Marie-Paule Muller

come

ABOUT THE ERD…S PAIRS

Marie-Paule M ULLER Abstract 1) A simpler and more transparent proof is given for earlier results by B. Bollobas and J. Jin 1) A simpler and more transparent proof is given for earlier results by B. Bollobas and J. Jin about the partitions of an interval of integers into two parts, related to a Ramsey-type problem of about the partitions of an interval of integers into two parts, related to a Ramsey-type problem of Erdšs.

2) A new construction allows to improve some of their results for the problem of Erdšs in the 2) A new construction allows to improve some of their results for the problem of Erdšs in the more general case of partitions into an arbitrary number of parts.

RŽsumŽ 1) On donne une preuve plus simple et plus transparente de rŽsultats de B. Bollobas et J. Jin 1) On donne une preuve plus simple et plus transparente de rŽsultats de B. Bollobas et J. Jin rŽpondant ˆ un probl•me de P. Erdšs sur les partitions en deux parties dÕun intervalle dÕentiers.

2) Une nouvelle construction permet dÕamŽliorer leurs rŽsultats pour le probl•me dÕErdšs dans le 2) Une nouvelle construction permet dÕamŽliorer leurs rŽsultats pour le probl•me dÕErdšs dans le cas plus gŽnŽral de partitions en un nombre quelconque de parties.

AMS Subject Classification : 11 B 75 (05 A 16, 05 D 99)

Introduction.

A few years ago, Paul Erdšs posed a problem about the partitions of a sequence of integers A few years ago, Paul Erdšs posed a problem about the partitions of a sequence of integers in the following terms.

Let us consider two natural integers m < n , and a partition into two parts A Let us consider two natural integers m < n , and a partition into two parts A if there are, in at least one part, some (distinct) elements whose sum is if there are, in at least one part, some (distinct) elements whose sum is equal to n . If every partition of [m, n[ is an Erdšs partition, (m, n) is an Erdšs pair. If (m, n) is an Erdšs pair, then (mÕ, n) for mÕ < m , and also (km, kn) for every positive If (m, n) is an Erdšs pair, then (mÕ, n) for mÕ < m , and also (km, kn) for every positive integer k , are clearly Erdšs pairs (it is enough to consider induced partitions). On the other integer k , are clearly Erdšs pairs (it is enough to consider induced partitions). On the other hand, (1, n) The first aim of the present note is to give a simpler and more ÒvisualÓ proof of these results (1) .

Moreover, the method which is used here allows us to determine easily the exact bound beyond Moreover, the method which is used here allows us to determine easily the exact bound beyond ___________________________ Some sections (in particular lemma 1 and its corollary 1 ) are presented so that they can be Some sections (in particular lemma 1 and its corollary 1 ) are presented so that they can be applied to the more general case of partitions into p parts of [m,n[ . For l applied to the more general case of partitions into p parts of [m,n[ . For

l p p (n) (n)
and its inferior and its inferior limit l p defined in the obvious way, the conjecture is that l p = 1/ 2p (see [B-J]).

The second aim of this note is to present (in section 4 ) a new construction which gives support

The second aim of this note is to present (in section 4 ) a new construction which gives support to this conjecture and allows us to improve the related results of [B-J] : we show that l to this conjecture and allows us to improve the related results of [B-J] : we show that l p p (n) < (n) < n/ 2p for every n , hence l p ≤ 1/ 2p .

The subject discussed here is linked to the general theme of the Ramsey-type problems (see The subject discussed here is linked to the general theme of the Ramsey-type problems (see [G-R-S] ) in additive number theory. Among the classical results to which it is related, let us which is not of Erdšs.

which is not of Erdšs.

Hence we exclude the partitions for which some part certainly contains elements whose sum is Hence we exclude the partitions for which some part certainly contains elements whose sum is n . It is natural to exclude first those for which some A n . It is natural to exclude first those for which some A i i contains contains two two elements whose sum is elements whose sum is n . We can easily ÒseeÓ these partitions : let us display the numbers m, É, n-1 so that two n . We can easily ÒseeÓ these partitions : let us display the numbers m, É, n-1 so that two numbers whose sum is n face each other :

if n = 2p+1 : m m+1 É k É p-1 p n-1 É n-m n-m-1 É n-k É p+2 p+1 if n = 2p+2 : m m+1 É k É p-1 p p+1 n-1 É n-m n-m-1 É n-k É p+3 p+2
Thus, the partitions first excluded are those for which some complete column is contained in a Thus, the partitions first excluded are those for which some complete column is contained in a part A part A i i

. Notice that the elements in the incomplete columns on the left cannot occur in a sum . Notice that the elements in the incomplete columns on the left cannot occur in a sum equal to n , hence no matter the parts to which they belong.

The display chosen above for the numbers [m, n[ also enables us to locate qualitatively three The display chosen above for the numbers [m, n[ also enables us to locate qualitatively three elements or more whose sum is n : we can see immediately that at most one of these elements is elements or more whose sum is n : we can see immediately that at most one of these elements is in the second row, and that this element must be on the right of the others : * * * *

When the 2-partition is not of Erdšs, if two numbers u and v are in the same part A When the 2-partition is not of Erdšs, if two numbers u and v are in the same part A n-u-v is distinct from u and v , then n-u-v is not in A n-u-v is distinct from u and v , then n-u-v is not in A i i . In particular, if u+v is in a column . In particular, if u+v is in a column which is complete and distinct from those of u and v , then u+v is also in A which is complete and distinct from those of u and v , then u+v is also in A i i : otherwise, the : otherwise, the element n-u-v which faces it would be in A i , and this is impossible.

These remarks allow us to prove easily that, for example, (1, n) is an Erdšs pair if (and only if) n ³ 12 ; or that (2, n) is an Erdšs pair for n = 15 or n ³ 17 . If the partition is not of Erdšs, we know that the sequence [p, p+1, p+2] necessarily contains If the partition is not of Erdšs, we know that the sequence [p, p+1, p+2] necessarily contains a flip, since p faces either p+1 or p+2 (according to the parity of n ), and two numbers with a flip, since p faces either p+1 or p+2 (according to the parity of n ), and two numbers with sum n do not belong to the same part.

Flips and pincers.

Definition

This remark can be generalized in the following way.

Lemma 1 Lemma 1 .

.

For j = 2 , 3 , 4 , ... with j < n/m , there is, surrounding n/j , a sequence of j+1

For j = 2 , 3 , 4 , ... with j < n/m , there is, surrounding n/j , a sequence of j+1 consecutive integers which contains j elements whose sum is n . With the convention that the consecutive integers which contains j elements whose sum is n . With the convention that the smallest number of the sequence must figure in that sum, this smallest number is equal to the smallest number of the sequence must figure in that sum, this smallest number is equal to the integer part of n/j -(j-1)/2 .

Proof Proof

. Denote by q the integer part of n/j (resp. n/j + . Denote by q the integer part of n/j (resp. n/j +

1/2 1/2
) if j is odd (resp. even). The ) if j is odd (resp. even). The sequence of length j+1 which is centered in q+1/2 (resp. q ) contains j integers whose sum sequence of length j+1 which is centered in q+1/2 (resp. q ) contains j integers whose sum is n ; the element which must be omitted in the sum is determined by the class of n mod. j .

is n ; the element which must be omitted in the sum is determined by the class of n mod. j .

Let us specify that the last element (i.e. the (j+1)

Let us specify that the last element (i.e. the (j+1) th th

) of the sequence has to be omitted when n

) of the sequence has to be omitted when n

(resp. n + j/ (resp. n + j/ 2
) is a multiple of j , with j odd (resp. even) : thus, in that case, we have j

) is a multiple of j , with j odd (resp. even) : thus, in that case, we have j consecutive integers whose sum is n .

Notation

Notation

. Let us denote by S . Let us denote by S j j the sequence of length j+1 or j (according to the case) the sequence of length j+1 or j (according to the case) which has been described in the proof of lemma 1 : it is the shortest sequence which contains j which has been described in the proof of lemma 1 : it is the shortest sequence which contains j elements with sum n . Example ( j = 2 ) : S 2 = [p, p+1, p+2] if n is even, and S 2 = [p, p+1] if n is odd. i) x i and y i belong to a same part A i (with distinct A 1 and A 2 ) ii) x 1 + y 1 = x 2 + y 2 £ n-m iii) the column of x i + y i is distinct from those of x 1 , y 1 , x 2 , y 2 .

ÓPincers extractionÒ of Erdšs 2-partitions.

In this section we shall only consider

In this section we shall only consider

2-partitions 2-partitions

(partitions into two parts only) of [m,n[ . In (partitions into two parts only) of [m,n[ . In this case, it is enough to find two Òwell placedÓ flips in order to make sure that the partition is of this case, it is enough to find two Òwell placedÓ flips in order to make sure that the partition is of Erdšs. Moreover, the sequences which contain a flip near to n/j (cf. lemma 1 ) can be Erdšs. Moreover, the sequences which contain a flip near to n/j (cf. lemma 1 ) can be shortened.

Lemma 2 Lemma 2 . . Given a 2-partition which is not of Erdšs, if two numbers x and y are in a same Given a 2-partition which is not of Erdšs, if two numbers x and y are in a same part A part A i i and if x+y is in a column which is complete and distinct from those of x and y , then and if x+y is in a column which is complete and distinct from those of x and y , then x+y is also in A i . Proof . The part A i does not contain the element n-x-y , thus it contains x+y . For the 2-partitions, the following lemma gives a sequence which is shorter than S For the 2-partitions, the following lemma gives a sequence which is shorter than S 4 , so it will , so it will allow us to situate more precisely a flip near n/4 . Notation . Denote by V the integer part of (n-3)/4 .

Proof

Proof

. In fact, V is the integer part of (p-1)/2 (let us recall that p , which is the smallest . In fact, V is the integer part of (p-1)/2 (let us recall that p , which is the smallest element of S element of S 2 , is itself the integer part of (n-1)/2 ). Hence, following lemma 1, [V, V+1, V+2] , is itself the integer part of (n-1)/2 ). Hence, following lemma 1, [V, V+1, V+2] contains two elements whose sum is p . As V must be one of these two elements (either p = contains two elements whose sum is p . As V must be one of these two elements (either p = V+(V+1), or p = V+(V+2) ), we can also find in [V, V+1, V+2] two elements whose sum is V+(V+1), or p = V+(V+2) ), we can also find in [V, V+1, V+2] two elements whose sum is p+1 .

Suppose now that the partition is not of Erdšs, the described sequence being contained in a Suppose now that the partition is not of Erdšs, the described sequence being contained in a part. If n is odd, we have p+1= n-p ; as p and p+1 are in the same part by lemma 2 , we part. If n is odd, we have p+1= n-p ; as p and p+1 are in the same part by lemma 2 , we arrive to a contradiction. If n is even, we have p+2 = n-p ; as p+2 is equal either to arrive to a contradiction. If n is even, we have p+2 = n-p ; as p+2 is equal either to (V+1)+(V+2) or to (V+1)+(V+3) (according to the parity of p ), lemma 2 leads once again to (V+1)+(V+2) or to (V+1)+(V+3) (according to the parity of p ), lemma 2 leads once again to a contradiction.

