Didier Dubois

Uncertainty Theories: a Unified View (Invited Paper)

The variability of physical phenomena and partial ignorance about them motivated the development of probability theory in the two last centuries. However, the mathematical framework of probability theory, together with the Bayesian credo claiming the inevitability of unique probability measures for representing agents beliefs, have blurred the distinction between variability and ignorance. Modern theories of uncertainty, by putting together probabilistic and set-valued representations of information, provide a better account of the various facets of uncertainty.

I. INTRODUCTION

The modeling of uncertainty is motivated by two concerns: taming the variability of observed phenomena and facing incomplete information in decision processes. These two concerns are related by the fact that in the face of variability, it is difficult to predict what the next state of the world will be. Yet the two concerns are distinct in the sense that variability is far from being the only cause of information incompleteness. Moreover, variability is an objective phenomenon, that supposedly corresponds to a property of the world. On the contrary, incompleteness of information refers to a human agent, and is thus irreducibly subjective. An agent cannot make prediction about the world behavior because of variability and because of lack of knowledge. But one may miss knowledge about quantities that are not subject to variability, like when guessing the birth date of a famous person, or the height of a mountain.

However, the development of probability theory, as witnessed by the Bayesian school especially, led to blur the distinction between variability and ignorance, suggesting that a unique probability distribution is enough to account for both randomness and incomplete information. More recently, new theories of uncertainty have emerged where partial ignorance is acknowledged and represented separately from randomness: the theories of imprecise probabilities, evidence, and possibility respectively. The aim of this paper is to suggest a unified view of these approaches.

II. SET-BASED REPRESENTATIONS OF PARTIAL

IGNORANCE

The basic tool for representing information incompleteness is set theory: an ill-known quantity is represented as a disjunctive set, i.e. a subset of mutually exclusive values, one of which is the real one. This kind of uncertainty is naturally accounted for in logical representations. In propositional logic, a set of formulas viewed as constraints on the joint Didier Dubois is with the Institut de Recherche en Informatique de Toulouse (IRIT), CNRS, Université de Toulouse, 31062 Toulouse Cedex 9, France (email: dubois@irit.fr).

values of Boolean variables, generally does not single out a unique assignment (a model). When a given proposition is not entailed by a belief base, it does not imply that its negation is necessarily entailed: there are propositions, the truth of which cannot be inferred. This is when knowledge is incomplete.

In the area of numerical modelling, the processing of incomplete information is basically carried out by interval analysis [START_REF] Moore | Methods and applications of Interval Analysis[END_REF] or constraint propagation methods. Incomplete information comes in the form of intervals assigned to unknown quantities. Assigning an interval [a, b] to a quantity x means that x is known to take one and only one value in [a, b], but it is nor known which one. Note that x is not necessarily a random quantity. It can be deterministic, yet unknown. For instance, the reader may not know the birth date of the president of Brazil, even if he or she can suggest a time interval. Note that such an interval like [a, b] is never an attribute of the world, since the real value of x is precise. It is an attribute of an agent, an observer, a sensor, etc. In this sense, the set-valued representation is subjective: two individuals may come up with different intervals for the value of x, both of them being correct, even if not equally informative. The wider the set, the less informative. Total ignorance corresponds to the whole domain of x being possible.

The two modalities attached to this representation of partial ignorance are the possibility (or plausibility) and the certainty (or necessity), not probability. Asserting x ∈ [a, b] comes down to declaring any value outside [a, b] as impossible for x. Moreover, for an agent only knowing x ∈ [a, b]:

• Any event A understood as the assertion x ∈ A is possible whenever A ∩ [a, b] is not empty. • Any such event A is certain for this agent, whenever [a, b] ⊆ A.
It is clear that possibility and certainty correspond to logical consistency and logical entailment from the available knowledge. This is the Boolean version of possibility theory [START_REF] Dubois | Possibility Theory[END_REF]. This type of uncertainty is captured by interval analysis. The basic problem in uncertainty analysis is: given a mathematical model of the form y = f (x 1 , . . . , x n ), find the range of the output y, when all is known about inputs

x i , i = 1, . . . , n is that x i ∈ [a i , b i ].
This type of approach to partial ignorance, although perfectly sound, is crude and potentially little informative if intervals are too wide, each bound being respectively overpessimistic and overoptimistic.

III. BAYESIAN PROBABILITY AS A REPRESENTATION OF PARTIAL IGNORANCE

That probability theory may account for the variability of observed phenomena is clear, once statistical data is used to infer probabilities, interpreted as limit frequencies. However, this interpretation of probability is not valid for nonrepeatable events. Several scholars in the XXth century, like Ramsey, De Finetti and Savage have introduced the notion of subjective probability so as to account for the fact that individuals entertain beliefs about non-repeatable situations. This trend has led to the Bayesian view of probability.

The operational definition of a degree of probability is then an amount of money an agent is ready to bet on the occurrence of an event. In such a betting experiment, the agent provides betting odds under an exchangeable bet assumption: it says that the agent is ready to buy a lottery ticket about the event at the same price as (s)he would sell it. On this basis, Bayesians claim that any state of incomplete knowledge of an agent can (and should) be modelled by a single probability distribution on the appropriate referential, and that degrees of belief coincide with probabilities that can be revealed by observing the betting behaviour of the agent. Failing to use a unique probability distribution, the agent is sure to lose money (this is called the Dutch book).

The idea that it is always possible to come up with a precise probability model, whatever the agent's state of knowledge, looks debatable. It is not clear that incomplete knowledge should be modelled by the same tool as variability (a unique probability distribution) [START_REF] Ferson | Different methods are needed to propagate ignorance and variability[END_REF]. One may argue, following e.g. Walley [START_REF] Walley | Statistical Reasoning with Imprecise Probabilities[END_REF], that the lack of knowledge is precisely reflected by the situation where the probability of events is ill-known, except maybe for a lower and an upper bound. In the face of ignorance, it is not clear that individuals would buy and sell a lottery tickets at the same price: the selling price would be higher than the buying price, the difference being all the higher as the ignorance is significant. Moreover one may also have incomplete knowledge about the variability of a non-deterministic quantity if the observations made were poor, or if only expert knowledge is available. This point of view may to some extent reconcile subjectivists and objectivists: it agrees with subjectivists that human knowledge matters in uncertainty judgements, but it concedes to objectivists that such knowledge is generally not rich enough to allow for a full-fledged probabilistic modelling.

Bayesian betting rates cannot distinguish between the situation of total ignorance and the situation of complete knowledge about a perfectly random phenomenon. The same uniform probability accounts for an unknown die and a die known as fair. In the face of ignorance this uniform probability is enough for a decision-maker accepting Savage decision theory as the way to follow when choosing among acts. It becomes debatable when a new information requires a belief revision process or when conservative risk analysis must be carried out. In a crime case involving suspects Peter Paul and Mary, if it is only known that the killer is a man with probability one-half, a Bayesian may attach probability one fourth to Peter and to Paul. If it is known later that Peter has an alibi, it sounds safer to bet again on the updated set of suspects (at equal odds on Mary and Paul), than to consider Mary as having more chance to be the killer due to Bayes rule attaching probability two-third to her. In the case of risk analysis, based on propagating uncertainty through a mathematical model, precise subjective probability assessments of ill-known inputs will lead to alter the variance of the output in a debatable way: part of this calculated variance will be due to ignorance, not to effective variability, but they cannot be told apart. Yet, variance due to ignorance can be reduced by collecting more information, while variance due to variability just reflects the behaviour of the phenomenon under concern.

IV. BLENDING SET-VALUED AND PROBABILISTIC

REPRESENTATIONS OF UNCERTAINTY

Modern uncertainty theories put together probabilistic and set-valued representations, which allows for a clear separation between randomness and incompleteness. Moreover, it makes set-valued representations much more expressive. The most general approach consists in moving from the use of a single probability to a set of probabilities, all the larger as information is poor. It may express imprecision about an ill-known (objective) probability model and induces upper and lower probabilities of events. The subjective view can be accommodated by giving up the exchangeable bet assumption. The minimal selling and maximal buying prices of a lottery ticket winning one pound if an event occurs correspond to the upper and lower probabilities of the event, respectively [START_REF] Walley | Statistical Reasoning with Imprecise Probabilities[END_REF]. Another way of blending probability and set representations is to randomize the latter. Random sets are the basis of the mathematical theory of evidence [START_REF] Shafer | A mathematical Theory of Evidence[END_REF]. The idea is to define a probability distribution not on a state space, but on the powerset thereof. Each probability weight attached to a set corresponds to an amount of probability that should be shared among the elements of this set, but is not by lack of information. Upper and lower probabilities induced by this random set representation are special cases of imprecise probabilities. Finally, the last representation framework consists in assuming that possibility is a matter of degree, values in the set containing the unknown quantity being more or less plausible (or surprising [START_REF] Shackle | Decision, Order and Time in Human Affairs[END_REF]). This is possibility theory [START_REF] Dubois | Fuzzy sets and probability: misunderstandings, bridges and gaps[END_REF], based on possibility distributions. In possibility theory, information is thus summarized by fuzzy sets [START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF]. It is important to figure out that the reason why fuzzy set theory is relevant for uncertainty handling is because it is a set-based approach, not just because it is fuzzy in the sense of Zadeh (i.e. gradual rather than abrupt). The bridge to imprecise probability comes from the fact that fuzzy sets representing possibility distributions are equivalent to consonant (nested) random sets [START_REF] Dubois | Fuzzy sets and probability: misunderstandings, bridges and gaps[END_REF].

A. Imprecise probabilities

The theory of imprecise probabilities has been systematized and popularized by Walley's book [START_REF] Walley | Statistical Reasoning with Imprecise Probabilities[END_REF]. In this theory, uncertainty is modeled by a family P of probability distributions. Lower and upper probability bounds are defined as follows :

P * (A) = inf P ∈P P (A) and P * (A) = sup P ∈P P (A).
These two measures are dual to each other (P * (A) = 1 -P * (A c )), and specifying one of them is enough to completely characterize the probability family. Let P * * = {P |∀A ⊆ X measurable, P * (A) ≤ P (A) ≤ P * (A)}. In general, we have P ⊂ P * * , since P * * can be seen as a projection of P on events. A family P can also be defined by a set of restrictions of the type

P (A) ≤ xi∈A p(x i ) ≤ P (A)
When it defines a non empty probability set P and each bound P (A) and P (A) is attained by one probability measure in P, the representation is said to be coherent.

B. Random disjunctive sets

Formally, a random set is a mapping from a probability space (Ω, A, P ) to the power set ℘(X) of another space X, also called a multi-valued mapping Γ. The set Γ(ω) represent incomplete knowledge about a random variable when the realization is ω in the probability space (as opposed to random sets which are conjoints of elements like an illknown region in a digital image). Then this multimapping induces a probability family on X representing all probability functions on X that could be found from all measurable mappings Ω → X compatible with Γ [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF]. Upper and lower probabilities probabilities on events in X are then generated. Such lower and upper probabilities are respectively called belief and plausibility functions by Shafer [START_REF] Shafer | A mathematical Theory of Evidence[END_REF] and denoted Bel and Pl respectively. He uses an alternative (and useful) representation of the random set consisting of a distribution of positive masses m over the power set ℘(X) s.t.

