The Markov process of total spins
The martingale problem with sticky reflection conditions, and a system of particles interacting at the boundary
The metastability threshold for modified bootstrap percolation in dimensions.
The ODE method for some self-interacting diffusions on ℝd
The aim of this paper is to study the long-term behavior of a class of self-interacting diffusion processes on ℝd. These are solutions to SDEs with a drift term depending on the actual position of the process and its normalized occupation measure μt. These processes have so far been studied on compact spaces by Benaïm, Ledoux and Raimond, using stochastic approximation methods. We extend these methods to ℝd, assuming a confinement potential satisfying some conditions. These hypotheses on the confinement...
The parabolic Anderson model in a dynamic random environment: Basic properties of the quenched Lyapunov exponent
In this paper we study the parabolic Anderson equation , , , where the -field and the -field are -valued, is the diffusion constant, and is the discrete Laplacian. The -field plays the role of adynamic random environmentthat drives the equation. The initial condition , , is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump...
The parabolic-parabolic Keller-Segel equation
I present in this note recent results on the uniqueness and stability for the parabolic-parabolic Keller-Segel equation on the plane, obtained in collaboration with S. Mischler in [11].
The posterior metric and the goodness of gibbsianness for transforms of Gibbs measures.
The renormalization transformation for two-type branching models
This paper studies countable systems of linearly and hierarchically interacting diffusions taking values in the positive quadrant. These systems arise in population dynamics for two types of individuals migrating between and interacting within colonies. Their large-scale space–time behavior can be studied by means of a renormalization program. This program, which has been carried out successfully in a number of other cases (mostly one-dimensional), is based on the construction and the analysis of...
The scaling limit of senile reinforced random walk.
The shape of a typical boxed plane partition.
The size of a pond in 2D invasion percolation.
The spatial -coalescent.
The spectral gap for a Glauber-type dynamics in a continuous gas
The statistical equilibrium of an isotropic stochastic flow with negative Lyapounov exponents is trivial
The survival probability for critical spread-out oriented percolation above dimensions. II. Expansion
The time constant and critical probabilities in percolation models.
The time constant vanishes only on the percolation cone in directed first passage percolation.
The time for a critical nearest particle system to reach equilibrium starting with a large gap.
The triangle and the open triangle
We show that for percolation on any transitive graph, the triangle condition implies the open triangle condition.
The Virgin Island model.