Adriana Neumann

Phase transition of a heat equation with Robin’s boundary conditions and exclusion process

For a heat equation with Robin’s boundary conditions which depends on a parameter α > 0, we prove that its unique weak solution ρα converges, when α goes to zero or to infinity, to the unique weak solution of the heat equation with Neumann’s boundary conditions or the heat equation with periodic boundary conditions, respectively. To this end, we use uniform bounds on a Sobolev norm of ρα obtained from the hydrodynamic limit of the symmetric slowed exclusion process, plus a careful analysis of boundary terms.

Phase transition in equilibrium fluctuations of symmetric slowed exclusion

We analyze the equilibrium fluctuations of density, current and tagged particle in symmetric exclusion with a slow bond. The system evolves in the one-dimensional lattice and the jump rate is everywhere equal to one except at the slow bond where it is αn−β, with α>0, β∈[0,+∞] and n is the scaling parameter. Depending on the regime of β, we find three different behaviors for the limiting fluctuations whose covariances are explicitly computed. In particular, for the critical value β=1, starting a tagged particle near the slow bond, we obtain a family of Gaussian processes indexed in α, interpolating a fractional Brownian motion of Hurst exponent 1/4 and the degenerate process equal to zero.

Hydrodynamical behavior of symmetric exclusion with slow bonds

We consider the exclusion process in the one-dimensional discrete torus with

Exclusion Process with Slow Boundary

N

A Thermodynamic Formalism for Continuous Time Markov Chains with Values on the Bernoulli Space: Entropy, Pressure and Large Deviations

points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance

Large deviations for the exclusion process with a slow bond

N^{-\beta}

Hydrodynamic Limit of Quantum Random Walks

, with

Occupation time of exclusion processes with conductances

\beta\in[0,\infty)

Equilibrium Fluctuations for the Slow Boundary Exclusion Process

. We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter

Large Deviations for Stationary Probabilities of a Family of Continuous Time Markov Chains via Aubry–Mather Theory

\beta