E⊆X m(E) = 1 and m(∅) = 0. Namely if E = Γ(ω), then let m(E) = p(ω).
A set E that receives strictly positive mass is called a focal set. The mass m(E) is interpreted as the probability of knowing only E as containing the actual solution to the problem under concern. We have :

Bel(A) = E,E⊆A m(E) P l(A) = 1 -Bel(A c ) = E,E∩A =∅ m(E).
There is a one-to-one correspondence between the mass distribution and the belief function since

m(E) = B⊂E (-1) |E-B| Bel(B).
This last equation is known as the Möbius inverse and can be applied to any kind of lower probability function. The positivity of the mass function obtained by the Möbius inverse is characteristic of the random set setting. In the finite case, it can be shown that the lower probability function is an ∞monotone capacity. The set P Bel = {P |∀A ⊆ X measurable, Bel(A) ≤ P (A) ≤ P l(A)} is coherent and forms the probability family induced by the belief function. Note that Shafer [START_REF] Shafer | A mathematical Theory of Evidence[END_REF] does not refer to an underlying probability space, nor does he uses the fact that a belief function is a lower probability: in his view, extensively taken over by Smets [START_REF] Smets | The transferable belief model[END_REF], Bel(A) is supposed to quantify an agent's belief per se with no reference to a probability. However, the basic mathematical tool common to Dempster's upper and lower probabilities and to the Shafer-Smets view is the notion of (generally finite) random disjunctive set.

C. Fuzzy sets as possibility distributions

A possibility distribution π [START_REF] Dubois | Possibility Theory[END_REF] is a mapping from X to the unit interval such that π(x) = 1 for some x ∈ X. Formally, a possibility distribution is equivalent to the membership function of fuzzy set µ(x) = π(x) ∀x. Possibility distributions were introduced by Zadeh [START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF] as flexible constraints induced by fuzzy natural language statements. Twenty years earlier, Shackle [START_REF] Shackle | Decision, Order and Time in Human Affairs[END_REF] had introduced an equivalent notion called distribution of potential surprise (corresponding to 1 -π(x)) with a view to represent nonprobabilistic uncertainty. Several set-functions can be defined from a possibility distribution π, especially [START_REF] Dubois | Possibility Theory[END_REF]:

• Possibility measures: Π(A) = sup x∈A π(x) • Necessity measures: N (A) = 1 -Π(A c )
The possibility degree of an event expresses the extent to which this event is plausible, i.e., consistent with a possible state of the world. Necessity degrees express the certainty of events, by duality. Under these measures, the possibility quantified by distribution π is potential (in the spirit of Shackle), i.e. π(x) = 1 does not guarantee the corresponding value x.

A (potential) possibility degree can be viewed as an upper bound of a probability degree [START_REF] Dubois | Fuzzy sets and probability: misunderstandings, bridges and gaps[END_REF]. Let P π = {P, ∀A ⊆ X measurable, N (A) ≤ P (A) ≤ Π(A)} be the set of probability measures encoded by a possibility distribution π. This representation is coherent since upper and lower probabilities induced by P π are precisely Π and N .

In the finite case, a possibility distribution is also equivalent to a random set whose focal elements are nested. Namely, a belief function (resp. a plausibility function) is a necessity measure (resp a possibility measure) if and only if they derive from a mass function with nested focal sets (already in [START_REF] Shafer | A mathematical Theory of Evidence[END_REF]). Their characteristic property is then N (A ∩ B) = min(N (A), N (B)) (resp. Π(A ∪ B) = max(Π(A), Π(B)). In this situation, the same amount of information is contained in the mass function m and the possibility distribution π(x) = P l({x}). In the general nonnested case, m cannot be reconstructed from P l({x}).

V. PRACTICAL REPRESENTATIONS OF INCOMPLETE

KNOWLEDGE

Imprecise probabilities are complex to represent, much more than probability measures, whether on finite sets or the real line. On a finite set X with N elements, a lower probability needs 2 N values to be specified (and a consistency check to make it sure that the corresponding family is not empty). It defines a convex polyhedron, which may have up to N ! vertices. If X is defined via a set of Boolean variables, there is a counterpart of Bayesian networks, called credal networks, which allow the use of probability bounds on much smaller subspaces. In general, to completely specify a probability family induced by a random set, one still needs to give 2 |X| different values, thus not reducing the complexity of the representation with respect to a capacity. However, simple belief functions having only a few positive focal sets do not exhibit such a complexity. From a computational perspective, the main advantage of random sets is that they can be seen as a probability distribution over subsets of X. Therefore, they can easily be simulated by some process such as Monte-Carlo sampling, which is not the case for other Choquet capacities. On the real line, a random set is often restricted to a finite collection of closed intervals with associated weights, and one can then easily extend results from interval analysis [START_REF] Moore | Methods and applications of Interval Analysis[END_REF] to random intervals [START_REF] Oberkampf | Challenge problems : uncertainty in system response given uncertain parameters[END_REF]. For continuous random intervals, the mass function is replaced by a mass density bearing on intervals. In the following we point out existing simple representations of probability families [START_REF] Destercke | On the relations between practical representations of imprecise probabilities[END_REF].

A. Fuzzy sets and intervals

At most |X| values are needed to completely specify a possibility distribution, making them easier to represent than general random sets. On the real line, continuous (or upper semi-continuous) unimodal possibility distributions on the real line encompass closed intervals and are called fuzzy intervals [START_REF] Dubois | Possibility Theory[END_REF]. They have a very natural interpretation as sets of nested confidence intervals [START_REF] Dubois | Possibility theory and statistical reasoning[END_REF] or probabilistic inequalities, like Chebyshev's [START_REF] Dubois | Probability-possibility transformations, triangular fuzzy sets, and probabilistic inequalities[END_REF]. Ill-known probability models where only some parameters are known, like the support and the mode, are liable to a possibilistic representation [START_REF] Baudrit | Practical representations of incomplete probabilistic knowledge[END_REF]. Possibility distributions are thus the simplest models of probability families, and they play a central role when modeling vague assessments of probabilities (see [START_REF] De Cooman | A behavioural model for vague probability assessments[END_REF]).

B. P-boxes and generalized p-boxes

A p-box [START_REF] Ferson | Construction of probability boxes and Dempster-Shafer structures[END_REF] is defined by a pair of cumulative distributions on the real line such that F ≤ F , bounding the cumulative distribution of an imprecisely known probability function with density p. It can be viewed as a generalized interval as well. The corresponding set of probabilities P p-box is representable by a belief function whose focal elements are of the form {x, F (x) ≥ α} \ {x, F (x) ≥ α}. A p-box is a covering approximation of a parameterized probability model whose parameters (like mean and variance) are only known to belong to an interval.

A p-box can be induced from a possibility distribution π, letting F (x) = N ((-∞, x]) and F (x) = Π((-∞, x]), but the probability family P p-box induced by this p-box is strictly larger than P π [START_REF] Baudrit | Practical representations of incomplete probabilistic knowledge[END_REF]. So, while a fuzzy interval induces a p-box, such generated p-boxes are less informative than the possibility distributions they are computed from.

Interestingly, the notion of cumulative distribution is based on the existence of the natural ordering of numbers. On a finite set, no obvious notion of cumulative distribution exists. In order to make sense of this notion over X, one must equip it with a complete preordering and define two cumulative distributions according to this ordering. it comes down to a family of nested confidence sets ∅ ⊆ A 1 ⊆ A 2 ⊆ . . . ⊆ A n ⊂ X. The family P p-box can then be represented by the following restrictions on probability measures

α i ≤ P (A i ) ≤ β i i = 1, . . . , n (1)
with

α 1 ≤ α 2 ≤ . . . ≤ α n ≤ 1 and β 1 ≤ β 2 ≤ . . . ≤ β n ≤ 1.
If we take X = and

A i = (-∞, x i ],
it is easy to see that we retrieve the usual definition of P-boxes. A family P p-box described by a generalized P-box can be encoded by a pair of possibility distributions π 1 , π 2 s.t. P p-box = P π1 ∩ P π2 where π 1 comes from constraints α i ≤ P (A i ) and π 2 from constraints P (A i ) ≤ β i . Again, it is representable by a belief function [START_REF] Destercke | On the relations between practical representations of imprecise probabilities[END_REF].

C. Probability Intervals

Probability intervals over the finite space X are defined as lower and upper probability bounds restricted to singletons x i [START_REF] Campos | Probability intervals : a tool for uncertain reasoning[END_REF]. They can be seen as a set of intervals L = {[l i , u i ], i = 1, . . . , n} defining the family

P L = {P |l i ≤ p(x i ) ≤ u i ∀x i ∈ X}
it is easy to see that P L is totally determined by 2|X| values. P L is non-empty provided that

n i=1 l i ≤ 1 ≤ n i=1 u i .
A set of probability intervals L will be called reachable if, for each x i , each bound u i and l i can be reached by at least one distribution of the family P L . Reachability is equivalent to the condition

j =i l j + u i ≤ 1 and j =i u j + l i ≥ 1
Lower and upper probabilities P * (A), P * (A) are calculated by the following expressions

P * (A) = max( xi∈A l i , 1 -xi / ∈A u i ), P * (A) = min( xi∈A u i , 1 -xi / ∈A l i ).
De Campos et al. [START_REF] Campos | Probability intervals : a tool for uncertain reasoning[END_REF] have shown that these bounds are Choquet capacities of order 2. It is easy to extract pboxes from probability intervals and conversely, but some information is lost in the process.

D. Clouds

A cloud [START_REF] Neumaier | Clouds, fuzzy sets and probability intervals[END_REF] can be seen as an Interval-Valued Fuzzy Set F such that (0, 1) ⊆ ∪ x∈X F (x) ⊆ [0, 1], where F (x) is an interval [δ(x), π(x)]. It implies that π(x) = 1 for some x and δ(y) = 0 for some y. A probability measure P on X is said to belong to a cloud F if and only if ∀α ∈ [0, 1]:

P (δ(x) ≥ α) ≤ 1 -α ≤ P (π(x) > α) (2)
under all suitable measurability assumptions. From this definition, a cloud (δ, π) is equivalent to the cloud

(1 -π, 1 -δ). If X is a finite space of cardinality n, let A i = {x i , π(x i ) > α i+1 } and B i = {x i , δ(x i ) ≥ α i+1 } A cloud
can thus be defined by the following restrictions [START_REF] Destercke | On the relations between practical representations of imprecise probabilities[END_REF]:

P (B i ) ≤ 1 -α i ≤ P (A i ) and B i ⊆ A i i = 1, . . . , n (3)
where

1 = α 0 > α 1 > α 2 > . . . > α n > α n+1 = 0 and ∅ = A 0 ⊂ A 1 ⊆ A 2 ⊆ . . . ⊆ A n ⊆ A n+1 = X; ∅ = B 0 ⊆ B 1 ⊆ B 2 ⊆ . . . ⊆ B n ⊆ B n+1 = X.
Let P δ,π be the probability family described by the cloud (δ, π) on a referential X. Clouds are closely related to possibility distributions and p-boxes as follows [START_REF] Destercke | On the relations between practical representations of imprecise probabilities[END_REF]:

• P δ,π = P π ∩P 1-δ using the probability families induced by the two possibility distributions π and 1 -δ. • A cloud is a generalized p-box iff the sets {A i , B i , i = 1, . . . , n} form a nested sequence (i.e. there is a complete order with respect to inclusion); in other words, it means that π and δ are comonotonic. So a comonotonic cloud generates upper and lower probabilities that are plausibility and belief functions. • When the cloud is not comonotonic, P δ,π generates lower probabilities that are not even 2-monotone. It is anyway possible to approximate upper and lower probabilities of events from the outside by possibility and necessity measures based on π and 1 -δ:

max(N π (A), N 1-δ (A)) ≤ P (A) ≤ min(Π π (A), Π 1-δ (A)).
The belief and plausibility functions of the random set s.t. m(A i \B i-1 ) = α i-1 -α i are inner approximations of P δ,π , which become exact when the cloud is monotonic.

When π = δ the cloud is said to be thin. In the finite case, P π,π = ∅. To make it not empty, we need a one-step index shift, such that (assuming the π i 's are decreasingly ordered) δ i = 1 -π i+1 (with π n+1 = 0). Then, P δ,π contains a single distribution p such that p i = π i -π i+1 . In the continuous case P π,π contains an infinity of probability measures and corresponds to a random set whose realizations are doubletons (the end-points of the cuts of π).

