Persistent Random Walks in Stationary Environment

We study the behavior of persistent random walks (RW) on the integers in a random environment. A complete characterization of the almost sure limit behavior of these processes, including the law of large numbers, is obtained. This is done in a general situation where the environmental sequence of random variables is stationary and ergodic. Szász and Tóth obtained a central limit theorem when the ratio /, of right- and left-transpassing probabilities satisfies /a<1 a.s. (for a given constant a). We consider the case where / has wider fluctuations; we shall observe that an unusual situation arises: the RW may converge a.s. to infinity even with zero drift. Then, we obtain nonclassical limiting distributions for the RW. Proofs are based on the introduction of suitable branching processes in order to count the steps performed by the RW.     

Space and time inversions of stochastic processes and Kelvin transform

Let X be a standard Markov process. We prove that a space inversion property of X implies the existence of a Kelvin transform of X‐harmonic, excessive and operator‐harmonic functions and that the inversion property is inherited by Doob h‐transforms. We determine new classes of processes having space inversion properties amongst transient processes satisfying the time inversion property. For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly. We treat in details the examples of free scaled power Bessel processes, non‐colliding Bessel particles, Wishart processes, Gaussian Ensemble and Dyson Brownian Motion.     

Boundary-Crossing Identities for Diffusions Having the Time-Inversion Property

We review and study a one-parameter family of functional transformations, denoted by (S ^(β)) _β∈ℝ, which, in the case β<0, provides a path realization of bridges associated to the family of diffusion processes enjoying the time-inversion property. This family includes Brownian motions, Bessel processes with a positive dimension and their conservative h-transforms. By means of these transformations, we derive an explicit and simple expression which relates the law of the boundary-crossing times for these diffusions over a given function f to those over the image of f by the mapping S ^(β), for some fixed β∈ℝ. We give some new examples of boundary-crossing problems for the Brownian motion and the family of Bessel processes. We also provide, in the Brownian case, an interpretation of the results obtained by the standard method of images and establish connections between the exact asymptotics for large time of the densities corresponding to various curves of each family.     

On a Fluctuation Identity for Random Walks and Levy Processes

In this paper, some identities in laws involving ladder processesfor random walks and Lévy processes are extended andunified. 2000 Mathematics Subject Classification 60G50, 60G51(primary), 60G17 (secondary).     

Interplay Between Steric and Electronic Effects: A Joint Spectroscopy and Computational Study of Nonheme Iron(IV)-Oxo Complexes

Iron is an essential element in nonheme enzymes that plays a crucial role in many vital oxidative transformations and metabolic reactions in the human body. Many of those reactions are regio- and stereospecific and it is believed that the selectivity is guided by second-coordination sphere effects in the protein. Here, results are shown of a few engineered biomimetic ligand frameworks based on the N4Py (N,N-bis(2-pyridylmethyl)-N-bis(2-pyridyl)methylamine) scaffold and the second-coordination sphere effects are studied. For the first time, selective substitutions in the ligand framework have been shown to tune the catalytic properties of the iron(IV)-oxo complexes by regulating the steric and electronic factors. In particular, a better positioning of the oxidant and substrate in the rate-determining transition state lowers the reaction barriers. Therefore, an optimum balance between steric and electronic factors mediates the ideal positioning of oxidant and substrate in the rate-determining transition state that affects the reactivity of high-valent reaction intermediates.     

On the first crossing times of a Brownian motion and a family of continuous curves

We review the analytic transformations allowing to construct standard Brownian bridges from a Brownian motion. These are generalized and some of their properties are studied. The new family maps the space of continuous positive functions into a family of curves which is the topic of our study. We establish a simple and explicit formula relating the distributions of the first hitting times of each of these curves by a standard Brownian motion. To cite this article: L. Alili, P. Patie, C. R. Acad. Sci. Paris, Ser. I 340 (2005).