Pierre Andreoletti

Bio: Pierre Andreoletti is an academic researcher from University of Orléans. The author has contributed to research in topics: Random walk & Heterogeneous random walk in one dimension. The author has an hindex of 10, co-authored 29 publications receiving 200 citations. Previous affiliations of Pierre Andreoletti include Aix-Marseille University & University of Chile.

The number of generations entirely visited for recurrent random walks on random environment

Abstract: In this paper we deal with a random walk in a random environment on a super-critical Galton–Watson tree. We focus on the recurrent cases already studied by Hu and Shi (Ann. Probab. 35:1978–1997, 2007; Probab. Theory Relat. Fields 138:521–549, 2007), Faraud et al. (Probab. Theory Relat. Fields, 2011, in press), and Faraud (Electron. J. Probab. 16(6):174–215, 2011). We prove that the largest generation entirely visited by these walks behaves like logn, and that the constant of normalization, which differs from one case to another, is a function of the inverse of the constant of Biggins’ law of large numbers for branching random walks (Biggins in Adv. Appl. Probab. 8:446–459, 1976).

On the estimation of the potential of Sinai’s RWRE

Abstract: We consider a one-dimensional random walk in random environment. We prove that the logarithm of the local time can be used as an estimator of the random environment. We give a constructive method allowing us to locally built, up to a translation, the random potential associated to the environment from a single trajectory of the random walk.

Spread of visited sites of a random walk along the generations of a branching process

Abstract: In this paper we consider a null recurrent random walk in random environment on a super-critical Galton-Watson tree. We consider the case where the log-Laplace transform $\psi$ of the branching process satisfies $\psi(1)=\psi'(1)=0$ for which G. Faraud, Y. Hu and Z. Shi have shown that, with probability one, the largest generation visited by the walk, until the instant $n$, is of the order of $(\log n)^3$. We already proved that the largest generation entirely visited behaves almost surely like $\log n$ up to a constant. Here we study how the walk visits the generations $\ell=(\log n)^{1+ \zeta}$, with $0 < \zeta <2$. We obtain results in probability giving the asymptotic logarithmic behavior of the number of visited sites at a given generation. We prove that there is a phase transition at generation $(\log n)^2$ for the mean of visited sites until $n$ returns to the root. Also we show that the visited sites spread all over the tree until generation $\ell$.

Limit law of the local time for Brox's diffusion

Abstract: We consider Brox's model: a one-dimensional diffusion in a Brownian potential W. We show that the normalized local time process (L(t;m_(log t) + x)=t; x \in R), where m_(log t) is the bottom of the deepest valley reached by the process before time t, behaves asymptotically like a process which only depends on W. As a consequence, we get the weak convergence of the local time to a functional of two independent three-dimensional Bessel processes and thus the limit law of the supremum of the normalized local time. These results are discussed and compared to the discrete time and space case which same questions have been solved recently by N. Gantert, Y. Peres and Z. Shi.

Branching random walks

Abstract: I Introduction.- II Galton-Watson trees.- III Branching random walks and martingales.- IV The spinal decomposition theorem.- V Applications of the spinal decomposition theorem.- VI Branching random walks with selection.- VII Biased random walks on Galton-Watson trees.- A Sums of i.i.d. random variables.- References.

Rough paths and 1d SDE with a time dependent distributional drift: application to polymers

Abstract: Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional stochastic differential equations, the drift of which is a distribution, by means of rough paths theory. Existence and uniqueness are established in the weak sense when the drift reads as the derivative of a Holder continuous function. Regularity of the drift part is investigated carefully and a related stochastic calculus is also proposed, which makes the structure of the solutions more explicit than within the earlier framework of Dirichlet processes.

Force-induced denaturation of RNA

Abstract: We quantitatively describe an RNA molecule under the influence of an external force exerted at its two ends as in a typical single-molecule experiment. Our calculation incorporates the interactions between nucleotides by using the experimentally determined free energy rules for RNA secondary structure and models the polymeric properties of the exterior single-stranded regions explicitly as elastic freely jointed chains. We find that despite complicated secondary structures, force-extension curves are typically smooth in quasi-equilibrium. We identify and characterize two sequence/structure-dependent mechanisms that, in addition to the sequence-independent entropic elasticity of the exterior single-stranded regions, are responsible for the smoothness. These involve compensation between different structural elements on which the external force acts simultaneously and contribution of suboptimal structures, respectively. We estimate how many features a force-extension curve recorded in nonequilibrium, where the pulling proceeds faster than rearrangements in the secondary structure of the molecule, could show in principle. Our software is available to the public through an "RNA-pulling server."

Rough paths and 1d sde with a time dependent distributional drift. Application to polymers

Abstract: Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional stochastic differential equations, the drift of which is a distribution, by means of rough paths theory. Existence and uniqueness are established in the weak sense when the drift reads as the derivative of a H{\"o}lder continuous function. Regularity of the drift part is investigated carefully and a related stochastic calculus is also proposed, which makes the structure of the solutions more explicit than within the earlier framework of Dirichlet processes.