The above representations of imprecise probabilities are easier to handle than general probability families. They often require less evaluations to be fully specified and they allow many mathematical simplifications that increase computational efficiency (except, perhaps, for non-comonotonic clouds).

VI. BASIC PROBLEMS IN UNCERTAINTY THEORIES

Probability theory is used either to capture stochastic features of a population of situations, or to measure degrees of belief of agents. Even if the two purposes converge for agents that derive their degrees of belief from the knowledge of frequencies, the two kinds of probability are not of the same nature. While frequentist probabilities refer to a population of situations and represent generic knowledge, subjective probabilities are elicited on the basis of single contingent events. This distinction is important to better understand various uncertainty management problems and how to to state and solve them when imprecise probabilities enter the picture. Especially, insofar as one may admit that reality is precise but only our perception of reality is imprecise, there cannot be such thing as a fully objective imprecise probability model. Even when capturing variability, i.e. when a set P of probability measures is assumed to contain the "true" probability measure P that governs a process, the imprecision itself is always subjective in the sense that it will vary from one observer to another. As for subjective upper and lower probabilities, they can be viewed as degrees of belief and plausibility of contingent events, without assuming the existence of a true probabilistic representation of belief (both Shafer [START_REF] Shafer | A mathematical Theory of Evidence[END_REF] and Smets [START_REF] Smets | The transferable belief model[END_REF] reject this assumption) and without reference to statistics. Shafer-Smets evidence theory is about uncertain testimonies, not about induction from data tainted with variability.

In the following, we briefly review basic problems of imprecise probability management:

• Learning imprecise models vs. eliciting degrees of belief Since the assumption of a unique probability distribution is often made in statistics, the basic learning problem is how to is to extract a probability distribution from data. When data is scarce, it would seem more natural to get imprecise probabilities. For instance, confidence intervals on distribution parameters naturally define probability families. Moreover the Imprecise Dirichlet model [START_REF] Bernard | An introduction to the imprecise Dirichlet model for multinomial data[END_REF] offers a tool to relate the number of observations to the width of probability intervals without resorting to so-called uninformative priors. Another important problem is to adapt classical methods for inferential statistics to the case of imprecise (interval or fuzzy) data. In the case of subjective probabilities, elicitation techniques are tailored to produce unique probability functions. Yet expert knowledge is imprecise, and more naturally comes under the form of nested intervals with various confidence levels. Some effort is needed to adapt existing elicitation techniques towards producing possibility distributions and generalized p-boxes, that naturally capture nested intervals. • Uncertainty propagation Given several inputs, some modeled by intervals, some modeled by probability distributions, some by possibility distributions, how to compute the output of a mathematical models? It is possible to use Monte-Carlo methods conjoined with interval analysis, when the imprecise probabilities derive from random sets [START_REF] Oberkampf | Challenge problems : uncertainty in system response given uncertain parameters[END_REF], [START_REF] Baudrit | Joint Propagation and Exploitation of Probabilistic and Possibilistic Information in Risk Assessment[END_REF]. However the important issue of modeling independence and accounting for partial knowledge about dependence in imprecise probability theories remains an open topic to a large extent [START_REF] Couso | A survey of concepts of independence for imprecise probabilities[END_REF]. [START_REF] Dubois | Representing partial ignorance[END_REF]. In evidence theory, this is Dempster rule of conditioning [START_REF] Shafer | A mathematical Theory of Evidence[END_REF]. It revises P and differs from the imprecise Bayesian conditioning rule, even if both rules coincide if P contains a single probability measure.

• Fusion The fusion problem consists in merging several bodies of uncertain information of the same nature issued from several sources. This is a problem that was of interest at the origins of probability theory (the merging of uncertain testimonies in legal procedures). It was revived in the late XXth century with the emergence of the computer, in robotics (merging sensor information), in reliability (merging expert opinions), in signal processing (merging digital images), etc. There are now many fusion rules especially in possibility theory [START_REF] Dubois | Possibility theory in information fusion[END_REF], evidence theory [START_REF] Shafer | A mathematical Theory of Evidence[END_REF], and in probability theory as well [START_REF] Cooke | Experts in Uncertainty[END_REF]. Fusion can be viewed as a preliminary step before inference (laying bare a 'sure input" from contradictory reports), and as a generalization of revision (Dempster rule of combination [START_REF] Shafer | A mathematical Theory of Evidence[END_REF] can be viewed as a generalization of Dempster rule of conditioning). • Statistics with imprecise data In the face of imprecise or fuzzy data, or for the purpose of summarizing the output of an uncertainty propagation or a fusion procedure in the framework of imprecise probabilities, it is useful to summarize information for the user. It is possible to compute mean values and variances, empirical or theoretical; however they will generally be imprecise, and can be difficult to compute. P-boxes can also be extracted [START_REF] Baudrit | Joint Propagation and Exploitation of Probabilistic and Possibilistic Information in Risk Assessment[END_REF]. But such p-boxes are generally not enough to address questions such as the probability for an output to lie between two bounds. A considerable effort is required to make imprecise statistics palatable for users.

VII. CONCLUSION

Imprecise probabilities are a natural concept for conjointly handling variability of phenomena and incomplete knowledge about them. Imprecise modeling is unusual. In classical approaches, a probabilistic model is an approximate but precise representation of variability. It is assumed that the precise results it produces are not too far from observations of a random reality. In contrast, an imprecise model is of higher order: it represents both knowledge about reality and knowledge about knowledge. There is a need to reconsider the foundations of statistics in this perspective. We reviewed simplified representations of imprecise probabilities, that should eventually help solving related computational problems at the practical level.

ACKNOWLEDGMENT

This paper is partially supported by the ANR project CRISCO2. It owes much to my colleagues and students, especially C. Baudrit, S. Destercke and H. Prade.

  B), where B captures the contingent observations, and A is the event of interest, when P varies in {P ∈ P, P (B) > 0}[START_REF] Walley | Statistical Reasoning with Imprecise Probabilities[END_REF]. This is a matter of querying the knowledge P by focusing it to the reference class B[START_REF] Dubois | Representing partial ignorance[END_REF].• Revision In the above problem, the set of probabilities P does not evolve because it is not of the same nature as the contingent information. In the case of revision. P represents prior incomplete uncertain information about a case, and B is of the same nature, yet certain. The input information tells us that P (B) = 1, so that some subjective probabilities in P can be ruled out. It leads to changing P into {P ∈ P, P (B) = 1}, or, if empty, {P ∈ P, P (B) maximal }. Then, in many cases, P * (A | B) = P

• Inference from contingent information A classical use of probabilistic models is the following: given a probabilistic model, representing generic knowledge, and deterministic observations on a contingent situation, predict whether some property is true in this situation (diagnosing a fault, assigning a class to an instance, etc...). This is done by Bayesian conditioning. In the case of imprecise probability models, this is done by an extension of Bayesian conditioning consisting in finding upper and lower conditional probabilities P (A | * (A∩B)

P * (A)


Microslip Induced Damping in the Contact of Nominally Flat Surfaces with Geometric Defects

Noussa Bouchaala, Jean-Luc Dion, and Nicolas Peyret

In this chapter, a rheological contact model (the EGM) is developed in order to quantify the energy dissipated by microslip in a jointed interface. The proposed rheological contact model is based upon a statistical description of contact between asperities. The key assumption for this model is that the coefficient of friction between the nominally flat surfaces is constant and that friction is the only source of energy dissipation. Measurements of static normal load and dynamic tangential load without any coupling between these two directions are used to inform and guide the model development.

Historical Perspectives on Dissipation Due to Microsliding

For many industrial applications in which the products are designed for dynamic solicitation (shock and vibration excitation), the vibration amplitudes of mechanical systems are not well predicted during the design phase. The source of inaccuracy in prediction of the vibration amplitudes is due to the lack of understanding of how to predict the amount of energy dissipated by an interface. Current approaches require that models be calibrated from measurements of existing systems, which precludes the prediction of the response of a system before it is fabricated for the first time. Thus, the ultimate goal of joints modeling is to predict the dynamic properties (mode shapes, natural frequencies, and damping in particular) of a jointed system accurately.

Sources of Energy Dissipation

The sources of damping in assembled structures are classified into two categories: material damping (which is often low) and damping in joints (which is more difficult to quantify due to its dependence on vibration level). Material damping, while difficult to model in inelastic solids, is typically low and easily modeled in engineering metals, which are the primary material of concern in this work. Damping due to joints, however, is significantly more difficult to model due to the complex interactions that occur at the interface surfaces (including wear, plasticity, etc.). Consequently, the measurement and prediction of damping in joints is an active research area [START_REF] Loch | Modélisation et Identification de l'Amortissement Dans les Structures Spatiales[END_REF][START_REF] Caignot | Prediction of damping in space launch vehicles using a virtual testing strategy[END_REF].

In assembled structures, the damping is due to both macroslip [START_REF] Berthillier | Blades forced response analysis with friction dampers[END_REF][START_REF] Poudou | Hybrid frequency-time domain methods for the analysis of complex structural systems with dry friction damping[END_REF] (i.e., the bulk movement of the entire interface) and microslip [START_REF] Goodman | Analysis of slip damping with reference to a turbine blade vibration[END_REF][START_REF] Beards | The damping of structural vibration by rotational slip in joints[END_REF] (i.e., the relative motion of sub-elements of the interface). In order to study the damping due to both microslip and macroslip, a number of different experimental setups have been developed (see, for instance, Chap. 5). One experimental setup, in particular, is a clamped beam with a longitudinal interface (see Goodman and Klumpp 1956 and Fig. 19.1).

Perspectives on Dissipation Due to Microslip

Contact between two bodies, either in motion or stationary, is a phenomenon that all interacting parts of a mechanism are subjected to. Regardless of the condition of the surface finish (whether cut, or polished to a mirror-like smoothness), contact at the nano-and microscale is rough. Often, a relative movement between the elements in contact occurs producing microslip at the interface of contact. Such slip should be well studied to evaluate frictional forces induced on the surface. Thus, the relative motion at the interface of a contact is a source of friction that is dependent upon the area of contact engaged in microslip, which plays an important role in energy dissipation.

As the adhesion area and microslip zone are not well known a priori, it should be obtained from experiments. This kind of problem is referred to in the literature as incipient sliding or quasi-static contact. The problem with partial slip was addressed for the first time by [START_REF] Cattaneo | Sul Contatto di due Corpi Elastici: Distribuzione Locale Degli Sforzi[END_REF]. Cattaneo considers the case of quasi-static contact of two spheres with the same elastic properties, loaded normally and tangentially. In the absence of microslip, Cattaneo shows that the tangential force is expected to generate an infinite tangential traction along the contact, which is physically inadmissible. So under the action of a tangential force, even very small, microslip is inevitable at the contact edges. Cattaneo assumes that the tangential force is equal to the normal force multiplied by the coefficient of friction, and the elliptical contact area is divided into two parts:

• Elliptical central area in which there is no relative movement between surfaces and where the tangential traction (tension) q.x/ satisfies the equation q.x/< p.x/ • An annular-elliptical zone of microslip where q.x/ D p.x/ This decomposition leads to a tangential traction q, shown in Fig. 19.2.

Building on the results of [START_REF] Cattaneo | Sul Contatto di due Corpi Elastici: Distribuzione Locale Degli Sforzi[END_REF], Mindlin investigated the compliance of two perfectly smooth elastic spheres, loaded normally and tangentially [START_REF] Mindlin | Compliance of elastic bodies in contact[END_REF]. Mindlin determined the distribution of tangential tractions (Fig. 19.3) assuming that both bodies have the same geometry and elastic properties. The results from both [START_REF] Cattaneo | Sul Contatto di due Corpi Elastici: Distribuzione Locale Degli Sforzi[END_REF][START_REF] Mindlin | Compliance of elastic bodies in contact[END_REF]) serve as a basis for modeling the contact of asperities.

