Badkobeh Golnaz

King'

Maxime Crochemore

Fewest repetitions in infinite binary words

Keywords: combinatorics on words, repetitions, word morphisms. MSC: 68R15 Combinatorics on words

A square is the concatenation of a nonempty word with itself. A word has period p if its letters at distance p match. The exponent of a nonempty word is the quotient of its length over its smallest period.

In this article we give a proof of the fact that there exists an infinite binary word which contains finitely many squares and simultaneously avoids words of exponent larger than 7/3.

Our infinite word contains 12 squares, which is the smallest possible number of squares to get the property, and 2 factors of exponent 7/3. These are the only factors of exponent larger than 2.

The value 7/3 introduces what we call the finite-repetition threshold of the binary alphabet. We conjecture it is 7/4 for the ternary alphabet, like its repetitive threshold.

Introduction

Repetitions in words is a basic question in Theoretical Informatics, certainly because it is related to many applications although it has first been studied by Thue at the beginning of the twentieth century [START_REF] Thue | Über unendliche Zeichenreihen[END_REF] with a pure theoretical objective. Related results apply to the design of efficient string pattern matching algorithm, to text compression methods and entropy analysis, as well as to the study of repetitions in biological molecular sequences among others.

The knowledge of the strongest constraints an infinite word can tolerate help for the design and analysis of efficient algorithms. The optimal bound on the maximal exponent of factors of the word has been studied by Thue and many other authors after him. One of the first discoveries was that an infinite binary word can avoid factors with an exponent larger than 2, called 2 + -powers. This has been extended by Dejean [START_REF] Dejean | Sur un théorème de Thue[END_REF] to the ternary alphabet and her famous conjecture on the repetitive threshold for larger alphabets has eventually been proved recently after a series of partial results by different authors (see [START_REF] Rao | Last cases of Dejean's conjecture[END_REF][START_REF] Currie | A proof of Dejean's conjecture[END_REF] and references therein).

Another constraint is considered by Fraenkel and Simpson [START_REF] Fraenkel | How many squares must a binary sequence contain?[END_REF]: their parameter to the complexity of binary infinite words is the number of squares occurring in them without any restriction on the number of occurrences. It is fairly straightforward to check that no infinite binary word can contain less than three squares and they proved that some of them contain exactly three. Two of these squares appear in the cubes 000 and 111 so that the maximum exponent is 3 in their word. In this article we produce an infinite word with few distinct squares and a smaller maximal exponent.

Fraenkel and Simpson's proof uses a pair of morphisms, one to get an infinite word by iteration, the other to produce the final translation on the binary alphabet. Their result has been proved with different pairs of morphisms by Rampersad et al. [START_REF] Rampersad | Avoiding large squares in infinite binary words[END_REF] (the first morphism is uniform), by Harju and Nowotka [START_REF] Harju | Binary words with few squares[END_REF] (the second morphism accepts any infinite square-free word), and by Badkobeh and Crochemore [START_REF] Badkobeh | An infinite binary word containing only three distinct squares[END_REF] (the simplest morphisms).

In this article we show that we can combine the two types of constraints for the binary alphabet: producing an infinite word whose maximal exponent of its factor is the smallest possible while containing the smallest number of squares. The maximal exponent is 7/3 and the number of squares is 12 to which can be added two words of exponent 7/3.

It is known from Karhumäki and Shallit [START_REF] Karhumäki | Polynomial versus exponential growth in repetitionfree binary words[END_REF] that if an infinite binary word avoids 7/3-powers it contains an infinite number of squares. Proving that it contains more than 12 squares is indeed a matter of simple computation.

Shallit [START_REF] Shallit | Simultaneous avoidance of large squares and fractional powers in infinite binary words[END_REF] has built an infinite binary word avoiding 7/3 + -powers and all squares of period at least 7. His word contains 18 squares.

Our infinite binary word avoids the same powers but contains only 12 squares, the largest having period 8. As before the proof relies on a pair of morphisms satisfying suitable properties. Both morphisms are almost uniform (up to one unit). The first morphism is weakly square-free on a 6-letter alphabet, and the second does not even correspond to a uniquely-decipherable code but admits a unique decoding on the words produced by the first. To get the morphisms, we first examined carefully the structure of long words satisfying the conditions and obtained by backtracking computation. Then, we inferred the morphisms from the regularities found in the words.

After introducing the definitions and main results in the next section, we provide a weakly square-free morphism and the infinite square-free word on 6 letters it generates in Section 3. Section 4 shows how this word is translated into an infinite binary word satisfying the constraints. In the conclusion we define the new notion of finite-repetition threshold and state a conjecture on its value for the 3-letter alphabet.

Repetitions in binary words

A word is a sequence of letters drawn from a finite alphabet. We consider the binary alphabet B = {0, 1}, the ternary alphabet A 3 = {a, b, c}, and the 6-letter alphabet A 6 = {a, b, c, d, e, f}.

A square is a word of the form uu where u is a nonempty (finite) word. A word has period p if its letters at distance p are equal. The exponent of a nonempty word is the quotient of its length over its smallest period. Thus, a square is any word with an even integer exponent.

In this article we consider infinite binary words in which a small number of squares occur.

The maximal length of a binary word containing less than three square is finite. It can be checked that it is 18, e.g. 010011000111001101 contains only 00 and 11. But, as recalled above, this length is infinite if 3 squares are allowed to appear in the word. A simple proof of it relies on two morphisms f and h 0 defined as follows. The morphism f is defined from

A * 3 to itself by    f (a) = abc, f (b) = ac, f (c) = b.
It is known that the infinite word f = f (a) ∞ it generates is square-free (see [START_REF] Lothaire | Combinatorics on Words[END_REF]Chapter 2]). The morphism h 0 is from A * 3 to B * and defined by

   h(a) = 01001110001101, h(b) = 0011, h(c) = 000111.
Then the result is a consequence of the next statement.

Theorem 1 ([1])

The infinite word h 0 = h 0 (f (a) ∞ ) contains the 3 squares 00, 11 and 1010 only. The cubes 000 and 111 are the only factors occurring in h and of exponent larger than 2.

It is impossible to avoid 2 + -powers and keep a bounded number of squares. As proved by Karhumäki and Shallit [START_REF] Karhumäki | Polynomial versus exponential growth in repetitionfree binary words[END_REF], the exponent has to go up to 7/3 to allow the property.

In the two following sections we define two morphisms and derive the properties that we need to prove the next statement.

Theorem 2 There exists an infinite binary word whose factors have an exponent at most 7/3 and that contains 12 squares, the fewest possible.

Our infinite binary word contain the 12 squares 0 2 , 1 2 , (01) 2 , (10) 2 , (001) 2 , (010) 2 , (011) 2 , (100) 2 , (101) 2 , (110) 2 , (01101001) 2 , (10010110) 2 , and the two words 0110110 and 1001001 of exponent 7/3.

Proving that it is impossible to have less than 12 squares in the previous statement results from the next table. It has been obtained by pruned backtracking sequential computation that avoids exhaustive search. It shows the maximal length of binary words whose factors have an exponent at most 7/3, for each number s of squares, 0 ≤ s ≤ 11.

s

= 0 1 2 3 4 5 6 7 8 9 10 11 ℓ(s) = 3 5 8 12 14 18 24 30 37 43 83 116 3 A weakly square-free morphism on six letters

In this section we consider a specific morphism used for the proof of Theorem 2. It is called g and defined from A * 6 to itself by:

             g(a) = abac, g(b) = babd, g(c) = eabdf, g(d) = fbace, g(e) = bace, g(f) = abdf.
We prove below that the morphism is weakly square-free in the sense that g = g ∞ (a) is an infinite square-free word, that is, all its finite factors have an exponent smaller than 2. Note that however it is not square-free since for example g(cf) = eabdfabdf contains the square (abdf) 2 . This prevents from using characterisation of square-freeness of the morphism, or equivalently of the fixed points of the morphism. As far as we know only an ad hoc proof is possible.

The set of codewords g(a)'s (a ∈ A 6 ) is a prefix code and therefore a uniquely-decipherable code. Note also that any occurrence of abac in g(w), for w ∈ A * 6 , uniquely corresponds to an occurrence of a in w. The proof below relies on the fact that not all doublets and triplets (words of length 2 and 3 respectively) occur in g, as the next statements show.

Lemma 1 The set of doublets occurring in g is D = {ab, ac, ba, bd, cb, ce, da, df, ea, fb}.

Proof. Note that all letters of A 6 appear in g. Then doublets ab, ac, ba, bd, ce, df, ea, fb appear in g because they appear in the images of one letter. The images of these doublets generate two more doublets, cb and da, whose images do not create new doublets.

Lemma 2

The set of triplets in g is T = {aba, abd, acb, ace, bab, bac, bda, bdf, cba, cea, dab, dfb, eab, fba}.

Proof. Triplets appear in the images of a letter or of a doublet. Triplets found in images of one letter are: aba, abd, ace, bab, bac, bdf, eab, fba. The images of doublets occurring in g, in set D of Lemma 1, contain the extra triplets: acb, bda, cba, cea, dab, dfb.

Table 1: Gaps of abac: words between consecutive occurrences of abac in g. They are images of gaps between consecutive occurrences of a.

g(b)

= babd 4 g(cb)

= eabdfbabd 9 g(bd)

= babdfbace 9 g(ce)

= eabdfbace 9 g(bdfb) = babdfbaceabdfbabd 17

To prove that the infinite word g is square-free we first show that it contains no square with less than four occurrences of the word g(a) = abac. Then, we show it contains no square with at least four occurrences of it. The word abac is chosen because its occurrences in g correspond to g(a) only, so they are used to synchronise the parsing of the word according to the codewords g(a)'s.

Lemma 3 No square in g can contain less than four occurrences of abac.

Proof. Assume by contradiction that a square ww in g contains less than four occurrences of abac. Let x be the shortest word whose image by g contains ww.

Then x is a factor of g that belongs to the set a((A 6 \ {a}) * a) 5 . Since two consecutive occurrences of a in g are separated by a string of length at most 4 (the largest such string is indeed bdfb as a consequence of Lemma 2), the set is finite.

The square-freeness of all these factors has been checked via an elementary implementation of the test, which proves the result.

Proposition 1 No square in g can contain at least four occurrences of abac.

Proof. The proof is by contradiction: let k be the maximal integer for which g k (a) is square-free and let ww be a square occurring in g k+1 (a) and containing at least 4 occurrences of abac. Distinguishing several cases according to the words between consecutive occurrences of abac (see Table 1), we deduce that g k (a) is not square-free, the contradiction.

The square ww can be written

v 0 (abac • • • abac)u 1 v 1 (abac • • • abac)u 2
where v 0 , u 1 , v 1 , and u 2 contain no occurrence of abac. It occurs in the image of a factor of g. The central part of w starting and ending with abac is the image of a unique word U factor of g k (a) due to the code property:

g(U ) = v -1 0 wu -1 1 = v -1 1 wu -1 2 .
We split the proof in two parts according to whether abac occurs in u 1 v 1 or not.

No abac in u 1 v 1 . We consider five cases according to the value of u 1 v 1 , the gap of abac (see Table 1).

1.

u 1 v 1 = babd corresponds to g(b) only. If either u 1 or v 1 is empty, then v 0 or u 2 is g(b)
, in either case we get bU bU or U bU b that are squares. Else v 0 has a suffix d so it belongs to g(b), and again bU bU is a square in g.

2. u 1 v 1 = eabdfbabd corresponds to g(cb)
only. An occurrence of cb always belongs to g(ab) therefore U has a prefix abd and a suffix aba, and the letter after aba is c. If v 1 is empty, u 2 has a prefix eabdfbabd so it is g(cb) and again U cbU cb is a square. If v 1 is not empty then v 0 has a suffix d, suffix of g(b), therefore bU cbU c is a square.

