≤j≤d E 1 ε x+ε x Y ρ (t j , y)dy -Y ρ (t j , x) (32)
Moreover, since the stochastic process {Y ρ (t, y), y ∈ R}, is almost surely continuous in y for every t, according to the dominated convergence theorem, (32) converges to zero as ε tends to zero independently of ρ.

We deal now with I ε,ρ 2 . Since the family of processes {B ρ (t), t ∈ [0, 1]} ρ>0 converges in distribution to the fBm B H(t 0 ) (t), t ∈ [0, 1] with Hurst parameter H(t 0 ), the second term converges to zero as ρ tends to 0 by Lemma 5.1 . The last term in (31) is treated in a similar way as the first and the proof of the finite dimensional convergence is complete .

Limit theorems

The following result is an immediate consequence of the lass property of mBm. Proposition 5.2. Let B be a multifractional Brownian motion, assume that H satisfies the condition (H β ) and denote by (t, x) the local time of the fBm with Hurst parameter H(t 0 ). Then for every f ∈ L 1 (R), locally Riemann integrable and with compact support such that R f (x)dx = 0, the following convergence in law holds

lim λ→∞ lim ρ→0 + 1 λ 1-H(t 0 ) λt 0 f B(ρs + t 0 ) -B(t 0 ) ρ H(t 0 ) ds = R f (x)dx • (t, 0).
If H is constant, B is a fBm. Being self similar with stationary increments, the term in left hand side of the previous formula is identical in law to 1 λ 1-H λt 0 f (B(s)) ds. We retrieve the result of [START_REF] Kasahara | A limit theorem for occupation times of fractional Brownian motion[END_REF] for the fBm.

Proof. The proof relies on the lass property and the self similarity of fBm. We sketch it for the sake of completeness. Combining the fact that f is locally Riemann integrable, the lass property of mBm and a result in Jeganathan [(2004), Theorem 6 and the comment after it], we obtain = denotes the equality in distribution. Then the result of the theorem follows from the continuity of (t, x, ω) with respect to the space variable.

However the presence of a double limit may be not convenient, and the result may be better if λ and ρ are dependent. We prove the following local limit theorem Theorem 5.3. Let B be a multifractional Brownian motion, assume that H satisfies the condition (H β ). Denote by (t, x) the local time of the fBm with Hurst parameter H(t 0 ) and consider f ∈ L 1 (R) such that : Proof. Using the occupation density formula we obtain

1 ψ(ρ) ρt+t 0 t 0 f B(s) -B(t 0 ) -ρ H(t 0 ) y θ(ρ) ds = R f (x)
L(ρt + t 0 , θ(ρ)x + ρ H(t 0 ) y + B(t 0 )) -L(t 0 , θ(ρ)x + ρ H(t 0 ) y + B(t 0 )) ρ 1-H(t 0 ) dx = L(ρt + t 0 , ρ H(t 0 ) y + B(t 0 )) -L(t 0 , ρ H(t 0 ) y + B(t 0 )) ρ 1-H(t 0 ) × R f (x)dx (34) + R f (x) L(I 0 , θ(ρ)x + ρ H(t 0 ) y + B(t 0 )) -L(I 0 , ρ H(t 0 ) y + B(t 0 )) ρ 1-H(t 0 ) dx,

where we denote in the last expression I 0 = [t 0 , t 0 + ρt] for simplicity. According to Theorem 5.1, the expression (34) converges in distribution to (t, y) × R f (x)dx. It suffices now to prove that (35) converges to 0 in some strong sense. We have

E R f (x)
L(I 0 , θ(ρ)x + ρ H(t 0 ) y + B(t 0 )) -L(I 0 , ρ H(t 0 ) y + B(t 0 )) ρ 1-H(t 0 ) dx ≤ R |f (x)| L(I 0 , θ(ρ)x + ρ H(t 0 ) y + B(t 0 )) -L(I 0 , ρ H(t 0 ) y + B(t 0 ))