Contact of Asperities

For contact between rough surfaces, [START_REF] Greenwood | Contact of nominally flat surfaces[END_REF] proposed a rough model, using several assumptions including:

• The asperities are spherical, with the same mean curvature at the top.

• Asperities are sufficiently far apart to behave independently.

• The heights of asperities follow a probability distribution '.y i /.

• Individual behavior of each asperity can be described by the Hertz theory [START_REF] Hertz | Über die Berührung Fester Elastischer Körper (On the contact of elastic solids)[END_REF][START_REF] Johnson | Contact Mechanics[END_REF].

This statistical representation for contact along an interface has been extensively developed over the last 50 years (for instance, see the review presented in Chap. 18).

In what follows, this branch of contact modeling is further extended to study the energy dissipation within a jointed interface. Therefore, the aim of this chapter is to quantify the dissipated energy by microsliding using both experimental and model based approaches. In Sect. 19.2, the experimental procedure is presented and experimental results are discussed. In Sect. 19.3, a mathematical model is developed considering the hypotheses of [START_REF] Greenwood | Contact of nominally flat surfaces[END_REF] for the contact between two nominally flat surfaces. The energy dissipated and the damping ratio are determined in Sect. 19.4. Numerical results are presented, discussed, and compared with those obtained experimentally.

Tribometer and Measurements

Tribometer Set

In order to perform accurate measurements, a tribometer with a hydraulic actuator is used. This kind of tribometer is described in [START_REF] Dion | A new test bench for friction measurements and the evolution of the Lugre model to the frozen model[END_REF]. The test fixture allows uncoupling normal and tangential loads over a wide frequency bandwidth to ensure good quality of measurements and test conditions (Fig. 19.4). This tribometer allows obtaining at 200 Hz and 1-100 m displacement range, inducing a 1.2-120 mm/s velocity range. The studied surface (approximately 2 cm 2 ) is a small planar contact (Fig. 19.4). The normal static load F n varies between 100 and 5000 N while the tangential load F t is dynamic and varies between 5000 N and +5000 N. The studied samples shown in Fig. 19.5 are in allied aluminum Al 2017 machined by milling. For this sample, the roughness R a is lower than 0.4 µm and the flatness error is lower than 0.02 mm (see Fig. 19.6). The effective contact area (measured after the test) is estimated to 10% of the apparent contact area with 2000 N normal load. For all the tests, the hydraulic jack movements are displacement controlled with the feedback signal of the internal displacement transducer (LVDT) (Fig. 19.4). Here, it is assumed that there is no coupling between normal and tangential loads. When the dynamic tangential load is applied, the static normal load keeps its nominal value with a 2% maximum deviation that stems from the dynamic tangential load. The displacement excitation is controlled and assumed to be sinusoidal. The harmonic ratio (between the fundamental and the 3rd harmonic) is around 5% in the worst case.

Experimental Result

The evolution of the normalized load T=N versus the tangential displacement is depicted in Fig. 19.7. This type of representation is often used to show a contact's dissipative behavior. The area inside the curve is the dissipated energy during microslip per cycle. The nature of the behavior (linear or not, dissipative or not) is clearly shown through the shape of the curve. The quasi-horizontal parts of the curve are indicative of macroslip and the oblique parts represent the tangential stiffness.

Microcontact Model

Consider a contact between a plane and a nominally flat surface covered with a large number of asperities. It is assumed that all asperity summits are spherical with the same radius and that their heights vary randomly '.y i /. Figure 19.8 shows schematically the considered type of contact.

Normal Distribution of the Load on the Asperities

The contact model between an individual spherical asperity and a rigid plane is shown in Fig. 19.9. It is assumed that the elastic proprieties of the contacting bodies are identical to an equivalent elastic material on a rigid plane [START_REF] Greenwood | Contact of nominally flat surfaces[END_REF]. The frictional load is parallel to the x-axis.

The behavior of an individual asperity can be derived from the Hertzian equations. For the contact between a sphere of radius and a rigid plane, the contact radius a i ,a r e aA i , and load N i can be expressed in terms of the compliance ı ni by [START_REF] Greenwood | Contact of nominally flat surfaces[END_REF], [START_REF] Johnson | Contact Mechanics[END_REF]

a i D p ı ni (19.1) A i D ı ni (19.2) N i D 4 3 p E ı 3=2 ni : (19.3)
Here, E is a composite modulus of elasticity given by

1 E D 1 2 1 E 1 C 1 2 2 E 2 ; (19.4)
where and E are Poisson's ratio and Young's modulus, with the subscripts referring to the two bodies in contact. For a distance ı n between the rigid plane and the mean plane of the rough surface (Fig. 19.8), there will be contact at any asperity whose height is greater than ı n . Thus, the probability of contact at any given asperity of height y i is

Prob.y i >ı n / D Z 1 ı n '.y i /dy i : (19.5)
The total number of asperities is N a D a S a , where a is the density of asperities distributed over the apparent area of contact S a . The number of asperities in contact is given by [START_REF] Greenwood | Contact of nominally flat surfaces[END_REF])

n a D N a Z 1 ı n '.y i /dy i : (19.6)
The total contact area is expressed by

S D N a Z 1 ı n .y i ı n /'.y i /dy i : (19.7)
For the compliance ı ni D y i ı n (the distance over which points outside the deforming zone move together during the deformation), the total load supported by asperities is expressed as [START_REF] Greenwood | Contact of nominally flat surfaces[END_REF])

N D 4 3 N a E p Z 1 ı n
.y i ı n / 3=2 '.y i /dy i : (19.8)

Distribution of Tangential Load on the Asperities

Consider an asperity in contact with the plane submitted first to a constant normal force N i then to a tangential displacement ı ti . Two slip conditions can be distinguished as partial slip and total slip in what follows.

The Partial Slip Condition

The contact area is split into a stick zone and slip zone (Fig. 19.10). The maximum tangential force T i max does not exceed in absolute value the product of the normal force by the coefficient of friction T i max <N i . The stick region is the circle of radius c whose value can be found from the magnitude of the tangential force

c a i D 1 T i N i 1=3 :
(19.9)

The force displacement relationship for an individual asperity is given by Gallégo (2007)

T i D N i 1 1 16a i ı ti 3N i G 3=2 ! ; (19.10)
where G is a composite shear modulus

1 G D 2 1 G 1 C 2 2 G 2 : (19.11)
The tangential force has maximum T i D N i for total slip. Correspondingly, from Eq. (19.10), the maximum tangential deflection before the inception of sliding is

ı Li D 3 16
G N i a i : (19.12)

The Total Slip Condition

In this state, there is no area that is permanently adhered. During the cycle, the maximum tangential force reaches the absolute product of the normal force by the coefficient of friction T i max D N i . From [START_REF] Johnson | Contact Mechanics[END_REF], the solutions of a circular contact initially charged by constant normal force P and subjected to a tangential load Q x (Q x <P 0 ) oscillating between the values ˙Q are given (see Fig. 19.11).

During unloading (between A and C), the distribution of tangential load on the asperities is [START_REF] Gallégo | Fretting et Usure Des Contacts Mécaniques: Modélisation Numérique[END_REF])

T id D 2N i 0 @ 0 @ 1 2 16a i ı ti 3N i G C 1 T i max N i 2=3 C 1 ! 3=2 1 A 1 1 A T i max :
(19.13)

The situation at Q x D Q is identical with that at Q x D Q , except for the reversal of sign. Hence, between (C and A) the distribution of tangential load on the asperities then becomes [START_REF] Gallégo | Fretting et Usure Des Contacts Mécaniques: Modélisation Numérique[END_REF])

D B O C A Q Q * d * d -d A(Q=Q * ) C(Q=-Q * ) B(Q=0) P 0 q(r) r -Q * Q a c
T ic D T id . ı/ D 2N i 0 @ 0 @ 1 2 16a 1 ı ti 3N i G 1 T i max N i 2=3 1 ! 3=2 1 A 1 1 A T i max :
(19.14)

In [START_REF] Greenwood | Contact of nominally flat surfaces[END_REF], a model based on a statistical distribution of asperities in contact in the state of partial sliding is proposed. In order to generalize the Greenwood model for damping in an assembly, stick, partial slip, and total sliding have to be modeled with independent initial conditions and a standalone set of equations for each asperity Eqs. (19.15)-(19.17).

If ı ti max >ı Li (total slip), the maximum tangential force reaches the absolute product of the normal force by the coefficient of friction T i max D N i and the asperity moves by an amount ı 0 D ı ti max ı Li . Thus, Eqs. (19.10), (19.13), and (19.14) are recast as

T i D 8 < : N i 1 1 16a i ı ti 3N i G eq 3=2 ;ı ti <ı Li N i ;ı ti ı Li (19.15) T id D 8 < : 2N i 1 2 16a i .ı ti ı 0 / 3N i G eq C 1 T i max N i 2=3 C 1 3=2 ! 1 ! C T i max ;
ı ti >ı 0 ı Li N i ;ı ti ı 0 ı Li (19.16)

T ic D 8 < : 2N i 1 2 16a i .ı ti Cı 0 / 3N i G eq 1 T i max N i 2=3 1 3=2 ! 1 ! T i max ; ı ti <ı Li ı 0 N i ;ı ti ı Li ı 0 (19.17)
Notice that this EGM allows partial slip and total sliding for two asperities in the same interface.

Model Properties

It is assumed that the coefficient of sliding friction between the surfaces is constant and that dry friction is modeled by Coulomb's law. In accordance with experimental results, numerical simulations will be performed with the following material data: D 0:55 and E 1 D E 2 D 69 GPa.

Changing Phases for a Single Asperity

In this section, an asperity in contact with the rigid plane is compressed by a constant normal force N i parallel to the y-axis (Fig. 19.9), to which a tangential displacement ı ti oscillating is subsequently applied.

Oscillating Tangential Displacement with a Constant Amplitude

First, an oscillating tangential displacement with a constant amplitude ˙ıti max is considered. The contact radius and the contact area due to N i are held constant and as given by Hertz. Figure 19.12 shows the evolution of the tangential displacement versus time.

The first application of ı ti in a positive direction P ı ti >0causes microslip in the annulus c r a i . The tangential force is given by Eq. (19.10), and shown by the curve OA in Fig. 19.13. Keeping the normal force constant, the tangential force is increased from zero and the stick region decreases in size according to Eq. (19.9).

At point A in Fig. 19.12, the tangential displacement begins to decrease P ı ti < 0, which is equivalent to the application of a negative increment in ı ti .D u r i n g unloading, the tangential force is given by Eq. (19.13), and shown by the curve ABC in Fig. 19.13. The slip region is then defined by Gallégo ( 2007)

c a i 3 D 1 2 1 C c a i 3 ! ; (19.18)
where c is the value of c in A. At point C, when the tangential displacement is completely reversed, substituting ı ti D ı ti max in Eq. (19.14)givesT ic D T i max . Thus, original slip is covered by the reversed slip and the state achieved is a complete reversal of that at ı ti D ı ti max . Further loading leads to a series of states similar to unloading from ı ti D ı ti max ,but of opposite sign, as depicted by the curve CDA in Fig. 19.13.

In fact, it is shown in what follows that the effect of a tangential displacement with a magnitude lower than the limit deflection ı Li does not give rise to a total slip motion but, nevertheless, induces a partial slip referred to as slip or microslip. Conversely, when ı ti max >ıLi, the tangential force increases from zero to a limiting value T i max D N i , the asperity moves by an amount ı 0 D ı ti max ı Li , and the stick region dwindles to a single point at the origin resulting in the bodies being on the verge of sliding. Taking into account Eqs. (19.15)-(19.17), the hysteretic forcedeflection relation for a contact is shown in Fig. 19.14.

