Emmanuel Jacob's Web Page
Research interests
Langevin process, excursions, Itô excursion theory, reflecting boundary, stationary processes, random walks, renewal theory on the whole line, stochastic partial differential equations, change of probability (Doob's h-transforms), superprocesses.
Publications
Excursions of the integral of the Brownian motion
Annales de l’Institut Henri Poincaré, Volume 46, Number 3, pages 869-887
;
arXiv link
Preprints
Langevin process reflected on a partially elastic boundary I
arXiv link
Langevin process reflected on a partially elastic boundary II
soon available
Thesis
Processus de Langevin réfléchis au second ordre
Second order reflected Langevin processes
TEL link
Summary
Résumé
This thesis proposes an encounter between a stochastic object, the Langevin process,
that is the integral of the Brownian motion, and a differential equation, the "second order
reflection", which, to my knowledge, has been studied until now almost exclusively in a
deterministic framework.
Historically, the Langevin process was a competing model of the Brownian motion for
the description of the erratic trajectories of particles such as the ones observed by Brown.
In the same way, the second order re°ected Langevin processes are a competing model
of reflected Brownian motions, which should always be understood as first order reflected
Brownian motions, according to our terminology.
The Langevin process and the second order reflected Langevin processes do not pretend
to vie with the Brownian motion and the reflected Brownian motion for their influence and
their range of applications in various domains. They still pretend to be a more relevant
physical model.
Besides, for the deterministic second order reflection, when the force has a strongly
oscillating behavior, the differential equation has generically several solutions. When a
Langevin process is reflected, we should consider the stochastic differential equation, when
the force is a white noise... We will show nonetheless the existence of a unique solution,
in a weak sense. This result is in sharp contrast with the non-uniqueness results in the
deterministic case.
This thesis has four chapters. The first one is a long introduction to the subject, written
in French in a discursive style. The other three ones are published or still unpublished
articles I wrote during my thesis, written in English.
In the first chapter I start with the description of the old and recent historical context
motivating the study. On the one hand I introduce second order reflection. On the other
hand I introduce the Langevin process and its excursions, recalling some results that we
will use later on.
Then I give an overview of various notions and tools that we will need. I mean, firstly,
in addition to the famous Itô excursion measure of a Markov process, the Pitman excursion
measure of a stationary process. I mean, next, the h-transforms principle, in the sense of
Doob, used to define conditioned Markov processes. Finally I give a detailed abstract, in
French, of the three following chapters.
The second chapter contains, first, an introduction to the stationary Langevin process,
then a study of its Pitman excursion measure. This work is applied to the study of the
Langevin process re°ected on a totally inelastic boundary.
The third chapter starts the study of the Langevin process reflected on a partially
elastic boundary. We highlight the existence of two clearly distinct regimes, according
to the value of the elasticity coe±cient of the reflection, compared to the critical value
0,163. In the supercritical and critical regimes, the main difficulty is
related to the case when the process starts from 0 with zero speed. We show that the
process stays uniquely defined.
The fourth chapter deals with the more difficult subcritical regime. In particular,
whatever the initial condition, after a finite time the process will be at zero with zero
speed. We still show the existence of a unique reflected process, described this time via its
Itô excursion measure.
Thesis carried out under the supervision of Jean Bertoin and defended on December 10, 2010
Other
Master thesis (under the supervision of Edwin Perkins):
Propriétés de connexité du support du superbrownien: Une nouvelle approche par des
outils élémentaires