3. u 1 v 1 = babdfbace corresponds to g(bd). The word abda is a factor of g(ba) only so U has a prefix aba and a suffix ba. If |u 1 | = 0, v 0 = babdfbace can only be g(bd) so bdU bdU is a square. Otherwise u 2 must have a prefix b; since U has a suffix ba the next letter after it is either b or c; as only g(b) is prefixed by b the letter is b so u 2 has a prefix or is a prefix of g(b), and we know that bab is always followed by d thus U bdU bd is a square.

4. u 1 v 1 = eabdfbace corresponds to g(ce) only. If u 1 is empty, v 0 is g(ce)
so ceU ceU is a square. Otherwise, u 2 has a prefix or is a prefix of g(c); the next letter after g(c) is either b or e; (see Lemma 1); if it is b the right-most U has a suffix aba but the left-most U has a suffix fba, which cannot be. Therefore the letter after c is e and U ceU ce is a square.

5

.

u 1 v 1 = babdfbaceabdfbabd. If |v 1 | > 12, v 0 has a suffix g(dfb)
and the letter before it is b, so bdfbU bdfbU is a square. If 0 < |v 1 | ≤ 12, then |u 1 | ≥ 5, so u 2 has a prefix or is a prefix of g(bd) so the next letter is either a or f. If it is a the right-most U has a suffix ba but v 0 is a suffix of or has a suffix g(b); the letter before it is either g(c) or g(f); if it is c then U has a prefix abd and bdfbabd is from the concatenation of g(c) and g(b) or g(dfb); in either case the left occurrence of U will have ea as a suffix, a contradiction since fbU bdfbU bd and U bdfbU bdfb are both squares.

An occurrence of abac in u 1 v 1 . Then the suffix of u 1 is either aba, ab or a while the respective prefix of v 1 is c, ac or bac. Note that c is followed either by b or e (Lemma 1) and that cb occurs only in the image of ab. Then if the occurrence of abac is followed by b, the occurrence of cb in v 0 is preceded by aba, and then there is a square starting 1, 2 or 3 positions before the occurrence of ww, which brings us back to the first case. Therefore, abac is followed by e.

The occurrence of abace comes from g(ac), and by Lemma 2 u 1 v 1 contains an occurrence of g(bac). So, the occurrence of abace is preceded by d, and since da occurs only in the image of ba, the occurrence of da in u 2 is followed by bac, which yields a square starting 1, 2 or 3 positions after the occurrence of ww. Again this takes us back to the first case.

In all cases we deduce the existence of a square in g k (a), which is a contradiction with the definition of k. Therefore there is no square in g containing at least four occurrences of abac.

The next corollary is a direct consequence of Lemma 3 and Proposition 1.

Corollary 1

The infinite word g is square-free, or equivalently, the morphism g is weakly square-free.

Binary translation

The second part of the proof of Theorem 2 consists in showing that the special infinite square-free word on 6 letters introduced in the previous section can be transformed into the desired binary word. This is done with a second morphism

h from A * 6 to B * defined by              h(a) = 10011, h(b) = 01100, h(c) = 01001, h(d) = 10110, h(e) = 0110, h(f) = 1001.
Note that the codewords of h do not form a prefix code, nor a suffix code, nor even a uniquely-decipherable code! We have for example g(ae) = 10011•0110 = 1001 • 10110 = g(fd). However, parsing the word h(y) when y is a factor of g is unique due to the absence of some doublets and triplets in it (see Lemmas 1 and 2). For example fd does not occur, which induces the unique parsing of 100110110 as 10011 • 0110.

Proposition 2

The infinite word h = h(g ∞ (a)) contains no factor of exponent larger than 7/3. It contains the 12 squares 0 2 , 1 2 , (01) The proof is based on the fact that occurrences of 10011 in h identify occurrences of a in g and on the unique parsing mentioned above. It proceeds by considering several cases according to the gaps between consecutive occurrences of 10011 (see Table 2), associated with gaps between consecutive occurrences of a in g, which leads to analyse paths in the graph of Figure 1.

Proof. We show that if h would contain a square not in the list it would come from a square in g, which cannot be since g is square-free (Corollary 1).

Suppose h contains the square w 2 . It is a factor of h(g k (a)), for some integer k and can be written v 0 (h(a) Table 2: Gaps between consecutive occurrences of 10011 in h.

• • • h(a))u 1 v 1 (h(a) • • • h(a))u 2 .
h(b) = 01100 5 h(cb) = 0100101100 10 h(bd) = 0110010110 10 h(ce) = 010010110 9 h(bdfb) = 0110010110100101100 19
w is the image of a unique square-free factor U of g k (a) due to the unique parsing mentioned above:

h(U ) = (h(a) • • • h(a)) = v -1 0 wu -1 1 = v -1 1 wu -1 2 .
We proceed through different cases as in the proof of Proposition 1.

No h(a) in u 1 v 1 .

1. u 1 v 1 = 01100 corresponds to h(b) only.

If |v 1 | > 1, then v 0 belongs to h(b), bU bU is a square. Else |u 1 | ≥ 4 so u 2 belongs to h(b), it cannot belong to h(e) since ae is not a factor of g, therefore U bU b is a square of g.

2. u 1 v 1 = 0110010110 corresponds to h(bd). v 0 (h(a) • • • h(a)) h(bd) (h(a) • • • h(a)) u 2
the word abda is a factor of g(ba) only, so U has a prefix abac and a suffix ba (Note that U cannot be aba since ababdaba is not a factor of g).

v 0 (h(abac) • • • h(ba)) h(bd) (h(abac) • • • h(ba)) u 2
If u 2 comes from or has a prefix h(b) then the letter after bab is always d so we have the square U bdU bd. Then u 2 is a prefix of or has a prefix h(c), the longest common prefix (LCP) of h(c) and h(b) is 01, so v 0 has a suffix 10010110, which is a suffix of h(bd) or h(ce). If v 0 comes from h(bd) then we have the square bdU bdU . So v 0 is a suffix of h(ce) The sign XX shows that the particular branch of the trie terminates because either a square occurs or the sequence is not a factor of g. The sign X on the other hand represents the termination of a particular branch as a consequence of the discontinuation of the corresponding branch in the other trie. If we continue these tries we will have: so we have the same situation as at the starting point; but U is shorter in this case, therefore if we continue this process we should have ce abac babd fbace abdf babd abac babd fbace abdf bace a but abdf bace is the image of fe that is not in D (Lemma 1).

h(ce) (h(abac) • • • h(ba)) h(bd) (h(abac) • • • h(ba)) h(c).
3. u 1 v 1 = 0100101100 corresponds to h(cb).

The word acba is a factor of g(ab) only, so U has a prefix abd and a suffix aba: v 0 (h(abd) . . . h(aba)) h(cb) (h(abd) . . . h(aba)) u 2

The word u 2 comes from or has a prefix h(c). If the letter after it is b, we have the square U cbU cb.

Otherwise u 2 comes from or has a prefix h(ce). If v 0 comes from or has a suffix h(b) then we have the square bU cbU c.

Therefore the letter before U is e preceded by c, i.e. the string before the left U is ce: Now we have the same situation as in the previous case h(g(ce)) (h(g(abac)) . . . h(g(ba))) h(g(bd)) (h(g(abac)) . . . h(g(ba))) h(g(c)).

h(ce) (h(abd) . . . h(aba)) h(cb) (h(abd) . . . h(aba)) h(ce).

u

1 v 1 = 010010110 corresponds to h(ce) only.
Before c is always ba (Lemma 2) and after e is ab (Lemma 2), so ab is a prefix of U and ba is a suffix of U :

v 0 (h(ab) . . . h(ba)) h(ce) (h(ab) . . . h(ba)) u 2 .
(i): u 2 belongs to h(cb) since we cannot have U ceU ce and the letter after c is b or e (Lemma 1):

v 0 (h(ab) . . .

h(ba)) h(ce) (h(ab) . . . h(ba)) h(cb)

The letter before bacb is a so: v 0 (h(ab) . . . h(aba)) h(ce) (h(ab) . . . h(aba)) h(cb).

NOTE: U is not aba since abaceabacb is not a factor of g k (a). Now abace is a prefix of the image of ac so U has a prefix abdf and the word before it is either ce or b; the first choice gives the square ceU ceU and the second choice: This is the same situation as the next case and we will see that after going one step back it brings us back to this case again. Now we are exactly in the same situation as at the beginning except that the length of the word X = abdf . . . a is shorter than U . Repeating this process enough times we should see that the word babd fbace abac babd abac eabdf bace abac babd aba which is the image of bdabaceaba, is not a factor of g k (a).

h(b) (h(abdf) . . . h(aba)) h(ce) (h(abdf) . . . h(aba)) h(cb).
(ii): u 2 belongs to h(b) (the LCP of h(c) and h(b) is 01) so v 0 must have a suffix 0010110, which belongs to h(bd) because if it belongs to h(ce) then ceU ceU is a square. If u 2 belongs to h(c), the LCP of h(c) and h(b) is 01 so u 0 must have a suffix 10010110100101100, therefore u 0 belongs to h(bdfb). But bdfbU bdfbU is a square and a factor of g k (a); a contradiction, so u 2 belongs to or has a prefix h(b). We have two choices here.

(i): the next word after the right occurrence of U is ba. The LCP of h(bd) and h(ba) is 10, u 0 has suffix of 110100101100, so it either belongs to h(dfb) or h(acb). The first case gives that dbfU bdbfU b is a square and a factor of g k (a), a contradiction. So u 0 belongs to h(acb): Now this is the next case so if we go back enough steps we should see that the length of U decreases and at the end we get ac babd abac eabdf babd abac eaba but this is not a factor of g k (a), a contradiction.

(ii): the word after U is bd. Now here the only possible letter after abd is a since if it is f it is a prefix of fb so we have U bdfbU bdfb, a contradiction. As the LCP of h(bdfb) and h(bda) is 01100101101001 u 0 must have a suffix 01100 so it can belong to h(ab) or h(acb).

(I):

h(ab) (h(a) . . . h(a)) h(bdfb) (h(a) . . . h(a)) h(bda).

Only using D, T and the Figure 1 we can continue building U , h(ab) (h(ace) . . . h(ba)) h(bdfb) (h(acea) . . . h(ba)) h(bda).

Continuing further we get:

h(abac eabdf babd abac . . . abac babd fbace abdf babd abac . . . abac babda).

This is the image of h(g(acb a . . . a bdfb a . . . a ba))

and we are back to the case above.

(II): h(acb) (h(a) . . . h(a)) h(bdfb) (h(a) . . . h(a)) h(bda).

Using the same method we build the word U : ac b abd . . . ba bd fb ace . . . ba bd a.

Here we cannot go further as U cannot have abd nor ace as prefixes at the same time.

An occurrence of h(a) in u 1 v 1 . Looking at Figure 1, the image of the concatenation of two connected nodes (distance 1 arrow) are the possibilities for u 1 v 1 h(a), but note that the second period of the square must start within h(a), starting point of the arrow, otherwise it is one of the cases above. If the lengths of both nodes are larger than 2 then by unique parsing we are bound to have a square in g k (a) and get a contradiction. So we have to consider only the four cases where one of the nodes is ba:

1. u 1 v 1 = h(bacb) = 01100100110100101100, so u 2 must have a prefix h(b) and u 0 a suffix of h(cb), before cb is always a, so acbU bacbU b is a square in g k (a).

2. u 1 v 1 = h(bace) = 0110010011010010110, so u 2 must have a prefix h(b) and u 0 a suffix h(ce), before ce is always a, so aceU baceU b is a square in g k (a).

3. u 1 v 1 = h(ceab) = 0100101101001101100, so u 2 must have a prefix of h(ce) and u 0 a suffix of h(b), after ce is always a, so bU ceabU cea is a square in g k (a).

4. u 1 v 1 = h(bdab) = 01100101101001101100, so using tries as before shows that after enough backward iteration we should have fbace abdf babd abac babd abac eabdf babd abac babd which contains a square.

