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We apply the penalization technique introduced by Roynette, Vallois, Yor for Brownian motion to Galton–Watson processes with a penalizing function of the form P ( x ) s x where P is a polynomial of degree p and s ∈ [ 0 , 1 ] . We prove that the limiting martingales obtained by this method are most of the time classical ones, except in the super-critical case for s = 1 (or s → 1 ) where we obtain new martingales. If we make a change of probability measure with this martingale, we obtain a multi-type Galton–Watson tree with p distinguished infinite spines.

At Cap Blanc Nez (Channel coast, France), the chalk of the Lower Cenomanian is very rich in glauconite. Glauconite is of authigenic origin and requires the mobilization of chemical elements for its growth: Si, Fe and K. If we already know thanks to elementary geochemistry (Ge/Si ratio) that the silica of the flint present in the chalk originates from the dissolution of sponges, is it the same for glauconite? This question only makes sense if glauconite is proved to be autochthonous and synsedimentary, and not reworked during the Cenomanian transgression. In addition, we wanted to know whether the study of the content of trace elements in glauconite could provide information on the conditions of authigenesis in glauconious chalk. The clay content of the chalk from the Lower Cenomanian has been examined and the green minerals were extracted from the rock to study their morphology, mineralogy, geochemistry (major & trace elements) and grain size. The chalk consists of calcium carbonate, smectite and true glauconite, in the form of pellets, sometimes with rare traces of quartz and some centimeter-scale phosphate gravel. The grain-size distribution of the glauconites varies from one sample to another and is always poorly sorted, which militates in favor of an autochthonous (not reworked) origin of these minerals. This origin is also suggested by the virtual absence of terrigenous minerals, except for smectite known for its potential for wide distribution in the marine environment. The geochemistry of the samples shows a very homogeneous composition of major and trace elements, with a K2O content greater than 8%. This characterizes these glauconites as being very evolved, which indicates a long authigenic formation time (> 100 ky) and therefore an extremely reduced or irregular sedimentation rate. Here, glauconites are very rich in germanium, which makes it impossible to identify a source of silica (unlike what is possible with flints). It cannot therefore be said that the silica results from the dissolution of sponges but this enrichment in Ge, coupled with that in vanadium and the absence of enrichment in molybdenum, indicates a slightly reducing deposition milieu (suboxic). Such conditions usually favor organic matter accumulation, but not here, due to protracted sedimentation hiatuses. Lastly, glauconite trapped relatively large amounts of Ge due to reducing conditions and long exposure time to seawater, which makes it a potential chronometer assessing the duration of authigenesis, and a possible compartment of the marine cycle of germanium.

International audience; Black shales, though laminated, are not systematically synonymous with quiet conditions of deposition. A number of papers report about black shales yielding sedimentary structures echoing relatively high hydrodynamic conditions. Here we examine two sedimentary sequences pertaining to the Late Jurassic Argiles de Châtillon Formation of the Boulonnais area (Northernmost France), each of them including a meter-thick black-shale horizon. The two laminated black shales contain sulfurized (i.e., organic S-rich) organic matter, reaching a maximum of 9%. However, some differences set the two black shales apart. The lower one was deposited under calm, suboxic to anoxic, bottom-water conditions (the « classic » way); the upper black shale was deposited under oxic, agitated bottom-water conditions, not compatible with stable sulfidic conditions required for organic-matter sulfurization. The upper black shale experienced hydrodynamically-induced concentration of sulfurized, recalcitrant organic matter of marine origin. The sulfurized organic matter could be preserved and quantitatively accumulated owing to its non-putrescible nature, leading to the formation of a potential hydrocarbon source rock. The association of pyrite inclusions with sulfurized organic matter probably modified the hydrodynamic behavior of organic particles. The storm-induced remobilization and concentration of sulfurized organic matter implies that the “sulfurization factory” operated in proximal, shallow environments such as estuaries, mud flats or mangrove environments. Such a model is in agreement with findings from modern coastal/shelf environments (notably mangroves) and fully transposable to many other sedimentary situations of any geological time period.