ρ 1-H(t 0 ) L 2 (Ω) dx. ( 36
)
Using essentially the same arguments as when proving expression (22) and Remark 3.7 in [START_REF] Boufoussi | On the local time of the multifractional Brownian motion[END_REF], but for the process X(t) = B(t) -B(t 0 ) instead of the mBm, we obtain L(I 0 , θ(ρ)x + ρ H(t 0 ) y + B(t 0 )) -L(I 0 , ρ H(t 0 ) y + B(t 0 )) ρ 1-H(t 0 )

L 2 (Ω)

≤ Ct 1-H(t 0 )(1+ξ) |x| ξ θ(ρ) ρ H(t 0 ) ξ ,

  Theorem 22.1 in Geman and Horowitz (1980), B H has a local time L([a, b], .) ∈ L 2 (R). Let's now prove the second point : According to Boufoussi et al. (2006), we have for any interval [a, b] ⊂ [0, ∞) with small length,

  On the other hand, since the local time exists on the interval [a, b], then according to Geman and Horowitz [(1980), Theorem 22.1 expression (22.3)] the following integral is finite

Proposition 4 . 3 .

 43 Let B be a multifractional Brownian motion, and assume that H satisfies the condition (H β ). The pointwise Hölder exponent α L of sup x∈R L(., x) at t satisfiesα L (t) = 1 -H(t) a.s.Proof. The lower bound was already given inBoufoussi et al. (2006, Corollary 3.6). The upper bound is a consequence of Chung's law of the mBm. Indeed, since the local time vanishes outside the range of B, we obtain

Lemma 5 . 1 . 0 1

 510 Let {X(t), t ∈ [0, 1]} be a stochastic process in the Skorohod space D([0, 1]) and define for a fixed interval I = [a, b] ⊂ R the map φ I (X)(t) = t {X(s)∈I} ds.

1

 1

  + t 0 ) -B(t 0 ) ρ H(t 0 )ds H(t 0 ) (s))ds as ρ → 0 + .Moreover, using the occupation density formula and the self similarity of fBm we obtain1 λ 1-H(t 0 ) λt 0 f (B H(t 0 ) (s))ds (d) = R f (x) (t, λ -H(t 0 ) x)dx,where(d)

Rf

  |f (x)||x| ξ dx < ∞ for some 0 < ξ < 1 2 sup t≥0 H(t) ) -B(t 0 ) -ρ H(t 0 ) (x)dx • (t, y), as ρ → 0 + ,where θ(.) and ψ(.) satisfy ψ(ρ) θ(ρ) = ρ 1-H(t 0 ) , and θ(ρ) ρ H(t 0 ) = o (1) .

  for h small enough. If H is constant, this property is called the one sided strong local nondeterminism; we refer to Monrad and Rootzén (1995 expression (2.2)) and the references therein for definition and applications. The property (22) is satisfied by many other processes with multifractal behavior, such as the Riemann-Liouville mBm {X(t), t ≥ 0} introduced in Lim (2001) by substituting, in the definition of the Riemann-Liouville fBm, the constant Hurst parameter H by a regular function as follows

Research supported by the program Volubilis : "Action intégrée MA/ 06/142".

for ρ small enough and all 0 < ξ < 1 2 sup t≥0 H(t) -1 2 . Hence, (36) is dominated by

This last integral is finite by assumption (33), and then (37) tends to zero as ρ tends to zero. This completes the proof.


Jean-Pierre Demailly
email: jean-pierre.demailly@univ-grenoble-alpes.fr

FANO MANIFOLDS WITH NEF TANGENT BUNDLES ARE WEAKLY ALMOST K ÄHLER-EINSTEIN

Keywords: 2010 Mathematics Subject Classification. 14J45, 14M17, 32C30, 32Q10, 32Q20 Fano manifold, numerically effective vector bundle, rational homogeneous manifold, Campana-Peternell conjecture, Kähler-Einstein metric, closed positive current, regularization of currents, Schauder fixed point theorem