The bound n

The bound n ³ ³ traduces the condition that the column of p must be disjoint from the 11 traduces the condition that the column of p must be disjoint from the considered sequence, in order to apply lemma 2 .

Remark 1 Remark 1

. As it is formulated, this proof may also be viewed as the starting point of a . As it is formulated, this proof may also be viewed as the starting point of a recursive process which could be useful for the study of the 3-partitions (of course, lemma 2 recursive process which could be useful for the study of the 3-partitions (of course, lemma 2 must be replaced by the corresponding lemma). deprived of its first and last elements : in this case, we see that the integer part of (n-6)/4 is deprived of its first and last elements : in this case, we see that the integer part of (n-6)/4 is equal to V-1 . But in case n = 2 mod. 4 , lemma 4 does not improve the result of lemma 1 : the equal to V-1 . But in case n = 2 mod. 4 , lemma 4 does not improve the result of lemma 1 : the integer parts of (n-6)/4 and (n-3)/4 are equal, thus [V , V+1 , V+2 , V+3] = S 4 in this case. The following lemma may be seen as a complement to lemma 4 . Picking out a few other

The following lemma may be seen as a complement to lemma 4 . Picking out a few other cases which force the 2-partition to be of Erdšs, it will allow us to precise the values of n such cases which force the 2-partition to be of Erdšs, it will allow us to precise the values of n such that the pair (V, n ) is of Erdšs.

Proof

Proof

. Similar to the proof of lemma 4 . Apply lemma 2 , after having noticed that two elements . Similar to the proof of lemma 4 . Apply lemma 2 , after having noticed that two elements which face each other (either p and n-p , or p-1 and n-p+1 , according to the case) can be which face each other (either p and n-p , or p-1 and n-p+1 , according to the case) can be obtained as sums of elements belonging to the same part.

Theorem . For n ³ 56 , every 2-partition of [m,n[ with m £ V is of Erdšs. ( n = 55 , m = V )
There is a 2-partition of [13,55[ which is not of Erdšs. For every n , there is a 2-partition of [V+1, n [ which is not of Erdšs.

Proof of the theorem. Let us suppose that the partition induced on [V, n[ is not of Erdšs.

By lemma 4 and corollary 1, there is at least one flip in each of the sequences SÕ By lemma 4 and corollary 1, there is at least one flip in each of the sequences SÕ 4 and S and S 3 . We . We shall distinguish two cases, depending on their types.

Let U be the smallest element of S Let U be the smallest element of S 3 . Recall that V is the integer part of (n-3)/4 , and that . Recall that V is the integer part of (n-3)/4 , and that U is the integer part of n/3 -1 (lemma 1 ).

Case 1. The sequences SÕ 4 and S 3 contain respectively flips with different types.

First, let us make sure that SÕ First, let us make sure that SÕ

(n = 0 mod.3 ) V+2U+6 < n (n-3)/4 + 2n/3+4 < n : n > 39 112 Ò V+2U+5 < n Ò Ò 112 2221 (n ¹ 0 mod.3 ) V+2U+7 < n (n-3)/4 + 2(n-1)/3+5 < n : n > 43 1112 (n=2 mod.4 ) Ò V+2U+8 < n (n-6)/4 + 2(n-1)/3+6 < n : n > 46
Case 2. The sequences SÕ 4 et S 3 do not contain flips of distinct types.

In this case, there is one (unique) flip in each sequence ; let us say, of type 12 . The fact that In this case, there is one (unique) flip in each sequence ; let us say, of type 12 . The fact that the two sequences are disjoint does no longer ensure the existence of pincers : it is enough to the two sequences are disjoint does no longer ensure the existence of pincers : it is enough to concatenate SÕ 4 of type 1112 with S 3 of type 1222 in order to be convinced of that.

First, notice that SÕ First, notice that SÕ

i) i)

If there is an element of type 2 between the two flips, then the flip of SÕ If there is an element of type 2 between the two flips, then the flip of SÕ 4 gives us the first gives us the first two elements of pincers whose fourth element is, at most, equal to U : therefore, the pincers two elements of pincers whose fourth element is, at most, equal to U : therefore, the pincers condition iii) is evidently fulfilled. Here are some examples : [1112] 2 [1222] , [1122][1222] , [112] 221.. [1112] .

ii) ii)

If there is no element of type 2 between the two flips and if SÕ If there is no element of type 2 between the two flips and if SÕ (for n = 2 mod.4 ) has not even to be studied anymore : it is (for n = 2 mod.4 ) has not even to be studied anymore : it is of Erdšs by lemma 5 . The previous analysis may also be exploited in order to construct a non-Erdšs partition of The previous analysis may also be exploited in order to construct a non-Erdšs partition of [13,55[ . As

Non-Erdšs p-partitions.

The following construction allows us to obtain very simply, for every integer n and whatever

The following construction allows us to obtain very simply, for every integer n and whatever the number of parts p , a non-Erdšs p-partition of [m, n[ , provided that m ³ n / 2p .

Notice that the example for p=3 and n divisible by 60 of [B-J] is a product of a different Notice that the example for p=3 and n divisible by 60 of [B-J] is a product of a different process (besides it has to be slightly modified in order to be conclusive).

We begin with the study of p-partitions of an interval [ m/n , 1[ of real numbers .

Proof.

Proof.

The parts A

The parts A 1 1 , ..., A , ..., A p p are constructed in order for the interior of each of these parts to are constructed in order for the interior of each of these parts to be stable by addition. Moreover, the parts to which the boundary points will be attributed must be stable by addition. Moreover, the parts to which the boundary points will be attributed must be chosen adequately. The choice which is made below will allow to improve (very slightly) the be chosen adequately. The choice which is made below will allow to improve (very slightly) the bound m when the proposition will be applied to the p-partitions of the integers [m, n[ . Let us define the following intervals, where i ( 1 Let us define the following intervals, where i ( 1

£ £ i i £ £
p ) is a counter for the parts, and k p ) is a counter for the parts, and k ( k ³ 0 ) is a counter for the connected components of each part : -for k = 0 ,

J i 0 = [ 1 2p , 1 2p-1 ] if i = 1 ] 1 2p-i+1 , 1 2p-i ] if 2 £ i £ p -for k ³ 1 with 2 k £ 2p-i , J i k = ] 2 k 2p-i+1 , 2 k 2p-i ] if 2 k 2p-i £ 1 2 ] 2 k 2p-i+1 , 2 k 2p-i [ if 2 k 2p-i+1 = 1 2 [ 2 k 2p-i+1 , 2 k 2p-i [ if 2 k 2p-i+1 > 1 2 Let A Let A i i
be the union of the J be the union of the J Of course, 1 is then a boundary-point too, and the partition is well-defined.

ii) Clearly, for elements taken in J i 0 and not all of them identical, the sum is different from 1 .

iii) From i), ii) and from the choice which has been made at the boundary-points of the intervals iii) From i), ii) and from the choice which has been made at the boundary-points of the intervals J i k , we deduce that the sum of elements of A i , all of them being distinct, is different from 1 .

The only case where the checking is not trivial is the one with all the elements taking the form 2 k 2p-i (i.e. ends of intervals). Denoting by k i the largest integer such that

2 k i 2p-i £ 1 2
(in fact, k i = K or k i = K-1 ) , the sum of the given elements is at most

1 2p-i 2 k i +1 -1 £ 1 2p-i 2p -i -1 < 1
Notice that the case where all the elements are at the beginning of intervals is immediate For some values of n mod. 2p , this bound obtained for m can be slightly improved (one For some values of n mod. 2p , this bound obtained for m can be slightly improved (one unit less), as a result of the discretisation : among the reals, the only elements which are to be unit less), as a result of the discretisation : among the reals, the only elements which are to be taken into account are the multiples of 1/n , and they have to be situated in relation to 1/ 2p .

For p = 2 for example :

-if n = 0 mod.4 or n = 3 mod.4 , the bound which has been given by proposition 2 is -if n = 0 mod.4 or n = 3 mod.4 , the bound which has been given by proposition 2 is optimal : the condition m ³ n / 4 is equivalent to m ³ V+1 (cf. ¤3.)

-if n = 1 mod.4 , the smallest element of the interval J -if n = 1 mod.4 , the smallest element of the interval J 1 0 is (n+3)/4 , that is V+2 in the is (n+3)/4 , that is V+2 in the considered case ; it is straightforward to check that the partition remains non-Erdšs if the considered case ; it is straightforward to check that the partition remains non-Erdšs if the

  m, n-1[ = {m, m+1, ..., n-1} of all the integers from m to n-1 . We shall call it the sequence [m, n-1[ = {m, m+1, ..., n-1} of all the integers from m to n-1 . We shall call

  is an Erdšs pair if (and only if) n hand, (1, n) is an Erdšs pair if (and only if) n ≥ ≥ (the remarks of section 1 below enable us 12 (the remarks of section 1 below enable us to check this easily). So we have, for fixed n to check this easily). So we have, for fixed n /n ) when n tends to infinity. The (n)/n ) when n tends to infinity. The problem posed by Paul Erdšs in 1992 was to determine the number l problem posed by Paul Erdšs in 1992 was to determine the number the value of l 2 (n) for n large enough (however without precising a bound).

1

  Had I known about the paper of B. Bollobas et G. Jin, I would certainly not have started this work ; so, many Had I known about the paper of B. Bollobas et G. Jin, I would certainly not have started this work ; so, many thanks to my colleague D. Dumont for having told me about the Erdšs problem before, and about the reference thanks to my colleague D. Dumont for having told me about the Erdšs problem before, and about the reference [B-J] afterwards ! which the general expression of l 2 (n) is valid.

  [G-R-S] ) in additive number theory. Among the classical results to which it is related, let us mention those of Davenport [D], Erdšs and Heilbronn [E-H]. More recent results have been mention those of Davenport [D], Erdšs and Heilbronn [E-H]. More recent results have been established by Bollobas, Erdšs and Jin ([B-E-J] , [B-J 2]). I am very grateful to Vilmos Komornik for his encouragement and his interest in this work. 1. A few simple remarks. Let us search for a 2-partition (two parts only) [m, n [ = A Let us search for a 2-partition (two parts only) [m, n [

Definition.

  Given a partition of [m,n[ , let us call a . Given a partition of [m,n[ , let us call a flip flip a sequence of two consecutive a sequence of two consecutive numbers which do not belong to the same part.

  ,n[ is equipped with a partition which is not of Erdšs, then in each If [m,n[ is equipped with a partition which is not of Erdšs, then in each sequence S j contained in [m,n[ there is at least one flip.Let us notice that asymptotically, i.e. for n large enough with the ratio n/m fixed, these Let us notice that asymptotically, i.e. for n large enough with the ratio n/m fixed, these sequences are contained in[m,n[ and are disjoint. of [m,n[ , let us say that four elements x Given a partition of [m,n[ , let us say that four elements x

Lemma 3 .

