On the genealogy of conditioned stable Lévy forests

A Lévy forest of size s > 0 is a Poisson point process in the set of Lévytrees which is defined on the time interval [0, s]. The total mass of this forest is defined by the sum of the masses of all its trees. We give a realization of the stableLévy forest of a given size conditioned on its total mass using the path of the unconditioned forest. Then, we prove an invariance principle for this conditioned forest by considering k independent Galton-Watson trees whose offspring distribution is in the domain of attraction of any stable law conditioned on their total progeny to be equal to n. We prove that when n and k tend towards +∞, under suitable rescaling, the associated coding random walk, the contour and height processes all converge in law on the Skorokhod space towards the first passage bridge and height process of a stable Lévy process with no negative jumps respectively

Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, Via Conditioning

Derived from extensive teaching experience in Paris, this second edition now includes over 100 exercises in probability. New exercises have been added to reflect important areas of current research in probability theory, including infinite divisibility of stochastic processes, past-future martingales and fluctuation theory. For each exercise the authors provide detailed solutions as well as references for preliminary and further reading. There are also many insightful notes to motivate the student and set the exercises in context. Students will find these exercises extremely useful for easing the transition between simple and complex probabilistic frameworks. Indeed, many of the exercises here will lead the student on to frontier research topics in probability. Along the way, attention is drawn to a number of traps into which students of probability often fall. This book is ideal for independent study or as the companion to a course in advanced probability theory

Reflection principle and Ocone martingales

Let $M =(M_t)_{t\geq 0}$ be any continuous real-valued stochastic process. We prove that if there exists a sequence $(a_n)_{n\geq 1}$ of real numbers which converges to 0 and such that $M$ satisfies the reflection property at all levels $a_n$ and $2a_n$ with $n\geq 1$, then $M$ is an Ocone local martingale with respect to its natural filtration. We state the subsequent open question: is this result still true when the property only holds at levels $a_n$? Then we prove that the later question is equivalent to the fact that for Brownian motion, the $\sigma$-field of the invariant events by all reflections at levels $a_n$, $n\ge1$ is trivial. We establish similar results for skip free $\mathbb{Z}$-valued processes and use them for the proof in continuous time, via a discretisation in space

On a Fluctuation Identity for Random Walks and Lévy Processes

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

On the Hausdorff dimension of regular points of inviscid Burgers equation with stable initial data

Consider an inviscid Burgers equation whose initial data is a Levy a-stable process Z with a > 1. We show that when Z has positive jumps, the Hausdorff dimension of the set of Lagrangian regular points associated with the equation is strictly smaller than 1/a, as soon as a is close to 1. This gives a negative answer to a conjecture of Janicki and Woyczynski. Along the way, we contradict a recent conjecture of Z. Shi about the lower tails of integrated stable processes

Optical determination and identification of organic shells around nanoparticles: application to silver nanoparticles

We present a simple method to prove the presence of an organic shell around silver nanoparticles. This method is based on the comparison between optical extinction measurements of isolated nanoparticles and Mie calculations predicting the expected wavelength of the Localized Surface Plasmon Resonance of the nanoparticles with and without the presence of an organic layer. This method was applied to silver nanoparticles which seemed to be well protected from oxidation. Further experimental characterization via Surface Enhanced Raman Spectroscopy (SERS) measurements allowed to identify this protective shell as ethylene glycol. Combining LSPR and SERS measurements could thus give proof of both presence and identification for other plasmonic nanoparticles surrounded by organic shells

AMORPHOUS PD-SI ALLOYS AND HYDRIDES PREPARED BY LOW-TEMPERATURE ION-IMPLANTATION

Ion implantation simultaneously produces compositional changes and radiation damage in the target. If the latter is not annealed, amorphization should ultimately result. Can implantation of a covalent solute into a transition metal host stabilize the damage and hence produce an amorphous alloy at lower concentrations than other techniques ? We have studied the composition-dependence of the resistivity and TCR of thin (600-800 Å) Pd films implanted at 6 K with Si ions : The results are compared to those obtained on the corresponding well-documented quench-condensed alloys, which are amorphous at Si concentrations ~.18. The resistivity of the implanted films saturates at about 90 µΩ·cm for Si concentrations above ~.18. Thus, the critical concentration for amorphization is presumably the same for the low-temperature implanted or quench-condensed Pd-Si alloy, confirming that local structure effects dominate amorphous alloy formation criteria. In a further experiment, hydrogen was implanted into the amorphous Pd-Si films (again at 6K). The resistivity increased sharply, doubling at H concentrations around 100 %. The resulting systems were superconducting ; their maximum critical temperature was 2.6 K

Invariance principles for local times at the maximum of random walks and Lévy processes

We prove that when a sequence of Lévy processes X(n) or a normed sequence of random walks S(n) converges a.s. on the Skorokhod space toward a Lévy process X, the sequence L(n) of local times at the supremum of X(n) converges uniformly on compact sets in probability toward the local time at the supremum of X. A consequence of this result is that the sequence of (quadrivariate) ladder processes (both ascending and descending) converges jointly in law toward the ladder processes of X. As an application, we show that in general, the sequence S(n) conditioned to stay positive converges weakly, jointly with its local time at the future minimum, toward the corresponding functional for the limiting process X. From this we deduce an invariance principle for the meander which extends known results for the case of attraction to a stable law

Self-similar processes and their applications

This volume contains some articles related to the conferenceSelf-similar processes and their applications which took place in Angers, from the 20th to the 24th of July 2009. Self-similarity is the property which certain stochastic processes have of preserving their distribution under a time-scale change. This property appears in all areas of probability theory and offers a number of fields of application. The aim of this conference is to bring together the main representatives of different aspects of self-similarity currently being studied in order to promote exchanges on their recent research and enable them to share their knowledge with young researchers. Self-similar Markov processes. Matrix valued self-similar processes. Self-similarity, trees, branching and fragmentation. Fractional and multifractional processes Stochastic Löwner evolution Selfsimilarity in financ