In all cases the conclusion is that we get a square in g k (a), a contradiction with the definition of k. This completes the proof of Proposition 2.

Theorem 2 follows immediately from Proposition 2.

Conclusion

The constraint on the number of squares imposed on binary words slightly differs from the constraint considered by Shallit [START_REF] Shallit | Simultaneous avoidance of large squares and fractional powers in infinite binary words[END_REF]. The squares occurring in his word have period smaller than 7. Our word contains less squares but their maximal period is 8.

Looking at repetitions in words on larger alphabets, the subject introduces a new type of threshold, that we call the finite-repetitions threshold (FRt). For the alphabet of a letters, FRt(a) is defined as the smallest rational number for which there exists an infinite word avoiding FRt(a) + -powers and containing a finite number of r-powers, where r is Dejean's repetitive threshold. Karhumäki and Shallit results as well as ours show that FRt(2) = 7/3. Our result additionally proves that the associated minimal number of squares is 12.

Computation shows that the maximal length of (7/4) + -free ternary word with only one 7/4-repetition is 102. This leads us state the following conjecture, which has been tested up to length 20000.

Conjecture 1

The finite-repetitions threshold of 3-letter alphabet is 7 4 and the associated number of 7 4 -powers is 2.

Values for larger alphabets remain to be explored.

Figure 1 :

 1 Figure 1: Graph showing immediate successors of gaps in the word g: a suffix of it following an occurrence of a is the label of an infinite path.

  ce abac babd fbace abdf babd abac eabdf bace abac babd abac eabdf . . .ba bd abac babd fbace abdf babd abac eabdf bace abac babd abac eabdf . . .ba c which is the image of eabdf bace abac . . . abac babd fbace abac . . . abac e itself image of ce a . . . a bd a . . . a c

  continue the above tries we get: b abd fbace abac babd abac eabdf babd abac babd fbace abdf ba . . .ba ce abdf bace abac babd abac eabdf babd abac babd fbace abdf ba . . .ba cb which is the image of bd abac babd fbace abdf . . .ba ce abac babd fbace abdf . . .ba b.

5 . u 1 v 1 =

 51 h(bd) (h(ab) . . . h(ba)) h(ce) (h(ab) . . . h(ba)) h(b). we have bd abac babd fbace a . . .ba ce abac babd fbace a . . .ba bd. This is factor of g(b abdf . . . a ce abdf . . . a cb) which is the previous case. 0110010110100101100 corresponds to h(bdfb) only. This case is dealt with the same method.

u 0

 0 (h(a) . . . h(a)) h(bdfb) (h(a) . . . h(a)) u 2 .

  h(acb) (h(abda) . . . h(a)) h(bdf b) (h(abda) . . . h(a)) h(ba).Prefixes and suffixes of U are determined only by looking at D and T . abac babd abac eabdf bace . . .abac babd fbace abdf babd abac eabdf bace . . .abac babd fbace abac which is the image of ab ace . . . a bdfb ace . . . a bda.


Hugo Boujut
email: hugo.boujut@labri.fr

Jenny Benois-Pineau

Remi Megret
email: remi.megret@ims-bordeaux.fr

Fusion of multiple visual cues for visual saliency extraction from wearable camera settings with strong motion

In this paper we are interested in the saliency of visual content from wearable cameras. The subjective saliency in wearable video is studied first due to the psycho-visual experience on this content. Then the method for objective saliency map computation with a specific contribution based on geometrical saliency is proposed. Fusion of spatial, temporal and geometric cues in an objective saliency map is realized by the multiplicative operator. Resulting objective saliency maps are evaluated against the subjective maps with promising results, highlighting interesting performance of proposed geometric saliency model.

Introduction

Since recently, the focus of attention in video content understanding, presentation, and assessment has moved toward incorporating of visual saliency information to drive local analysis process. Hence in the paper from A. Benoit [START_REF] Ionescu | A color-action perceptual approach to the classification of animated movies[END_REF], the task is to classify animated movies from low motion content. The global saliency is therefore expressed by rhythm and other motion descriptors. In another task such as content viewing on a mobile screen, the most salient regions are selected according to the perceptual model of L. Itti and C. Koch [START_REF] Itti | Computational modelling of visual attention[END_REF]. If we simplify the concept of saliency to its very basic definition, we can reasonably say that visual saliency is what attracts human gaze. Numerous psycho-visual studies which have been conducted since the last quarter of 20th century uncovered some factors influencing it. Considering only signal features, the sensitivity to color contrasts, contours, orientation and motion observed in image plane has been stated by numerous authors [START_REF] Itti | Quantifying the contribution of low-level saliency to human eye movements in dynamic scenes[END_REF][START_REF] Le Meur | Predicting visual fixations on video based on low-level video features[END_REF]. Nevertheless, only these features are not sufficient to delimit the area in the image plane which is the strongest gaze attractor. In [START_REF] Tatler | The central fixation bias in scene viewing: Selecting an optimal viewing position independently of motor biases and image feature distributions[END_REF], the author states, for still images, that observers show a marked tendency to fixate the center of the screen when viewing scenes on computer monitors. The authors of [START_REF] Dorr | Variability of eye movements when viewing dynamic natural scenes[END_REF] come to the same conclusion for dynamic general video content such as movies and Hollywood trailers. This is why the authors of [START_REF] Brouard | Cartes de Saillance Spatio-Temporelle basées Contrastes de Couleur et Mouvement Relatif[END_REF] propose the third cue which is the geometrical saliency modelled by a 2D Gaussian located at the image center. While signal based cues remain valuable saliency indicators, we claim that geometrical saliency depends on global motion and camera settings in the dynamic scene. Nowadays, the attention of computer vision community is more and more turned to the new forms of video content: such as wearable video cameras, or "egocentric" view of the world [START_REF] Lee | Discovering important people and objects for egocentric video summarization[END_REF][START_REF] Starner | Visual contextual awareness in wearable computing[END_REF]. Some attempts to identify visual saliency mainly on the basis of the frequency of repetition of visual objects and regions in the wearable video content have recently been made in [START_REF] Ren | Egocentric recognition of handled objects: Benchmark and analysis[END_REF]. We are specifically interested in building visual saliency maps by fusion of all cues in the pixel domain for the case of "egocentric" video content recorded with wearable cameras. Hence in this paper, we propose an automatic method of spatio-temporal saliency extraction for wearable camera videos with a specific accent on geometrical saliency dependent on strong wearable camera motion. We evaluate the proposed method with regard to subjective saliency maps obtained from gaze tracking.

The rest of the paper is organized as follows. In section 2 we will introduce the context and motivation of our work. In section 3 we report the method and our psycho-visual experiments for reference subjective saliency map construction. The objective saliency map will be presented in Section 4. The evaluation of the latter will be described in Section 5. Section 6 will conclude our work and outline its perspectives.

Motivations

The context of actual work is the multi-disciplinary research on Alzheimer disease [START_REF] Karaman | Human Daily Activities Indexing in Videos from Wearable Cameras for Monitoring of Patients with Dementia Diseases[END_REF][START_REF] Szolgay | Detection of moving foreground objects in videos with strong camera motion[END_REF]. The goal here is to ensure an objective assessment of the capacity of patients to conduct the IADL (Instrumental Activities of Daily Living). In [START_REF] Karaman | Human Daily Activities Indexing in Videos from Wearable Cameras for Monitoring of Patients with Dementia Diseases[END_REF] the framework for video monitoring with wearable camera was designed. The wearable camera is fixed in an ergonomic position on the patient's shoulder (see Fig. 1) and the recording is realized at patient's home. The computer vision task consists in an automatic recognition and indexing of IADLS from a taxonomy proposed to patients in each recording. We seek to limit the automatic analysis of the observed dynamic scene to the area of interest which is visually salient for the medical practitioner. Hence in the next section we study the subjective saliency of this type of content. Then we propose an automatic, objective saliency model from video data and assess it with regard to the built subjective saliency ground truth.

Subjective saliency map

In this section we present our approach for subjective saliency extraction on wearable video.

Eye-tracker experiment

The subjective saliency maps expressing user attention are obtained on the basis of psycho-visual experiment consisting in measuring the gaze positions on videos from wearable video camera. The map of the visual attention has to be built on each frame of these videos. Videos from wearable camera differ from traditional video scenes: the camera films the user point of view, including his hands. Unlike traditional videos, wearable camera videos have a very high temporal activity due to the strong ego-motion of the wearer.

The gaze positions are recorded with an eye-tracker. We used HS-VET 250Hz from Cambridge Research Systems Ltd. This device is able to record 250 eye positions per second. The videos we display in this experiment have a framerate of 29.97 frames per second. A total of 28 videos filming the activities of daily living of patients and healthy volunteers are displayed to each participant of the experiment. This represents 17 minutes and 30 seconds of video. The resolution of the videos is 1280x960 pixels and the storage format is raw YUV 4:2:0. The experiment conditions and the experiment room is compliant to the recommendation ITU-R BT.500-11 [START_REF]Methodology for the subjective assessment of the quality of television pictures[END_REF]. Videos are displayed on a 23 inches LCD monitor with a native resolution of 1920x1080 pixels. To avoid image distortions, videos are not resized to screen resolution. A mid-gray frame is inserted around the displayed video. 25 participants were gathered for this experiment, 10 women and 15 men. For 5 participants some problems occurred in the eye-tracking recording process. So we decided to exclude these 5 records.

After looking at gaze position records on video frames, we stated that gaze anticipated camera motion and user actions. This phenomenon has been already reported by M. Land et al. in [START_REF] Land | The roles of vision and eye movements in the control of activities of daily living[END_REF]. They state that visual fixation does precede motor manipulation, putting eye movements in the vanguard of each motor act, rather than as adjuncts to it.

Nevertheless, gaze positions cannot directly be applied as ground truth to compare automatic saliency model we aim at. They must be processed in order to get the subjective saliency map. The next section describes how to build subjective saliency maps from gaze tracking records.

Subjective saliency map build method

Any objective human visual perception model has to be validated and evaluated with regard to a ground truth. The ground truth is the subjective saliency in this case. The subjective saliency is built from eye position measurements. There are two reasons for which eye positions cannot be directly used to represent the visual attention. First, the eye positions are only spots on the frame and do not represent the field of view. Secondly, to get accurate results, the saliency map is not built with the eye tracking data from one subject, but from many subjects. So the subjective saliency map should provide an information about the density of eye positions.

The method proposed by D. S. Wooding [START_REF] Wooding | Eye movements of large populations: Ii. deriving regions of interest, coverage, and similarity using fixation maps[END_REF] fulfils these two constraints. In the case of video sequences, the method is applied on each frame I of a video sequence K. The process result is a subjective saliency map S subj (I) for each frame I. With this method, the saliency map is computed in three steps. In the first step, for each eye measure m of frame I, a two dimensional Gaussian is applied at the center of the eye measure (x 0 , y 0 ) m . The two dimensional Gaussian depicts the fovea projection on the screen. The fovea is the central retina part where the vision is the most accurate. In the Sensibility to Light [START_REF] Hood | Sensitivity to light[END_REF] book chapter from D.C. Hood and M.A. Finkelstein (1986), the authors stated that the fovea covers an area from 1.5 • to 2 • in diameter at the retina center. D.S. Wooding proposed to set the Gaussian spread σ to an angle of 2 • .

For the eye measure m of the frame I, a partial saliency map S subj (I, m) is computed [START_REF] Ionescu | A color-action perceptual approach to the classification of animated movies[END_REF].

S subj (I, m) = Ae - (x-x 0m ) 2 2σ 2 x + (y-y 0m ) 2 2σ 2 y with σ x = σ y = σ and A = 1 (1)
Then, at the second step, all the partial saliency maps S subj (I, m) of frame S i I are added into S subj ′(I) [START_REF] Itti | Computational modelling of visual attention[END_REF].