published or not. The documents may come

Fano manifolds with nef tangent bundles are weakly almost Kähler-Einstein

Introduction

Recall that a holomorphic vector bundle E on a projective manifold X is said to be numerically effective (nef) if the line bundle O P(E) (1) is nef on Y = P(E). Clearly, every homogeneous projective manifold X has its tangent bundle T X generated by sections, so T X is nef; morever, by [START_REF] Demailly | Compact complex manifolds with numerically effective tangent bundles[END_REF], every compact Kähler manifold with T X nef admits a finite étale cover X that is a locally trivial fibration over its Albanese torus, and the fibers are themselves Fano manifolds F with T F nef. Hence, the classification problem is essentially reduced to the case when X is a Fano manifold with T X nef. In this direction, Campana and Peternell [START_REF] Campana | Projective manifolds whose tangent bundles are numerically effective[END_REF] conjectured in 1991 that Fano manifolds with nef tangent bundles are rational homogeneous manifolds G/P , namely quotients of linear algebraic groups by parabolic subgroups. Although this can be checked up to dimension 3 by inspecting the classification of Fano 3-folds by Manin-Iskovskii and Mukai, very little is known in higher dimensions. It would be tempting to use the theory of VMRT's developed by J.M. Hwang and N. Mok ([Hwa01], [START_REF] Mok | Geometric structures on uniruled projective manifolds defined by their varieties of minimal rational tangents[END_REF]), since the expected homogeneity property should be reflected in the geometry of rational curves. Even then, the difficulties to be solved remain formidable; see e.g. [START_REF] Mok | On Fano manifolds with nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents[END_REF] and also [MOSWW] for a recent account of the problem.

On the other hand, every rational homogeneous manifold X = G/P carries a Kähler metric that is invariant by a compact real form G R of G (cf. [START_REF] Alekseevskii | Invariant Kähler-Einstein metrics on compact homogeneous spaces[END_REF]), and the corresponding Ricci curvature form (i.e. the curvature of -K G/P ) is then a Kähler-Einstein metric. A stronger condition than nefness of T X is the existence of a Kähler metric on X whose holomorphic bisectional curvature is nonnegative. N. Mok [START_REF] Mok | The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature[END_REF] characterized those manifolds, they are exactly the products of hermitian symmetric spaces of compact type by flat compact complex tori C q /Λ and projective spaces P n j with a Kähler metric of nonnegative holomorphic bisectional curvature. However, hermitian symmetric spaces of compact type are a smaller class than rational homogeneous manifolds (for instance, they do not include complete flag manifolds), thus one cannot expect these Kähler-Einstein metrics to have a nonnegative bisectional curvature in general.

It is nevertheless a natural related question to investigate whether every Fano manifold X with nef tangent bundle actually possesses a Kähler-Einstein metric. Our main observation is the following (much) weaker statement, whose proof is based on regularization techniques for closed positive currents [START_REF] Demailly | Regularization of closed positive currents and Intersection Theory[END_REF], [START_REF] Demailly | Pseudoconvex-concave duality and regularization of currents, Several Complex Variables[END_REF]. We denote here by PC 1,1 (X) the cone of positive currents of bidegree (1, 1) on X and put n = dim C X. Though it is a convex set of infinite dimension, the intersection PC 1,1 (X) ∩ c 1 (X) is compact and metrizable for the weak topology (see § 2), and therefore it carries a unique uniform structure that can be defined by any compatible metric.

Theorem 1.1. Let X be a Fano manifold such that the tangent bundle T X is nef. Then (i) there exists a family of smoothing operators (J ε ) ε∈]0,1] that map every closed positive current T ∈ PC 1,1 (X) ∩ c 1 (X) to a smooth closed positive definite (1, 1)-form

α ε = J ε (T ) ∈ c 1 (X),
in such a way that J ε (T ) converges weakly to T as ε → 0 (uniformly for all T ), and T → J ε (T ) is continuous with respect to the weak topology of currents and the strong topology of C ∞ convergence on smooth (1, 1)-forms ; (ii) for every ε ∈ ]0, 1], there exists a Kähler metric

ω ε on X such that Ricci(ω ε ) = J ε (ω ε ),
in other words ω ε is "weakly almost Kähler-Einstein".