On glacial to interglacial time scales, northern Africa fluctuated between arid to hyperarid states and much wetter conditions called African Humid Periods (AHP). These AHP are characterized by a major transformation of the Saharan hydrological cycle, favoring the development of vast fluvial networks, tropical flora and fauna in a region previously hyperarid. In the present-day context of global warming, it is crucial to understand the environmental mechanisms and responses associated with these dramatic swings between two extreme climatic states in order to improve the climatic projections. Numerous studies have been focused on the last AHP, which occurred at the beginning of the Holocene and corresponds to a period when insolation - governed by precession – and obliquity both reached their maximum almost synchronously, thus complicating the distinction of their respective roles. The study of older AHP corresponding to different orbital configurations is likely to provide some answers. However, finding climatic archives allowing the reconstruction of past changes in the Saharan hydrological cycle on longer timescales remains challenging (e.g., discontinuity of continental archives, preservation of tracers…). In this study, we propose to circumvent this difficulty by studying the Saharan dust deposited in marine sediments of the northeastern Atlantic tropical ocean. In fact, past modifications of Saharan dust deposited off West Africa can provide precious information on changes in environmental conditions in their source areas (aridity, weathering), as well as on changes in the characteristics of their atmospheric transport (pathways and strength). Here, we present a unique high-resolution (1 sample/200yrs) multi-proxy characterization of the dust deposited continuously through the last 240ka - a period punctuated by eight AHP - in the marine core MD03-2705 (18°05N; 21°09W; 3085 mbsl) retrieved from a bathymetric dome, 300 meters above the surrounding seafloor. Considering this particular environmental setting, the terrigenous fraction in this record is assumed to be predominantly of eolian origin. We combine the 230Th-normalized dust flux1 together with grain-size distribution, clay mineralogy and geochemical compositions in order to explore changes in the Saharan hydroclimate and atmospheric circulation over North Africa on millennial to orbital timescales, with a particular focus on the mechanisms associated with the recurrence of the AHP.1Skonieczny et al., 2019 – Science Advances 5 (1) - eaav1887

We consider the genealogical tree of a stationary continuous state branching process with immigration. For a sub-critical stable branching mechanism, we consider the genealogical tree of the extant population at some fixed time and prove that, up to a deterministic time-change, it is distributed as a continuous-time Galton-Watson process with immigration. We obtain similar results for a critical stable branching mechanism when only looking at immigrants arriving in some fixed time-interval. For a general sub-critical branching mechanism, we consider the number of individuals that give descendants in the extant population. The associated processes (forward or backward in time) are pure-death or pure-birth Markov processes, for which we compute the transition rates., Comment: 31 pages, 1 figure

International audience; We give an invariance principle for very general additive functionals of conditioned Bienaymé-Galton-Watson trees in the global regime when the offspring distribution lies in the domain of attraction of a stable distribution, the limit being an additive functional of a stable Lévy tree. This includes the case when the offspring distribution has finite variance (the Lévy tree being then the Brownian tree). We also describe, using an integral test, a phase transition for toll functions depending on the size and height.

International audience; The goal of this paper is the presentation of a post-processing method allowing to remove impulse noise in binary images, while preserving thin structures. We use a grain filter as in [5]. We propose a method to automatically determine the required threshold using Galton-Watson processes. We present numerical results and a complete analysis on a synthetic image. We end the numerical section considering a specific application to granite samples crack detection: here we deal with X-tomography images that have been binarized via preprocessing techniques and we want to remove residual impulse noise while keeping cracks and micro-cracks structure.

We consider a marking procedure of the vertices of a tree where each vertex is marked independently from the others with a probability that depends only on its out-degree. We prove that a critical Galton–Watson tree conditioned on having a large number of marked vertices converges in distribution to the associated size-biased tree. We then apply this result to give the limit in distribution of a critical Galton–Watson tree conditioned on having a large number of protected nodes.

International audience; Under minimal condition, we prove the local convergence of a critical multi-type Galton-Watson tree conditioned on having a large total progeny by types towards a multi-type Kesten's tree. We obtain the result by generalizing Neveu's strong ratio limit theorem for aperiodic random walks on Z^d .

International audience; We deal with a severe ill posed problem, namely the reconstruction process of an image during tomography acquisition with (very) few views. We present different methods that we have been investigated during the past decade. They are based on variational analysis. This is a survey paper and we refer to the quoted papers for more details. Mathematics Subject Classification (2010). 49K40, 45Q05,65M32.

