Approximation du temps local des surfaces gaussiennes

Summary LetX be a centered stationary Gaussian stochastic process with ad-dimensional parameter (d2),F its spectral measure, (x denotes the Euclidean norm ofx). We consider regularizations of the trajectories ofX by means of convolutions of the formX (t)=( *X)(t) where stands for an approximation of unity (as tends to zero) satisfying certain regularity conditions.The aim of this paper is to recover the local time ofX at a given levelu, as a limit of appropriate normalizations of the geometric measure of theu-level set of the regular approximating processesX . A part of the difficulties comes from the fact that the geometric behavior of the covariance of the Gaussian processX can be a complex one as approaches O.The results are onL ²-convergence and include bounds for the speed of convergence.L ^presults may be obtained in similar ways, but almost sure convergence or simultaneous convergence for the various values ofu do not seem to follow from our methods. In Sect. 3 we have included examples showing a diversity of geometric behaviors, especially in what concerns the dependence on the thickness of the set in which the covariance of the original processX is irregular.Some technical results of analytic nature are included as appendices in Sect. 4.     

Nonperturbative renormalization-group study of reaction-diffusion processes

We generalize nonperturbative renormalization group methods to nonequilibrium critical phenomena. Within this formalism, reaction-diffusion processes are described by a scale-dependent effective action, the flow of which is derived. We investigate branching and annihilating random walks with an odd number of offspring. Along with recovering their universal physics (described by the directed percolation universality class), we determine their phase diagrams and predict that a transition occurs even in three dimensions, contrarily to what perturbation theory suggests.     

On Tightness and Weak Convergence in the Approximation of the Occupation Measure of Fractional Brownian Motion

In this paper we consider approximations of the occupation measure of the Fractional Brownian motion by means of some functionals defined on regularizations of the paths. In a previous article Berzin and León proved a cylindrical convergence to a Wiener process of conveniently rescaled functionals. Here we show the tightness of the approximation in the space of continuous functions endowed with the topology of uniform convergence on compact sets. This allows us to simplify the identification of the limit.     

On the sequence of partial maxima of some random sequences

Let {X_n, n ? 1} be a sequence of identically distributed random variables, Z_n = max {X₁,…, X_n} and {u_n, n ? 1 } an increasing sequence of real numbers. Under certain additional requirements, necessary and sufficient conditions are given to have, with probability one, an infinite number of crossings of {Z_n} with respect to {u_n}, in two cases: (1) The X_n's are independent, (2) {X_n} is stationary Gaussian and satisfies a mixing condition.     

A new method to solve the non-perturbative renormalization group equations

We propose a method to solve the non-perturbative renormalization group equations for the n-point functions. In leading order, it consists in solving the equations obtained by closing the infinite hierarchy of equations for the n-point functions. This is achieved: (i) by exploiting the decoupling of modes and the analyticity of the n  -point functions at small momenta: this allows us to neglect some momentum dependence of the vertices entering the flow equations; (ii) by relating vertices at zero momenta to derivatives of lower order vertices with respect to a constant background field. Although the approximation is not controlled by a small parameter, its accuracy can be systematically improved. When it is applied to the O(N)$O(N)$ model, its leading order is exact in the large-N limit; in this case, one recovers known results in a simple and direct way, i.e., without introducing an auxiliary field.     

Non-perturbative renormalization group calculation of the scalar self-energy

We present the first numerical application of a method that we have recently proposed to solve the Non Perturbative Renormalization Group equations and obtain the n-point functions for arbitrary external momenta. This method leads to flow equations for the n-point functions which are also differential equations with respect to a constant background field. This makes them, a priori, difficult to solve. However, we demonstrate in this paper that, within a simple approximation which turns out to be quite accurate, the solution of these flow equations is not more complicated than that of the flow equations obtained in the derivative expansion. Thus, with a numerical effort comparable to that involved in the derivative expansion, we can get the full momentum dependence of the n-point functions. The method is applied, in its leading order, to the calculation of the self-energy in a 3-dimensional scalar field theory, at criticality. Accurate results are obtained over the entire range of momenta.     

Approximation of the occupation measure of Lévy processes

Consider a real-valued Lévy process with non-zero Gaussian component and jumps with locally finite variation. We obtain an invariance principle theorem for the speed of approximation of its occupation measure by means of functionals defined on regularizations of the paths. This theorem is a first extension to processes with jumps of previous results for semimartingales with continuous paths. To cite this article: E. Mordecki, M. Wschebor, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail

We study the probability distribution F(u)$F\left(u\right)$ of the maximum of smooth Gaussian fields defined on compact subsets of R^d${\mathbb{R}}^d$ having some geometric regularity.     

Calculation of the pressure of a hot scalar theory within the Non-Perturbative Renormalization Group

We apply to the calculation of the pressure of a hot scalar field theory a method that has been recently developed to solve the Non-Perturbative Renormalization Group. This method yields an accurate determination of the momentum dependence of n  -point functions over the entire momentum range, from the low momentum, possibly critical, region up to the perturbative, high momentum region. It has therefore the potential to account well for the contributions of modes of all wavelengths to the thermodynamical functions, as well as for the effects of the mixing of quasiparticles with multi-particle states. We compare the thermodynamical functions obtained with this method to those of the so-called Local Potential Approximation, and we find extremely small corrections. This result points to the robustness of the quasiparticle picture in this system. It also demonstrates the stability of the overall approximation scheme, and this up to the largest values of the coupling constant that can be used in a scalar theory in 3+1$3+1$ dimensions. This is in sharp contrast to perturbation theory which shows no sign of convergence, up to the highest orders that have been recently calculated.