S subj ′(I) = N I m=0 S subj (I, m) (2)
where N I is the number of eye measures recorded on all the subjects for the frame I. Finally, at the third step, the saliency map S subj ′(I) is normalized by the highest value argmax of S subj ′(I). The normalized subjective saliency map is stored in S subj (I). An example of a subjective saliency map is presented in Fig. 2. Subjective saliency maps cannot be used for real-world applications in video analysis, as requiring psycho-visual experiments for each video to process. We are thus interested in an automatic objective saliency maps. The subjective saliency maps will be used as the ground truth to asses the objective maps automatically built.

Objective saliency map

To delimit the area of video analysis in video frames to the regions which are potentially interesting to human observers we need to model visual saliency on the basis of video signal features. Here we follow the results of community research we reported in Section 1 proposing fusion of spatial, temporal and geometric cues. We extend the state-of-the art approaches by a specific modelling of geometrical saliency and propose multiplicative fusion of all three cues.

Spatial saliency map

The spatial saliency map S sp is mainly based on color contrasts [START_REF] Aziz | Fast and robust generation of feature maps for regionbased visual attention[END_REF]. We used the method from O. Brouard, V. Ricordel and D. Barba [START_REF] Brouard | Cartes de Saillance Spatio-Temporelle basées Contrastes de Couleur et Mouvement Relatif[END_REF]. The spatial saliency map extraction is based on seven color contrast descriptors. These descriptors are computed in the HSI color space. On the contrary to RGB color system, the HSI color space is well suited to describe color interpretation by humans.

S ′ sp (I, i) = 1 7 7 ς=1 V ς (I, i) (3)
Finally, S ′ sp (I, i) is normalized between 0 and 1 to S sp (I, i)according to its maximum value.

Temporal saliency map

The objective spatio-temporal saliency map model requires a temporal saliency dimension. This section will describe how to build temporal saliency maps. The temporal saliency map S t models the attraction of attention to motion singularities in a scene. The visual attention is not grabbed by the motion itself. The gaze is attracted by the motion difference between the real motion scene and the global motion scene. The motion difference is called the residual motion. O. Brouard et al. [START_REF] Brouard | Cartes de Saillance Spatio-Temporelle basées Contrastes de Couleur et Mouvement Relatif[END_REF] and S. Marat [START_REF] Marat | Modelling spatio-temporal saliency to predict gaze direction for short videos[END_REF] propose a temporal saliency map model that takes advantage of the residual motion. In this paper, we have implemented the model from O. Brouard et al. [START_REF] Brouard | Cartes de Saillance Spatio-Temporelle basées Contrastes de Couleur et Mouvement Relatif[END_REF].

The temporal saliency map is computed in three steps. The first one is the optical flow estimation. Then the global motion is estimated in order to get the residual motion. Finally a psycho-visual filter is applied on the residual motion.

To compute the optical flow, we have applied the Lucas Kanade method from OpenCV library [START_REF] Bouguet | Pyramidal implementation of the lucas kanade feature tracker[END_REF]. The optical flow was sparsely computed on 4x4 blocks, as good results were reported in [START_REF] Boujut | A metric for no-reference video quality assessment for hd tv delivery based on saliency maps[END_REF] when using 4 × 4 macro-block motion vectors from the H.264 AVC compressed stream. The next step in temporal saliency computation is the global motion estimation.

The goal here is to estimate a global motion model to differentiate then local motion from camera motion. In this work, we follow the preliminary study from [START_REF] Boujut | A metric for no-reference video quality assessment for hd tv delivery based on saliency maps[END_REF] and use a complete first order affine model (4) :

dx i = a 1 + a 2 x + a 3 y dy i = a 4 + a 5 x + a 6 y (4)
Here θ = (a 1 , a 2 , . . . , a 6 ) T is the parameter vector of the global model ( 4) and (dx i , dy i ) T is the motion vector of a block. To estimate this model, we used robust least square estimator presented in [START_REF] Kraemer | Scene Similarity Measure for Video Content Segmentation in the Framework of Rough Indexing Paradigm[END_REF]. We denote this motion vector V θ (I, i). Our goal is now to extract the local motion in video frames i.e. residual motion with regard to model [START_REF] Le Meur | Predicting visual fixations on video based on low-level video features[END_REF]. We denote the macro-block optical flow motion vector V c (I, i). The residual motion V r (I, i) is computed as a difference between block motion vectors and estimated global motion vectors.

Finally, the temporal saliency map S t (I, i) is computed by filtering the amount of residual motion in the frame. The authors of [START_REF] Brouard | Cartes de Saillance Spatio-Temporelle basées Contrastes de Couleur et Mouvement Relatif[END_REF] reported, as established by S. Daly, that the human eye cannot follow objects with a velocity higher than 80 • /s [START_REF] Daly | Engineering observations from spatiovelocity and spatiotemporal visual models[END_REF]. In this case, the saliency is null. S. Daly has also demonstrated that the saliency reaches its maximum with motion values between 6 • /s and 30 • /s. According to this psycho-visual constraints, the filter proposed in [START_REF] Brouard | Cartes de Saillance Spatio-Temporelle basées Contrastes de Couleur et Mouvement Relatif[END_REF] is given by ( 5).

S t (s i ) =          1 6 V r (I, i), if 0 ≤ V r (I, i) < v 1 1, if v 1 ≤ V r (I, i) < v 2 -1 50 V r (I, i) + 8 5 , if v 2 ≤ V r (I, i) < v max 0, if v max ≤ V r (I, i) (5)
with v 1 = 6 • /s, v 2 = 30 • /s and v max = 80 • /s. We follow this filtering scheme in temporal saliency map computation.

Geometric saliency map

As stated in the introduction, many studies have showed that the observers are attracted by the screen center. In [START_REF] Brouard | Cartes de Saillance Spatio-Temporelle basées Contrastes de Couleur et Mouvement Relatif[END_REF], the geometrical saliency map is a 2D Gaussian located at the screen center with a spread σ x = σ y = 5 • . In our psycho-visual experiments we stated that in a shoulder-fixed wearable camera video the gaze is always located in the first upper third of video frames, see the scattered plot of subjective saliency peaks in Fig. 3. Therefore, we have set the 2D Gaussian center at x 0 = width 2 and y 0 = height

3

. The geometrical saliency S g map equation is given by [START_REF] Dorr | Variability of eye movements when viewing dynamic natural scenes[END_REF].

S g (I) = e - (x-x 0 ) 2 2σ 2 x + (y-y 0 ) 2 2σ 2 y (6)
However, this attraction may change with the camera motion. This is explained by the anticipation phenomenon [START_REF] Land | The roles of vision and eye movements in the control of activities of daily living[END_REF], see section 3.1. Hence we propose to simulate this phenomenon by moving the 2D Gaussian centred on initial "geometric saliency point" in the direction of the camera motion projected in the image plane. A rough approximation of this projection is the motion of image center computed with the global motion estimation model, equation [START_REF] Le Meur | Predicting visual fixations on video based on low-level video features[END_REF], where x = width

Saliency map fusion

In the previous sections we explained how to compute spatial temporal and geometric saliency maps. In this section we describe the method that merges these three saliency maps in the target objective saliency map. The fusion result is a spatio-temporal-geometric saliency map. In [START_REF] Boujut | A metric for no-reference video quality assessment for hd tv delivery based on saliency maps[END_REF], several fusion methods for the spatio-temporal saliency without geometric component were proposed. We have tested these fusion methods on wearable video database. The results show that the multiplicative fusion performs the best. So for the full spatio-temporalgeometric saliency we compute multiplicative S mul sp-t-g [START_REF] Brouard | Cartes de Saillance Spatio-Temporelle basées Contrastes de Couleur et Mouvement Relatif[END_REF].

S mul sp-t-g (I) = S sp (I) × S t (I) × S g (I) [START_REF] Brouard | Cartes de Saillance Spatio-Temporelle basées Contrastes de Couleur et Mouvement Relatif[END_REF] 5 Evaluation

Normalized Scanpath Saliency

In this section, we compare the objective spatio-temporal saliency maps with subjective saliency map obtained from gaze tracking S subj .

Here, we use the Normalized Scanpath Saliency (NSS) metric that was proposed in [START_REF] Marat | Modelling spatio-temporal saliency to predict gaze direction for short videos[END_REF]. The N SS is a Z-Score that expresses the divergence of the subjective saliency maps from the objective saliency maps. The N SS computation for a frame I is depicted by [START_REF] Lee | Discovering important people and objects for egocentric video summarization[END_REF]. Here, S N obj denotes the objective saliency map S obj normalized to have a zero mean and a unit standard deviation, X means an average. When S subj × S N obj is higher than the average objective saliency, the N SS is positive; it means that the gaze locations are inside the saliency depicted by the objective saliency map. In other words, higher the N SS is, more objective and subjective saliency maps are similar.

N SS = S subj × S N obj -S obj σ(S obj ) (8)
The N SS score for a video sequence is obtained by computing the average of N SS for all frames as in [START_REF] Marat | Modelling spatio-temporal saliency to predict gaze direction for short videos[END_REF]. Then the overall N SS score on each video database is the average N SS of all video sequences. Results are presented in the next section.

Results

In this section, we compare the correlation of three automatic saliency maps with the subjective saliency. These three saliency maps are the spatio-temporal saliency map, the spatio-temporal-geometrical without camera motion, and the proposed method the spatio-temporal-geometrical with camera motion, expressing the anticipation phenomenon. The 28 video sequences described in Section 2 from wearable cameras are all characterized by strong camera motion which is up to 50 pixels magnitude in the center of frames. As it can be seen from the Fig. 4 the proposed method with moving of geometrical Gaussian almost systematically outperforms the base-line spatio-temporal saliency model and the spatio-temporal-geometrical saliency with a fixed Gaussian. For few sequences (e.g. number 2), the performance is poorer than obtained by geometric saliency with a fixed Gaussian. In these visual scenes, the distractors appear in the field of view. The resulting subjective saliency map then contains multiple maxima due to the unequal perception of scenes by the subjects. This is more "semantic saliency" phenomenon (faces, etc) which can not be handled with the proposed model. The average NSS on the whole database also shows the interest of proposed moving geometrical saliency. The mean NSS scores are respectively 1.832 for spatio-temporal, 2.607 for spatio-temporal with still geometrical Gaussian, and 2.791 with moving geometrical Gaussian. Which means 52.37% improvement of correspondence with subjective visual saliency map, which was our goal.

Conclusion

Hence in this work we proposed new objective visual saliency model and computation method by fusing spatial, temporal and geometric cues in video frames. The method was proposed for video with a strong motion recorded with cameras worn by subjects in the context of recording of Instrumental Activities of Daily Living. To our knowledge, this is the first attempt to define saliency in such a content taking into account psycho-visual models known for other types of traditional video content. First of all, conducting psycho-visual experiment on a representative set of test subjects, we stated the anticipation phenomenon in gaze positioning which was obviously transmitted to the subjective saliency map built according to Wooding method. While in previous research, the geometrical saliency was systematically frame centred, we recorded experimental evidence of the dependence of the geometric saliency of camera fixation on a body. Finally, in order to incorporate the anticipation phenomenon into the automatic construction of objective saliency map we expressed it by moving geometric saliency Gaussian in direction of camera motion projected into the image plane. These

Fig. 1 .Fig. 2 .

 12 Fig. 1. Wearable camera setup

2 and y = height 2 .Fig. 3 .

 223 Fig. 3. Scattered plot of subjective saliency peaks for all database frames

Fig. 4 .

 4 Fig. 4. Average NSS computed on 28 videos from a wearable camera

  The spatial saliency is defined according to the following seven local color contrasts V in the HSI domain : the Contrast of Saturation, Contrast of Intensity, Contrast of Hue, Contrast of Opponents, Contrast of Warm and Cold Colors, Dominance of Warm Colors, and Dominance of Brightness and Saturation. The spatial saliency value S ′ sp (I, i) for pixel i from frame I is computed by mean fusion operator from seven color contrast descriptors (3) :

Acknowledgments

This research is supported by the EU FP7 PI Dem@Care project #288199.