We consider the Brownian tree introduced by Aldous and the associated Q-process which consists in an infinite spine on which are grafted independent Brownian trees. We present a reversal procedure on these trees that consists in looking at the tree downward from its top: the branching points becoming leaves and leaves becoming branching points. We prove that the distribution of the tree is invariant under this reversal procedure, which provides a better understanding of previous results from Bi and Delmas (2016)., Comment: This is the second version of the preprint arXiv:1612.03715 that has been splitted into two papers

We consider a super-critical Galton-Watson tree $\tau $ whose non-degenerate offspring distribution has finite mean. We consider the random trees $\tau _n$ distributed as $\tau $ conditioned on the $n$-th generation, $Z_n$, to be of size $a_n\in{\mathbb N} $. We identify the possible local limits of $\tau _n$ as $n$ goes to infinity according to the growth rate of $a_n$. In the low regime, the local limit $\tau ^0$ is the Kesten tree, in the moderate regime the family of local limits, $\tau ^\theta $ for $\theta \in (0, +\infty )$, is distributed as $\tau $ conditionally on $\{W=\theta \}$, where $W$ is the (non-trivial) limit of the renormalization of $Z_n$. In the high regime, we prove the local convergence towards $\tau ^\infty $ in the Harris case (finite support of the offspring distribution) and we give a conjecture for the possible limit when the offspring distribution has some exponential moments. When the offspring distribution has a fat tail, the problem is open. The proof relies on the strong ratio theorem for Galton-Watson processes. Those latter results are new in the low regime and high regime, and they can be used to complete the description of the (space-time) Martin boundary of Galton-Watson processes. Eventually, we consider the continuity in distribution of the local limits $(\tau ^\theta , \theta \in [0, \infty ])$.

Let τn be a random tree distributed as a Galton-Watson tree with geometric offspring distribution conditioned on {Zn = an} where Zn is the size of the nth generation and (an, n ∈ ℕ*) is a deterministic positive sequence. We study the local limit of these trees τn as n →∞ and observe three distinct regimes: if (an, n ∈ ℕ*) grows slowly, the limit consists in an infinite spine decorated with finite trees (which corresponds to the size-biased tree for critical or subcritical offspring distributions), in an intermediate regime, the limiting tree is composed of an infinite skeleton (that does not satisfy the branching property) still decorated with finite trees and, if the sequence (an, n ∈ ℕ*) increases rapidly, a condensation phenomenon appears and the root of the limiting tree has an infinite number of offspring.

19 pages; International audience; In this article, we focus on tomographic reconstruction. The problem is to determine the shape of the interior interface using a tomographic approach while very few X-ray radiographs are performed. We use a multi-marginal optimal transport approach. Preliminary numerical results are presented.

We give an explicit construction of the increasing tree-valued process introduced by Abraham and Delmas using a random point process of trees and a grafting procedure. This random point process will be used in companion papers to study record processes on Lévy trees. We use the Poissonian structure of the jumps of the increasing tree-valued process to describe its behavior at the first time the tree grows higher than a given height, using a spinal decomposition of the tree, similar to the classical Bismut and Williams decompositions. We also give the joint distribution of this exit time and the ascension time which corresponds to the first infinite jump of the tree-valued process.

28 pages To appear in: Applied Math. Optim. (2008); International audience; We propose a variational method for tomographic reconstruction of blurred and noised binary images based on a penalization process of a minimization problem settled in the space of bounded variation functions. We prove existence and/or uniqueness results and derive a penalized optimality system. Numerical simulations are provided to demonstrate the relevance of the approach.

In this paper, we propose an edge detection technique based on some local smoothing of the image followed by a statistical hypothesis testing on the gradient. An edge point being defined as a zero-crossing of the Laplacian, it is said to be a significant edge point if the gradient at this point is larger than a threshold s(@e) defined by: if the image I is pure noise, then the probability of @[email protected]?I(x)@?>=s(@e) conditionally on @DI(x)=0 is less than @e. In other words, a significant edge is an edge which has a very low probability to be there because of noise. We will show that the threshold s(@e) can be explicitly computed in the case of a stationary Gaussian noise. In the images we are interested in, which are obtained by tomographic reconstruction from a radiograph, this method fails since the Gaussian noise is not stationary anymore. Nevertheless, we are still able to give the law of the gradient conditionally on the zero-crossing of the Laplacian, and thus compute the threshold s(@e). We will end this paper with some experiments and compare the results with those obtained with other edge detection methods.