 3 If a 2-partition contains pincers, then it is of Erdšs.ProofProof. Consider a set of pincers (x . Consider a set of pincers (whose sum is n in its part. By condition iii), , a triple whose sum is n in its part. By condition iii), these three elements are distinct.The following task is to obtain effectively pincers from adequate flips. In fact, two flips whichThe following task is to obtain effectively pincers from adequate flips. In fact, two flips which are respectively situated near n/3 and n/4 will suffice (for n large enough : n are respectively situated near n/3 and n/4 will suffice (for n large enough : n condition iii) ), in fact in a column which is further right, as x (cf. condition iii) ), in fact in a column which is further right,

.

  Given a 2-partition of[V, n[ , if the sequence [V , V+1 , 11 . Given a 2-partition of[V, n[ , if the sequence [V , V+1 , V+2 , V+3] is contained in a part, then the partition is of Erdšs. In the case n V+2 , V+3] is contained in a part, then the partition is of Erdšs. In the case n ¹ ¹ mod.4 , it is 2 mod.4 , it is enough for [ V , V+1 , V+2 ] to be contained in a part.

  length 3 or 4 (according to the case) which has the sequence of length 3 or 4 (according to the case) which has been described in lemma 4 .DefinitionDefinition . A flip [x, x+1] is said to be of . A flip [x, x+1] is said to be of type 12 type 12 if x is an element of the part A if x is an element of the part A sequence is the corresponding sequence of the of a given sequence is the corresponding sequence of the indices of the parts which contain the given elements.

  of the first four or five (according to the case) elements of Consider the sequence of the first four or five (according to the case) elements of[V, n[ , with n > 18 . If its type falls within one of the following cases, then the 2-partition is of [V, n[ , with n > 18 . If its type falls within one of the following cases, then the 2-partition is of Erdšs : for n = 0 mod.4 , type 11#1 for n = 1 mod.4 , type 11#1 , 1#11 or 11##1 for n = 2 mod.4 , type 11##1 or 1#1#1 where # is a generic character for 1 or 2 .

  that V+3 £ U . If n = 2 mod.4 , then V = (n-6)/4 , hence V+4 £ U .Hence, the two flips provide four elements x Hence, the two flips provide four elements only remains to verify condition iii), in order for n -( x pincers. It only remains to verify condition iii), in order for n -The most ÒunfavourableÓ situation is the one in which x < n . The most ÒunfavourableÓ situation is the one in which as possible, are as large as possible, the flips being situated at the ends of the sequences : let us say SÕ the flips being situated at the ends of the sequences : let us say or 221, according to the class of n mod.12 . The pincers of type 2221 or 221, according to the class of n mod.12 . The pincers condition iii) is expressed by an inequation involving V , U and n . Now, the expressions of condition iii) is expressed by an inequation involving V , U and n . Now, the expressions of V and U as functions of n provide a bound which is a sufficient condition : n V and U as functions of n provide a bound which is a sufficient condition : n ³ ³ is valid no 47 is valid no matter the class of n mod.12 . Let us summarize this little study in the following table :

  V+5 £ U , according to the length of SÕ 4 ).

.

  the configuration is of type[112] contiguous, then the configuration is of type[112] .2] . Four elements are likely to form [1..2] . Four elements are likely to form pincers (of type 2,1,1,2 , now) . The most ÒunfavourableÓ situation is the one in which the flip pincers (of type 2,1,1,2 , now) . The most ÒunfavourableÓ situation is the one in which the end of the sequence : the extremal elements of the pincers are the last elements of is at the end of the sequence : the extremal elements of the pincers are the last elements It is enough to adapt the study which has been made in case 1 . It may be noticed . It is enough to adapt the study which has been made in case 1 . It may be noticed that the configuration [

  non-Erdšs p-partition of the real interval [ 1 / 2p , 1[ : none There exists a non-Erdšs p-partition of the real interval [ 1 / 2p , 1[ : none of the p parts contain a finite subfamily whose sum is equal to 1 .

  short, if we leave aside the boundary points, a 0 ) . In short, if we leave aside the boundary points, a partition of ] 1/ 2p , 1/p [ is extended homothetically to ] 1/ 2p , 1[ . An elementary checking may (and has to) be made : 1/2 is a boundary-point. i) There are a value k = K , and a value i = I desired value of i (with 1 £ i £ p ).

  , then there exists a non-Erdšs partition into p parts of the 1/ 2p , then there exists a non-Erdšs partition into p parts of the interval of natural numbers[m, n[ .

  a consequence of both case 2 above and lemma 2, the partition is (and must be) of the following type :

	It is easy to give a 2-partition of [V+1, n[ which is not of Erdšs. But a more general
		13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
	type:	1	1	2	1	1	1	1	2	2	2	2	2	2	2	1
		... 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28
	type: ... 2	2	1	2	2	2	2	1	1	1	1	1	1	1	2
	It is easy to give a 2-partition of [V+1, n[ which is not of Erdšs. But a more general
	construction will be presented in the next section.

preceding integer (n-1)/4 is joined to A 1 .

-if n = 2 mod.4 , the smallest element of the interval J -if n = 2 mod.4 , the smallest element of the interval J 1 1 0 0 is V+2 too . As before, the is V+2 too . As before, the preceding integer, (n-2)/4 , can be joined to A 1 .

Hence we have, for every value n , a non-Erdšs 2-partition of [V+1, n[ .

References.

[B-J] B. Bollobas, G. Jin ,


M Émery

W Schachermayer

A REMARK ON TSIRELSON'S STOCHASTIC DIFFERENTIAL EQUATION

Tsirelson's stochastic differential equation is called "celebrated and mysterious" by Rogers and Williams 15 . This note aims at making it a little more celebrated and a little less mysterious.

Using a deterministic time-change, we translate the study of Tsirelson's equation into the study of "eternal" Brownian motion on the circle. This allows us to show that the filtration generated by any solution of Tsirelson's equation is also generated by some Brownian motion (which, however, cannot be the Brownian motion driving the equation, because the equation has no strong solution).

Introduction

The so-called innovation problem is a remarkable phenomenon in the theory of filtered probability spaces; see for instance § 5.4 of von Weizsäcker 23 . When the answer to the innovation problem is negative, some kind of creation of information occurs. This may happen in discrete, or continuous time (by discrete time, we refer to processes parametrized by ).

In discrete time, a paradigmatic example, well-known in ergodic theory, is obtained from an independent sequence (U n ) n∈¡ of random variables uniformly distributed on {-1, 1}. Call (G n ) n∈¡ the natural filtration of U and (F n ) n∈¡ the filtration generated by the "innovations" V n = U n-1 U n (= U n /U n-1 ). Both G -∞ and F -∞ are trivial (by Kolmogorov's zero-one law), and both filtrations have V n as innovations: V n is independent of F n-1 (resp. G n-1 ), and, together with it, generates F n (resp. G n ). Yet, the filtration G strictly contains F , because each random variable U n is independent of F ∞ . More precisely, in the filtration F, the process U is observed up to multiplication by an unknown factor ±1 only. This example has been independently discovered by several authors; the earliest reference we know is Vershik 21 , see also von Weizsäcker 23 and Exercise 4.12 in Williams 24 . In this example, the filtrations G and F are isomorphic, because they are generated by processes (U n ) n∈¡ and (V n ) n∈¡ with the same law.

In continuous time, there is an example very similar to the above discrete-time example. Replace the discrete time by the line ¢ and the state space {-1, 1} by the circle £ = ¢ / . Consider a £ -valued Brownian motion (U t ) t∈¤ : each U t is uniformly distributed on £ and U moves Brownianly. For each t ∈ ¢ , call G t the σ-algebra generated by all the past positions U s for s ¥ t and F t the σ-algebra generated by all the past innovations U t -U s for s ¥ t. As above, each U t is independent of F ∞ ; this should be rather obvious on an intuitive level (and a formal proof will be provided below). As above, both filtrations G (generated by U ) and F have trivial tail σ-fields at -∞. By analogy with the previous example, a natural question is whether both filtrations are isomorphic. More precisely, it is easy to see that, after transforming the time-axis ¢ into (0, ∞) by the time-change log t → t, F becomes the natural filtration of some real Brownian motion (first part of Proposition 3). The question becomes, is the time-changed filtration (G log t ) t 0 also generated by some Brownian motion? The next section will show that the answer is positive (second part of Proposition 3). Another continuous-time example pertaining to the innovation problem is Tsirelson's stochastic differential equation [START_REF] Tsirel'son | An example of a stochastic differential equation having no strong solution[END_REF]. This equation is driven by a Brownian motion B, and (weak) solutions X exist and all have the same law, but cannot be adapted to the natural filtration of B. Our third section will establish that, as far as filtrations are concerned, this situation is exactly the same as the preceding one: given such a pair (X, B) on the time-interval 0, ∞), the natural filtrations of X and B become, after a deterministic time-change from 0, ∞) to -∞, ∞), those of a £ -valued Brownian motion and its innovations (Proposition 4). As a consequence, the natural (non time-changed) filtration of every solution X of Tsirelson's stochastic differential equation is generated by some Brownian motion (Corollary 2). This result should be appreciated in the context of the recent literature; in the remarkable paper 7 , Dubins, Feldman, Smorodinsky and Tsirelson construct a variant of Tsirelson's equation whose solution X generates a non-Brownian filtration. The question remained open, whether Tsirelson's original equation also has this property. As this note shows, the answer is no; in retrospect, this makes the-technically very involved-construction of 7 still more interesting. For further recent pertinent examples of Brownian and non-Brownian filtrations, we refer to 9 , 10 , 17 , 19 , 2 , 5 , 16 , 22 , 3 , 8 . Most of these examples feature the above-mentioned phenomenon of creation of information: there is a decreasing sequence of σ-fields, all of them containing some common information, but with trivial intersection. And Tsirelson's equation does exhibit this phenomenon, so much so, that it inspired Rogers and Williams to comment in 15 : "somehow, magically, this independent random variable has appeared from somewhere! Indeed, it really has appeared from thin air, because . . . it is not present at time 0!" Throughout this note, the usual hypotheses are in force: the probability spaces are complete, the filtrations contain all negligible events and are right-continuous.

Circular Brownian motions

The circle ¢ / is called £ ; the symbol 1 = means equality modulo 1 (between two real numbers, or between a real number and an element of £ ). For x ∈ ¢ or x ∈ £ , {x} denotes the fractional part of x, that is, the real number {x} such that 0 ¥ {x} < 1 and {x} 1 = x. If U is a continuous, £ -valued process, we shall distinguish between the increment U t -U s , which takes its values in £ , and the real random variable t s dU r , defined by the following three (lifting) conditions: it depends continuously on t, it vanishes for t = s, and

t s dU r 1 = U t -U s .
Definition. -Given a filtration H = (H t ) t∈¤ , a £ -valued process (U t ) t∈¤ will be called a circular Brownian motion for H (abbreviated H-CBM) if it is continuous and adapted to H and if for each s ∈ ¢ , the process t → t s dU r , defined on the interval s, ∞), is a real-valued Brownian motion for the filtration (H t ) t∈ s,∞) .

(Equivalently, U is a Markov process for H, with transition probabilities the Brownian semigroup on the circle.)