S Vakeroudis
email: stavros.vakeroudis@upmc.fr

Keywords: AMS 2010 subject classification: Primary: 60J65, 60J60, 60-02, 60G07; secondary: 60G15, 60J25, 60G46, 60E10, 60J55, 30C80, 44A10 Bougerol's identity, time-change, hyperbolic Brownian motion, subordination, Gauss-Laplace transform, planar Brownian motion, Ornstein-Uhlenbeck processes, two-dimensional Bougerol's identity, local time, multi-dimensional Bougerol's identity, Bougerol's diffusion, peacock, convex order, Bougerol's process Jacobi processes . . . . . . . . . . . . . .

come

Introduction

Bougerol's celebrated identity in law has been the subject of research for several authors since first formulated in 1983 [START_REF] Ph | Exemples de théorèmes locaux sur les groupes résolubles[END_REF]. A reason for this study is on the one hand its interest from the mathematical point of view and on the other hand its numerous applications, namely in Finance (pricing of Asian options etc.)-see e.g. [START_REF] Yor | On some Exponential Functionals of Brownian Motion[END_REF][START_REF] Dufresne | Laguerre Series for Asian and Other Options[END_REF][START_REF] Yor | Exponential Functionals of Brownian Motion and Related Processes[END_REF]. However, one still feels that some better understanding remains to be discovered.

This paper is essentially an attempt to collect all the known results (up to now) and to give a (full) survey of the several different equivalent expressions and extensions (to other processes, multidimensional versions, etc.) in a concise way. We also provide a bibliography, as complete as possible. For the extended proofs we address the reader to the original articles.

Bougerol's remarkable identity states that (see e.g. [START_REF] Ph | Exemples de théorèmes locaux sur les groupes résolubles[END_REF][START_REF] Alili | Sur l'identité de Bougerol pour les fonctionnelles exponentielles du mouvement Brownien avec drift[END_REF] and [START_REF] Yor | Exponential Functionals of Brownian Motion and Related Processes[END_REF] (p. 200)), with (B u , u ≥ 0) and (β u , u ≥ 0) denoting two independent linear Brownian motions § , we have:

for fixed t, sinh(B t ) (law) = β At(B) , (1)
where A u (B) = u 0 ds exp(2B s ) is independent of (β u , u ≥ 0). For a first approach of (1), see e.g. the corresponding Chapters in [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] and in [START_REF] Chaumont | Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, via Conditioning[END_REF]. In what follows, § When we simply write: Brownian motion, we always mean real-valued Brownian motion starting from 0. For 2-dimensional Brownian motion we indicate planar or complex BM.

sometimes for simplicity we will use the notation A u instead of A u (•). Alili, Dufresne and Yor [START_REF] Alili | Sur l'identité de Bougerol pour les fonctionnelles exponentielles du mouvement Brownien avec drift[END_REF] obtained the following simple proof of Bougerol's identity (1):

Proof. On the one hand, we define S t ≡ sinh(B t ); then, applying Itô's formula we have:

S t = t 0 1 + S 2 s dB s + 1 2 t 0 S s ds . (2)
On the other hand, a time-reversal argument for Brownian motion yields: for fixed t ≥ 0,

β At(B) = t 0 e Bs dγ s (law) = e Bt t 0 e -Bs dγ s ≡ Q t , (3)
where (γ s , s ≥ 0) denotes another 1-dimensional Brownian motion, independent from (B s , s ≥ 0). Applying once more Itô's formula to Q t , we have:

dQ t = 1 2 Q t dt + (Q t dB t + dγ t ) = 1 2 Q t dt + Q 2 t + 1 dδ t , (4)
where δ is another 1-dimensional Brownian motion, depending on B and on γ. From (2) and (4) we deduce that S and Q satisfy the same Stochastic Differential Equation with Lipschitz coefficients, hence, we obtain (1).

With some elementary computations, from (1) (e.g. identifying the densities of both sides, for further details see [START_REF] Vakeroudis | Nombres de tours de certains processus stochastiques plans et applications à la rotation d'un polymère[END_REF][START_REF] Bertoin | Some two-dimensional extensions of Bougerol's identity in law for the exponential functional of linear Brownian motion[END_REF]), we may obtain the Gauss-Laplace transform of the clock A t : for every x ∈ R, with a(x) ≡ arg sinh(x)

≡ log x + √ 1 + x 2 E 1 √ A t exp - x 2 2A t = a ′ (x) √ t exp - a 2 (x) 2t . ( 5
)
where a ′ (x) = (1 + x 2 ) -1/2 . For further use, we note that Bougerol's identity may be equivalently stated as:

sinh(|B u |) (law) = |β| Au(B) . (6)
Using now the symmetry principle (see [START_REF] André | Solution directe du problème résolu par M. Bertrand[END_REF] for the original note and [START_REF] Gallardo | Mouvement Brownien et calcul d[END_REF] for a detailed discussion):

sinh( Bu ) (law) = βAu(B) , (7)
where, e.g. Bu ≡ sup 0≤s≤u B s .

In the remainder of this article we give several versions and generalizations of Bougerol's identity (1). In particular, in Section 2 we give extensions of this identity to other processes (i.e. Brownian motion with drift, hyperbolic Brownian motion, etc.). Section 3 is devoted to some results that we obtain from subordination and some applications to the study of Bougerol's identity in terms of planar Brownian motion and of complex-valued Ornstein-Uhlenbeck processes. In Section 4 we give some 2 and 3 dimensional extensions of Bougerol's identity, first involving the local time at 0 of the Brownian motion B, and second by studying the joint law of 2 and 3 specific processes. In particular, in Subsection 4.2 we give a new 2-dimensional extension. In Section 5 we generalize Bougerol's identity for the case of diffusions, named "Bougerol's diffusions", followed by some studies in terms of Jacobi processes. Section 6 deals with Bougerol's identity from the point of view of "peacocks" (see this Section for the precise definition, as introduced in e.g. [START_REF] Hirsch | Peacocks and associated martingales, with explicit constructions[END_REF]). In Section 7 we propose some possible directions for further investigation of this "mysterious" identity in law with its versions and extensions and we give an as full as possible list of references (to the best of author's knowledge) up to now. Finally, in the Appendix, we present several tables of Bougerol's identity and all the equivalent forms and extensions that we present in this survey. These tables can be read independently from the rest of the text.

We also note that (sometimes) the notation used from Section to Section may be independent.

2 Extensions of Bougerol's identity to other processes

Brownian motions with drifts

Alili, Dufresne and Yor, in [START_REF] Alili | Sur l'identité de Bougerol pour les fonctionnelles exponentielles du mouvement Brownien avec drift[END_REF], showed the following result: Proposition 2.1. With µ, ν two real numbers, for every x fixed, the Markov process:

X (µ,ν) t ≡ (exp(B t + µt)) x + t 0 exp (-(B s + µs)) d(β s + νs) , (8)
for every t ≥ 0, has the same law as (sinh(Y (µ,ν) t

), t ≥ 0), where (Y (µ,ν) t

, t ≥ 0) is a diffusion with infinitesimal generator:

1 2 d 2 dy 2 + µ tanh(y) + ν cosh(y) d dy , (9)
starting from y = arg sinh(x).

Proof. It suffices to apply Itô's formula to both processes X (µ,ν) and sinh(Y (µ,ν) ).

It follows now:

Corollary 2.2. For every t fixed,

sinh(Y (µ,ν) t ) (law) = t 0 exp(B s + µs)d(β s + νs). ( 10
)
In particular, in the case µ = 1 and ν = 0:

sinh(B t + εt) (law) = t 0 exp(B s + s)dβ s , (11)
with ε denoting a symmetric Bernoulli variable taking values in {-1, 1}. Hence:

M t ≡ sinh(Y t ) = β t 0 ds(cosh 2 (Ys)) ≡ β t 0 ds(1+sinh 2 (Ys)) , (12)
and for this Markovian martingale, we have:

M t = sinh(Y t ) = t 0 cosh(Y s )dB s = t 0 1 + M 2 s dB s . ( 13
)
It can also be seen directly from (8) that X

(-1/2,0) t , t ≥ 0 is the product of two orthogonal martingales. This property is true because:

X (-1/2,0) t = B u R u u=A (1/2) t , (14) with A
(ν) t = t 0 ds exp(2B (ν) s ), (B (ν)
t , t ≥ 0) denoting a Brownian motion with drift, and (R t , t ≥ 0) a 2-dimensional Bessel process started at 0. Further details about this ratio are discussed in Sections 5 and 7. We also remark that, with the notation of Section 1, A (0) t ≡ A t .

Hyperbolic Brownian motion

Alili and Gruet in [START_REF] Alili | Exponential Functionals and Principal Values related to Brownian Motion. A collection of research papers[END_REF] generalized Bougerol's identity in terms of hyperbolic Brownian motion:

Proposition 2.4. We use the notation introduced in the previous Subsection, that is: (R t , t ≥ 0) is a 2-dimensional Bessel process with R 0 = 0 and wee also denote by Ξ an arcsine variable such that B (ν) , R and Ξ are independent. Let φ be the function defined by:

φ(x, z) = 2e x cosh(z) -e 2x -1, f or z ≥ |x|. ( 15
)
Then, for fixed t, we have:

β A (ν) t (law) = (2Ξ -1)φ B (ν) t , R 2 t + (B (ν) t ) 2 . ( 16
)
In particular, with ν = 0, we recover Bougerol's identity:

β At (law) = (2Ξ -1)φ B t , R 2 t + B 2 t (law) = sinh(B t ). (17)
This is an immediate consequence of the following:

Lemma 2.5. (i) The law of the functional A (ν) t is characterized by: for all u ≥ 0,

E exp - u 2 2 A (ν) t = e -ν 2 t/2 R dx e νx +∞ |x| dz z √ 2πt 3 e -z 2 t/2 J 0 (uφ(x, z)), (18)
where J 0 stands for the Bessel function of the first kind with parameter 0 [START_REF] Lebedev | Special Functions and their Applications[END_REF].

(ii) In particular, taking ν = 0, for u ≥ 0 and x ∈ R we have:

exp - x 2 2t E exp - u 2 2 A t B t = x = +∞ |x| dz z t e -z 2 t/2 J 0 (uφ(x, z)). ( 19
)
Proposition 2.4 follows now immediately from Lemma 2.5 by using the classical representation of the Bessel function of the first kind with parameter 0 (see e.g. [START_REF] Lebedev | Special Functions and their Applications[END_REF]):

J 0 (z) = 1 π +1 -1 dr √ 1 -r 2 cos(zr), (20)
and remarking that (with Ξ denoting again an arcsine variable), for all real ξ:

J 0 (ξ) = E [exp (iξ(2Ξ -1))] . (21)
Proof. (Lemma 2.5) With I µ and K µ denoting the modified Bessel functions of the first and the second kind respectively with parameter µ = ρ 2 + ν 2 (for ρ and ν reals), we define the function G µ : R 2 → R + by:

G µ (u, v) = 2I µ (u)K µ (v), u ≤ v; 2I µ (v)K µ (u), u ≥ v. (22)
First, using the skew product representation of planar Brownian motion, the following formula holds (for further details we address the interested reader to [START_REF] Alili | Exponential Functionals and Principal Values related to Brownian Motion. A collection of research papers[END_REF]):

∞ 0 dt exp - ρ 2 2 t E exp - u 2 2 A (ν) t = +∞ -∞ dy e νy G µ (u, ue y ). (23)
Using the integral representation (see e.g. [START_REF] Lebedev | Special Functions and their Applications[END_REF], problem 8, p. 140):

I µ (x)K µ (y) = 1 2 ∞ log(y/x) dr e -µr J 0 2 cosh(r)xy -x 2 -y 2 , y ≥ x. ( 24
)
we can invert (23) in order to obtain part i) of Lemma 2.5. Part ii) follows with the help of Cameron-Martin relation.