This paper proposes a method to detect spheres in 3D images of high density materials. This method needs to be fast because of the high number of spheres in images as well as accurate. In order to do this, an algorithm based on centres detection is used, which permits to obtain results in a low computation time. However, we need to improve algorithm accuracy, and for that, a circular Hough Transform algorithm is used on planes, and a sphere verification technique are added to the base method. Finally, we obtain an algorithm which provides accurate results in a relatively low computation time.

International audience; We study the pruning process developed by Abraham and Delmas (2012) on the discrete Galton-Watson sub-trees of the L\'{e}vy tree which are obtained by considering the minimal sub-tree connecting the root and leaves chosen uniformly at rate $\lambda$, see Duquesne and Le Gall (2002). The tree-valued process, as $\lambda$ increases, has been studied by Duquesne and Winkel (2007). Notice that we have a tree-valued process indexed by two parameters the pruning parameter $\theta$ and the intensity $\lambda$. Our main results are: construction and marginals of the pruning process, representation of the pruning process (forward in time that is as $\theta$ increases) and description of the growing process (backward in time that is as $\theta$ decreases) and distribution of the ascension time (or explosion time of the backward process) as well as the tree at the ascension time. A by-product of our result is that the super-critical L\'{e}vy trees independently introduced by Abraham and Delmas (2012) and Duquesne and Winkel (2007) coincide. This work is also related to the pruning of discrete Galton-Watson trees studied by Abraham, Delmas and He (2012).

We consider a Brownian snake (W s ,s≥0) with underlying process a reflected Brownian motion in a bounded domain D. We construct a continuous additive functional (L s ,s≥0) of the Brownian snake which counts the time spent by the end points Ŵ s of the Brownian snake paths on ∂D. The random measure Z=∫δŴ sdL s is supported by ∂D. Then we represent the solution v of Δu=4u 2 in D with weak Neumann boundary condition φ≥0 by using exponential moment of (Z,φ) under the excursion measure of the Brownian snake. We then derive an integral equation for v. For small φ it is then possible to describe negative solution of Δu=4u 2 in D with weak Neumann boundary condition φ. In contrast to the exit measure of the Brownian snake out of D, the measure Z is more regular. In particular we show it is absolutely continuous with respect to the surface measure on ∂D for dimension 2 and 3.

We consider the exit measure of super Brownian motion with a stable branching mechanism of a smooth domain D of ℝ d . We derive lower bounds for the hitting probability of small balls for the exit measure and upper bounds in the critical dimension. This completes results given by Sheu [22] and generalizes the results of Abraham and Le Gall [2]. Because of the links between exits measure and partial differential equations, those results imply bounds on solutions of elliptic semi-linear PDE. We also give the Hausdorff dimension of the support of the exit measure and show it is totally disconnected in high dimension. Eventually we prove the exit measure is singular with respect to the surface measure on ∂D in the critical dimension. Our main tool is the subordinated Brownian snake introduced by Bertoin, Le Gall and Le Jan [4].

The modelling of three-dimensional composite carbon fibers-resin materials for a multi-scale use requires the knowledge of the carbon fibers localization and orientation. We propose here a mathematical method exploiting tomographic data to determine carbon localization with a Markov Random Field (MRF) segmentation, identify carbon straight cylinders, and accurately determine fibers orientation.

The paper deals with a path-valued Markov process: the reflecting Brownian snake. It is a particular case of the path-valued process previously introduced by Le Gall. Here the spatial motion is a reflecting Brownian motion in a domain D of R d . Using this probabilistic tool, we construct an explicit function v solution of an integral equation which is, under some hypotheses on the regularity of v , equivalent to a semi-linear partial differential equation in D with some mixed Neumann–Dirichlet conditions on the boundary. When the hypotheses on v are not satisfied, we prove that v is still solution of a weak formulation of the Neumann–Dirichlet problem.

