Multi-scaling of moments in stochastic volatility models

We introduce a class of stochastic volatility models $(X_t)_{t \geq 0}$ for which the absolute moments of the increments exhibit anomalous scaling: \E\left(|X_{t+h} - X_t|^q \right) scales as $h^{q/2}$ for $q < q^*$, but as $h^{A(q)}$ with $A(q) q^*$, for some threshold $q^*$. This multi-scaling phenomenon is observed in time series of financial assets. If the dynamics of the volatility is given by a mean-reverting equation driven by a Levy subordinator and the characteristic measure of the Levy process has power law tails, then multi-scaling occurs if and only if the mean reversion is superlinear

The role of disorder in the dynamics of critical fluctuations of mean field models

The purpose of this paper is to analyze how the disorder affects the dynamics of critical fluctuations for two different types of interacting particle system: the Curie-Weiss and Kuramoto model. The models under consideration are a collection of spins and rotators respectively. They both are subject to a mean field interaction and embedded in a site-dependent, i.i.d. random environment. As the number of particles goes to infinity their limiting dynamics become deterministic and exhibit phase transition. The main result concern the fluctuations around this deterministic limit at the critical point in the thermodynamic limit. From a qualitative point of view, it indicates that when disorder is added spin and rotator systems belong to two different classes of universality, which is not the case for the homogeneous models (i.e., without disorder).Comment: 41 page

Logarithmic Sobolev Inequality for Zero-Range Dynamics: independence of the number of particles

We prove that the logarithmic-Sobolev constant for Zero-Range Processes in a box of diameter L may depend on L but not on the number of particles. This is a first, but relevant and quite technical step, in the proof that this logarithmic-Sobolev constant grows as L^2, that will be presented in a forthcoming paper

Logarithmic Sobolev inequality for zero-range Dynamics

We prove that the logarithmic Sobolev constant for zero-range processes in a box of diameter $L$ grows as $L^2$.Comment: Published at http://dx.doi.org/10.1214/009117905000000332 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Hitting times for special patterns in the symmetric exclusion process on Z^d

We consider the symmetric exclusion process {\eta_t,t>0} on {0,1}^{Z^d}. We fix a pattern A:={\eta:\sum_{\Lambda}\eta(i)\ge k}, where \Lambda is a finite subset of Z^d and k is an integer, and we consider the problem of establishing sharp estimates for \tau, the hitting time of A. We present a novel argument based on monotonicity which helps in some cases to obtain sharp tail asymptotics for \tau in a simple way. Also, we characterize the trajectories {\eta_s,s\le t} conditioned on {\tau>t}.Comment: Published at http://dx.doi.org/10.1214/009117904000000487 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The dynamics of critical fluctuations in asymmetric Curie-Weiss models

We study the dynamics of fluctuations at the critical point for two time-asymmetric version of the Curie-Weiss model for spin systems that, in the macroscopic limit, undergo a Hopf bifurcation. The fluctuations around the macroscopic limit reflect the type of bifurcation, as they exhibit observables whose fluctuations evolve at different time scales. The limiting dynamics of fluctuations of slow observable is obtained via an averaging principle.Comment: 27 page

McKean-Vlasov limit for interacting systems with simultaneous jumps

Motivated by several applications, including neuronal models, we consider the McKean-Vlasov limit for mean-field systems of interacting diffusions with simultaneous jumps. We prove propagation of chaos via a coupling technique that involves an intermediate process and that gives a rate of convergence for the $W_1$ Wasserstein distance between the empirical measures of the two systems on the space of trajectories $\mathbf{D}([0,T],\mathbb{R}^d)$

Fast Mixing for the Low Temperature 2D Ising Model Through Irreversible Parallel Dynamics

We study tunneling 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

Collective periodicity in mean-field models of cooperative behavior

We propose a way to break symmetry in stochastic dynamics by introducing a dissipation term. We show in a specific mean-field model, that if the reversible model undergoes a phase transition of ferromagnetic type, then its dissipative counterpart exhibits periodic orbits in the thermodynamic limit.Comment: 19 pages, 3 figure