Keeping N i constant and increasing the amplitude of the tangential displacement per iteration (Fig. 19.15a), the hysteretic force-deflection relations for a contact are shown in Fig. 19.15b.

Oscillating Tangential Displacement with Variable Amplitude

The second case considered is the case where the tangential displacement oscillates with variable amplitude acting along the x-direction, as shown in Fig. 19.16.

In this case, the force-deflection relationship between the relative tangential displacement ı ti of the contact surface and the corresponding tangential load in the case of constant normal load are given by

T i D N i 1 1 16a i ı ti 3N i G 3=2 ! : (19.19)
• During the unloading between A and B ( P ı ti <0)

T id D 2N i 0 @ 0 @ 1 2 16a i ı ti 3N i G C 1 T i max N i 2=3 C 1 ! 3=2 1 A 1 1 A C T i max :
(19.20) • During the reloading between B and C ( P ı ti >0)

T ic D 2N i 0 @ 0 @ 1 2 16a i .ı ti ı 0 / 3N i G 1 max jT id j N i 2=3 1 ! 3=2 1 A 1 1 A max jT id j ; (19.21)
where

ı 0 D ı ti .B/ ı Li C ı Li 1 max jT id j N i 2=3 ! ; (19.22)
and ı ti .B/ is the value of ı ti in B.

• During the unloading between C and D ( P ı ti <0) (19.24) and ı ti .C/ is the value of ı ti in C.

T id D 2N i 0 @ 0 @ 1 2 16a i .ı ti C ı 0 / 3N i G C 1 T ic max N i 2=3 C 1 ! 3=2 1 A 1 1 A CT ic max ; (19.23) with ı 0 D ı Li ı Li 1 max.T ic / N i 2=3 ! ı ti .C/;
If the limit tangential deflection ı Lt (total slip) is reached, the forcedeflection relationships during unloading T id and reloading T ic for an individual asperity are given by Eqs. (19.16) and (19.17). (19.23),(19.16),and (19.17), the corresponding hysteretic force-deflection relations for a contact are shown in Fig. 19.17.

Case of N Asperities

In this section, a Gaussian (or normal) distribution of asperity heights '.y i / is used. Several experimental results [START_REF] Greenwood | Contact of nominally flat surfaces[END_REF] show that for many

φ(y i ) φ(y i )
surfaces the height distribution is Gaussian to a very good approximation. The probability density function of a Gaussian distribution (see Fig. 19.18) is given by (19.25) where parameter m (location of the peak) is the mean and 2 is the variance (the measure of the width of the distribution). The cumulative distribution function (Fig. 19.18b) is

'.y i / D 1 p 2 e .y i m/ 2 =2 2 ;
'.y i / D 1 2 erf 1 p 2 y i m : (19.26)
The total normal load N applied to the contact between a plane and a nominally flat surface (Fig. 19.8) can be expressed as the sum of each elementary normal load on each asperity Keeping N constant, the loading cycle for the tangential displacement ı t oscillates between ı t max and ı t max . The total tangential force T that can be supported by N a asperities is

T D X T i : (19.28)
The evolution of total tangential force T versus tangential displacement ı t is shown in Fig. 19.19. The shape of this curve is very similar to that obtained experimentally (see Fig. 19.7).

Relations Between the Iwan, Coulomb, and Extended Greenwood Models

In this section, N is kept constant and the total number of asperities N a in contact with the flat surface is varied. The hysteretic force-deflection relations for a contact are depicted in Fig. 19.20. It is interesting to mention that the variation of the total number of asperities influences significantly the load-deflection curve, although with an increase of the parameter N a the stiffness characteristics of the model increase.

Extended Greenwood Model Identification

In Fig. 19.22, the theoretical model is compared with experimental results. It is observed that the shape of the curve obtained by the theoretical model is very similar to that obtained experimentally.

Damping Ratio

This section aims to define the dissipated energy in order to compute the damping ratio. The area enclosed by the curves T id [Eq. (19.16)] and T ic [Eq. (19.17 where E s D 4ı t max T max . Figure 19.24 shows the evolution of as a function of the tangential displacement ı t .

It is noticed that the damping ratio increases as the tangential displacement increases. Figure 19.24 also shows the nonlinear character of this evolution.

Summary of the Extended Greenwood Model

In this chapter, an extension of the Greenwood model is proposed (the Extended Greenwood Model, EGM) that is based on a statistical distribution of asperities in contact between two assembled parts. These micro-contacts produce a damping effect in both total sliding and partial sliding. This new model is studied and identified with experimental results. The numerical results show that for a cyclic

  Laboratoire EA 7393, Saint-Ouen, France Ecole Nationale d'Ingénieurs de Sfax, Sfax, Tunisia J.-L. Dion ( )•N.Peyret SUPMECA Paris -QUARTZ Laboratoire EA 7393, Saint-Ouen, France e-mail: jean-luc.dion@supmeca.fr

Fig. 19 .

 19 Fig. 19.1 A press-fit joint subjected to a clamping pressure p and a vertical shear load F.( a)A bolted structure-A free-free beam with a lap joint subjected to an axial load F. (b) A press-fit joint subjected to a clamping pressure p and a vertical shear load F. (c) A lap-shear joint subjected to a clamping pressure p and an axial load F

Fig

  Fig. 19.2 Partial slip distribution of tangential traction by[START_REF] Cattaneo | Sul Contatto di due Corpi Elastici: Distribuzione Locale Degli Sforzi[END_REF]

Fig. 19 . 4

 194 Fig. 19.4 Global description of the tribometer

Fig

  Fig. 19.5 Thesamplein allied aluminum Al 2017 machined by milling

  Fig. 19.6 Roughness of the surface. (a) Position along surface, mm. (b) Probability of asperity heights, %

  Fig. 19.7 Slipping behavior under normalized load T=N versus tangential displacement

Fig. 19 . 9

 199 Fig. 19.9 The contact model for a single asperity

Fig. 19 .

 19 Fig. 19.10 Contact area

  Fig. 19.11 Load-displacement cycle[START_REF] Johnson | Contact Mechanics[END_REF]

Fig. 19 .Fig. 19 .

 1919 Fig. 19.12The loading path for the constant amplitude case

Fig. 19 .

 19 Fig. 19.14 Tangential force evolution according to the tangential displacement (total slip)

Fig. 19 .

 19 Fig.19.17 Hysteretic force-deflection from partial slip to total slip

Fig

  Fig. 19.18 (a) Normal distribution of asperity heights, and (b) cumulative distribution function

  Fig. 19.19 Slipping behavior under tangential force T versus tangential displacement ı t

  Fig. 19.20 Evolution of Extended Greenwood Model (EGM) against the number of asperities

  )]

  Fig. 19.22 The theoretical model and measured result for the material aluminum allied Al 2017

  Fig. 19.23 Dissipated energy as a function of tangential displacement

  When the number of asperities in contact increases, the EGM tends to the Coulomb friction model. Notice that this result is different from that obtained by the Iwan model (Segalman 2005; Argatov and Butcher 2011). The Iwan model is based on several cells composed of a stiffness and Coulomb friction model. When the number of cells increases, the Iwan model is close to the EGM composed with a single asperity (Fig. 19.21).

tangential displacement, the dissipated energy increases with the tangential displacement. The shape of the hysteretic force-deflection for a contact is very similar to that obtained experimentally.


Keywords: Bessel-McDonald functions, Sturm-Liouville equation, perpetuities AMS Classification 60 J 25, 60

We obtain the Laplace transform and integrability properties of the integral over R + of the call quantity associated with geometric Brownian motion with negative drift, thus adding a new element to the list of already studied Brownian perpetuities.

1 Motivation, introduction 1.1 Let (B t , t ≥ 0) denote 1-dimensional Brownian motion starting from 0 ; associated to B, one considers the geometric Brownian motion :

E t := exp B t - t 2 , t ≥ 0
which is a positive martingale converging a.s. to 0 as t → ∞. In recent years, the following questions have been asked to the second author, in connection with European option pricing :

i) to express as simply as possible the quantity :

∞ 0 λ e -λt E (E t -k) + dt (k ≥ 0)
ii) to find the law of s 0 (E t -k) + dt, for fixed s iii) to find the law of ∞ 0 λ e -λt (E t -k) + dt i) may be solved quite explicitly, and indeed this has led the authors, jointly with D.

Madan, to write a series of papers relating European option prices and laws of last passage times of continuous positive martingales converging to 0 as t → ∞ see, e.g. [START_REF] Madan | Yor An alternative expression for the Black-Scholes formula in terms of Brownian first and last passage times[END_REF], [START_REF] Madan | Yor Unifying Black-Scholes type formulae which involve Brownian last passage times up to a finite horizon[END_REF], [START_REF] Madan | Yor Put option prices as joint distribution functions in strike and maturity : the Black-Scholes case[END_REF] .

Questions ii) and iii) are harder to solve explicitly, as, indeed, one may look for double Laplace transforms of either quantities, e.g. :

∞ 0 e -µt E(e -λ t 0 (Es-k) + ds )dt

In the present paper, we are studying thoroughly the law of ∞ 0 (E t -k) + dt which seems to be slightly less difficult than either question ii) or iii). However, the results we obtain are not particulary simple.

1.2 These motivations having been presented, we now concentrate exclusively on the law of ∞ 0 (E t -k) + dt. It will be convenient to consider the two-parameter process :

E (x) t := x exp B t - t 2 = exp log x + B t - t 2 , t ≥ 0 (1.1) (E (x)
t , t ≥ 0, x > 0) is a Markov process taking values in R + which, most often, we shall denote as : (E t , t ≥ 0 ; P x , x > 0 since E (x) 0 = x. To simplify notation, we shall write P for P 1 and E t for E

(1) t . For any k > 0, let us define :

Π (x) k := ∞ 0 (E (x) t -k) + dt (1.2)
Again, to simplify, we shall write Π k for Π

(1)

k . Since E (x) t
-→ t→∞ 0 a.s., the integral which defines (1.2) is a.s. convergent, as (E

(x) t -k) + = 0 for t ≥ G k := sup{t ; E (x) t
= k}, and G k < ∞ a.s.

1.3

We now explain about some reductions of the study of the laws of Π (x) k , to closely related problems. i) From Itô's formula, we deduce :

E (x) t = x + t 0 E (x) s dB s = β A (x) t (1.3)
where (β u , u ≥ 0) denotes the Dubins-Schwarz Brownian motion associated with (E

(x)
t , t ≥ 0) (and β 0 = x) and :

A (x) t =<E (x) > t = t 0 (E (x) s ) 2 ds (1.4) Hence : Π (x) k = ∞ 0 (β A (x) s -k) + ds = T 0 (β) 0 (β v -k) + β 2 v dv (1.5)
after the change of variable A (x) s = v and where T 0 (β) = inf{u ≥ 0 ;

β u = 0} = ∞ k (y -k) y 2 L y T 0 (β) dy = ∞ k dz ∞ z dy y 2 L y T 0 (β) (1.6)
from the density of occupation formula, and where L y T 0 denotes the local time at time T 0 and level y of Brownian motion (β u , u ≥ 0).

ii) When x = k, the first Ray-Knight Theorem allows us to write, from (1.6) :

Π (k) k = ∞ k (y -k) y 2 λ y-k dy (1.7)
where (λ z , z ≥ 0), conditionally on λ 0 = l , is a 0-dimensional squared Bessel process starting at l, and where λ 0 is an exponential variable with parameter 1 2k , i.e. with expectation 2k.