International audience; We provide a complete picture of the local convergence of critical or subcritical Galton-Watson tree conditioned on having a large number of individuals with out-degree in a given set. The generic case, where the limit is a random tree with an infinite spine has been treated in a previous paper. We focus here on the non-generic case, where the limit is a random tree with a node with infinite out-degree. This case corresponds to the so-called condensation phenomenon.

International audience; We give a necessary and sufficient condition for the convergence in distribution of a conditioned Galton-Watson tree to Kesten's tree. This yields elementary proofs of Kesten's result as well as other known results on local limit of conditioned Galton-Watson trees. We then apply this condition to get new results, in the critical and sub-critical cases, on the limit in distribution of a Galton-Watson tree conditioned on having a large number of individuals with out-degree in a given set.

International audience; We perform a pruning procedure on a Lévy tree and instead of throwing away the removed sub-tree, we regraft it on a given branch (not related to the Lévy tree). We prove that the tree constructed by regrafting is distributed as the original Lévy tree, generalizing a result where only Aldous's tree is considered. As a consequence, we obtain that the quantity which represents in some sense the number of cuts needed to isolate the root of the tree, is distributed as the height of a leaf picked at random in the Lévy tree.

We present an extension of the Gromov-Hausdorff metric on the set of compact metric spaces: the Gromov-Hausdorff-Prokhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the non-compact case by describing a metric on the set of rooted complete locally compact length spaces endowed with a boundedly finite measure. We prove that this space with the extended Gromov-Hausdorff-Prokhorov metric is a Polish space. This generalization is needed to define Lévy trees, which are (possibly unbounded) random real trees endowed with a locally finite measure.[br/]

Tribe proved in a previous paper that a typical point of the support of super Brownian motion considered at a fixed time is a.s. disconnected from the others when the space dimension is greater than or equal to 3. We give here a simpler proof of this result based on Le Gall's Brownian snake. This proof can then be adapted in order to obtain an analogous result for the support of the exit measure of the super Brownian motion from a smooth domain of Rd when d is greater than or equal to 4.

Nous presentons une nouvelle procedure d’elagage d’arbres discrets en ajoutant des marques sur les noeuds de l’arbre. Cette procedure nous permet de definir un processus de Markov $\{\mathcal{G}(u)\}$ a valeurs arbres en elaguant un arbre de Galton–Watson. Nous definissons egalement de maniere analogue un processus $\{\mathcal{G}^{*}(u)\}$ en elaguant un arbre de Galton–Watson critique ou sous-critique conditionne a etre infini. Sous de faibles hypotheses sur la loi de reproduction, nous montrons que le processus $\{\mathcal{G}(u)\}$ arrete en son temps d’ascension admet une representation en terme du processus $\{\mathcal{G}^{*}(u)\}$. Un resultat similaire a ete obtenu par Aldous et Pitman (Ann. Inst. H. Poincare Probab. Statist. 34 (1998) 637–686) dans le cas particulier de lois de reproductions poissoniennes en considerant un elagage uniforme sur les branches de l’arbre.

We study some properties of the exit measure of super Brownian motion from a smooth domainD inR d . In particular, we give precise estimates for the probability that the exit measure gives a positive mass to a small ball on the boundary. As an application, we compute the Hausdorff dimension of the support of the exit measure. In dimension 2, we prove that the exit measure is absolutely continuous with respect to the Lebesgue measure on the boundary. In connection with Dynkin's work, our results give some information on the behavior of solutions of Δu=u 2 inD, and are related to the characterization of removable singularities at the boundary. As a consequence of our estimates, we give a sufficient condition for the uniqueness of the positive solution of Δu=u 2 inD that tends to ∞ on an open subsetO of ϖD and to 0 on the complement in ϖD of the closure ofO. Our proofs use the path-valued process studied in [L2, L3].

International audience; Given a general critical or sub-critical branching mechanism, we define a pruning procedure of the associated Lévy continuum random tree. This pruning procedure is defined by adding some marks on the tree, using Lévy snake techniques. We then prove that the resulting sub-tree after pruning is still a Lévy continuum random tree. This last result is proved using the exploration process that codes the CRT, a special Markov property and martingale problems for exploration processes. We finally give the joint law under the excursion measure of the lengths of the excursions of the initial exploration process and the pruned one.