The construction of J ε that we have is unfortunately not very explicit, and we do not even know if J ε can be constructed as a natural linear (say convolution-like) operator. The proof of (ii) is based on the use of the Schauder fixed point theorem. Let us denote by

0 < ρ 1,ε (x) ≤ . . . ≤ ρ n,ε (x)
the eigenvalues of Ricci(ω ε ) with respect to ω ε at each point x ∈ X, and ζ 1,ε , . . . , ζ n,ε ∈ T X,x a corresponding orthonormal family of eigenvectors with respect to ω ε . We know that ( * )

X n j=1 ρ j,ε ω n ε = n X Ricci(ω ε ) ∧ ω n-1 ε = n c 1 (X) n ,
in particular the left-hand side is bounded. Also, as Ricci(ω ε ) -

ω ε = J ε (ω ε ) -ω ε converges weakly to 0, we know that X n j=1 (ρ j,ε -1)ζ * 1,ε ∧ ζ * 1,ε ∧ u
converges to 0 for every smooth (n -1, n -1)-form u on X, hence the eigenvalues ρ j,ε "converge weakly to 1" in the sense that n j=1 (ρ j,ε -1)ζ * 1,ε ∧ ζ * 1,ε converges weakly to 0 in the space of smooth (1, 1)-forms. Therefore, it does not seem too unrealistic to expect that the family (ω ε ) is well behaved in the following sense.

Definition 1.2. We say the family (ω ε ) of weak almost Kähler-Einstein metrics is well behaved in L p norm if

( * * ) p X (ρ n,ε -ρ 1,ε ) p ω n ε converges to 0 as ε tends to 0.
Standard curvature inequalities then yield the following result.

Theorem 1.3. Assume that X is Fano with T X nef, and possesses a family of weak almost Kähler-Einstein metrics that is well behaved in L p norm. Then

(i) if p ≥ 1, T X is c 1 (X)-semistable ; (ii) if p ≥ 2, the Guggenheimer-Yau-Bogomolov-Miyaoka Chern class inequality nc 1 (X) 2 -(2n + 2)c 2 (X) • c 1 (X) n-2 ≤ 0 is satisfied.
The author is indebted to the referee and to Yury Ustinovskiy for pointing out some issues in an earlier version of this work. He expresses warm thanks to Ngaiming Mok, Jun-Muk Hwang, Thomas Peternell, Yum-Tong Siu and Shing-Tung Yau for inspiring discussions or viewpoints around these questions in the last 25 years (or more!).

Proofs

Existence of regularization operators in 1.1 (i). Let us fix once for all a Kähler metric β ∈ c 1 (X). It is well known that K = PC 1,1 (X) ∩ c 1 (X) is compact for the weak topology of currents; this comes from the fact that currents T ∈ K have a uniformly bounded mass

X T ∧ β n-1 = c 1 (X) n .
We also know from [START_REF] Demailly | Regularization of closed positive currents and Intersection Theory[END_REF], [START_REF] Demailly | Pseudoconvex-concave duality and regularization of currents, Several Complex Variables[END_REF] that every such current T admits a family of regularizations T ε ∈ C ∞ (X) ∩ c 1 (X) converging weakly to T , such that T ε ≥ -εβ. Here, the fact that we can achieve an arbitrary small lower bound uses in an essential way the assumption that T X is nef. After replacing T ε by T ′ ε = (1 -ε)T ε + εβ ≥ ε 2 β, we can assume that T ε is a Kähler form. An important issue is that we want to produce a continuous operator J ε (T ) = T ε defined on the weakly compact set K. This is clearly the case when the regularization is produced by convolution, as is done [START_REF] Demailly | Regularization of closed positive currents of type (1, 1) by the flow of a Chern connection, Actes du Colloque en lhonneur de P. Dolbeault[END_REF], but in that case one needs a more demanding condition on T X , e.g. that T X possesses a hermitian metric with nonnegative bisectional curvature (i.e. that T X is Griffiths semipositive). However, the existence of a continuous global operator J ε is an easy consequence of the convexity of K. In fact, the weak topology of K is induced by the L 2 topology on the space of potentials ϕ when writing T = β + dd c ϕ (and taking the quotient by constants) -it is also induced by the L p topology for any p ∈ [1, ∞[, but L 2 has the advantage of being a Hilbert space topology. By compactness, for every δ ∈ ]0, 1] we can then find finitely many currents (T j ) 1≤j≤N (δ) such that the Hilbert balls B(T j , δ) cover K. Let K δ be the convex hull of the family (T j ) 1≤j≤N (δ) and let p δ : K → K δ be induced by the (nonlinear) Hilbert projection L 2 /R → K δ . Since K δ is finite dimensional and is a finite union of simplices, we can take regularizations T j,ε of the vertices T j and construct a piecewise linear operator Jδ,ε : K δ → K ∩ C ∞ (X) + to Kähler forms, simply by taking linear combinations of the T j,ε 's. Then Jδ,ε • p δ (T ) is the continuous regularization operator we need. The uniform weak convergence to T is guaranteed if we take J ε := Jδ(ε),ε with ε ≪ δ(ε) → 0, e.g. with a step function ε → δ(ε) that converges slowly to 0 compared to ε. These operators have the drawback of being non explicit and a priori non linear.