If H equals the natural filtration G of U , we shall simply say that U is a CBM. Note that this notion only depends on the law of U . It is easy to see that circular Brownian motions exist: as the uniform probability on £ is invariant for the Brownian semigroup, a stationary, £ -valued Brownian motion with the uniform law at each fixed time is easy to construct. Uniqueness in law of CBM is less straightforward, but follows (among other possible proofs) from Proposition 1 below.

We have phrased the definition of a CBM U with an arbitrary filtration H (instead of only considering the natural filtration G of U ), as we shall encounter situations where G is strictly contained in another filtration H. It then follows from the predictable representation property of (usual) Brownian motion that a CBM U is an H-CBM if and only if every G-martingale is an H-martingale.

Associated to a CBM U are two filtrations: its natural filtration, generated by the past values U t , and the innovation filtration, generated by the past increments U t -U s or by their liftings t s dU r (which amounts to the same). Clearly, the innovation filtration is included in the natural filtration; Proposition 1 will show that it is always strictly smaller. All CBM's have the same law, as shown by the next proposition, whose proof is borrowed from Proposition (6.13) of Stroock and Yor 18 (see also § IX.3 of 14 and § V.18 of 15 ). Proposition 1. -Let U be a CBM, with innovation filtration F = (F t ) t∈¤ . For each t ∈ ¢ the random variable U t is uniformly distributed on £ and independent of the σ-field F ∞ (generated by all increments).

Proof. -Call G = (G t ) t∈¤ the natural filtration of U and for p ∈ denote by e p the character on £ defined by e p (u

) = exp (2πipx) if u ∈ £ , x ∈ ¢ and u 1 = x. For s < t, t s dU r is independent of G s , with law N (0, t-s), so e p (U t )|G s = e p (U s ) e p (U t -U s )|G s = e p (U s ) e p (U t -U s )|G s = e p (U s ) exp 2πip t s dU r G s = e p (U s ) exp -1 2 4π 2 p 2 (t-s) . This implies e p (U t ) = e p (U t )|G s ¥ | e p (U t )|G s | = exp -1 2 4π 2 p 2 (t-s)
, and, by letting s tend to -∞, e p (U t ) = 0 for p = 0. As a consequence, by the Stone-Weierstraß theorem, U t has a uniform law on £ .

For each s, the σ-field F s = σ(U t -U s , t ¡ s) of increments after s is independent of G s and a fortiori of U s . Thus, for s < t and p = 0,

e p (U t )|F s = e p (U s )|F s e p (U t -U s ) = e p (U s ) e p (U t -U s ) = 0 ,
since we have just seen that e p (U s ) = 0. Now, s∈¤ F s = F ∞ , and, when s ↓ -∞, e p (U t )|F s → e p (U t )|F ∞ by martingale convergence; so e p (U t )|F ∞ = 0 too, and U t is not only uniform, but also independent of F ∞ .

Definitions. -A regular time-change is an increasing (deterministic) bijection a from ¢ to (0, ∞) such that both a and its inverse a -1 are absolutely continuous functions.

A filtration (F t ) t 0 will be called Brownian if it is the natural filtration of some real Brownian motion (B t ) t 0 issued from the origin. (Only one-dimensional Brownian filtrations will be considered, so we simply call them Brownian.) Proposition 2 and definition. -Let (F t ) t∈¤ be a filtration indexed by ¢ . The following are equivalent: (i) for some regular time-change a, the filtration (F t ) t 0 defined by F t = F a -1 (t) for t > 0 and by right-continuity for t = 0 is Brownian; (ii) for every regular time-change a, the filtration (F t ) t 0 defined by F t = F a -1 (t) for t > 0 and by right-continuity for t = 0 is Brownian.

When these conditions are met, the filtration (F t ) t∈¤ is called Brownian.

For instance, if (B t ) t 0 is a real Brownian motion issued from the origin, the stationary Ornstein-Uhlenbeck process (X t ) t∈¤ defined by X t = e -t/2 B e t generates a Brownian filtration on ¢ ; and conversely, every Brownian filtration on ¢ is the natural filtration of such a stationary Ornstein-Uhlenbeck process.

Proof of Proposition 2. -We have to show (i) ⇒ (ii); this reduces to checking that, if a and b are regular time-changes, the homeomorphism φ = a•b -1 from (0, ∞) to itself has the following property: if B is a Brownian motion on 0, ∞), started at 0, the process X t = B φ(t) generates a Brownian filtration. As φ and its inverse are absolutely continuous, the Lebesgue derivative ψ of φ is almost everywhere defined and strictly positive. The process X = B•φ is a gaussian martingale, with quadratic variation φ(t). The martingale β t = t 0 ψ -1/2 (s) dX s is a Brownian motion; as X t = t 0 ψ 1/2 (s) dβ s , both X and β generate the same filtration.

Definitions. -A chopping sequence is a sequence (t k ) k∈¡ of real numbers such that t k < t k+1 , lim k→-∞ t k = -∞ and lim k→+∞ t k = +∞.
Given a chopping sequence (t k ) k∈¡ , a chopped Brownian motion (respectively a £ -valued chopped Brownian motion ) is a càdlàg process Z whose restriction Z k to each interval t k , t k+1 ) is a real Brownian motion (respectively a £ -valued Brownian motion) started from 0 at time t k , all the Z k 's being independent. Remark. -The converse also holds (but we shall not need it): every Brownian filtration indexed by ¢ is the natural filtration of a chopped Brownian motion. More generally, if some càdlàg process Z generates a Brownian filtration, every Brownian filtration is generated by a process with the same law as Z.

Proof of Lemma 1. -A £ -valued chopped Brownian motion V clearly generates the same filtration as the real chopped Brownian motion Z defined by Z t = t t k dV s for t ∈ t k , t k+1 ); so it suffices to prove the lemma for real processes.

The chopping sequence (t k ) k∈¡ and the chopped Brownian motion Z are given. Introduce a sequence (c k ) k∈¡ of strictly positive constants such that

k<0 c k (t k+1 -t k ) < ∞ and k 0 c k (t k+1 -t k ) = ∞; define a regular time-change by a(t) = t -∞ k∈¡ c k 1l (t k ,t k+1 (s) ds and a process (B t ) t>0 by B a(t) = √ c k Z t + k √ c -1 Z t - for t k ¥ t < t k+1 .
By scaling,

B k t = B t -B a(t k ) = √ c k Z k a -1 (t)
is a Brownian motion on the interval a(t k ), a(t k+1 ) ; and these Brownian motions B k are independent. So the series in the definition of B is convergent and B is a Brownian motion. As

Z t = 1 √ c k (B a(t) -B a(t k ) ) for t k ¥ t <
Z t = k∈¡ 1l t k ,t k+1 ) (t) t t k dU r generates F , so F is Brownian by Lemma 1.
The proof for G is less straightforward. In fact, it involves a key idea of the present paper, a certain coupling lemma (for a vivid presentation of the use of coupling in the theory of Markov processes, see Diaconis 6 ).

Lemma 2.
-If (X t ) t 0 is a £ -valued Brownian motion defined on some Ω, A, ¡ , (H t ) t 0 , there exists a £ -valued Brownian motion (Y t ) t 0 , defined on the same Ω, A, ¡ , (H t ) t 0 , satisfying the following four properties:

(i) Y 0 = 0 ∈ £ ; (ii) Y is independent of X 0 ; (iii) both processes (X t ) t 0 and (X 0 , Y t ) t 0 generate the same filtration; (iv) calling S the H-stopping time inf {t : X t = Y t }, one has ¡ S ¡ t ¥ 1/(4t) and Y = X on S, ∞ . Proof of Lemma 2. -Introduce the H-stopping time S = inf {t : X t + X t = X 0 }. The process Y t = X 0 -X t if t ¥ S X t if t ¡ S.
is continuous by definition of S; it starts from 0, is H-adapted and verifies

t 0 dY s = t 0 (-1l 0,S +1l S,∞ )(s) dX s ,
so it is a £ -valued H-Brownian motion. As it starts from 0, it is independent of H 0 , whence (ii). Replacing H by the natural filtration of X shows that (X 0 , Y ) is adapted to that filtration ; to show (iii), it suffices to reconstruct X from X 0 and Y . This is easy: S is also the first time when Y + Y = X 0 , and X is equal to X 0 -Y up to S, and equal to Y from S on. Clearly, S is the same as the one defined in (iv). Last, to establish the estimate ¡ S ¡ t ¥ 1/(4t) in (iv), define

T = inf t : t 0 dX s = 1
2 . When t ranges from 0 to T , X t -X 0 visits all points of one of the two arcs linking 0 and 1 2 in the circle £ ; so during this time-interval (X t -X 0 ) + (X t -X 0 ) assumes all possible values on the circle, in particular the value -X 0 . This implies S ¥ T , whence the majoration ¡ S ¡ t ¥ ¡ T ¡ t . Now the Brownian estimate

T = T 0 dX s 2 = (±1/2) 2 = 1/4 yields ¡ T ¡ t ¥ T /t = 1/(4t).
End of the proof of Proposition 3. -To show that the natural filtration G of U is Brownian, it suffices by Lemma 1 to exhibit a £ -valued chopped Brownian motion V that generates G. To this end, choose any chopping sequence (t k ) k∈¡ such that

k<0 1 t k+1 -t k < ∞ ;
for instance, t k = k 3 is a possible choice. For each k, Lemma 2 applied to the £ -valued Brownian motion (U t ) t t k provides us with a £ -valued Brownian motion V k defined on t k , ∞), issued from 0 at time t k , verifying condition (iii) of Lemma 2, and equal to U after some G-stopping time S k such that ¡ S k -t k ¡ t ¥ 1/(4t). We shall establish that the process (V t ) t∈¤ equal to V k on t k , t k+1 ) is a £ -valued chopped Brownian motion and generates the filtration G. Clearly, V is G-adapted.

To see that V is a £ -valued chopped Brownian motion, we only have to show that the processes (V k t ) t∈ t k ,t k+1 ) are independent; it suffices to establish that V k is independent of G t k . This can be obtained by writing, for a real-valued, bounded Borel functional f ,

f (V k )|G t k (1) = f •φ k (U t ) t t k G t k (2) = f •φ k (U t ) t t k U t k = f (V k )|U t k (3) = f (V k ) ,
where (1) stems from the fact that V k is a functional of (U t ) t t k (by Property (iii) of Lemma 2), (2) from the Markov property of U , and (3) from the independence of V k and U t k (by Property (ii) of Lemma 2). It remains to see that V generates G, or equivalently that U is adapted to V . By Property (iii) of Lemma 2, there are some adapted Borel functionals ψ k such that

(U t ) t∈ t k ,t k+1 ) = ψ k U t k , (V t ) t∈ t k ,t k+1 )
a. s. for each k.

