Abstract : This document contains an overview about the research performed between
1996 and 2005, after the Ph. D. Thesis of the author, and concerns the sharp study of some
stochastic processes : linear or planar Brownian motion, diffusion processes, fractional Brownian
motion, solutions of stochastic differential equations or stochastic partial differential equations.
The thesis contains six chapters each corresponding to one of the following subjects :
study of integrals with respect to local time of some diffusions, large deviations for
a process obtained as a Brownian perturbation of a dynamical system without uniqueness
of solutions, stochastic calculus for the Gaussian non-Markov non-semimartingale
fractional Brownian motion process, study of Itô's and Tanaka's type formulae for the stochastic
heat equation, study of the lifetime of the planar Brownian motion reflected inside of
a domain having an absorbing boundary and finally, non-parametric estimation and
construction of a goodness-of-fit test based on discrete time observations for the diffusion
coefficient of a stochastic differential equation.
The approaches of all these subjects are probabilistic and they are based upon the
stochastic analysis. Some tools of differential equations, partial differential equations
and analysis are also employed.