Question 2.1. Is it possible to construct a linear regularization operator J ε with the same properties as above, e.g. by means of a convolution process ?

In fact, such a "linear" regularization operator J ε is constructed in [START_REF] Demailly | Regularization of closed positive currents of type (1, 1) by the flow of a Chern connection, Actes du Colloque en lhonneur de P. Dolbeault[END_REF] by putting T = β + dd c ϕ and J ε (T ) = β + dd c ϕ ε where

ϕ ε (z) = ζ∈T X,z ϕ(exph z (ζ)) χ(|ζ| 2 h /ε 2 ) dV h (ζ)
for a suitable hermitian metrics h on T X , taking exph to be the holomorphic part of the exponential map associated with the Chern connection ∇ h . However, the positivity of J ε (T ) is in general not preserved, unless one assumes that (T X , h) is (say) Griffiths semipositive.

As the relation between nefness and Griffiths semipositivity is not yet elucidated, one would perhaps need to extend the above formula to the case of Finsler metrics. This is eventually possible by applying some of the techniques of [START_REF] Demailly | Pseudoconvex-concave duality and regularization of currents, Several Complex Variables[END_REF] to produce suitable dual Finsler metrics on T X (notice that any assumption that T X ⊗ L is ample for some Q-line bundle L translates into the existence of a strictly plurisubharmonic Finsler metric on the total space of (T X ⊗ L) * {0}, so one needs to dualize).

Use of the Schauder fixed point theorem for 1.1 (ii). By Yau's theorem [START_REF] Yau | On the Ricci curvature of a complex Kähler manifold and the complex Monge-Ampère equation I[END_REF], for every closed (1, 1)-form ρ ∈ c 1 (X), there exists a unique Kähler metric γ(ρ) ∈ c 1 (X) such that Ricci(γ(ρ)) = ρ. Moreover, by the regularity theory of nonlinear elliptic operators, the map ρ → γ(ρ) is continuous in C ∞ topology. We consider the composition

γ • J ε : K → K ∩ C ∞ (X) + → K ∩ C ∞ (X) + ⊂ K, T → ρ = J ε (T ) → γ(ρ).
Since J ε is continuous from the weak topology to the strong C ∞ topology, we infer that γ • J ε is continuous on K in the weak topology. Now, K is convex and weakly compact, therefore γ • J ε must have a fixed point T = ω ε by the theorem of Schauder. By construction ω ε must be a Kähler metric in c 1 (X), since γ • J ε maps K into the space of Kähler metrics contained in c 1 (X). This proves Theorem 1.1.

Semistability of T X (1.3 (i)). Let F ⊂ O(T X ) be a coherent subsheaf such that O(T X )/F is torsion free. We can view F as a holomorphic subbundle of T X outside of an algebraic subset of codimension 2 in X. The Chern curvature tensor Θ T X ,ωε = i 2π ∇ 2 satisfies the Hermite-Einstein condition

Θ T X ,ωε ∧ ω n-1 ε = 1 n ρ ε ω n ε ,
where ρ ε ∈ C ∞ (X, End(T X )) is the Ricci operator (with the same eigenvalues ρ j,ε as Ricci(ω ε )).