(Adaptedness means that for t ∈ t k , t k+1 ), the restriction of ψ k (x, v) to t k , t is a function of x and of the restriction of v to t k , t .) Using those ψ's, it is possible for each k ∈ to define inductively a £ -valued, càdlàg, V -adapted process (U k t ) t∈¤ by

U k t = 0 for t ∈ (-∞, t k ), (U k t ) t∈ t k ,t k+1 ) = ψ k V t k -, (V t ) t∈ t k ,t k+1 ) , (U k t ) t∈ t ,t +1 ) = ψ U k t -, (V t ) t∈ t ,t +1 ) for > k.
(If the left-limit U k t -is not defined, put for instance U k t = 0 for t ¡ t .) On the event

E k = {S k-1 < t k }, one has V t k -= U t k -= U t k ; by definition of ψ k , this gives U k = U on t k , t k+1 )×E k . Now if, for some ¡ k, U k is equal to U on the set t , t +1 )×E k , then U k t +1 -= U t +1 -= U t
k<0 ¡ E c k < ∞.
The Borel-Cantelli lemma then says that lim sup k→-∞ E c k is negligible, and E k converges almost surely to Ω when k → -∞. So t k , ∞)×E k tends to ¢ ×Ω, and consequently U k converges (stationarily) to U . As each U k is V -adapted, so is also U .

Remark (not used in the sequel). -When £ is replaced by a compact, connected Riemannian manifold , it remains true that Brownian motion in indexed by ¢ has a Brownian filtration (but the innovation filtration can no longer be defined, unless is endowed with some extra structure, e.g. Lie group or symmetric space). The definition of a Brownian filtration must of course be modified, with d-dimensional Bronian motions instead of real ones (where d is the dimension of ). We have not found a simple proof of this fact; M. Arnaudon establishes it in an appendix to this note by an argument using a mirror-coupling and some hypoelliptical diffusions. But the case when is a d-dimensional sphere is much simpler (the mirror-coupling can be defined so as to avoid all cut-locus difficulties); as an illustration, we show here how that case can be dealt with, and refer to Arnaudon's appendix 1 for the general case.

It suffices to extend Lemma 2 to , the rest of the proof carries over almost verbatim. There are only two small differences: first, in (i) the point 0 ∈ £ has to be replaced with an arbitrary origin O ∈ ; second, in (iv), any estimate of the form ¡ S ¡ t ¥ f (t) will do, provided f (t) → 0 when t → ∞ (the condition k<0 (t k+1 -t k ) -1 < ∞ in the proof will become k<0 f (t k+1 -t k ) < ∞). To prove the extended Lemma 2, consider as the subset of ¢ d+1 made of all points at distance R from the origin 0 ∈ ¢ d+1 ; for x ∈ call H x the perpendicular bissector of the segment Ox and define the mirror-map Φ x as the symmetry with respect to the hyperplane H x ; Φ x is an isometry of exchanging O and x (when x = O, just take Φ x = Id¡ ). It then suffices to put S = inf {t :

X t ∈ H X 0 } (take S = 0 if X 0 = O) and Y t = Φ X 0 X t if t ¥ S, Y t = X t if t ¡ S. This Y is a -valued X -Brownian motion satisfying (in ¢ d+1 ) Y t = O + t 0 (1l 0,S Φ X 0 + 1l S,∞ Id)(dX s ) ,
whence properties (i), (ii) and (iii). To establish the estimate (iv), it suffices to show that, if E denotes a fixed hemisphere and T the first hitting time of the boundary ∂E by a Brownian motion in E, then ¡ x T ¡ t tends to 0 uniformly in x ∈ E when t → ∞. But the function g(t, x) = ¡ x T ¡ t is continuous on (0, ∞)×E (it is the solution to the heat equation ∂g/∂t = 1 2 ∆g with boundary conditions g = 1 for t = 0 and g = 0 on ∂E), and for each fixed x, it decreases to zero when t → ∞; so uniformity in x is a consequence of Dini's lemma.

Corollary 1. -Let Ω, A, ¡ , (H t ) t∈¤ be a filtered probability space. There exists an H-CBM if and only if for some (or equivalently for every) regular time-change a, the filtration (H t ) t 0 defined by H t = H a -1 (t) admits an H -Brownian motion (i.e., some H -martingale is a Brownian motion).

Proof. -If U = (U t ) t∈¤ is an H-CBM, its natural filtration G is Brownian by Proposition 3, and the time-changed filtration G is generated by a Brownian motion B with B 0 = 0. As U is a Markov process for H, every G-martingale is an H-martingale, and, by time-change, every G -martingale is an H -martingale. So B is an H -Brownian motion.

Conversely, if B is an H -Brownian motion, call G the natural filtration of B -B 0 and notice that every G -martingale is an H -martingale. The time-change transforms G into the natural filtration G of some CBM U ; as every G-martingale is an H-martingale, U is also an H-CBM.

Tsirelson's stochastic differential equation

In 1975, Tsirelson 20 has constructed the first example of a stochastic differential equation (T) dX t = dB t + g t, (X s , s¥ t) dt X 0 = 0 having the following properties: B = (B t ) t 0 is a Brownian motion, g is a bounded, measurable function of the past (X s , 0¥ s¥ t) of the solution X = (X t ) t 0 , (T) has some (weak) solutions, but no solution X can be adapted to the filtration generated by B.

Tsirelson's equation has been extensively studied; see for instance 4 ,

To define g, Tsirelson introduces a sequence (t k ) k 0 of instants verifying t k < t k+1 and lim k→-∞ t k = 0. The function g he considers is given by

g t, (X s , s¥ t) = k<0 X t k -X t k-1 t k -t k-1 1l (t k ,t k+1 (t)
(recall that {x} denotes the fractional part of x).

A solution to (T) is a system (Ω, A, ¡ , H, B, X), where (Ω, A, ¡ , H) is a filtered probability space, B is an H-Brownian motion started at 0, and X is H-adapted and verifies (T). Given such a solution, if G denotes the sub-filtration of H generated by X, then B is clearly adapted to G, so it is also a G-Brownian motion, and (Ω, A, ¡ , G, B, X) is a solution too. The aim of this section is to establish that G is generated by some real Brownian motion started at the origin. This will be done by reducing the problem to the one addressed in Proposition 3; to do so, we shall need some notation.

Fix once and for all the sequence (t k ) k 0 and by this choice the function g. Define two functions f and α on (0, ∞) by

f (t) = k<0 1 t k+1 -t k 1l (t k ,t k+1 (t) + 1l (t 0 ,∞) (t) α(t) = t t 0 f (s) 2 ds .
As

t 0 0 f (s) 2 ds = k<0 1 t k+1 -t k = +∞ and ∞ t 0 f (s) 2 ds = ∞ t 0 ds = +∞ ,
α is a homeomorphism from (0, ∞) to ¢ ; its inverse a is a regular time-change.

Proposition 4. -Fix the probability space (Ω, A, ¡ ). a) If, on (Ω, A, ¡ ), (H, B, X) is a solution to (T), define a £ -valued process (U t ) t∈(0,∞) by Then (H, B, X) is a solution to (T). c) The two maps defined in a) and b) are inverse to each other; they establish a bijection between the solutions (H, B, X) to (T) and the CBM's ( H, U ). d) Given a corresponding pair (H, B, X) ←→ ( H, U ), call G the natural filtration of X, F that of B, G that of U and F the innovation filtration of U . These filtrations are time-changed from one another:

U t 1 =        B t -B t k t k+1 -t k + X t k -X t k-1 t k -t k-1 for t ∈ (t k , t k+1 B t -B t 0 + X t 0 -X t -1 t 0 -t -
G t = G a(t) G t = G α(t) ; F t = F a(t) F t = F α(t) .
In particular, H is the natural filtration of X if and only if H is the natural filtration of U .

Remark. -This long statement can be summarized as follows. For fixed (Ω, A, ¡ ), there exists a bijection between the solutions (H, B, X) to (T) on the one hand and the pairs ( H, U ) where U is a H-CBM on the other hand, with the following properties: The time-changes a and α exchange the filtrations H and H; they also exchange the natural filtrations G of X and G of U , as well as the natural filtration F of B and the innovation filtration F of U . So the process X corresponds to U while B corresponds to the innovations of U . Remark that the triple (H, B, X) is redundant, for B is a functional of X; consequently the proposition could be rephrased so as to state a correspondence between the pairs (H, X) and ( H, U ); this would look more symmetric.

Proof of Proposition 4. -a) If (H, B, X) is a solution to (T), for each k < 0 one has

X t k+1 -X t k = B t k+1 -B t k + t k+1 t k g t, (X s , s¥ t) dt = B t k+1 -B t k + X t k -X t k-1 t k -t k-1 (t k+1 -t k ) ;
dividing both sides by t k+1 -t k and working modulo 1 to strip off the braces yields

X t k+1 -X t k t k+1 -t k 1 = B t k+1 -B t k t k+1 -t k + X t k -X t k-1 t k -t k-1 .
Considered as elements of £ , the right-hand side is U t k+1 and the left-hand one is the limit of U t when t tends to t k+1 from above. This shows that the process U is continuous at point t k+1 and hence everywhere, and gives a meaning to t s dU r . For s ¥ t with s and t in the same interval (t k , t k+1 or (t 0 , ∞), the definition of U implies By additivity, this formula remains valid for all pairs s ¥ t in (0, ∞); notice that the right-hand side is an H-martingale in t on the interval s, ∞). By time-change, one gets, for s

¥ t in ¢ , t s d U r = a(t) a(s) f (r) dB r
and the right-hand side is an H-Brownian motion since

a(t) a(s) f 2 (r) dr = t -s. As U is adapted to H, it is an H-CBM.
b) For each s > 0, the process (B t ) t s is a continuous H-martingale, with quadratic variation

t s 1 f 2 (r) d U, U r = α(t) α(s) 1 f 2 α(r) dr = t -s , hence a Brownian motion. Consequently, (B t ) t 0 is an H-Brownian motion, starting at B 0 = 0. Since 0 ¥ {U t k } < 1, the integral in the definition of X is convergent. For k ¥ 0, X t k -X t k-1 = B t k -B t k-1 + {U t k-1 } (t k -t k-1 ) = t k t k-1 1 f (r) dU r + {U t k-1 } (t k -t k-1 ) = (t k -t k-1 ) t k t k-1 dU r + {U t k-1 } , so X t k -X t k-1 t k -t k-1 1 = U t k , wherefrom {U t k } = X t k -X t k-1 t k -t k-1 and X verifies (T). c) We first show that the composed map (H, B, X) → ( H, U ) → (H , B , X ) is identity: (H , B , X ) = (H, B, X). That H = H is trivial: two inverse time-changes cancel. Then, B is defined by dB t = dU t /f (t) and U verifies dU t = f (t) dB t , giving B = B. Last, the right-continuity of U at t k yields U t k 1 = (X t k -X t k-1 )/(t k -t k-1 ), whence dX t = dB t + g t, (X s , s¥ t) dt = dX t , giving X = X.
Now the other way round: ( H, U ) → (H, B, X) → ( H , U ) is identity too. Again, H = H is trivial. Time-change U and U to get a U and a U . For t ∈ t k , t k+1 ) (with the convention t 1 = ∞),

U t 1 = t t k f (s) dB s + X t k -X t k-1 t k -t k-1 = t t k dU s + X t k -X t k-1 t k -t k-1 .
But we have seen in the proof of b) that The rest of the corollary is straightforward from Propositions 3 and 4.