1.4

From the elementary relations :

(x e Bt-t 2 -k) + = x e Bt-t 2 - k x + = k x k e Bt-t 2 -1 + (1.8)
valid for every x, k > 0, we deduce that the law of Π k under P x is that of x Π k x under P , or that of kΠ 1 under P x k . In other words, for every Borel positive function ϕ, we have :

E x ϕ(Π k ) = E ϕ(x Π k x ) = E x k ϕ(kΠ 1 ) (1.9)
These relations allow us to reduce our study of the law of Π k under P x to that of Π 1 under P x k . We might as well limit ourselves to the study of Π k under P .

1.5 It is proven in Dufresne [Duf] and Yor [Y] see also P. Salminen and M. Yor [SY1] that, for every a = 0 and ν > 0 :

∞ 0 exp (aB t -νt)dt (law) = 2 a 2 γ 2ν a 2 (1.10)
where γ b denotes a gamma variable with parameter b, i.e. :

P (γ b ∈ dt) = 1 Γ(b) e -t t b-1 dt (t ≥ 0) (1.11)
In particular, for a = 1 and ν

= 1 2 Π 0 (law) = 2 γ 1 = 2 e (1.12)
where e is a standard exponential variable.

1.6 Let ν = 0 and :

E (x,ν) t := exp ν (log x + B t ) - ν 2 t 2 = x ν exp νB t - ν 2 t 2
and define, for k > 0 :

Π (x,ν) k := ∞ 0 (E (x,ν) s -k) + ds (1.13)
Since, by scaling, (E

(x,ν) t , t ≥ 0) (law) = (x ν E ν 2 t , t ≥ 0), we have : Π (x,ν) k (law) = ∞ 0 (x ν E ν 2 s -k) + ds = x ν ν 2 ∞ 0 E v - k x ν + dv (law) = x ν ν 2 Π k x ν (1.14)
Thus, the study of the law of Π (x,ν) k may be reduced very simply to that of Π k x ν . This is the reason why we have chosen, in this paper, to limit ourselves to ν = 1.

1.7 Here are our results : Theorem 1.1 Let α ≥ 0. Then, for every x > 0, E x (Π 1 ) α < ∞ if and only if α < 1 Theorem 1.2 Let α < 0 and x > 0. Then :

i) For every x > 1, E x (Π 1 ) α < ∞. ii) For every x < 1, P x (Π 1 = 0) = 1 -x ; hence E x (Π 1 ) α = +∞. iii) For x = 1, E 1 (Π 1 ) α < ∞ if and only if |α| < 1 3 • Theorem 1.3 (Laplace transform of Π 1 )
. For every θ ≥ 0 :

E x (e -θ 2 Π 1 ) =                √ x K γ ( √ 4θx) 1 2 K γ ( √ 4θ) - √ θ K γ ( √ 4θ) if x ≥ 1 1 + x 1 2 K γ ( √ 4θ) + √ θK γ ( √ 4θ) 1 2 K γ ( √ 4θ) - √ θK γ ( √ 4θ) if 0 < x ≤ 1 (1.15)
where K γ denotes the Bessel-McDonald function with index γ see [Leb], p. 108 and where

γ = √ 1 -4θ if 4θ ≤ 1 and γ = i √ 4θ -1 if 4θ ≥ 1.
We shall also prove, as a consequence of Theorem 1.3 : Theorem 1.4 Let (λ x , x ≥ 0) denote a squared Bessel process with dimension 0 started at l and denote its law by Q (0) l . Then :

• If 4θ ≥ 1 : Q (0) l exp - θ 2 ∞ 1 (x -1) x 2 λ x-1 dx = Q (0) l exp - θ 2 ∞ 0 x (x + 1) 2 λ x dx = exp l 2 1 2 + √ θK iν ( √ 4θ) K iν ( √ 4θ) (1.16) with ν = √ 4θ -1 • If 4θ ≤ 1 : Q (0) l exp - θ 2 ∞ 1 x -1 x 2 λ x-1 dx = exp l 2 1 2 + √ θ K ν ( √ 4θ) K ν ( √ 4θ) with ν = √ 1 -4θ.
In Section 3 of this work, we shall study the asymptotic behavior, as θ → ∞, of E(e -θ 2 Π 1 ) and we shall obtain :

E(e -θ 2 Π 1 ) ∼ θ→∞ C θ 1 3 (C > 0) (1.17)
Finally, in a short Section 4, we shall indicate how Theorems 1.1, 1.2 and 1.3 extend when we replace Π k by Π (ρ) k , with : 2 Proofs of Theorems 1.1, 1.2 and 1.3

Π (ρ) k := ∞ 0 e ρ(Bt-t 2 ) -k + dt (ρ, k > 0) (law) = 1 ρ 2 ∞ 0 (e (Bu-
2.1 A first proof of Theorem 1.1 2.1 i) We now prove that, for every x > 0, E x (Π 1 ) = ∞ Indeed, from (1.6) :

E 1 (Π 1 ) = ∞ 1 (y -1) y 2 E 1 (L y T 0 )dy = ∞ 1 (y -1) y 2 E 1 (L 1 T 0 )dy = 2 ∞ 1 (y -1) y 2 dy = +∞ (2.1) since, for y ≥ 1, E 1 L y T 0 = E 1 L 1 T 0 = 2, as L 1
T 0 is an exponential variable with parameter 1 2 .

ii)

We now prove that, for every x > 0,

E x (Π 1 ) = ∞ • For x ≥ 1, this is clear since Π (x)
k is an increasing function of x (and a decreasing function of k). Hence :

E x (Π 1 ) ≥ E 1 (Π 1 ) = +∞ from (2.1)
• For x < 1, from the Markov property :

Π (x) 1 (law) = 1 T (-1 2 ) log( 1 x ) <∞ • Π 1 (2.2) with T (-1 2 ) log( 1 x ) := inf t ≥ 0 ; B t - t 2 = log 1 x
, and in (2.2), Π 1 and T (-1 2 ) log( 1x ) are assumed independent. Hence :

E(Π (x) 1 ) = P (T (-1 2 ) log( 1 x ) < ∞) E(Π 1 ) = +∞ from (2.1) and since P (T (-1 2 ) log( 1 x ) < ∞) = x (see Lemma 2.1 below). 2.1 iii) We prove that, for 0 ≤ α < 1, E x (Π 1 ) α < ∞ We have : E x (Π 1 ) α = x α E (Π 1 x ) α from (1.9) ≤ x α E (Π 0 ) α (since Π k is a decreasing function of k) ≤ x α ∞ 0 2 y α e -y dy < ∞ from(1.12)
2.2 Second proof of Theorem 1.1

It hinges upon :

Lemma 2.1 Let G (ν) a := sup{u ≥ 0 ; B u + νu = a} (= 0 if this set is empty) and T (ν) a := inf{u ≥ 0 ; B u + νu = a} (= +∞ if this set is empty). i)
If ν and a have the same sign :

• E e µG (ν) a < ∞ if and only if µ < ν 2 2 (2.3) • for every real α, E (G (ν) a ) α < ∞ (2.4)
ii) If ν and a have opposite signs :

• E e µG (ν) a < ∞ if and only if µ < ν 2 2 (2.5) • for every real α > 0, E (G (ν) a ) α < ∞ (2.6) • P (T (ν) a = ∞) = P (G (ν) a = 0) = 1 -e 2νa (2.7) Hence, for α < 0, E (G (ν) a ) α = +∞
The proof of this Lemma is obvious. It hinges on the well-known formulae :

• If ν, a > 0 :

P (G (ν) a ∈ dt) = ν √ 2πt exp - 1 2t (a -νt) 2 dt • If ν, a > 0 : P (G (-ν) a ∈ dt) = ν √ 2πt exp - 1 2t (a + νt) 2 dt
• If ν and a have opposites signs :

P (G (ν) a > 0) = P (T (ν) a < ∞) = e 2νa • G (ν) a (law) = G (-ν) (-a) (law) = 1 T (a) ν
We now give a second proof of Theorem 1.1.

• We first show that, for 0 ≤ α < 1, E x (Π 1 ) α < ∞. From the relation :

Π (x) 1 = x ∞ 0 E t - 1 x + dt = x G ( -1 2 ) -log x 0 E t - 1 x + dt ≤ x sup t≥0 E t • G (-1 2 ) -log x
we deduce that, for 0 ≤ α < 1, αp < 1 and

1 p + 1 q = 1 (p, q > 1) : E x (Π 1 ) α ≤ x α E sup t≥0 E t α G (-1 2 ) -log x α ≤ x α E sup t≥0 E t αp 1 p E G (-1 2 ) (-log x) αq 1 q ≤ C x α E sup t≥0 E t αp 1 p (from Lemma 2.1)
As (E t , t ≥ 0) is a positive martingale, starting from 1, and converging a.s. to 0 as t → ∞, we get see [NY] or [RY], Chap. II, Ex. 3.12, p. 73 :

sup s≥0 E s (law) = 1 U (2.8)
with U uniform on [0, 1]. Thus :

E sup t≥0 E t αp = 1 0 1 u αp du < ∞ (2.9)
since αp < 1.

• We then show that E x (Π 1 ) = ∞. First of all, it follows from [START_REF] Madan | Yor An alternative expression for the Black-Scholes formula in terms of Brownian first and last passage times[END_REF] that :

E 1 (E t -k) + = P G ( 1 2 ) log k ≤ t (2.10)
Hence :

E 1 (Π 1 ) = ∞ 0 P G ( 1 2 ) 0 ≤ t dt = E ∞ G ( 1 2 ) 0 dt = +∞ since G ( 1 2 ) 0 < +∞ a.s. Likewise : E x (Π 1 ) = x E 1 (Π 1 x ) from (1.9) = x E   ∞ G ( 1 2 ) log 1 x dt   = ∞ from (2.10) and since G ( 1 2 ) log 1 x < ∞ a.s.
2.3 Proof of Theorem 1.2 i) We already prove point i) : Let x > 1 and x such that 1 < x < x. It is then obvious that :

Π (x) 1 ≥ (x -1)T (-1 2 ) log( x x ) (2.11) with T (-1 2 ) a := inf{t ≥ 0 ; B t -t 2 = a}, since, if t < T (-1 2 ) log x x
, then e log x+Bt-t 2 -1 ≥ x -1 . Hence, with γ > 0 :

E x 1 (Π 1 ) γ = ∞ 0 P x 1 (Π 1 ) γ ≥ t dt = ∞ 0 P x (Π 1 < v) γ v 1+γ dv ≤ ∞ 0 γ v 1+γ dv P T (-1 2 ) log x x ≤ v x -1 dv from (2.11) = ∞ 0 γ v 1+γ dv P G (log( x x )) 1 2 ≥ x -1 v dv since T (ν) a (law) = 1 G (a) ν and G (a) ν (law) = G (-a) -ν = ∞ 0 γ (x -1) γ u γ-1 du P G (log ( x x )) 1 2 ≥ u du = 1 (x -1) γ E (G (log x x ) 1 2 ) γ < ∞ (2.12) from point i) of Lemma 2.1 .
ii) We now prove point ii) : It is clear that, for x < 1 :

{Π (x) 1 = 0} = T (-1 2 ) log 1 x = ∞ Thus, P {Π (x) 1 = 0} = P T (-1 2 ) log 1 x = ∞ = P G (log x) 1 2 = 0} = 1 -x > 0 from Lemma 2.1 and
(1.26).

iii) We now prove point iii) : For this purpose, we write, for γ > 0 :

E 1 1 (Π 1 ) γ = 1 Γ(γ) ∞ 0 E(e -θ Π 1 )θ γ-1 dθ (2.13)
and we show, in the next Section 3, that :

E 1 (e -θΠ 1 ) ∼ θ→∞ C θ 1 3 (C > 0) Thus, E 1 (Π 1 ) γ < ∞ if and only if ∞ 1 θ γ-1-1 3 dθ < ∞, that is, if and only if γ < 1 3 • 2.4 Proof of Theorem 1.3 a) A useful Lemma Lemma 2.2 Let θ ≥ 0 and K γ the Bessel McDonald function with index γ, such that γ = √ 1 -4θ if θ ≤ 1 4 and γ = i √ 4θ -1 if θ ≥ 1 4 • 1)
Define the function ϕ θ : R + → R by :

ϕ θ (y) = √ y K γ ( 4θy) (y ≥ 0) (2.

14)

Then :

i) ϕ θ is a real valued function which satisfies :

ϕ θ (y) + - θ y + θ y 2 ϕ θ (y) = 0 (2.15)
ii) ϕ θ , restricted to [1, ∞[ is positive, convex, bounded and decreasing.

2) We define the function ϕ θ : R + → R + by :

ϕ θ (y) =    ϕ θ (y) if y ≥ 1 ϕ θ (1) -ϕ θ (1) + yϕ θ (1) if 0 ≤ y ≤ 1 (2.16)
Then ϕ θ is a bounded, positive, convex, decreasing function which satisfies :

ϕ θ (y) + - θ y + θ y 2 1 y≥1 • ϕ θ (y) = 0 (2.17) b) Proof of Lemma 2.2 i) Relation (2.
15) as well as relation (2.17) follows from a direct computation, using the [Leb], p. 110 and the fact that γ 2 = 1 -4θ see Petiau [Pet], p. 306, formula (8), which needs to be corrected by replacing a by -a, or Kamke [Kam], p. 440 .

equation K γ (z) + 1 z K γ (z) -1 + γ 2 z 2 K γ (z) = 0 see
We distinguish two cases :

Case 1 : 4θ ≤ 1, γ = √ 1 -4θ. The function K γ , hence also ϕ θ , is positive. Furthermore ϕ θ is bounded on R + since : ϕ θ (y) ∼ y→0 C y 1-γ 2 , ϕ θ (y) ∼ y→∞ C y 1 4 e - √ 4θy
see [Leb], p. 123 and 136

On the other hand, from (2.15), the function ϕ θ is convex on the interval [1, ∞[. As it is convex, positive, and bounded, it is decreasing. Lemma 2.2 is thus proven in this case. we deduce that K iν (y) is real valued, hence so is ϕ θ (y).

On the other hand, ϕ θ is bounded on [0, ∞[. Indeed, for y ≥ 0, from (2.18) :

√ 4θ ϕ θ y 2 4θ = y K iν (y) ≤ y ∞ 0 e -y e u 2 du ≤ y ∞ y 2 e -v dv v Thus : √ 4θ ϕ θ y 2 4θ ≤ e -y 4
for y large enough, hence

√ 4θ ϕ θ y 2 4θ -→ y→∞ 0 and √ 4θ ϕ θ y 2 4θ ≤ y C + | log(y)| -→ y→0 0
On the other hand, it is clear, from (2.18), that K iν (y) > 0 hence ϕ θ (y) > 0, for y large enough.

• We now show that K iν is decreasing on [ν, ∞[. If not, there would exist a point y 0 > ν such that :

K iν (y 0 ) > 0, K iν (y 0 ) = 0, K iν (y 0 ) ≤ 0.
However,

K iν (y 0 ) + 1 y 0 K iν (y 0 ) = 1 - ν 2 y 2 0 K iν (y 0 )
Thus K iν (y 0 ) > 0, which is absurd. Since K iν is decreasing on [ν, ∞[, and positive near ∞,

K iν is positive on [ν, ∞[. Thus, ϕ θ is positive on [1, ∞[ since, if y ≥ 1, √ 4θy ≥ √ 4θ ≥ √ 4θ -1 = ν .
From the relation (2.15), we then deduce that ϕ θ is convex on [1, ∞[. Since it is bounded, convex and positive, it is decreasing. Lemma 2.2 is proven. c) End of the proof of Theorem 1.3.

Let M θ t := ϕ θ (B t ) exp - θ 2 t 0 ϕ θ ϕ θ (B s )ds, t ≥ 0 . Then (M θ t , t ≥ 0) is a local martingale.
It is equal, from (2.17) to :

M θ t = ϕ θ (B t ) exp - θ 2 t 0 (B s -1) + B 2 s ds (2.19)
and, from Lemma 2.2, for every x ≥ 0, (M θ t∧T 0 , t ≥ 0) is bounded. Thus, from Doob's optional stopping Theorem :

ϕ θ (x) = E x (M θ 0 ) = E x ϕ θ (B T 0 ) exp - θ 2 T 0 (B) 0 (B s -1) + B 2 s ds = ϕ θ (0)E x exp - θ 2 Π 1 from (1.5)
Thus :

E x e -θ 2 Π 1 = ϕ θ (x) ϕ θ (0)
This is precisely Theorem 1.3, owing to formula (2.16) which yields ϕ θ explicitly.

Remark 2.3 Theorem 1.3 allows to recover formula (1.9) : Π 0

(law) = 2 e
• Indeed, on one hand :

E 1 e -θ 2 Π 0 = E e -θ 2 2 e = ∞ 0 e -θ z -z dz = 2 √ θ K 1 ( √ 4θ
) see [Leb], p. 119 and, on the other hand :

E 1 (e -θ 2 Π 0 ) = lim ↓0 E 1 (e -θ 2 Π ) = lim ↓0 E exp - θ 2 ∞ 0 (e Bt-t 2 -) + dt = lim ↓0 E e -θ 2 ∞ 0 ( 1 (e B t -t 2 -1) + dt = lim ↓0 E1 (e -θ 2 Π 1 ) = lim ↓0 1 √ K √ 1-4θ ( √ 4θ) 1 2 K √ 1-4θ ( √ 4θ ) -1 2 √ 4θ K √ 1-4θ ( √ 4θ )
(from (1.5), replacing θ by θ and x by 1 )

= lim

↓0 1 √ K 1 ( √ 4θ) + O(1) 1 2 K √ 1-4θ ( √ 4θ ) 1 - √ 1 -4θ + √ θ K √ 1-4θ +1 ( √ 4θ ) since zK ν (z) = νK ν (z) -z K ν+1 (z); see [Leb], p. 110 = lim ↓0 1 √ K 1 ( √ 4θ) + O(1) O(1) + √ θ 1 2θ + O(1) = 2 √ θ K 1 ( √ 4θ)
2.5 Proof of Theorem 1.4

From Theorem 1.3, we know that, for θ ≥ 1 4 :

E 1 (e -θ 2 Π 1 ) = 2K iν ( √ 4θ) K iν ( √ 4θ) - √ 4θ K iν ( √ 4θ) = 1 1 2 - √ θ K iν ( √ 4θ) K iν ( √ 4θ) = 1 2 ∞ 0 e -l 2 + l 2 1 2 + √ θ K iν ( √ 4θ) K iν ( √ 4θ) dl (2.20)
On the other hand, from (1.7) :

E 1 (e -θ 2 Π 1 ) = E 1 exp - θ 2 ∞ 1 (y -1) + y 2 λ y-1 dy = 1 2 ∞ 0 e -l 2 dl E exp - θ 2 ∞ 1 (y -1) + y 2 λ y-1 dy λ 0 = l = 1 2 ∞ 0 e -l 2 dl Q (0) l exp - θ 2 ∞ 0 y (1 + y) 2 λ y dy (2.21)
where Q (0) l denotes the expectation relative to a squared Bessel process of dimension 0, starting from l. The comparison of (2.21) and (2.20) implies Theorem 1.4 in the case

θ ≥ 1 4
, since the Laplace transform is one-to-one. The proof of Theorem 1.4 in the case θ ≤ 1 4 is the same.

Remark 2.4 Let µ denote a positive σ-finite measure on R + . Let Φ denote the unique decreasing, positive solution on R + of the Sturm-Liouville equation Φ = µΦ and such that Φ(0) = 1 . It is well known see [RY], Chap. IX, p. 444 that :

Q (0) l exp - 1 2 ∞ 0 λ y µ(dy) = exp l 2 Φ (0 + ) (2.22)
Observe that, as (λ y , y ≥ 0) has compact support a.s.,

∞ 0 λ y µ(dy) < ∞ if µ is σ-finite on R + .
Theorem 1.4 may be recovered easily by applying formula (2.22) with µ(dy) = y 1 + y 2 dy. On the other hand, if µ(dy) = a(y)dy, with

c 1 1 [0,γ 1 ] (y) ≤ a(y) ≤ c 2 1 [0,γ 2 ] (y) (0 < c 1 ≤ c 2 , 0 < γ 1 ≤ γ 2 ),
we deduce from [RY], Chap. XI, Corollary 1.8, that :

exp - l 2 c 2 √ θ ≤ E (0) l e -θ 2 ∞ 0 λy µ(dy) ≤ exp - l 2 c 1 √ θ (2.23)
for θ large enough, whereas, as we shall show in Section 3 :

Q (0) l exp - θ 2 ∞ 0 y 1 + y 2 λ y dy ∼ θ→∞ C e -l 2 θ 1 3 (2.24) 3 Asymptotic behavior of E 1 (e -θ 2 Π 1 ) as θ → ∞
We shall now end the proof of Theorem 1.2 by showing : Theorem 3.1 There is the equivalence result:

E 1 e -θ 2 Π 1 ∼ θ→∞ C θ 1 3 (3.1)
Proof of Theorem 3.1

We recall that, from Theorem 1.3 :

E 1 e -θ 2 Π 1 = 2K iν ( √ 4θ) K iν ( √ 4θ) - √ 4θ K iν ( √ 4θ) (3.2) with ν = √ 4θ -1 θ ≥ 1 4
• We shall successively find an equivalent of the numerator and the denominator of (3.2), the difficulty arising from the fact that, in K iν ( √ 4θ) and

K iν ( √ 4θ
) the argument √ 4θ and index iν = i √ 4θ -1 tend both to infinity as θ → ∞. To overcome this difficulty, we shall use some results about Bessel functions found in Watson [Wat], p. 245-248 , which we now recall.

i) Let H

(1)

iν the first Hankel function see [Leb], p. 120 ; it is related to K iν via the formula :

K iν (z) = iπ 2 e -νπ 2 H (1) iν (z e iπ 2 ) (3.3)
We define by the formula : iν = iz(1 -) and assume that, as z → ∞, remains bounded (with, of course, ν depending on z). Then, there is the second order asymptotic expansion :

H (1) ν (z) = - 2 3π    e 2 3 πi sin π 3 Γ 1 3 1 6 z 1 3 + e 4 3 πi ( z) sin 2π 3 Γ 2 3 1 6 z 2 3 + o 1 z 2 3    (3.4)
ii) Let us study the numerator of (3.2) :

N = 2K iν ( √ 4θ) = 2 iπ 2 e -νπ 2 H (1) iν e iπ 2 √ 4θ) = (iπ)e -νπ 2 - 2 3π    C 1 e 2 3 πi 1 e iπ 2 √ 4θ 1 3 + o 1 θ 1 6    (3.5) with C 1 = sin π 3 Γ 1 3 6 1 3 ∼ θ→∞ 2 2 3 3 C 1 e -νπ 2 1 θ 1 6 (3.6)
Here, we have used the first order expansion (3.4) with : e

2 3 iπ •e -iπ 6 = i 4π 6 - π 6 = 3π 6 = π 2 and the fact that here z = z -ν = i √ 4θ -i √ 4θ -1 = i √ 4θ 1 -1 - 1 4θ -→ θ→∞ 0
iii) We now study the denominator of (3.2) [Leb], p. 110 . Since we have already studied the asymptotic behavior of K iν ( √ 4θ), it remains to study that of