It is well known that the curvature of a subbundle is always bounded above by the restriction of the full curvature tensor, i.e. Θ F ,ωε ≤ Θ T X ,ωε|F (say, in the sense of Griffiths positivity, viewing the curvature tensors as hermitian forms on T X ⊗ F ). By taking the trace with respect to ω ε , we get

Θ F ,ωε ∧ ω n-1 ε ≤ 1 n ρ ε|F ω n ε .
By the minimax principle, if r = rank F , the eigenvalues of ρ ε|F are bounded above by

ρ n-r+1,ε ≤ . . . ≤ ρ n,ε , hence X c 1 (F ) ∧ ω n-1 ε = X tr F Θ F ,ωε ∧ ω n-1 ε ≤ 1 n X r j=1 ρ n-r+j,ε ω n ε .
The left hand side is unchanged if we replace ω ε by β ∈ c 1 (X), and our assumption ( * * ) p for p = 1 (cf. Definition 1.2) implies

X c 1 (F ) ∧ β n-1 ≤ lim ε→0 1 n X r j=1 ρ n-r+j,ε ω n ε = lim ε→0 r n 2 X n j=1 ρ j,ε ω n ε = r n X c 1 (X) ∧ β n-1
by ( * ). This means that T X is c 1 (X)-semistable.

Chern class inequality (1.3 (ii)). Let us write

Θ T X ,ωε = (θ αβ ) 1≤α,β≤n = j,k θ αβjk dz j ∧ dz k 1≤α,β≤n
as an n×n matrix of (1, 1)-forms with respect to an orthonormal frame of T X that diagonalizes the Ricci operator ρ ε (with respect to the metric ω ε ). A standard calculation yields

2 n c 1 (X) 2 h -(2n + 2) c 2 (X) ∧ ω n-2 ε (n -2)! = - α =β,j =k |θ ααjk -θ ββjk | 2 -(n + 1) α,β,j,k, pairwise = |θ αβjk | 2 -(4n + 8) α =j<k =α |θ ααjk | 2 -4 α =j |θ αααj | 2 - α =β =j =α |θ ααjj -θ ββjj | 2 - α =β |θ αααα -2θ ααββ | 2 + n α ρ 2 α,ε - α ρ α,ε 2 ω n ε
where all terms in the summation are nonpositive except the ones involving the Ricci eigenvalues ρ α,ε . However, the assumption ( * * ) p for p ≥ 2 implies that the integral of the difference appearing in the last line converges to 0 as ε → 0. Theorem 1.3 is proved.

Our results lead to the following interesting question.

Question 2.2. Assuming that a "sufficiently good" family of regularization operators J ε is used, can one infer that the resulting family (ω ε ) of fixed points such that Ricci(ω ε ) = J ε (ω ε ) (or some subsequence ) is well behaved in L 1 , resp. L 2 norm ?

Remark 2.3. A strategy to attack the Campana-Peternell conjecture could be as follows.

The first step would be to prove that H 0 (X, T X ) = 0 (if dim X > 0). Assume therefore H 0 (X, T X ) = 0 and try to reach a contradiction. It is well known in that case that there are much fewer obstructions to the existence of Kähler-Einstein metrics, for instance one has the Bando-Mabuchi uniqueness theorem [START_REF] Bando | Uniqueness of Einstein Kähler metrics modulo connected group actions, Algebraic geometry[END_REF] and (obviously) the vanishing of the classical Futaki invariant [START_REF] Futaki | An obstruction to the existence of Einstein Kähler metrics[END_REF]; hence we could expect our family of weakly almost Kähler-Einstein metrics to converge to a genuine Kähler-Einstein metric. But then the resulting known Chern class inequalities (the ones of Theorem 1.3 (ii) combined with the Fulton-Lazarsfeld inequalities [START_REF] Fulton | Positive polynomials for ample vector bundles[END_REF], [START_REF] Demailly | Compact complex manifolds with numerically effective tangent bundles[END_REF]) might help to contradict H 0 (X, T X ) = 0, e.g. by the Riemann-Roch formula or by ad hoc nonvanishing theorems such as the generalized hard Lefschetz theorem [START_REF] Demailly | Pseudo-effective line bundles on compact Kähler manifolds[END_REF]. Now, the existence of holomorphic vector fields on X implies that X has a non trivial group of automorphisms H = Aut(X), and one could then try to apply induction on dimension on a suitable desingularization Y of X // H, if Y is not reduced to a point and one can achieve Y to have T Y nef.