X t k -X t k-1 t k -t k-1 1 = U t k , so U t = U t .
Remark.

-If the reader does not care about CBM's and is only interested in knowing that the filtration of X is Brownian, the proofs given above can be shortened. First, in Proposition 4, only a) and d) are needed, so half of the computations can be dispensed with.

More important, a long detour we have taken can be bypassed. Our proof consisted in time-changing (T) to get a CBM; and the proof that the filtration of a CBM is Brownian was done by time-changing back the CBM to work on ¢ + (with a time-change provided by Lemma 1). By choosing in the proof of Proposition 3 the same (modulo the time-change) sequence (t k ) k 0 that is used to define g, it is possible to show directly that the filtration of X is Brownian, with a proof quite similar to that of Proposition 3, that uses the time-changed CBM U featuring in Proposition 4 a).

The reason for this detour was that we found it instructive, on an intuitive level, that the "mysterious" Tsirelson example may be one-to-one translated into Brownian motion on the circle, an object which to us does not seem mysterious at all.

Lemma 1 .

 1 -The filtration generated by a chopped Brownian motion, or by a £ -valued chopped Brownian motion, is Brownian.

  ) dB r .

  d) As X and B are G-adapted, the definition given in a), how to obtain U from B and X, shows that U is G-adapted. Conversely, the definitions given in b), how to obtain B and X from U , show that X is adapted to the natural filtration of U . So U has G as natural filtration; by time-change, this gives the relation between G and G.The formulae dB = (1/f ) dU and dU = f dB show that the increments of U generate the filtration F ; by time-change, this gives the relation between F and F . Proposition 4 reduces the study of Tsirelson's equation to that of CBM's. Transferring to (T) what we know about CBM's gives the following statement (where only the last sentence is new): Corollary 2. -Solutions (Ω, A, ¡ , H, B, X) to Tsirelson's equation exist. More precisely, on a given Ω, A, ¡ , (H t ) t 0 , a solution (B, X) exists if and only if there exists an H-Brownian motion. The law of a solution (B, X) depends only on the sequence (t k ) k 0 . Given any solution, the natural filtration of X is Brownian. Proof. -If (H, B, X) is a solution, B is an H-Brownian motion. Conversely, if an H-Brownian motion exists, the filtration ( H t ) t∈¤ defined by H t = H a(t) contains some CBM U by Corollary 1 and a solution (H, B, X) exists by Proposition 4.b.

  t k+1 , the natural filtrations of Z and B are time-changed from each other, and Z generates a Brownian filtration.

Proposition 3. -The natural filtration G and the innovation filtration F of a CBM are Brownian. Proof. -Let U = (U t ) t∈¤ be a CBM, G its natural filtration, and F its innovation filtration. Choose any chopping sequence (t k ) k∈¡ ; for instance, t k = k is a possible choice. The chopped Brownian motion

  +1 on E k and so, by definition of ψ +1 , U k agrees with U on the next interval t +1 , t +2 )×E k . Consequently, U k and U are equal on t k , ∞)×E k . Now take t = t k+1 -t k in the estimate ¡ S k -t k ¡ t ¥ 1/(4t) and use the condition on the t k 's, to get

  1 for t > t 0 and time-change both U and H by α: for t ∈ ¢ , set U t = U a(t) and H t = H a(t) . The process U is a H-CBM. b) Conversely, if ( H t ) t∈¤ is a filtration on (Ω, A, ¡ ) and if ( U t ) t∈¤ is an H-CBM, time-change them to a filtration (H t ) t 0 and a process (U t ) t>0 by H t = H α(t) and U t = U α(t) ; for t > 0 set {U t k } 1l (t k ,t k+1 (s) ds .

	B t =	0	t	1 f (s)	dU s ;	X t = B t +	t 0 k<0


Franc -Ois Peres
email: francois.peres@enit.fr

,Ã , Leı ¨la Bouzaı Jean-Claude Bocquet
email: jean-claude.bocquet@lgi.ecp.fr

Franc -Ois Billy
email: francois.billy@edf.fr

Andre ´lannoy
email: andre.lannoy@laposte.net

Philippe Haı
email: philippe.haik@edf.fr

Anticipating aging failure using feedback data and expert judgment

Keywords: Maintenance, Aging, Anticipation, Potential failures, Life cycle management, Expert judgment

This paper presents a methodology for anticipating failures in a component up to the end of its life cycle. Often, feedback data is not sufficient and must be complemented by the analysis of expert judgment. The methodology developed aims at anticipating the degradation mechanisms responsible for aging, and evaluating their relevance and related uncertainties. This is necessary information for risk analysis related to the operating of a component up to the end of its life cycle. Lastly, the methodology is applied to a nuclear component.

Introduction

In the framework of good management of the life cycle of an industrial installation and a possible extension of its service life, it is necessary to be able to predict the effects of aging. To do this, one must analyze potential failures that may occur and propose solutions for avoiding them or reducing the seriousness of their effects [START_REF] Bouzaı¨ene | Expert judgment methodology for failure anticipation in nuclear power plants[END_REF]. This paper proposes to examine this industrial problem. Anticipation is by its nature based on feedback. However, the data available is not always complete or sufficient to carry out the needed analysis. To complement and enrich the data, we recommend exploiting the information available on ''analogous'' equipment, ''third party'' feedback from other companies and expert judgment. Our objective is therefore to develop a methodology by which it is possible to anticipate potential failures due to aging using both feedback and expert judgment.

The anticipation methodology that we have developed brings into play three distinct overlapping processes, outlined in the diagram below (Cf. Fig. 1).

The different phases of the methodology represent a procedure, which can be qualified as a Global Anticipation Process (GAP). The third phase of this process (seen in the first column of Fig. 1) can itself be subdivided into a series of 5 steps, which we call the Surveying process. Finally, the entire Surveying process is assisted by a parallel process called the Expert Elicitation Process. In order to improve the understanding of the structure we will label the steps in the GAP as 'phases' keeping the words ''step'' for the surveying process and ''stage'' for the expert elicitation process.

After having introduced the context of the studies and the players involved in the production and exploitation of the results, we will describe the three processes and will apply the methodology in an industrial context.

Context and background

In any predictive approach, it is essential to be able to identify the basic data, which can provide useful and complete information needed for decision-making. We call this information ''anticipation data''. It is also important to define clearly the objectives of the method and the end results one wishes to obtain. As these results are destined for decision-makers, we call them ''decision-support data''. Furthermore, it is equally essential to be able to localize the different types of information and identify information flows among the various players and knowledge sources.

We have identified three types of players in the ''anticipation'' process: analyst, expert and decision-maker.

It is the analyst who coordinates the anticipation process. His role is to gather the information needed for anticipation, and to synthesize it to provide data for decision-making. To enrich this data concerning the/ or the state of the equipment, the analyst may also gather feedback data on equipment analogous to that being investigated as well as the experts opinion (Fig. 2).

It is the expert who has the knowledge and the expertise to anticipate failures or degradation mechanisms. To do so, he will need a certain amount of data and influencing variables on the basis of which he can make an assessment.

The decision-maker is the one who, on the basis of the synthesized data provided by the analyst, chooses the ''best'' solutions and compromises in accordance with decision-making processes, while remaining faithful to corporate strategic guidelines. The optimum solution will often be a compromise.

In order to facilitate comprehension of the text we give the definitions of a number of key words in Table 1 below.

Global anticipation process

We start with the description of the essential features of the global anticipation process (Cf. Fig. 3), the different phases it implies and the various methods and tools required.

Phase 1. Definition of the context and objectives of the anticipation process: This phase is dedicated to the description of the context and the aim of the study as well

Expert(s) Analyst

Knowledge Information

Existing data (design, maintenance,...)

Feedback concerning similar equipment

Available objectives and data concerning equipment

Decision-maker

Decision

Equipment data completed by expert survey

Fig. 2. Context of use of expert knowledge.

Table 1 Definitions of main scientific terms used within the framework of this study Word De´finition Anticipation Identification of events which might be detrimental in terms of safety, availability or cost, before they occur, so as to evaluate the risks they represent and to prepare and implement the appropriate measures of preventive or unscheduled maintenance.

Degradation mechanism

Physio-chemical change in the characteristics of a material under specific conditions of temperature or pressure or in a specific chemical environment. This progressive alteration over time may cause degradation of the material with measurable effects (a crack, for example). With time, this degradation may further deteriorate and, beyond a certain threshold, may cause a partial alteration or total loss of function in the component; this is a failure. It can be identified by a failure mode (e.g. a leak).

Failure

Loss of capacity on the part of a set of components to perform the required function(s) at the level of performance defined in the technical specifications.

Aging

Process in which the characteristics of a component change with time or with use.

Unscheduled maintenance

Set of tasks, performed on a structure, which are generally preventive and non-repetitive; they may be generic; they are economically costly.

Passive equipment

Part of the structure that does not involve dynamic component. Such equipment does not normally undergo preventive maintenance checks.

Active equipment

Sub system of the installation that directly participates in the production of the expected service or product. This type of equipment is subject to preventive maintenance.

as the results that are hoped for. To this end, a document is provided to the experts in order to specify the environment and objectives of the survey, the important definitions and the different steps of the methodology. Phase 2. Identification and gathering of information needed for anticipation: This phase will enable the analyst, generally not an expert on the component, to determine the information that will help him communicate with the equipment experts, and to ask the right questions. For the next phase, this information will provide input data, which will help stimulate expert responses. During this phase, one type of information is particularly needed: failure history of equipment different from the one studied but made of the same materials, with similar design and manufacturing procedures, found in the same operating or maintenance environment or fulfilling the same functions. We call such equipment ''analogous'' equipment.

Phase 3. Surveying process: This phase will be described later on as a specific subprocess. Let us just point out that at each step, one must set objectives for the survey, identify the experts to be questioned and choose the questioning procedure: individual or collective questioning. A certain number of tools and methods have been chosen or implemented to guide or facilitate the work at this level, depending on the end objective. Let us mention for instance the use of Bayesian networks to evaluate the relevance of the mechanisms, or the choice of the TRIZ method to identify potential preventive solutions.

Phase 4. Synthesis and exploitation of the results: This phase is dedicated to the treatment of the results of the anticipation process, but also the analysis of the different phases that led to these results. To exploit this information, an easy-to-use format is required which allows one to simply to retrieve the data needed so as to facilitate their reuse and updating. For a given piece of equipment, each component/degradation mechanism pair has to be analyzed in order to capitalize the information concerning the material affected by the mechanism, the operating conditions that promoted the appearance of the mechanism, the measurable effects, the failure modes, the existing feedback and the monitoring and preventive maintenance procedures. This data will be useful in future studies, to retrieve feedback on ''analogous'' equipment. Furthermore, at the end of the study, one must provide the results to the decision-makers and participating experts. This can be done in a document that sums up the results of the anticipation process.

Surveying process

This paragraph is dedicated to the description of the third phase of the global anticipation process, which represents the heart of the methodology since it leads to the identification and the evaluation of the degradation mechanisms. A descending approach appeared to be the best suited for describing this process. Consequently, the process to be followed to anticipate an event can be: identify potentially detrimental events, evaluate their consequences and seriousness, prepare and implement the appropriate maintenance measures.

Because our work deals with life cycle management, we shall concentrate on anticipating failures due to aging. Any equipment, which ages is subject to mechanisms of degradation which act on its component in specific ways. To anticipate failure due to aging, the degradation mechanism which may initiate it (trigger) has to be identified. Its relevance must also be evaluated, considering the evolution over time of the measurable effects it may have (kinetics). Next, the corresponding failure modes and their consequences are determined. It is then possible to identify the maintenance measures needed to avoid or slow the mechanism. In the context of life cycle management, the efficiency of these maintenance measures from the perspective of safety and costs has to be considered.

The process of anticipation would seem to be more suited to passive components. Generally, such equipment is subject to only one degradation mechanism. The kinetics of degradation are slow and progressive. If the equipment cannot be replaced, its lifetime may condition that of the installation as a whole. Replacement of such components can only be exceptional and occasional. However, active components important for safety or critical in terms of maintenance costs or availability may also deserve anticipatory measures. Unlike passive components, active components are those subject to significant wear, often quite rapid, due to several compounded degradation mechanisms; they are subject to regularly scheduled preventive maintenance operations, and in some cases, replacement. For active components, the degradation mechanisms which trigger the failure cannot be isolated; one must therefore identify and anticipate the degradation or failure modes understanding that a mode may be attributable to several mechanisms. Within this framework, the process of anticipating potential failures due to aging (Fig. 4) involves the following steps1 :

3.1 Identification of the potential degradation mechanisms (for a passive component) and the potential failure modes (for an active component). 3.2 Assessment of the relevance of these mechanisms. 3.3 Analysis and ranking of the relevant mechanisms (or modes): potential failures (kinetics, seriousness, measurable effects, consequences). 3.4 Identification of potential avoidance solutions. 3.5 Assessment of the suitability of these solutions in terms of efficiency and cost.

Expert elicitation process

Tools and methods

Functional analysis, design documents and, most particularly, Reliability Centered Maintenance (RCM) studies can inform about the potential failures of a piece of equipment. Feedback can provide information on failures actually observed in equipment. However, if the environmental, operating or maintenance conditions or the duration of operation of the equipment change in relation to those considered at the time of design, only expert judgment is capable of predicting the impact of the changes. Drawing on existing data, the expert can identify the potential failures, which might occur subsequent to these changes. He is therefore an player in the anticipation process.

As it can be seen on Fig. 4, each step of the surveying process can be provided with information resulting from a process of expert questioning. Such a process is called elicitation.

Ballay [START_REF] Ballay | Capitaliser and transmettre les savoir-faire de l'entreprise[END_REF] defines the expert as being the ''person who has the knowledge''. In [START_REF] Lannoy | L'utilisation du jugement d'experts en suˆreted e fonctionnement[END_REF], Lannoy and Procaccia list four situations, which require recourse to expert judgment in the field of dependability: completing, validating, interpreting and integrating existing data; assessing the impact of a change (in design, manufacturing, operation, maintenance or environment), predicting the occurrence of future events and the consequences of a decision, determining the present state of knowledge in the field, providing the elements needed for decision-making in the presence of several options. ''Anticipation'' relates to three of these situations. First, the problem is to complete and validate feedback data, which will enable the identification of a certain number of observed failures. Next, one must evaluate the impact of any changes in relation to the initial design (for example, an extension of the service life beyond the period planned at the time of design) and estimate the probability of occurrence of future events, which the expert must identify and evaluate in terms of their seriousness. Lastly, the expert must provide decision-makers with the elements needed to make their decisions and choose the alternative, which will enable the avoidance or delay of potential failures identified.

As expert judgment has often been used in dependability applications, the choice of an ascending method was chosen to handle this aspect of the methodology to be developed. A certain number of methods for soliciting expert knowledge were examined to identify or build up a method adapted to the needs of anticipation [START_REF] Bouzaı¨ene | Expert judgment methodology for failure anticipation in nuclear power plants[END_REF].

Seven of these methods have been used essentially in nuclear safety applications [START_REF] Cojazzi | Benchmark Exercise on Expert Judgment Techniques in PSA Level 2[END_REF] (Table 2):

In addition, we looked at the methods for questioning experts used in these different approaches: LCM (Life Cycle Management) [START_REF] Melvin | Demonstration of life cycle management planning for systems and components with Pilot applications at Oconee and Prairie Island Nuclear Stations[END_REF], TRIZ-AFD (for Anticipatory Failure Determination, as well as for identifying solutions and design-support) [START_REF] Guarnieri | Une nouvelle me´thode pour l'identification and la maıˆtrise des de´faillances: pre´sentation, illustration and perspective[END_REF][START_REF] Kaplan | New tools for failure and risk analysis: anticipatory failure determination (AFD) and the theory of scenario structuring[END_REF], RIPBR (Risk-Informed, Performance-Based Regulation in nuclear safety and maintenance support) [START_REF] Golay | Improved nuclear power plant operations and safety through performance-based safety regulation[END_REF], PMDA-PIRT (Proactive Materials Degradation Assessment-Phenomena Identification and Ranking Table , for anticipating degradation of materials due to aging) [13].

A description of these methods can be found in [START_REF] Bouzaı¨ene | Expert judgment methodology for failure anticipation in nuclear power plants[END_REF], except for PMDA-PIRT, identified in January 2004. PMDA-PIRT is a method in the course of development since 2003 by the US-Nuclear Regulatory Commission (NRC). It is a method for anticipating degradation mechanisms related to equipment aging in nuclear plants. The objectives of the method are to: identify and rank the degradation mechanisms likely to affect critical plant equipment, evaluate the efficiency of periodic inspections or implement new procedures for corrective maintenance or replacement, enable improved understanding of observed, but also potential degradation mechanisms.

Classification of methods for surveying experts

In order to compare the 11 methods studied, we proposed to classify them in terms of their suitability for the purposes of anticipation and the efforts required, using an anticipation/effort diagram. The objective was to identify the methods most useful for anticipation and which do not require major efforts for implementation (see Fig. 5).

The X-axis, representing the effort needed for implementation, expresses the ease of application of the method. This effort takes into account the number of experts to be surveyed, the number of times they are to be questioned, the time needed for the process and the degree of complexity of the different phases of questioning and of the tools used. The Y-axis, more specific to anticipation, enables the evaluation of the suitability of the method to this context. This aspect takes into account the objectives of the method, the different applications made, the multidisciplinary nature of expert teams and the suitability of the tools from the perspective of anticipation.

Six experts from different disciplines contributed to the classification of the methods. In the paper the experts are considered as having the knowledge. The experts were chosen for the survey because they had already applied one or several methods we wanted to compare.

This classification points out the existence of three families of methods:

Family A falls in quadrants 1 and 4 of the diagram and is composed of the CTN-UPM, NNC, TUD and FEJ-GRS methods. These methods are of no real interest for anticipation.

Family B falls in quadrant 2 of the diagram and is composed of the TRIZ-AFD and LCM methods. These two methods seem well suited to anticipation and do not require much effort for implementation. However, each corresponds only partially to the needs of anticipation: TRIZ-AFD enables the identification of failures and maintenance solutions but gives no means of evaluating them.

LCM enables the identification and evaluation of maintenance solutions in relation to known failures. It recommends systematic monitoring of critical equipment and proposes solutions in the event of an observed failure. This reactive method does not, however, enable the anticipation of failures, which have never occurred. This type of approach, which presents safety risks and, in some cases can generate significant expenditure, is inadequate in any field where safety is a factor of prime importance.

Family C falls in quadrant 3 of the diagram and is composed of the PMDA-PIRT, RIPBR, STUCK-VTT, NUREG-1150, KEEJAM methods. All of these methods have benefits for anticipation but require major efforts for implementation. It was therefore necessary, for each Table 2 Methods for expert elicitation in the context of nuclear safety NUREG-1150 [START_REF] Keeney | Use of expert judgement in NUREG-1150[END_REF], STUK-VTT [START_REF] Pulkkinen | A method for using expert judgement in PSA[END_REF], NNC FEJ-GRS KEEJAM [START_REF] Cojazzi | KEEJAM: A knowledge engineering methodology for expert judgment acquisition and modelling in probabilistic safety assessment, actes du ESREL'97[END_REF] CTN-UPM TUD [START_REF] Cooke | Procedure guide for the use of expert judgement in uncertainty analyses, actes du ESREL'96-PSAM III[END_REF] method, to identify (1) the positive points, which could be integrated in a global anticipation method and (2) the weak points that must be avoided.

Assessment of the methods in relation to anticipation

The different methods likely to deal with anticipation are presented in Table 3. The benefits from the perspective of anticipation as well as the weaknesses of each method are also discussed.

Lessons learned from the analysis

This overview showed that none of the methods studied really provides a global solution to the objective of anticipation. However, it did point out certain characteristics beneficial for building up a method for surveying experts with a view to anticipation:

1 Drawing on methods to stimulate creativity (a concept found in TRIZ-AFD). 2 Surveying in two phases: one collective and one individual (NUREG-1150). 3 Breaking down the problem into several steps and questioning experts from various disciplines (FEJ-GRC). 4 Formalizing exchange among the experts and exploiting existing knowledge (KEEJAM, STUK-VTT). 5 Simplification of the ''probability'' aspect for the experts and structuring of results (STUK-VTT).

Moreover, this bibliographical review of existing methods revealed the generic stages in the process of surveying experts. They are introduced in Fig. 6.

Application to a passive component in a nuclear plant: the pressurizer

To validate the proposed approach, it is now being practically applied to a passive component: the pressurizer, a component, which regulates the pressure in the primary cooling system of a nuclear power plant. The test will enable the evaluation of the feasibility of the method and identifying any difficulties encountered in actual practice (Fig. 7).

Step 1. Definition of the context and objectives of the anticipation process: Was carried out using a document that summarized the industrial context of anticipation, as well as the objectives. Various definitions and the methodology phases were presented. This document was sent to the selected experts who then accepted to participate in the practical application.

Step 2. Identification and gathering of information needed for anticipation: A manual for finding pertinent data was produced. It will make this phase easier in future practical applications. The result of this phase was presented in a document containing all the information required for anticipating the condition of the pressurizer. The following information was presented:

Functional data: Functional data includes the global equipment functions and the function-equipment breakdown, which breaks equipment down into groups of components performing the same function.

Design data: This data includes all elements relative to the design of the component. It groups the equipment components, the technical diagrams, the geometric dimensions of the different components, the materials used, the manufacturing procedures and the related costs.

Data on materials: This data is related to the different materials used to make up the component. It groups the chemical and mechanical characteristics of the materials, their properties and a description of any welds (if appropriate).

Effort required for implementation

Operating parameters: Operating parameters include the temperatures, pressures, flow rates and chemical environments of the component in question.

External environment data: It gives information about the ambient environment in which the equipment operates, and its interactions with other components.

Operating data: This data is related to the different operating modes for the equipment, the number of cycles and the number of hours of operation.