Bougerol's identity and subordination

In this Section, we consider (Z t = X t + iY t , t ≥ 0) a standard planar Brownian motion (BM) starting from x 0 + i0, x 0 > 0 (for simplicity and without loss of generality, we suppose that x 0 = 1). Then, a.s., (Z t , t ≥ 0) does not visit 0 but it winds around it infinitely often, hence θ t = Im( t 0 dZs Zs ), t ≥ 0 is well defined [START_REF] Itô | Diffusion Processes and their Sample Paths[END_REF]. There is the well-known skew-product representation:

log |Z t | + iθ t ≡ t 0 dZ s Z s = (B u + iγ u ) u=Ht= t 0 ds |Zs| 2 , ( 25
)
where (B u + iγ u , u ≥ 0) is another planar Brownian motion starting from log 1 + i0. Thus:

H -1 u ≡ inf{t : H t > u} = u 0 ds exp(2B s ) := A u (B).
For further study of the Bessel clock H, see e.g. [START_REF] Yor | Loi de l'indice du lacet Brownien et Distribution de Hartman-Watson[END_REF]. We also define the first hitting times T θ c ≡ inf{t :

θ t = c} and T |θ| c ≡ inf{t : |θ t | = c}.

General results

Bougerol's identity in law combined with the symmetry principle of André [START_REF] André | Solution directe du problème résolu par M. Bertrand[END_REF][START_REF] Gallardo | Mouvement Brownien et calcul d[END_REF] yields the following identity in law (see e.g. [START_REF] Bertoin | Retrieving information from subordination[END_REF][START_REF] Bertoin | Some two-dimensional extensions of Bougerol's identity in law for the exponential functional of linear Brownian motion[END_REF]): for every fixed l > 0,

H τ l (law) = τ a(l) (26)
where (τ l , l ≥ 0) stands for a stable (1/2)-subordinator. An example of this kind of identities in law is given for the planar Brownian motion case in the next Subsection.

The main point in [START_REF] Bertoin | Retrieving information from subordination[END_REF] is that (26) is not extended in the level of processes indexed by l ≥ 0.

Bougerol's identity in terms of planar Brownian motion

Vakeroudis [START_REF] Vakeroudis | On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol's identity[END_REF] investigated Bougerol's identity in terms of planar Brownian motion and obtained some striking identities in law:

Proposition 3.1. Let (β u , u ≥ 0) be a 1-dimensional Brownian motion independent of the planar Brownian motion (Z u , u ≥ 0) starting from 1. Then, for any b ≥ 0 fixed, the following identities in law hold:

i) H T β b (law) = T B a(b) ii) θ T β b (law) = C a(b) iii) θT β b (law) = |C a(b) |,
where C A is a Cauchy variable with parameter A and θu = sup 0≤s≤u θ s .

Proof. i) We identify the laws of the first hitting times of a fixed level b by the processes on each side of (7) and we obtain:

T B a(b) (law) = H T β b , which is i). ii) It follows from i) since: θ u (law)
= γ Hu , with (γ s , s ≥ 0) a Brownian motion independent of (H u , u ≥ 0) and (C u , u ≥ 0) may be represented as (γ T B u , u ≥ 0). iii) follows from ii) again with the help of the symmetry principle.

Using now these identities in law, we can apply William's "pinching" method [START_REF] Williams | A simple geometric proof of Spitzer's winding number formula for 2-dimensional Brownian motion[END_REF][START_REF] Messulam | On D. Williams' "pinching method" and some applications[END_REF] and recover Spitzer's celebrated asymptotic law which states that [START_REF] Spitzer | Some theorems concerning two-dimensional Brownian Motion[END_REF]:

2 log t θ t (law) -→ t→∞ C 1 , (27)
with C 1 denoting a standard Cauchy variable (for other proofs, see also e.g. [Wil74, Dur82, MeY82, BeW94, Yor97, VaY11a]). One can also find a characterization of the distribution of T θ c and of T |θ| c in [START_REF] Vakeroudis | On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol's identity[END_REF]. First, applying Bougerol's identity (1) in terms of planar Brownian motion we have:

Proposition 3.2. For fixed c > 0, sinh(C c ) (law) = β (T θ c ) (law) = T θ c N , (28)
where N ∼ N (0, 1) and the involved random variables are independent.

Furthermore, we can obtain the following Gauss-Laplace transforms which are equivalent to Bougerol's identity exploited for planar Brownian motion:

Proposition 3.3. For x ≥ 0 and m = π 2c , c E π 2T θ c exp - x 2T θ c = 1 √ 1 + x c 2 (c 2 + log 2 ( √ x + √ 1 + x)) ; (29) c E 2 πT |θ| c exp - x 2T |θ| c = 1 √ 1 + x 2 ( √ 1 + x + √ x) m + ( √ 1 + x - √ x) m . ( 30
)
Proof. For the proof of (29), it suffices to identify the densities of the two parts of (28) and to recall that the density of a Cauchy variable with parameter c equals:

c π(c 2 + y 2 )
.

For (30), we apply Bougerol's identity with u = T |γ| c ≡ inf{t : |γ t | = c} and we obtain:

sinh(B T |γ| c ) (law) = β (T |θ| c ) (law) = T |θ| c N . (31)
Once again we identify the densities of the two parts. For the left hand side, we use the following Laplace transform: for λ ≥ 0, E e -λ 2 2 T |γ| b = 1 cosh(λb) (see e.g. Proposition 3.7, p. 71 in Revuz and Yor [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF]). We also use the well-known result [START_REF] Lévy | OEuvres de Paul Lévy, Processus Stochastiques[END_REF][START_REF] Biane | Valeurs principales associées aux temps locaux browniens[END_REF]:

E exp(iλB T |γ| c ) = 1 cosh(λc) = 1 cosh(πλ c π ) = ∞ -∞ e i( λc π )x 1 2π 1 cosh( x 2 )
dx .

(32)

Changing now the variables y = cx/π, we obtain the density of B T |γ| c which equals:

2c cosh( yπ 2c ) -1 = c(e yπ 2c + e -yπ 2c ) -1
, and finishes the proof.

Vakeroudis and Yor in [START_REF] Vakeroudis | Integrability properties and Limit Theorems for the first exit times from a cone of planar Brownian motion[END_REF][START_REF] Vakeroudis | Some infinite divisibility properties of the reciprocal of planar Brownian motion exit time from a cone[END_REF] investigated further the law of these random times.

The Ornstein-Uhlenbeck case

Vakeroudis in [START_REF] Vakeroudis | On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol's identity[END_REF][START_REF] Vakeroudis | On the windings of complex-valued Ornstein-Uhlenbeck processes driven by a Brownian motion and by a Stable process[END_REF] investigated also the case of Ornstein-Uhlenbeck processes. In particular, we consider now a complex valued Ornstein-Uhlenbeck (OU) process:

Z t = z 0 + Zt -λ t 0 Z s ds, (33)
where Zt is a complex valued Brownian motion, z 0 ∈ C (for simplicity and without loss of generality, we suppose again z 0 = 1), λ ≥ 0 and

T (λ) c ≡ T |θ Z | c ≡ inf t ≥ 0 : θ Z t = c (θ Z
t is the continuous winding process associated to Z) denoting the first hitting time of the symmetric conic boundary of angle c for Z. Then, we have the following: Proposition 3.4. Consider (Z λ t , t ≥ 0) and (U λ t , t ≥ 0) two independent Ornstein-Uhlenbeck processes, the first one complex valued and the second one real valued, both starting from a point different from 0, and define T (λ) b (U λ ) = inf t ≥ 0 : e λt U λ t = b , for any b ≥ 0. Then, an Ornstein-Uhlenbeck extension of identity in law ii) in Proposition 3.1 is the following:

θ Z λ T (λ) b (U λ ) (law) = C a(b) , (34)
where a(x) = arg sinh(x) and C A is a Cauchy variable with parameter A.

Proof. First, for Ornstein-Uhlenbeck processes, is well known that [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], with (B t , t ≥ 0) denoting a complex valued Brownian motion starting from 1, Dambis-Dubins-Schwarz Theorem yields:

Z t = e -λt 1 + t 0 e λs d Zs = e -λt (B αt ) , (35)
Let us consider a second Ornstein-Uhlenbeck process (U λ t , t ≥ 0) independent of the first one. Taking now equation (35) for U λ t (1-dimensional case) we have:

e λt U λ t = δ ( e 2λt -1 2λ ) , (36)
where

(
θ Z t = θ B αt . (37)
By inverting α t , it follows now that:

T (λ) c = 1 2λ ln 1 + 2λT |θ| B c . (38)
Similarly, for the 1-dimensional case we have:

T (λ) b (U λ ) = 1 2λ ln 1 + 2λT δ b . (39)
Equation (37) for t = 1 2λ ln 1 + 2λT δ b , equivalently: α(t) = T δ b becomes:

θ Z λ T (λ) b (U λ ) = θ Z λ 1 2λ ln(1+2λT δ b ) = θ B u=T δ b (law) = C a(b) ,
where the last equation in law follows precisely from statement ii) in Proposition 3.1.

Multidimensional extensions of Bougerol's identity

4.1 The law of the couple (sinh(β t ), sinh(L t ))

A first 2-dimensional extension of Bougerol's identity was obtained by Bertoin, Dufresne and Yor in [START_REF] Bertoin | Some two-dimensional extensions of Bougerol's identity in law for the exponential functional of linear Brownian motion[END_REF] (for a first draft, see also [START_REF] Dufresne | A two dimensional extension of BougerolŠs identity in law for theexponential of Brownian motion[END_REF]). With (L t , t ≥ 0) denoting the local time at 0 of B, we have:

Theorem 4.1. For fixed t, the 3 following 2-dimensional random variables are equal in law:

(sinh(B t ), sinh(L t ))

(law) = (β At , exp(-B t ) λ At ) (law) = (exp(-B t ) β At , λ At ), (40)
where (λ u , u ≥ 0) is the local time of β at 0.

Remark 4.2. Theorem 4.1 can be equivalently stated as: for fixed t, the 3 following 2-dimensional random variables are equal in law:

(sinh(|B t |), sinh(L t )) (law) = (|β| At , exp(-B t ) λ At ) (law) = (exp(-B t ) |β| At , λ At ). (41)
Using now Paul Lévy's celebrated identity in law (see e.g. [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF]):

( Bt -B t , Bt ), t ≥ 0 (law) = ((|B t |, L t ), t ≥ 0) , (42)
we can reformulate (40) or (41), and we obtain:

(sinh( Bt -B t ), sinh( Bt )) (law) = ( β -β) At , exp(-B t ) βAt (law) = exp(-B t ) ( β -β) At , βAt . ( 43
)
The latter is particularly interesting when compared with the Wiener-Hopf factorization for Brownian motion. In particular, if we consider e q an independent exponential random variable of parameter q, then Beq is independent of B eq -Beq . This tells that the two random variables appearing on the right hand side of (43), when taken at e q , are independent.

Remark 4.3. Considering only the second processes of the first and the third part of (40) (or equivalently of (41)), we obtain a "local time" version of Bougerol's identity:

sinh(L t ) (law) = λ At , (44)
which (as was shown in [START_REF] Bertoin | Retrieving information from subordination[END_REF]), similar to the Brownian motion case, is true only for fixed t and not in the level of processes.

