Search

We obtain sharp asymptotics for the first time a “macroscopic” density fluctuation occurs in a system of independent simple symmetric random walks on Zd. Also, we show the convergence of the moments of the rescaled time by establishing tail estimates.

We consider a system of urns of Polya-type, with balls of two colors; the reinforcement of each urn depends both on the content of the same urn and on the average content of all urns. We show that the urns synchronize almost surely, in the sense that the fraction of balls of a given color converges almost surely, as the time goes to infinity, to the same limit for all urns. A normal approximation for a large number of urns is also obtained.

We investigate solutions to the equation $\partial_t{\cal E} - {\cal D}\Delta {\cal E} = \lambda S^2{\cal E}$, where $S(x,t)$ is a Gaussian stochastic field with covariance $C(x-x',t,t')$, and $x\in {\mathbb R}^d$. It is shown that the coupling $\lambda_{cN}(t)$ at which the $N$-th moment $$ diverges at time $t$, is always less or equal for ${\cal D}>0$ than for ${\cal D}=0$. Equality holds under some reasonable assumptions on $C$ and, in this case, $\lambda_{cN}(t)=N\lambda_c(t)$ where $\lambda_c(t)$ is the value of $\lambda$ at which $$ diverges. The ${\cal D}=0$ case is solved for a class of $S$. The dependence of $\lambda_{cN}(t)$ on $d$ is analyzed. Similar behavior is conjectured when diffusion is replaced by diffraction, ${\cal D}\to i{\cal D}$, the case of interest for backscattering instabilities in laser-plasma interaction., Comment: 19 pages, in LaTeX, e-mail addresses: asselah@gyptis.univ-mrs.fr, daipra@pop2.mate.polimi.it, lebowitz@sakharov.rutgers.edu, Philippe.Mounaix@cpht.polytechnique.fr

Stochastic resetting is simple enough to be approached analytically, yet modifies stochastic processes in a non-trivial way. This modification entails making it possible that the stochastic process has to restart from its initial position at any point in time. In part I of the thesis we study the effect it has on the statistical properties of additive functionals of the Ornstein-Uhlenbeck process and Brownian motion. We are particularly interested in the change of the probabilities of rare events. The Kuramoto model has been used to model synchronization for decades, yet the effect of the underlying structure of the interactions has only recently received attention. In Part II of the thesis we study the effect of community structure in the interaction network analytically in two simple cases, namely, a hierarchical network and a two-community network. The community structure significantly enriches the model in both cases.

We propose a simple stochastic volatility model which is analytically tractable, very easy to simulate and which captures some relevant stylized facts of financial assets, including scaling properties. In particular, the model displays a crossover in the log-return distribution from power-law tails (small time) to a Gaussian behavior (large time), slow decay in the volatility autocorrelation and multiscaling of moments. Despite its few parameters, the model is able to fit several key features of the time series of financial indexes, such as the Dow Jones Industrial Average, with a remarkable accuracy., 32 pages, 5 figures. Substantial revision, following the referee's suggestions. Version to appear in Adv. in Appl. Probab

We study metastability and mixing time for a non-reversible probabilistic cellular automaton. With a suitable choice of the parameters, we first show that the stationary distribution is close in total variation to a low temperature Ising model. Then we prove that both the mixing time and the time to exit a metastable state grow polynomially in the size of the system, while this growth is exponential in reversible dynamics. In this model, non-reversibility, parallel updatings and a suitable choice of boundary conditions combine to produce an efficient dynamical stability.