Maintenance data: Maintenance data is in relation to preventive maintenance programs, but also includes the various costs of maintenance, data on obsolescence, regulations and reports on safety, reliability and aging. Training the experts with regard to probability concepts is a constraint.

STUK-VTT

The use of bayesian networks to facilitate questioning of experts unfamiliar with probability concepts is an interesting idea.

The method is more suited to quantitative estimation of parameters (e.g. physical parameters).

NNC

The use of groups of experts from different disciplines corresponds well to the objective of anticipation.

The method is not very formalized.

FEJ-GRC

The breakdown of objectives and the use of groups of experts from different disciplines correspond well to the objective of anticipation.

The method is complex and difficult to implement, as it involves several groups of experts.

KEEJAM

This approach based on knowledge engineering is constructive for anticipation from the perspective of exploitation of knowledge.

The method is complex and costly to implement (long 15step process).

CTN-UPM

The method enables obtaining quantitative estimations; it appears better suited to an analysis process.

The method is experimental, and only now being developed.

TUD

The method enables obtaining quantitative estimations; it is also more appropriate for an analysis process.

The method is complex and costly to implement.

LCM

The objectives are similar (managing aging) but the anticipation aspect is insufficiently developed.

A systematic method, which leaves little room for creativity and does not enable the identification of new failures. A very simple, inexpensive method.

TRIZ-AFD

This method of anticipation is above all well suited for choosing solutions for dealing with a failure.

The method is experimental, and still being developed.

It does not enable answering questions related to the relevance and seriousness of potential failures.

RIPBR

This method of maintenance optimization is actually quite close to our objectives. It clearly expresses the need for integrating anticipation based on expert judgment.

The method is still being developed.

PMDA-PIRT

This method corresponds perfectly to the objective of anticipation, though it goes no further than identification and ranking of potential degradation mechanisms. The results constitute input data for managing maintenance and the life cycle of critical equipment in nuclear plants. Feedback data: It includes traditional feedback from the company, feedback on ''analogous'' equipment and from plants outside France, and reports on reviews of the state of the art.

In our application, it became clear that the information most useful for drawing up a list of the potential mechanisms and modes were technical diagrams, operating and environmental data and feedback data.

Step 3. Surveying process: The process is now under way. The first step (identifying potential degradation mechanisms) is completed. The second step (study of the relevance of the mechanisms identified) is being performed. The following paragraphs describe these two steps.

Step 3.1. Stimulation of expert knowledge to identify potential degradation mechanisms: This anticipatory step is very important, as one must identify the degradation mechanisms, which could be the source of potential failures. The analyst knows the degradation mechanisms identified in Reliability Centred Maintenance studies on the equipment in question (where these exist) and those observed in the component and reported in feedback. To identify new mechanisms, he may make use of a list of all degradation mechanisms. To build up this list, he will consult several studies on degradation mechanisms. In our study, the most exhaustive list possible was drawn up.

Feedback on analogous equipment is another helpful source. A component made of the same materials may have had observed failures due to degradation mechanisms which had not been identified for the component being studied. In this case, one must see whether or not the conditions under which they appeared suggest that one might anticipate failures in the equipment being studied.

In this step, the analyst will ask the experts to try to imagine what the component's potential degradation mechanisms might be. Knowing that exchange among the experts can stimulate the emergence of ideas, collective questioning would appear to be best suited to this step. A structured method has been fine tuned to enable stimulat-ing creative thinking on the part of the experts and helping them to make optimum use of the information gathered and their own knowledge [START_REF] Bouzaı¨ene | Stimulation croise´e d'expertise de maintenance: application au parc nucle´aire francais, Actes des confe´rences ESREL 2002[END_REF]. To prepare this method, a bibliographical review of methods to stimulate creativity and group thinking was carried out.

Preparation of the group meeting

Starting with a global list of degradation mechanisms and failure modes, the method developed provides for several successive filters (see Fig. 8). These filters oblige the experts to go over the global list and eliminate irrelevant mechanisms in accordance with a predetermined logic. To complete the global list and ensure that the final list is exhaustive, the experts are asked to do a preliminary exercise: each must note down the potential degradation mechanisms and failure modes for the pressurizer. The new mechanisms are integrated in the global list before the group survey. Six experts with varying professional backgrounds accepted to participate in this first survey step: a pressurizer expert, a reliability and feedback expert, an expert in equipment materials, an expert in degradation mechanisms, a chemical expert and an expert in mechanics and plant operation. The two documents produced in phases 1 and 2 were sent to the experts. Two facilitators presented the information and questioned the experts; two scribes reported on the process and provided the results of the questioning.

Results and evaluation of the method

The group meeting enabled the identification of 18 potential ''degradation mechanism/zone'' pairs and 16 potential ''aging effects/zone'' pairs for the pressurizer over a given period of operation. Six degradation mechanisms (e.g. stratification due to thermal fatigue) and two aging effects (e.g. thermal cracking) were added to the initial list by the experts. The ''multidisciplinary'' aspect was much appreciated by the experts, and judged to be important and necessary. The method was judged efficient with regard to long-term and accidental degradations. The method was judged exhaustive thanks to the use of the global list, which prevented omissions.

The experts are motivated to pursue the experiment.

Step 3.2. Study of the relevance of the degradation mechanisms: example of the Bayesian network: Among the degradation mechanisms identified in the first assessment step, a certain number may be found to be irrelevant. To verify their relevance, we must consider all the parameters, which may lead to their appearance. The tool needed to facilitate the expert's job must identify any cause-effect relationship among several parameters, which might promote the appearance of the degradation mechanism in question.

A causal diagram fills this need perfectly but the expert cannot always say with certainty that the degradation mechanism is relevant; he can only give an approximate opinion, since he is projecting into the future. To quantify these approximations, there is a tool, which links probability values to a causal diagram: the Bayesian network.

This graphic representation of the cause-effect relationships among the different parameters and the relevance of the degradation mechanisms allows the expert to visualize links, which were only tacit in his mind. However, he is not always able to associate probabilities with these different links, but may give more or less favorable opinions. It is then the role of the analyst to pose the appropriate questions, which will later enable the establishment of probabilities.

Fig. 9 shows one example of a network enabling the evaluation of the relevance of thermal fatigue. Other networks were constructed, to study the relevance of the corrosion mechanism, for example, or evaluate the efficiency of an existing or new avoidance solution.

The first step is to build the network structure. To do this, we must identify the parameters with a bearing on the relevance of thermal fatigue. One expert helped to build the network and another validated the structure. This network may be considered to be generic and may now be applied to any component subject to thermal fatigue.

The output node is the relevance of thermal fatigue after N-years, with the possibilities: yes/no. It gives the probability that the fatigue mechanism is relevant or not. The network is composed of 7 input variables in 2 or 3 modes and of 4 intermediate variables in 2 or 3 modes (see Fig. 9). The a priori probabilities are given by feedback or expert judgment. The conditional probabilities were most often given by expert judgment. As the expert is not always able to associate probabilities with the different variables, a questionnaire was drawn up for the experts. The analyst can then express the experts' qualitative responses in terms of probabilities.

Example of one question: Do you think that the number of transients after N-years will be ''low'' or ''high''? The possible answers for each mode are 1: impossible, 2: possible, 3: probable, 4: very probable, 5: certain. The analyst then associates probabilities to this scale.

Conclusion

In the framework of this study, a method was developed to anticipate potential failures due to aging. It consists in four phases: definition of the context and objectives, identification and gathering of the information needed for anticipation, surveying of experts, and synthesizing and exploiting results.

When questioning the experts to obtain input for anticipation, five steps are proposed to identify potential degradation mechanisms (for passive components) or potential failure modes (for active components), (1) study the relevance of these mechanisms (or modes), ( 2) analyze and rank the relevant mechanisms (or modes) (3) analyze and rank potential failures (kinetics, seriousness, effects of aging, consequences) (4) identify potential avoidance solutions and (5), examine the relevance of these solutions in terms of efficiency and cost.

Having reviewed the state of the art in methods for questioning experts, we identified a generic process. At each step described above, we then referred to this generic process which consists in defining the objectives behind the questions, choosing the experts to be surveyed, preparing the questionnaire (selecting appropriate questions), surveying the experts, aggregating and modeling the experts' responses and finally synthesizing and exploiting the question results.

One application of the methodology developed is now under way. It relates to a passive component in a nuclear power plant: the pressurizer. This application will enable the evaluation of the feasibility of the method and the identification of any practical difficulties encountered. This article presents the initial results of this application: implementation of a method for stimulating creative responses in the framework of a meeting of experts from different disciplines to identify potential degradation mechanisms; and the use of Bayesian networks to assess the relevance of the mechanisms identified.

This approach was developed for a specific industrial context, which is the nuclear industry. It could be applied to other sectors which make use of installations and components with long lifetimes or which use specific technologies and are subject to high safety constraints. Generally, there is little feedback (and it is therefore all the more valuable) and the benefits of anticipation are therefore great. We might point to the example of the aerospace or chemical industries. In these particular sectors, certain specific equipment can be very expensive, and replacement poses significant technical problems and can paralyze operations.

The approach developed here enables the identification of potential problems, which may be caused by aging, and adapting maintenance so as to limit the consequences. It should therefore be an essential first step in any predictive technico-economic study, as it enables the assessment of the physical condition of components.

Finally, only failures due to physical aging were considered in this study. The framework could be broadened to take into account other causes of failure such as human error or diagnosis-support, for example, or to conduct risk analyses or predictive studies.

Fig. 1 .

 1 Fig. 1. Structure of the work.

Fig. 3 .

 3 Fig. 3. Description of the global anticipation process.

Fig. 5 .

 5 Fig. 5. Classification of methods for surveying experts.

Fig. 6 .

 6 Fig. 6. Description of the expert elicitation process.

Fig. 7 .

 7 Fig. 7. Pressurizer of a nuclear plant.

Fig. 9 .

 9 Fig. 9. Example of a Bayesian network to study the relevance of thermal fatigue in the pressurizer after N-years.

Table 3

 3 Assessment of the method in relation to anticipation

	Method	Benefits from the perspective of anticipation	Drawbacks
	NUREG -1150	Experts surveyed in two phases, one collective and one
		individual; corresponds quite well to the objective of
		anticipation (creativity and analysis).

  In your opinion, what are the degradation mechanisms and modes which can affect the pressurizer and its surge line? Fig.8. Survey of experts to identify potential degradation mechanisms.

	Global list of degradation mechanisms and
	aging effects
	Expanded list of degradation mechanisms and
	aging effects
		Without giving your reasons, eliminate any mechanisms and
	Sub-list 1	modes from the list which are not important for the component.
	Sub-list 2	For these materials, eliminate non-sensitive mechanisms and modes.
	List of potential mechanisms	For the materials in this operating environment, eliminate the
	and modes	mechanisms and modes not important after N years of operation.
	Potential pairs:	For the mechanisms retained, identify the zones affected by
	mechanisms/zone,	these mechanisms and modes.
	mode/zone

The digit 3, which appears at the beginning of each step is to remind that this process belongs to the third step of the global anticipation process described earlier in Section 3.