Proof. (Theorem 4.1) From Remark 4.2 it suffices to prove (41). First, we denote S p , p ≥ 0 an exponential variable with parameter p independent from B and g t = sup{u < t : B u = 0}. We know that B u , u ≤ g Sp and B g Sp +u , u ≤ S pg Sp are independent, hence L Sp and B Sp are also independent. We also know that L t and |B t | have the same law. Hence, using the following computation: for every l ≥ 0, with (τ l , l ≥ 0) denoting the time L reaches l,

P L Sp ≥ l = P (S p ≥ τ l ) = E [exp(-pτ l )] = exp(-l 2p),
we deduce that the common density of L Sp and |B Sp | is:

2p exp(-u 2p), u ≥ 0.
Equivalently, we have:

√ 2e(|β(1)|, λ(1)) (law)
= (e, e ′ ), where on the left hand side e and e ′ are two independent copies of S 1 independent from β.

For the second identity in law in Theorem 4.1, it suffices to remark that

(β At , exp(-B t ) λ At ) (law) = ( A t β 1 , exp(-B t ) A t λ 1 ),
and use a time reversal argument. For the first identity in law we use an exponential time S p and we compute the joint Mellin transforms in both sides in order to show that:

√ 2e(sinh(|B| Sp ), sinh(L Sp )) (law) = √ 2e(exp(-B Sp ) A Sp |β 1 |, A Sp λ At ).
For further details we address the reader to [START_REF] Bertoin | Some two-dimensional extensions of Bougerol's identity in law for the exponential functional of linear Brownian motion[END_REF].

Using now Tanaka's formula we can also obtain the following identity in law for 2dimensional processes:

Corollary 4.4. (sinh(B t ), L t ) t≥0 (law) = exp(-B t ) β At , t 0 exp(-B s )dλ As t≥0 , (45)
where, in each part, the second process is the local time at level 0 and time t of the first one.

1 ∂ 2 ∂x 2 1 + 1 + 4x 2 2 ∂ 2 ∂x 2 2 + 4x 1 x 2 ∂ 2 ∂x 1 ∂x 2 + x 1 2 ∂ ∂x 1 + 2x 2 ∂ ∂x 2 . ( 54
)
Let us now study the couple:

(x (1) t , x (2)
t ) = sinh(B (1) t ), 1 2 sinh(2B (2) t ) , (55)
where (B

(1)

t , t ≥ 0) and (B (2)
t , t ≥ 0) are two dependent Brownian motions. By Itô's formula we have:

x (1) t = sinh(B (1) t ) = t 0 cosh(B (1) v ) dB (1) v + 1 2 t 0 sinh(B (1) v ) dv = t 0 1 + (x (1) v ) 2 dB (1) v + 1 2 t 0 x (1) v dv, (56) and:
x

(2) t

Remark 4.8. We remark that with:

α(x) = tanh(x) -1 cosh(x) 1 cosh(x) tanh(x) , (63)
we have:

dB ′ t dG ′ t = α(B t ) dB t dβ t , (64)
and

B ′ t G ′ t , t ≥ 0 is a 2-dimensional Brownian motion.
Proof. (Proposition 4.7) First proof: Using Itô's formula, we deduce easily that each of these triplets is a Markov process with infinitesimal generator (in C 2 (R 3 )):

1 2 (1 + x 2 ) d 2 dx 2 + 1 2 d 2 dy 2 + 1 2 d 2 dz 2 + x d 2 dxdy + d 2 dxdz + x d dx . ( 65
)
The proof finishes by the uniqueness (in law) of the solutions of the corresponding martingale problem.

Second proof: First, we admit that the identity in law is true. Then, if we replace on the left hand side (B s ) by (B ′ s ) and (β s ) by (G ′ s ), we have necessarily:

sinh(B t ) (law) = e B ′ t t 0 e -B ′ u dG ′ t , (66)
which is essentially a (partial) inversion formula of the transformation (64). Equation (66) can be proved by using Itô's formula on the right hand side.

Gruet in [ADY97] also remarked that:

Proposition 4.9. There exist two independent linear Brownian motions V and W and a diffusion J starting from 0 satisfying the following equation

dJ t = dW t + 1 2 tanh(J t )dt, (67)
such that,

dβ t dB t = α(-J t ) dV t dW t . ( 68
)
Hence, the two following 3-dimensional processes:

exp B t + t 2 t 0 exp -B s - s 2 dβ s , B t , β t ; t ≥ 0 and (sinh(J t ), B t , β t ; t ≥ 0) , are equal.
Proof. This result follows from a geometric proof and it is essentially an explanation of the second proof of Proposition 4.7, at least for ν = 0. For this purpose, we can compare the writing of a hyperbolic Brownian motion in the half-plane of Poincaré, decomposed in rectangular coordinates with the equidistant coordinates [START_REF] Vinberg | Geometry II, Spaces of constant curvature[END_REF]. For further details, see the Appendix in [START_REF] Alili | Sur l'identité de Bougerol pour les fonctionnelles exponentielles du mouvement Brownien avec drift[END_REF] due to Gruet.

5 The diffusion version of Bougerol's identity

Bougerol's diffusion

Bertoin, Dufresne and Yor in a recent work [START_REF] Bertoin | A relationship between Bougerol's generalized identity in law and Jacobi processes[END_REF] generalized Bougerol's identity in terms of diffusions. First, we remark that from Proposition 2.1 we have that (see also [START_REF] Alili | Sur l'identité de Bougerol pour les fonctionnelles exponentielles du mouvement Brownien avec drift[END_REF]):

(sinh(B t ), t ≥ 0) (law) = exp(-B t )β A (0) t , t ≥ 0 . ( 69
)
In particular, using Lamperti's relation (see e.g. [START_REF] Lamperti | Semi-stable Markov processes I[END_REF] or [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF]) we can invoke a Bessel process R (δ) independent from B in order to replace the right hand side of (69) by:

exp -B (ν) t R (δ) A (ν) t , t ≥ 0 ,
which turns out to be a diffusion (named Bougerol's diffusion) with a certain infinitesimal generator. Hence, we obtain the following:

Theorem 5.1. With Z = Z (δ) and Z ′ = Z (δ ′ ) denoting two independent squared Bessel processes of dimension δ = 2(1 + µ) and δ ′ = 2(1 + ν) respectively, starting from z and z ′ , the process:

X t ≡ X (ν,δ) t ≡ exp -2B (ν) t Z A (ν) t = Z u Z ′ u u=A (ν) t , t ≥ 0, (70)
is a diffusion with infinitesimal generator:

2x(1 + x)D 2 + (δ + (4 -δ ′ )x) D, (71)
where δ ′ = 2(1 + ν).

Remark 5.2. There is a discussion in [START_REF] Jakubowski | On hyperbolic Bessel processes and beyond[END_REF] concerning the particular case where the diffusion with generator given in (71) is the hyperbolic sine of the radial part of a hyperbolic Brownian motion (or equivalently the hyperbolic sine of a hyperbolic Bessel process) of index α ∈ (-1/2, ∞) (see [START_REF] Jakubowski | On hyperbolic Bessel processes and beyond[END_REF] Theorem 2.25, formula (46), p.15). In that case, with R t denoting this hyperbolic Bessel process starting from x and Y t = e Bt-(α+1/2)t , for any w ≥ 0, t ≥ 0, (sinh(R t ), t ≥ 0)

(law) = Y -1 t S t 0 Y 2 u du , t ≥ 0 , ( 72
)
where S is a Bessel process of dimension 2(1 + α) independent of B, and S 0 = sinh(x).

H (ν) u = u 0 ds Z ′ s ; H (µ) u = u 0 ds Z s . (80)
Moreover, we remark that (H

(ν) t ) is the inverse of (A (ν) t ) and (H (µ) t ) is the inverse of (A (µ)
t ). We also need to use:

H (ν,µ) u = u 0 ds X (ν,µ) s ; H (µ,ν) u = u 0 ds X (µ,ν) s . (81)
Simple calculations now yield:

H (ν,µ) t = H (µ) A (ν) t , (82)
Finally, using (70) we have:

H (µ) A (ν) t = t 0 ds X (ν,µ) A (ν) s , (83)
and we obtain easily the result.

Relations involving Jacobi processes

In this Subsection, we present a particular study of Theorem 5.1 in terms of the Jacobi processes Y (δ,δ ′ ) as introduced in Warren and Yor [START_REF] Warren | The brownian burglar : conditioning brownian motion by its local time process[END_REF] (see also the references therein for Jacobi processes), due to Bertoin, Dufresne and Yor [START_REF] Bertoin | A relationship between Bougerol's generalized identity in law and Jacobi processes[END_REF]. First, we recall some results involving Jacobi processes:

Proposition 5.4. (Warren and Yor [START_REF] Warren | The brownian burglar : conditioning brownian motion by its local time process[END_REF], Proposition 8) With T ≡ inf{u : Z u + Z ′ u = 0}, there exists a diffusion process Y u ≡ Y δ,δ ′ u , u ≥ 0 on [0, 1], independent from Z + Z ′ such that:

Z u Z u + Z ′ u = Y u 0 ds Zs+Z ′ s , u < T. (84)
We remark that Y ′ = 1 -Y is the Jacobi process with dimensions (δ ′ , δ), and Y has infinitesimal generator:

2y(1 -y)D 2 + (δ -(δ + δ ′ )y) D . (85)
Now, X defined in (70) and Y can be related as following:

Proposition 5.5. The following relation holds:

Y w 1 -Y w = X w 0 dv Y ′ v = X w 0 dv 1-Yv , (86)
or equivalently:

X k = Y w 1 -Y w w= k 0 dv 1+Xv . (87)
Proof. (Proposition 5.5) First, from (70), we have:

X t = Z u Z ′ u u=A (ν) t .
Conversely,

Z u Z ′ u = X H (ν) u , (88)
where

H (ν) u = u 0 ds Z ′ s
is the inverse of A (ν) . However, using the Jacobi process Y ,

Z u Z ′ u = Y w 1 -Y w w=Hu= u 0 ds Zs+Z ′ s , (89)
and moreover:

H (ν) u = u 0 ds Z ′ s = u 0 ds (Z s + Z ′ s ) 1 (1 -Y Hs ) = Hu 0 dv Y ′ v . ( 90
)
Plugging now (90) to (88) and comparing to (89), we obtain (86). For (87), it suffices to remark that k →

E[ψ(W )] ≤ E[ψ(V )], (91)
and we write:

W (c) ≤ V . A process (G t , t ≥ 0) is a peacock if, for every s ≤ t, G s (c)
≤ G t . Kelleler's Theorem now (see e.g. [START_REF] Kellerer | Markov-Komposition und eine Anwendung auf Martingale[END_REF][START_REF] Hirsch | Peacocks and associated martingales, with explicit constructions[END_REF][START_REF] Hirsch | A new proof of Kellerer's theorem[END_REF]) states that, to every peacock, we can associate a martingale (defined possibly on another probability space than G). In other words, there exists a martingale (M t , t ≥ 0) such that, for every fixed t ≥ 0,

G t (law) = M t . (92)
The main subject of [START_REF] Hirsch | Peacocks and associated martingales, with explicit constructions[END_REF] is to give several examples of peacocks and the associated martingales.

We return now to Bougerol's identity and we remark that (see also [START_REF] Hirsch | Peacocks and associated martingales, with explicit constructions[END_REF], paragraph 7.5.4, p. 322), for every λ ≥ 0, (sinh(λB t ), t ≥ 0) is a peacock with associated martingale λ t 0 e λβs dγ s , t ≥ 0 (see e.g. (3)). Moreover, for every λ real, e -λ 2 2 t sinh(λB t ), ≥ 0 is obviously a peacock, as it is a martingale. This is generalized in the following:

Further extensions and open questions

In this Section, we propose some possible directions to continue studying and possibly extending Bougerol's celebrated identity in law (for fixed time or as a process).

First, the natural question posed is wether this identity can be extended to higher dimensions. This very challenging question has already been attempted to be dealt with, and in this paper we've presented several extensions, at least for the 2-dimensional (and partly for the 3-dimensional) case.