International audience; We consider the height process of a Levy process with no negative jumps, and its associated continuous tree representation. Using Levy snake tools developed by Duquesne and Le Gall, with an underlying Poisson process, we construct a fragmentation process, which in the stable case corresponds to the self-similar fragmentation described by Miermont. For the general fragmentation process we compute a family of dislocation measures as well as the law of the size of a tagged fragment. We also give a special Markov property for the snake which is interesting in itself.

A multitype urn scheme with random replacements is considered. Each time a ball is picked, another ball is added, and its type is chosen according to the transition probabilities of a reducible Markov chain. The vector of frequencies is shown to converge almost surely to a random element of the set of stationary measures of the Markov chain. Its probability distribution is characterized as the solution to a fixed point problem. It is proved to be Dirichlet in the particular case of a single transient state to which no return is possible. This is no longer the case, however, as soon as returns to transient states are allowed.

A paraitre dans "Journal of Theoretical Probability"; International audience; We consider the exploration process associated to the continuous random tree (CRT) built using a Levy process with no negative jumps. This process has been studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is a useful tool to study CRT as well as super-Brownian motion with general branching mechanism. In this paper we prove this process is Feller, and we compute its infinitesimal generator on exponential functionals and give the corresponding martingale.

We consider an initial population whose size evolves according to a continuous state branching process. Then we add to this process an immigration (with the same branching mechanism as the initial population), in such a way that the immigration rate is proportional to the whole population size. We prove this continuous state branching process with immigration proportional to its own size is itself a continuous state branching process. By considering the immigration as the apparition of a new type, this construction is a natural way to model neutral mutation. It also provides in some sense a dual construction of the particular pruning at nodes of continuous state branching process introduced by the authors in a previous paper. For a critical or sub-critical quadratic branching mechanism, it is possible to explicitly compute some quantities of interest. For example, we compute the Laplace transform of the size of the initial population conditionally on the non-extinction of the whole population with immigration. We also derive the probability of simultaneous extinction of the initial population and the whole population with immigration.

International audience; This paper deals with a method of tomographic reconstruction of radially symmetric objects from a single radiograph, in order to study the behavior of shocked material. The usual tomographic reconstruction algorithms such as generalized inverse or filtered back-projection cannot be applied here because data are very noisy and the inverse problem associated to single view tomographic reconstruction is highly unstable. In order to improve the reconstruction, we propose here to add some a priori assumptions on the looked after object. One of these assumptions is that the object is binary and consequently, the object may be described by the curves that separate the two materials. We present a model that lives in BV space and leads to a non local Hamilton-Jacobi equation, via a level set strategy. Numerical experiments are performed (using level sets methods) on synthetic objects.

International audience; We consider the fragmentation at nodes of the Lévy continuous random tree introduced in a previous paper. In this framework we compute the asymptotic for the number of small fragments at time $\theta$. This limit is increasing in $\theta$ and discontinuous. In the $\alpha$-stable case the fragmentation is self-similar with index $1/\alpha$, with $\alpha \in (1,2)$ and the results are close to those Bertoin obtained for general self-similar fragmentations but with an additional assumtion which is not fulfilled here.

We consider a system of fully coupled forward-backward stochastic differential equations. First we generalize the results of Pardoux-Tang [7] concerning the regularity of the solutions with respect to initial conditions. Then, we prove that in some particular cases this system leads to a probabilistic representation of solutions of a second-order PDE whose second order coefficients depend on the gradient of the solution. We then give some examples in dimension 1 and dimension 2 for which the assumptions are easy to check.

We consider a path-valued process which is a generalization of the classical Brownian snake introduced by Le Gall. More precisely we add a drift term b to the lifetime process, which may depends on...

Our main object that we call the Poisson snake is a Brownian snake as introduced by Le Gall. This process has values which are trajectories of standard Poisson process stopped at some random finite lifetime with Brownian evolution. We use this Poisson snake to construct a self-similar fragmentation as introduced by Bertoin. A similar representation was given by Aldous and Pitman using the Continuum Random Tree. Whereas their proofs used approximation by discrete models, our representation allows continuous time arguments.

