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Nicolas Bouleau

Publications and source records attributed to Nicolas Bouleau.

At least 19 recordsLinked to original sources
arXiv2026

Energy decomposition by potential level

In the first part we study excessive functions for a Markov process that are continuous semimartingales along the sample paths. The property then follows from the theory of local times of semimartingales. We next treat the case of Dirichlet spaces and show that, if u is a quasi-continuous version of a function u belonging to a regular Dirichlet space, then the image under u of the local energy measure of u is absolutely continuous with respect to Lebesgue measure. Finally, in a third part, we establish the occupation-time density property for certain Dirichlet processes.

math.PR↗
arXiv2025

Desargues and the "trait à preuves"

Summary. The 'trait {à} preuves' is a 17th century expression that refers to drawings rigorously justified by mathematics. This work is a reflection on the principle of purity of methods from a historical, mathematical and epistemological point of view. Our starting point is the problematic that arose in the 17th century on the occasion of a dispute between stonemasons and geometers who were theorists of stereotomy. Girard Desargues was involved in this controversy, and we can assume that Desargues' theorem was a key element in his argument. It clearly raises the question of the purity of methods. We will illustrate this with examples. And this leads us to David Hilbert's fundamental work on the subject.

math.HO↗
arXiv2025

Mental Geometry

This article illustrates pedagogy through training in the handling of abstractions. Mental arithmetic is not limited to numerical calculation; one can mentally calculate primitives and simplify analytical expressions. Even if there is software that does this very well, this training retains its pedagogical value. Can we go further and consider geometric mental arithmetic: mentally proceeding with transformations of simple figures allowing the calculation of areas or volumes? It turns out that the intuition that allowed Archimedes to obtain his main geometric results, if we take only the ideas without the old-fashioned style, provides the opportunity for a pleasant and quite rich mental game that I present here in the form of a short narrative dialogue, not a philosophical tale because it does not bring any thesis, simply a story to be classified among the invitations to exercise the mind. It starts with the area of a triangle and ends with Guldin's two theorems.

math.HO↗
arXiv2014

Some Historical Aspects of Error Calculus by Dirichlet Forms

We discuss the main stages of development of the error calculation since the beginning of XIX-th century by insisting on what prefigures the use of Dirichlet forms and emphasizing the mathematical properties that make the use of Dirichlet forms more relevant and efficient. The purpose of the paper is mainly to clarify the concepts. We also indicate some possible future research.

math.HO↗
arXiv2013

Improving Monte Carlo simulations by Dirichlet forms

Equipping the probability space with a local Dirichlet form with square field operator Γand generator A allows to improve Monte Carlo simulations of expectations and densities as soon as we are able to simulate a random variable X together with Γ[X] and A[X]. We give examples on the Wiener space, on the Poisson space and on the Monte Carlo space. When X is real-valued we give an explicit formula yielding the density at the speed of the law of large numbers.

math.PR↗
arXiv2013

On error operators related to the arbitrary functions principle

The error on a real quantity Y due to the graduation of the measuring instrument may be asymptotically represented, when the graduation is regular and fines down, by a Dirichlet form on R whose square field operator does not depend on the probability law of Y as soon as this law possesses a continuous density. This feature is related to the "arbitrary functions principle" (Poincar'e, Hopf). We give extensions of this property to Rd and to the Wiener space for some approximations of the Brownian motion. This gives new approximations of the Ornstein-Uhlenbeck gradient. These results apply to the discretization of some stochastic differential equations encountered in mechanics.

math.PR↗
arXiv2013

The lent particle method for marked point processes

Although introduced in the case of Poisson random measures, the lent particle method applies as well in other situations. We study here the case of marked point processes. In this case the Malliavin calculus (here in the sense of Dirichlet forms) operates on the marks and the point process doesn't need to be Poisson. The proof of the method is even much simpler than in the case of Poisson random measures. We give applications to isotropic processes and to processes whose jumps are modified by independent diffusions.

math.PR↗
arXiv2013

Drichlet forms for Poisson measures and Lévy processes : the lent particle method

We present a new approach to absolute continuity of laws of Poisson functionals. The theoretical framework is that of local Dirichlet forms as a tool to study probability spaces. The method gives rise to a new explicit calculus that we show first on some simple examples : it consists in adding a particle and taking it back after computing the gradient. Then we apply it to SDE's driven by Poisson measure.

math.PR↗
arXiv2013

How to specify an approximate numerical result

The Dirichlet forms methods, in order to represent errors and their propagation, are particularly powerful in infinite dimensional problems such as models involving stochastic analysis encountered in finance or physics, cf. [5]. Now, coming back to the finite dimensional case, these methods give a new light on the very classical concept of 'numerical approximation' and suggest changes in the habits. We show that for some kinds of approximations only an Ito-like second order differential calculus is relevant to describe and propagate numerical errors through a mathematical model. We call these situations strongly stochastic. The main point of this work is an argument based on the arbitrary functions principle of Poincaré-Hopf showing that the errors due to measurements with graduated instruments are strongly stochastic. Eventually we discuss the consequences of this phenomenon on the specification of an approximate numerical result.

math.PR↗
arXiv2013

Iteration of the lent particle method for existence of smooth densities of Poisson functionals

In previous works we have introduced a new method called the lent particle method which is an efficient tool to establish existence of densities for Poisson functionals. We now go further and iterate this method in order to prove smoothness of densities. More precisely, we construct Sobolev spaces of any order and prove a Malliavin-type criterion of existence of smooth density. We apply this approach to SDE's driven by Poisson random measures and also present some non-trivial examples to which our method applies.

math.PR↗
arXiv2012

Chaotic extensions and the lent particle method for Brownian motion

In previous works, we have developed a new Malliavin calculus on the Poisson space based on the lent particle formula. The aim of this work is to prove that, on the Wiener space for the standard Ornstein-Uhlenbeck structure, we also have such a formula which permits to calculate easily and intuitively the Malliavin derivative of a functional. Our approach uses chaos extensions associated to stationary processes of rotations of normal martingales.

math.PR↗
arXiv2010

The Lent Particle Method, Application to Multiple Poisson Integrals

We give a extensive account of a recent new way of applying the Dirichlet form theory to random Poisson measures. The main application is to obtain existence of density for thelaws of random functionals of Lévy processes or solutions of stochastic differential equations with jumps. As in the Wiener case the Dirichlet form approach weakens significantly theregularity assumptions. The main novelty is an explicit formula for the gradient or for the "carré du champ' on the Poisson space called the lent particle formula because based on adding a new particle to the system, computing the derivative of the functional with respect to this new argument and taking back this particle before applying the Poisson measure. The article is expository in its first part and based on Bouleau-Denis [12] with several new examples, applications to multiple Poisson integrals are gathered in the last part which concerns the relation with the Fock space and some aspects of the second quantization.

math.PR↗
arXiv2010

Une Structure Uniforme sur un Espace F(E,F)

Let E be a topological space and F a uniform space. We introduce a new topology (in fact a uniform structure) called the V-congergence on the space of applications from E to F such that C(E,F) is closed for this topology and the restriction of this topology to C(E,F) is equivalent to pointwise convergence. In other words this topology is the coarsest preserving continuity. We give a criterion of convergence for this topology not involving the limit. Among properties preserved are mesurability and alpha-borelianity for a countable ordinal alpha.

math.GN↗
arXiv2009

Application of the lent particle method to Poisson driven SDE's

We apply the Dirichlet forms version of Malliavin calculus to stochastic differential equations with jumps. As in the continuous case this weakens significantly the assumptions on the coefficients of the SDE. In spite of the use of the Dirichlet forms theory, this approach brings also an important simplification which was not available nor visible previously : an explicit formula giving the carré du champ matrix, i.e. the Malliavin matrix. Following this formula a new procedure appears, called the lent particle method which shortens the computations both theoretically and in concrete examples.

math.PR↗
arXiv2009

Stochastic approach for the subordination in Bochner sense

It is possible to construct a double indexed process with sample paths a surface of a family of subordinators obtained by subordination. We study here a branch of this subordination process. This opens martingale methods on symbolic calculus questions.

math.PR↗
arXiv2007

Five Conferences on Undecidability

These five lectures on undecidability were given to students with a good level in mathematics but with no special knowledge on logic. The first conference presents the formalization of mathematics with a short historical survey, the language of first order predicates and the axioms of set theory. The second and third lectures explain the incompleteness phenomena from the Hilbert program until Gödel's theorems with a presentation of the sequent calculus of Gentzen.The fourth talk deepens model theory reasoning in the case of the continuum hypothesis, and the last conference gives examples of effective computability results.

math.LO↗