Another natural question is wether we can generalize Bougerol's identity to other processes. For this purpose, we may think in terms of a diffusion, as introduced in Section 5. It seems more intelligent to start from the right hand side of (1) and try to see, e.g. in (70), for every particular ratio of processes, which is the corresponding process on the left hand side (this process could be named "Bougerol's process").

In particular, it seems interesting to investigate a possible extension in the case of Lévy or stable processes. To that end, we could replace the ratio of the two squared independent Bessel processes in (70) by e.g. the ratio of two exponentials of Lévy processes, and investigate the process obtained after the time-change. However, this perspective is not in the aims of the present work.

Finally, another aspect which could be further studied is the applications that one may obtain by the subordination method, as presented in Section 3. Following the lines of this Section, one may retrieve further results and applications, others than for the planar Brownian motion case (see also [START_REF] Bertoin | Retrieving information from subordination[END_REF][START_REF] Bertoin | Some two-dimensional extensions of Bougerol's identity in law for the exponential functional of linear Brownian motion[END_REF]).

A Appendix: Tables of Bougerol's Identity and other equivalent expressions

Using now the notations introduced in the whole text, we can summarize all the results in the following tables (u > 0, wherever used is considered as fixed).

A.1 Table : Bougerol's Identity in law and equivalent expressions (u > 0 fixed)

With a(x) ≡ arg sinh(x), and B, β denoting two independent real Brownian motions, for u > 0 fixed, we have:

1) sinh(B u ) (law) = β (Au(B)≡ u 0 ds exp(2Bs)) (Bougerol's Identity) 2) sinh(|B u |) (law) = |β| (Au(B))
3) sinh( Bu ) : Bougerol's Identity for other 1-dimensional processes (u > 0 fixed)

(law) = β(Au(B)) , Bu = sup s≤u β s 4) E 1 √ 2πAu(B) exp -x 2Au(B) = 1 √ 2πu 1 √ 1+x exp -(a( √ x)) 2 2u , x ≥ 0 A.2 Table
We use µ, ν reals and we define: starting from y = arg sinh(x). B (µ) , β (ν) , ε, Ξ and R are independent. Then, for u > 0 fixed:

B (µ) t = B t + µt, β (ν) s = β t + νt, A (ν) t = t 0 ds exp(2B (ν) s ), ε: a Bernoulli variable in {-1, 1}, (R t , t ≥
9) sinh(C c ) (law) = β (T θ c ) (law) = T θ c N, c > 0 fixed 10) H T β b (law) = T B a(b) , T B y = inf{t : B t = y}, a(x) = arg sinh(x), b > 0 fixed 11) θ T β b (law) = C a(b) , b > 0 fixed 12) θT β b (law) = |C a(b) |, θu = sup s≤u θ s 13) E 1 √ 2πT θ c exp -x 2T θ c = 1 √ 1+x c π(c 2 +log 2 ( √ x+ √ 1+x)) , b > 0 fixed, x ≥ 0 14) E 1 √ 2πT |θ| c exp(-x 2T |θ| c ) = 1 c 1 √ 1+x 1 ( √ 1+x+ √ x) ζ +( √ 1+x- √ x) ζ , x ≥ 0, ζ = π 2c 15) θ Z λ T (λ) b (U λ ) (law)
= C a(b) (OU version)

A.4 Table: Multi-dimensional extensions of Bougerol's Identity

In the following table, (L t , t ≥ 0) and (λ t , t ≥ 0) denote the local times at 0 of B, β respectively and: where (ξ

X (1) u , X (2)
(1) v , v ≥ 0), (ξ (2)
v , v ≥ 0) and (B u , u ≥ 0) are three independent Brownian motions. Moreover, we denote by (B (1) , B (2) ) and (β (1) , β (2) ) two couples of dependent Brownian motions (independent from B), such that:

d < B (1) , B (2) > v = tanh(B (1) v ) tanh(2B (2) v ) dv,
= β

(1)

( u 0 dv exp(2Bv )) , β (2)
( u 0 dv exp(4Bv ))

23) e Let Z ≡ Z (δ) and Z ′ ≡ Z (δ ′ ) be two independent squared Bessel process of dimension δ = 2(1 + µ) and δ ′ = 2(1 + ν) respectively, starting from z and z ′ , and X t ≡ X (ν,δ) t a diffusion (named "Bougerol's diffusion"), with infinitesimal generator:

2x(1 + x)D 2 + (δ + (4 -δ ′ )x) D,
and Y ≡ Y δ,δ ′ the Jacobi process. Then, for t, w, k > 0:

25) X (ν,δ) t ≡ exp -2B (ν) t Z A (ν) t = Zu Z ′ u u=A (ν) t 26) Yw 1-Yw = X w 0 dv Y ′ v = X w 0 dv 1-Yv 27) X k = Yw 1-Yw w= k 0 dv 1+Xv

k 0 dv

 0 1+Xv is the inverse of the increasing process w → w 0 dv 1-Yv .6 Bougerol's identity and peacocksHirsch, Profeta, Roynette and Yor in[START_REF] Hirsch | Peacocks and associated martingales, with explicit constructions[END_REF], studied the processes which are increasing in the convex order, named peacocks (coming from the French term: Processus Croissant pour l'Ordre Convexe, which yields the acronym PCOC). Let us first introduce a notation: for W and V two real-valued random variables, W is said to be dominated by V for the convex order if, for every convex function ψ : R → R such that E[|ψ(W )|] < ∞ and E[|ψ(V )|] < ∞, we have:

  v exp(B v ), exp(-2B u ) u 0 dξ (2) v exp(2B v ) ,

  dv exp(4Bv )) .

0

  tanh(B s )dG s , with (G t , t ≥ 0) denoting another Brownian motion, independent from B and J a diffusion starting from 0 satisfying: dJ t = dW t + 1 2 tanh(J t )dt, where W stands for an independent Brownian motion. Hence, for u > 0 fixed:16) sinh(L u ) (law) = λ Au 17) (sinh(B u ), sinh(L u )) (law) = (β Au , exp(-B u ) λ Au ) (law) = (exp(-B u ) β Au , λ Au ) 18) (sinh(|B u |) sinh(L u )) (law) = (|β| Au , exp(-B u ) λ Au ) (law) = (exp(-B u ) |β| Au , λ Au )19) (sinh( Bu -B u ), sinh( Bu ))

  β) Au , exp(-B u ) βAu

  (-B u ) ( ββ) Au , βAu ) 20) (sinh(B t ), L t , t ≥ 0) (law) = exp(-B t ) β At , t 0 exp(-B s )dλ As , t ≥

  Remark 2.3. With µ = -1/2 and ν = 0, we have that sinh Y

									(-1/2,0) t	is a martingale.
	(-1/2,0) t sinh(Y t ) = Indeed, with Y t ≡ Y t , Itô's formula yields: 0 cosh(Y s ) dY s + t 1 0 2	sinh(Y s ) ds
	=	0	t	cosh(Y s ) dB s -	1 2	tanh(Y s ) ds +	1 2	0	t	sinh(Y s ) ds
			t
	=			cosh(Y s )dB s .
		0

  δ t , t ≥ 0) is a real valued Brownian motion starting from 1. Second, applying Itô's formula to (35) and dividing by Z s , we obtain (α t =

	e 2λt -1 2λ ):				t 0 e 2λs ds =
	Im	dZ s Z s	= Im	dB αs B αs	,
	hence:

  Proposition 6.1. ([HPRY11], Proposition 7.2) The process (e µt sinh(λB t ), ≥ 0) is a peacock if and only if µ ≥ -λ 2 2 . Proof. i) First, we suppose µ ≥ -λ 2 2 . Then, for s < t: However, x → |x| is convex and if (e µt sinh(µB t ), t ≥ 0) was a peacock, then E [|e µt sinh(µB t )|] would increase on t, which is a contradiction.

	e µt sinh(λB t ) = e (µ+ λ 2 2 )t sinh(λB t )e -λ 2 2 t	(c) ≥ e (µ+ λ 2 2 )s sinh(λB t )e -λ 2 2 t
	(c) ≥ e (µ+ λ 2 2 )s sinh(λB s )e -λ 2 2 s .
	ii) Conversely, Itô-Tanaka's formula yields:
	E [| sinh(λB t )|] = E [sinh(λ|B t |)] = e	λ 2 2 t λ	0	t	ds √ 2πs	e -λ 2 2 s ,
	hence:
	E |e µt sinh(µB t )|	t→+∞ ∼ λe ( λ 2 2 +µ)t λ	0	+∞	ds √ 2πs	e -λ 2 2 s ,
	which means that if µ < -λ 2 2 ,
	E |e µt sinh(µB t )|

t→+∞ -→ 0 .

  Bt t 0 e Bu dβ u , B t , β t ; t ≥ 0 exp -B s -s 2 dβ s , B t , β t ; t ≥ 0 = (sinh(J t ), B t , β t ; t ≥ 0) A.5 Table: Diffusion version of Bougerol's Identity (relations involving the Jacobi process)

		(law) = (sinh(B t ), B ′ t , G ′ t ; t ≥ 0)
	24) exp B t + t 2	t 0 (law)

Acknowledgements

The author is indebted to Professor Marc Yor for useful comments and advice, and to Professor Ron Doney for his invitation as a Post Doc fellow at the University of Manchester, where he prepared this survey. The author would also like to thank an anonymous referee for providing useful comments and for pointing out the last part of Remark 4.2 concerning the comparison with the Wiener-Hopf factorization of Brownian motion and references [AMSh01] and [JaW12].

Another two-dimensional extension

In this Subsection we will study the joint distribution of:

where (ξ

v , v ≥ 0) and (B u , u ≥ 0) are three independent Brownian motions. Hence, we obtain a new 2-dimensional extension which states the following: Proposition 4.5. We consider (B (1) t , t ≥ 0) and (B

(2) t , t ≥ 0) two real dependent Brownian motions, such that:

For the two-dimensional process X

(1)

, we have:

(i) In the level of processes:

( u 0 dv exp(4Bv )) .

(49)

Proof. Let us define:

where α = 1, 2. By Itô's formula, we have:

Hence:

and

v , v ≥ 0) and (η

v , v ≥ 0) are two dependent Brownian motions, with quadratic variation:

Thus, we deduce that the infinitesimal generator of X

(1) u , X

(2) u is:

Moreover, using (47):

Finally, we have that (x

u ) has the same infinitesimal generator with X

(1)

. Hence, we get part (i) of the Proposition.

For part (ii), we fix u and we have:

where (β

v , v ≥ 0) are two dependent Brownian motions and (B v , v ≥ 0) is another Brownian motion independent from them. Now, from (59), we obtain (60).

Remark 4.6. From (59), with p u (x, y) denoting now the density function of the couple (sinh(B

u )), we have:

In theory, we should be able to compute this probability density as we know the joint distribution of the couple of exponential functionals (see e.g. [START_REF] Alili | On a triplet of exponential Brownian functionals[END_REF]).

A three-dimensional extension

Alili, Dufresne and Yor, in [START_REF] Alili | Sur l'identité de Bougerol pour les fonctionnelles exponentielles du mouvement Brownien avec drift[END_REF], obtained a 3-dimensional extension of Bougerol's identity:

Proposition 4.7. The two following processes have the same law:

where:

with (G t , t ≥ 0) denoting another Brownian motion, independent from B.

Proof. (Proposition 5.1) Applying Itô's formula to the process X, we obtain:

For the second integral in (73), Itô's formula once more yields:

Thus:

For the first integral in (73), we recall that, with γ denoting another Brownian motion independent from B (thus independent also from Z):

Hence:

with γ denoting another Brownian motion, depending on γ and on B.

The proof finishes by some elementary computations from (73), using (74) and (76). Finally, using Lamperti's relation, which states that:

we obtain the last identity in (70).

We may continue a little further in order to obtain the following result relating the diffusion X with its reciprocal (recall that:

Corollary 5.3. The following relation holds:

Proof. It follows easily by some relations involving the changes of time: