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Search Results: 1 - 10 of 1860 matches for " Bertrand Cloez "
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Wasserstein decay of one dimensional jump-diffusions
Bertrand Cloez
Mathematics , 2012,
Abstract: This work is devoted to the Lipschitz contraction and the long time behavior of certain Markov processes. These processes diffuse and jump. They can represent some natural phenomena like size of cell or data transmission over the Internet. Using a Feynman-Kac semigroup, we prove a bound in Wasserstein metric. This bound is explicit and optimal in the sense of Wasserstein curvature. This notion of curvature is relatively close to the notion of (coarse) Ricci curvature or spectral gap. Several consequences and examples are developed, including an $L^2$ spectral for general Markov processes, explicit formulas for the integrals of compound Poisson processes with respect to a Brownian motion, quantitative bounds for Kolmogorov-Langevin processes and some total variation bounds for piecewise deterministic Markov processes.
Limit theorems for some branching measure-valued processes
Bertrand Cloez
Mathematics , 2011,
Abstract: We consider a particle system in continuous time, discrete population, with spatial motion and nonlocal branching. The offspring's weights and their number may depend on the mother's weight. Our setting captures, for instance, the processes indexed by a Galton-Watson tree. Using a size-biased auxiliary process for the empirical measure, we determine this asymptotic behaviour. We also obtain a large population approximation as weak solution of a growth-fragmentation equation. Several examples illustrate our results.
Exponential ergodicity for Markov processes with random switching
Bertrand Cloez,Martin Hairer
Mathematics , 2013, DOI: 10.3150/13-BEJ577
Abstract: We study a Markov process with two components: the first component evolves according to one of finitely many underlying Markovian dynamics, with a choice of dynamics that changes at the jump times of the second component. The second component is discrete and its jump rates may depend on the position of the whole process. Under regularity assumptions on the jump rates and Wasserstein contraction conditions for the underlying dynamics, we provide a concrete criterion for the convergence to equilibrium in terms of Wasserstein distance. The proof is based on a coupling argument and a weak form of the Harris theorem. In particular, we obtain exponential ergodicity in situations which do not verify any hypoellipticity assumption, but are not uniformly contracting either. We also obtain a bound in total variation distance under a suitable regularising assumption. Some examples are given to illustrate our result, including a class of piecewise deterministic Markov processes.
A stochastic approximation approach to quasi-stationary distributions on finite spaces
Michel Bena?m,Bertrand Cloez
Mathematics , 2014,
Abstract: This work is concerned with the analysis of a stochastic approximation algorithm for the simulation of quasi-stationary distributions on finite state spaces. This is a generalization of a method introduced by Aldous, Flannery and Palacios. It is shown that the asymptotic behavior of the empirical occupation measure of this process is precisely related to the asymptotic behavior of some deterministic dynamical system induced by a vector field on the unit simplex. This approach provides new proof of convergence as well as precise rates for this type of algorithm. We then compare this algorithm with particle system algorithms.
Quantitative results for the Fleming-Viot Particle system and quasi-stationary distributions in discrete space
Bertrand Cloez,Marie-Noémie Thai
Mathematics , 2013,
Abstract: We show, for a class of discrete Fleming-Viot (or Moran) type particle systems, that the convergence to the equilibrium is exponential for a suitable Wassertein coupling distance. The approach provides an explicit quantitative estimate on the rate of convergence. As a consequence, we show that the conditioned process converges exponentially fast to a unique quasi-stationary distribution. Moreover, by estimating the two-particle correlations, we prove that the Fleming-Viot process converges, uniformly in time, to the conditioned process with an explicit rate of convergence. We illustrate our results on the examples of the complete graph and of N particles jumping on two points.
Minimum Penalized Hellinger Distance for Model Selection in Small Samples  [PDF]
Papa Ngom, Bertrand Ntep
Open Journal of Statistics (OJS) , 2012, DOI: 10.4236/ojs.2012.24045
Abstract: In statistical modeling area, the Akaike information criterion AIC, is a widely known and extensively used tool for model choice. The φ-divergence test statistic is a recently developed tool for statistical model selection. The popularity of the divergence criterion is however tempered by their known lack of robustness in small sample. In this paper the penalized minimum Hellinger distance type statistics are considered and some properties are established. The limit laws of the estimates and test statistics are given under both the null and the alternative hypotheses, and approximations of the power functions are deduced. A model selection criterion relative to these divergence measures are developed for parametric inference. Our interest is in the problem to testing for choosing between two models using some informational type statistics, when independent sample are drawn from a discrete population. Here, we discuss the asymptotic properties and the performance of new procedure tests and investigate their small sample behavior.
Relativistic Gauge Invariant Wave Equation of the Electron-Neutrino  [PDF]
Claude Daviau, Jacques Bertrand
Journal of Modern Physics (JMP) , 2014, DOI: 10.4236/jmp.2014.511102
Abstract:

With the right and the left waves of an electron, plus the left wave of its neutrino, we write the tensorial densities coming from all associations of these three spinors. We recover the wave equation of the electro-weak theory. A new non linear mass term comes out. The wave equation is form invariant, then relativistic invariant, and it is gauge invariant under the U(1)×SU(2), Lie group of electro-weak interactions. The invariant form of the wave equation has the Lagrangian density as real scalar part. One of the real equations equivalent to the invariant form is the law of conservation of the total current.

A Wave Equation including Leptons and Quarks for the Standard Model of Quantum Physics in Clifford Algebra  [PDF]
Claude Daviau, Jacques Bertrand
Journal of Modern Physics (JMP) , 2014, DOI: 10.4236/jmp.2014.518210
Abstract:

A wave equation with mass term is studied for all fermionic particles and antiparticles of the first generation: electron and its neutrino, positron and antineutrino, quarks u and d with three states of color and antiquarks \"\" and \"\". This wave equation is form invariant under the \"\" group generalizing the relativistic invariance. It is gauge invariant under the U(1)×SU(2)×SU(3) group of the standard model of quantum physics. The wave is a function of space and time with value in the Clifford algebra Cl1,5. Then many features of the standard model, charge conjugation, color, left waves, and Lagrangian formalism, are obtained in the frame of the first quantization.

Geometry of the Standard Model of Quantum Physics  [PDF]
Claude Daviau, Jacques Bertrand
Journal of Applied Mathematics and Physics (JAMP) , 2015, DOI: 10.4236/jamp.2015.31007
Abstract:

General relativity links gravitation to the structure of our space-time. Nowadays physics knows four types of interactions: Gravitation, electromagnetism, weak interactions, strong interactions. The theory of everything (ToE) is the unification of these four domains. We study several necessary cornerstones for such a theory: geometry and mathematics, adapted manifolds on the real domain, Clifford algebras over tangent spaces of these manifolds, the real Lagrangian density in connection with the standard model of quantum physics. The geometry of the standard model of quantum physics uses three Clifford algebras. The algebra \"\"?of the 3-dimensional physical space is sufficient to describe the wave of the electron. The algebra \"\"of space-time is sufficient to describe the wave of the pair electron-neutrino. A greater space-time with two additional dimensions of space generates the algebra \"\" . It is sufficient to get the wave equation for all fermions, electron, its neutrino and quarks u and d of the first generation, and the wave equations for the two other generations. Values of these waves allow defining, in each point of space-time, geometric transformations from one intrinsic manifold of space-time into the usual manifold. The Lagrangian density is the scalar part of the wave equation.

Left Chiral Solutions for the Hydrogen Atom of the Wave Equation for Electron + Neutrino  [PDF]
Claude Daviau, Jacques Bertrand
Journal of Modern Physics (JMP) , 2015, DOI: 10.4236/jmp.2015.611166
Abstract: The resolution of our wave equation for electron + neutrino is made in the case of the H atom. From two non-classical potentials, we get chiral solutions with the same set of quantum numbers and the same energy levels as those coming from the Dirac equation for the lone electron. These chiral solutions are available for each electronic state in any atom. We discuss the implications of these new potentials.
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