We investigate the asymptotic behaviour of the probability that a normalized $d$-dimensional Brownian snake (for instance when the life-time process is an excursion of height 1) avoids 0 when starting at distance $\varepsilon$ from the origin. In particular we show that when $\varepsilon$ tends to 0, this probability respectively behaves (up to multiplicative constants) like $\varepsilon^4$, $\varepsilon^{2\sqrt{2}}$ and $\varepsilon^{(\sqrt {17}-1)/2}$, when $d=1$, $d=2$ and $d=3$. Analogous results are derived for super-Brownian motion started from $\delta_x$ (conditioned to survive until some time) when the modulus of $x$ tends to 0.

We consider an initial Eve-population and a population of neutral mutants, such that the total population dies out in finite time. We describe the evolution of the Eve-population and the total population with continuous state branching processes, and the neutral mutation procedure can be seen as an immigration process with intensity proportional to the size of the population. First we establish a Williams’ decomposition of the genealogy of the total population given by a continuum random tree, according to the ancestral lineage of the last individual alive. This allows us to give a closed formula for the probability of simultaneous extinction of the Eve-population and the total population.

In this thesis we are interested in three types of problems: 1-Existence and uniqueness of a positive harmonic function associated with an inhomogeneous random walk confined in an orthant. 2-Study of convergence in distribution of critical Galton Watson trees conditioned to have a large enoughnumber of protected nodes. 3-Study of the convergence in distribution of Galton Watson trees conditioned to have a large generation.; Dans cette thèse nous nous sommes intéressés de trois types de problèmes : 1 -Existence et unicité d’une fonction harmonique strictement positive associée à une marche aléatoire inhomogène confinée dans un orthant. 2 -Etude de la convergence en loi des arbres de Galton Watson critiques conditionnés à avoir un nombre assez grand de noeuds protégés. 3 -Etude de la convergence en loi des arbres de Galton Watson conditionnés à avoir une génération anormalement grande.

Digital simulation is a wide-spreading tool for composite materials design and choice. Indeed it allows to generate and test digitally various structures more easily and quickly than with real manufacturing and tests processes. A feedback is needed following the choice and the fabrication of a virtual material in order to simultaneously validate the simulation and the fabrication process. With this aim, models similar to generated virtual structures are obtained by digitization of manufacturing materials. The same simulation algorithms can then be applied and allow to verify the forecasts. This thesis is also about the modelling of composite materials from 3D images, in order to rediscover in them the original virtual material. Image processing methods are applied to images to extract material structure data, i.e. each constituent localization, and orientation if applicable. These knowledge theoretically allow to simulate thermal and mechanical behavior of structures constituted of studied material. However to accurately represent composites requires practically a very small discretization step. Therefore behavior simulation of a macroscopic structure needs too much discretization points, and then time and memory. Hence a part of this thesis focuses also on determination of equivalent homogeneous material problem, which allows, when resolved, to lighten calculation time for simulation algorithms., La simulation numérique est un outil de plus en plus utilisé pour la conception et le choix de matériaux composites. Celle-ci permet en effet de générer et tester numériquement des structures très diverses plus facilement et plus rapidement qu’avec des procédés de fabrication et de tests réels. Suite au choix d’un matériau virtuel et sa fabrication tangible, un retour sur expérience est nécessaire afin de valider simultanément la simulation et le procédé de fabrication. Pour cela, la numérisation des matériaux fabriqués permet de renvoyer une modélisation comparable aux structures virtuelles générées. Il devient possible d’appliquer les mêmes algorithmes de simulation et de vérifier les prévisions.Le sujet de cette thèse consiste donc en la modélisation de matériaux composites réels à partir d’images 3D, afin d’y retrouver le matériau virtuel originel. Des méthodes de traitement d’images sont appliquées aux images afin d’en extraire les informations sur la structure du matériau, c’est-à-dire la localisation de chaque constituant et, s’il y a lieu, de leurs orientations. Ces connaissances permettent théoriquement de simuler le comportement thermique et mécanique de structures constituées du matériau étudié.Cependant, en pratique, représenter fidèlement les composites demande de prendre une discrétisation très fine. Par conséquent, une structure macroscopique demande beaucoup trop de points de discrétisation et donc de temps de calcul et de mémoire pour simuler son comportement. Un aspect de la thèse consiste donc aussi en la détermination d’un matériau homogène équivalent, permettant d’alléger la charge de calcul pour les algorithmes de simulation.