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.

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.

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.

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.

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.

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.

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
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 Cl_{1,5}. Then many features of the standard model, charge
conjugation, color, left waves, and Lagrangian formalism, are obtained in the
frame of the first quantization.

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.

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.