D = K iν ( √ 4θ) - √ 4θ K iν ( √ 4θ) = K iν ( √ 4θ) + √ 4θ K iν-1 ( √ 4θ) + i √ 4θ -1 K iν ( √ 4θ) (3.7) after using d dz z µ K µ (z) = -z µ K µ-1 (z) ;
∆(θ) := √ 4θ K iν-1 ( √ 4θ) + i √ 4θ -1 K iν ( √ 4θ) (3.8) Now, since √ 4θ K iν-1 ( √ 4θ) + i √ 4θ -1 K iν ( √ 4θ) is real from (2.18) and since i √ 4θ -1 K iν ( √ 4θ) is purely imaginary, the development of ∆(θ) as θ → ∞ is that of the real part of √ 4θ K iν-1 ( √ 4θ
). Then, we obtain :

√ 4θ K iν-1 ( √ 4θ) = √ 4θ iπ 2 e iπ 2 (iν-1) H (1) iν-1 e iπ 2 √ 4θ = √ 4θ π 2 e -νπ 2 H (1)
iν-1 e √ 4θ and :

iν -1 = z(1 -) = i √ 4θ(1 -), i.e. : z = e iπ 2 √ 4θ or z = e iπ 2 √ 4θ = 1 + 1 4 √ θ + o 1 √ θ ν = √ 4θ -1 (3.10)
We obtain :

√ 4θ K iν-1 ( √ 4θ) = √ 4θ π 2 e -νπ 2 - 2 3 π × • • • • • •    C 1 e 2iπ 3 1 e iπ 2 √ 4θ 1 3 + C 2 e 4iπ 3 1 + i 4 √ θ • 1 e iπ 2 √ 4θ 2 3 + o 1 θ 1 3    (3.11) with C 1 = sin π 3 Γ 1 3 6 1 3 and C 2 = sin 2π 3 Γ 2 3 6 2 3 (C 1 , C 2 > 0) = - 1 3 √ 4θ e -νπ 2 C 1 i 1 ( √ 4θ) 1 3 -C 2 1 + i 4 √ θ 1 ( √ 4θ 2 3 + o 1 θ 1 3
(3.12)

Considering now the real part of (3.12) to obtain :

∆(θ) = Re √ 4θ K iν-1 ( √ 4θ) + i √ 4θ -1 K iν ( √ 4θ) = 1 3 √ 4θ e -νπ 2 C 2 1 ( √ 4θ) 2 3 + o 1 θ 1 3 ∼ θ→∞ 2 1 3 C 2 3 e -νπ 2 θ 1 6
(3.13)

• We then gather (3.2), (3.6) and (3.13), we obtain :

E 1 e -θ 2 Π 1 ∼ θ→∞
from the explicit formulae for C 1 and C 2 . This is Theorem 3.1.

Remark 3.2 Using Theorem 1.3, we obtain, for 0 < x ≤ 1 :

E x e -θ 2 Π 1 = 1 + x 1 2 K i √ 4θ-1 ( √ 4θ) + √ θ K i √ 4θ-1 ( √ 4θ) 1 2 K i √ 4θ-1 ( √ 4θ) - √ θ K i √ 4θ-1 ( √ 4θ)
It now follows easily from (3.6) and (3.13) that :

E x e -θ 2 Π 1 -→ θ→∞ 1 -x (3.15)
Now, on the other hand :

E x e -θ 2 Π 1 -→ θ→∞ P x (Π 1 = 0) (3.16)
We recall (and recover here) that P x (Π 1 = 0) = 1 -x. This is point ii) of Theorem 1.2.

4 Extending the preceding results to the variables Π 1 under P x is given by :

E x (e -θ 2 Π (ρ) 1 )                  √ x K γ √ 4θ ρ x ρ 2 1 2 K γ ( √ 4θ ρ ) - √ θ K γ ( √ 4θ ρ ) if x ≥ 1 1 + x 1 2 K γ ( √ 4θ ρ ) + √ θK γ ( √ 4θ ρ ) 1 2 K γ ( √ 4θ ρ ) - √ θ K γ ( √ 4θ ρ ) if 0 < x ≤ 1 (4.2) with γ = √ 1 -4θ ρ if θ ≤ 1 4 and γ = i √ 4θ -1 ρ if θ > 1 4 .
The proof of this Theorem 4.1 is quite similar to that of Theorem 1.3. It hinges upon the following Lemma 4.2.

Lemma 4.2 i) Let, for θ ≥ 0 and ρ > 0, the function ϕ (ρ) θ : R + → R defined by :

ϕ (ρ) θ (y) = √ y K γ √ 4θ ρ y ρ 2 (y ≥ 0) (4.3)
Then, it satisfies :

ϕ (ρ) θ (y) + θ -y ρ-2 + 1 y 2 ϕ (ρ) θ (y) = 0 (4.4)
see [Pet], p. 306, with ρ -2 = m, formula (8), after taking care of replacing a by -a .

ii) Let ϕ (ρ)

θ : R + → R defined by :

ϕ (ρ) θ (y) =      ϕ (ρ) θ (y) if y ≥ 1 ϕ (ρ) θ (1) -ϕ (ρ) θ (1) + y ϕ (ρ) θ (1) if y ≤ 1 (4.5) Then ϕ (ρ)
θ is positive, decreasing, convex and satisfies :

ϕ (ρ) θ (y) + θ -y ρ-2 + 1 y 2 1 y≥1 ϕ (ρ)
θ (y) = 0 (4.6)

Remark 4.1 As a check, we note that formula (4.2) allows to recover the identity (1.8) :

∞ 0 e ρ Bt-ρt 2 dt (law) = 2 ρ 2 γ 1 ρ • Indeed, on one hand : E e -θ 2 2 ρ 2 γ 1 ρ = 1 Γ 1 ρ ∞ 0 e -θ ρ 2 z -z z 1 ρ -1 dz = 2 Γ 1 ρ K 1 ρ √ 4θ ρ θ ρ 2 1 2ρ
see [Leb], p. 115

On the other hand :

E 1 e -θ 2 Π (ρ) 0 = lim ↓0 E 1 exp - θ 2 ∞ 0 e ρBt-ρt 2 -ε + dt = lim ↓0 E 1 exp - θε 2 ∞ 0 e ρ log 1 ε ρ +Bt -ρt 2 -1 + dt = lim ↓0 1 ε 1 2ρ K √ 1-4θε ρ √ 4θε ρ 1 ε 1 ρ ρ 2 1 2 K √ 1-4θε ρ √ 4θε ρ - √ θε K √ 1-4θε ρ √ 4θε ρ from (4.
2), replacing θ by θε and x by 1 ε

1 ρ ∼ ε→0 1 ε 1 2ρ K 1 ρ √ 4θ ρ 1 2 K √ 1-4θε ρ √ 4θε ρ -ρ 2 √ 4θε ρ K √ 1-4θε ρ √ 4θε ρ ∼ ε→0 1 ε 1 2ρ K 1 ρ √ 4θ ρ ρ 2 √ 4θε ρ K 1 ρ +1 √ 4θε ρ since zK ν = νK ν (z) -z K ν+1 (z) ∼ ε→0 1 ε 1 2ρ K 1 ρ √ 4θ ρ ρ 2 Γ( 1 ρ + 1) √ 4θε ρ -1 ρ since K ν (z) ∼ 1 2 Γ(ν) z 2 -1 = 1 ε 1 2ρ K 1 ρ √ 4θ ρ Γ 1 ρ ε 1 2ρ θ 1 2ρ ρ 1 ρ -→ ε↓0 θ ρ 2 1 2ρ 2 Γ 1 ρ K 1 ρ √ 4θ ρ = E e -θ 2 Π (ρ) 0 Remark 4.2
Taking up again the arguments of the proof of Theorem 3.1, it is not difficult to see that :

E 1 e -θ 2 Π (ρ) 1 ∼ θ→∞ C(ρ) θ 1 3
where C(ρ) is a strictly positive constant, depending on ρ. We then deduce that, for α < 0 :

E 1 (Π (ρ)
1 ) α < ∞ if and only if |α| < 1 3 On the other hand, it is not difficult to see that :

• If x > 1, for all α < 0, E 1 (Π (ρ) 1 ) α < ∞. • If x < 1, for all α < 0, E 1 (Π (ρ) 1 ) α = +∞. Concerning the positive moments of Π (ρ) 1 :

• If 0 < α < 1 ρ , then E x (Π (ρ) 1 ) α < ∞ (4.7)
Indeed (4.7), for x = 1, follows from :

E 1 (Π (ρ) 1 ) α ≤ E 1 (Π (ρ) 0 ) α = 1 Γ 1 ρ ∞ 0 2 ρ 2 z α e -z z 1 ρ -1 dz < ∞ if α < 1 ρ since Π 0 (law) = 2 ρ 2 γ 1 ρ , from (1.10)
The fact that E x (Π (ρ) 1 ) α < ∞ for every x > 0, and every α < 1 ρ may be obtained by using arguments close to those used in the proof of Theorem 1.1. We believe that, for all ρ > 0 : (4.9) Indeed, from Theorem 4.1, we have :

E x (Π (
E 1 exp - θ 2 Π (ρ) 1 = 1 + 1 2 K γ √ 4θ ρ + √ θ K γ √ 4θ ρ 1 2 K γ √ 4θ ρ - √ θ K γ √ 4θ ρ with γ = √ 1 -4θ ρ and θ ≤ 1 4 )• Thus : 1 -E 1 e -θ 2 Π (ρ) 1 = - 1 2 K γ √ 4θ ρ + ρ 2 √ 4θ ρ K γ √ 4θ ρ 1 2 K γ √ 4θ ρ -ρ 2 √ 4θ ρ K γ √ 4θ ρ = - 1 2 K √ 1-4θ ρ √ 4θ ρ (1 - √ 1-4θ ρ ) - √ θ ρ K √ 1-4θ ρ -1 √ 4θ ρ 1 2 K √ 1-4θ ρ √ 4θ ρ (1 + √ 1-4θ ρ ) + √ θ ρ K √ 1-4θ ρ -1 √ 4θ ρ
after using zK µ (z) = -µ K µ (z) -z K µ-1 (z) . Since ρ > 1, we replace K √ 1-4θ ρ -1 by K 1- √ 1-4θ ρ (since K µ = K -µ ) and we deduce, from :

K 1- √ 1-4θ ρ -→ θ→0 K 1-1 ρ , with 1 - 1 ρ > 0 :
that :

1

-E 1 e -θ 2 Π (ρ) 1 ∼ θ→0 a θ 1-1 2ρ + b θ 1 2ρ a θ -1 2ρ + b θ 1 2ρ
since, for µ > 0 K µ (z) ∼ z→0 C µ z -µ ; see [Leb], p. 136

∼ θ→0 b a θ 1 ρ since ρ ≥ 1 implies 1 - 1 2ρ ≥ 1 2ρ
• From (4.9) we deduce :

∞ 0 e -θt 2 P (Π

Figure 1 :

 1 Figure 1: Graphs of ϕ θ and ϕ θCase2 : 4θ ≥ 1, γ = i √ 4θ -1 and here ϕ θ (y) = √ y K iν ( √ 4θy) with ν = √ 4θ -1.From the integral representation formula see[Leb], p. 119 :

  by replacing iν by iν -1. This time, we use the second order development (3.4) with here : z = e iπ 2

  x, k > 0). In the preceding Sections, we have studied the case ρ = 1. The analogue of Theorem 1.3 may be stated as : Theorem 4.1 The Laplace transform of Π (ρ)

  (4.8), when ρ > 1 It then suffices, by using the arguments of the proof of Theorem 1.1, to see that (4.8) is true when x = 1.• We first show that : 1 -E 1 e -

Hence, from the Tauberian Theorem :

and

1 ≥ t ρ dt = +∞, from (4.10).

We note that, for ρ < 1, the preceding argument cannot be applied since :