On obtient un théorème général concernant la perturbation multiplicative par un opérateur (linéaire borné, mais pas forcément d’inverse borné), du générateur d’un semi-groupe fortement continu sur un espace de Banach. On en déduit un résultat intimement lié au changement de temps dans les processus de Markov, qui étend un théorème de Dorroh (et résout par l’affirmative la seule situation qui restait en doute dans le contexte du théorème de Dorroh cité). Comme exemple d’autres possibilités d’application,...

In analogy to the analyticity condition $\parallel A{e}^{tA}\parallel \le C{t}^{-1}$, t > 0, for a continuous time semigroup ${\left(e^{tA}\right)}_{t\ge 0}$, a bounded operator T is called analytic if the discrete time semigroup ${\left(T^n\right)}_{n\in \mathbb{N}}$ satisfies $\parallel (T-I)T^n\parallel \le C{n}^{-1}$, n ∈ ℕ. We generalize O. Nevanlinna’s characterization of powerbounded and analytic operators T to the following perturbation result: if S is a perturbation of T such that $\parallel R({\lambda }_0,T)-R({\lambda }_0,S)\parallel$ is small enough for some ${\lambda }_0\in \varrho \left(T\right)\cap \varrho \left(S\right)$, then the type $\omega$ of the semigroup $\left(e^{t(S-I)}\right)$ also controls the analyticity of S in the sense that $\parallel (S-I)S^n\parallel \le C(\omega +n^{-1})e^{\omega n}$, n ∈ ℕ. As an application we generalize...

Lors de cet exposé, nous nous intéressons à l’étude de perturbations stochastiques de certaines inclusions différentielles du premier ordre  : les processus de rafle par des ensembles uniformément prox-réguliers. Ce travail nous amène à combiner la théorie des processus de rafle et celle traitant de la reflexion d’un mouvement brownien sur la frontière d’un ensemble. Nous donnerons des résultats traitant du caractère bien-posé de ces inclusions différentielles stochastiques et de leur stabilité.

We study linear rough differential equations and we solve perturbed linear rough differential equations using the Duhamel principle. These results provide us with a key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach algebras more general than tensor algebras and to consider extensions of classical results such as the Magnus...

We study a class of rotation invariant determinantal ensembles in the complex plane; examples include the eigenvalues of Gaussian random matrices and the roots of certain families of random polynomials. The main result is a criterion for a central limit theorem to hold for angular statistics of the points. The proof exploits an exact formula relating the generating function of such statistics to the determinant of a perturbed Toeplitz matrix.

We consider the hexagonal circle packing with radius $1/2$ and perturb it by letting the circles move as independent Brownian motions for time $t$. It is shown that, for large enough $t$, if ${𝛱}_t$ is the point process given by the center of the circles at time $t$, then, as $t\to \infty$, the critical radius for circles centered at ${𝛱}_t$ to contain an infinite component converges to that of continuum percolation (which was shown – based on a Monte Carlo estimate – by Balister, Bollobás and Walters to be strictly bigger than...

We consider a left-transient random walk in a random environment on $\mathbb{Z}$ that will be disturbed by cookies inducing a drift to the right of strength 1. The number of cookies per site is i.i.d. and independent of the environment. Criteria for recurrence and transience of the random walk are obtained. For this purpose we use subcritical branching processes in random environments with immigration and formulate criteria for recurrence and transience for these processes.