We prove a law of large numbers for the empirical density of one-dimensional, boundary driven, symmetric exclusion processes with different types of non-reversible dynamics at the boundary. The proofs rely on duality techniques.
A fractional Ficks law and fractional hydrostatics for the one dimensional exclusion process with long jumps in contact with infinite reservoirs at different densities on the left and on the right are derived.
We consider a one-dimensional, weakly asymmetric, boundary driven exclusion process on the interval $[0,N]cap Z$ in the super-diffusive time scale $N^2 epsilon^{-1}_N$, where $1ll epsilon^{-1}_N ll N^{1/4}$. We assume that the external field and the
chemical potentials, which fix the density at the boundaries, evolve smoothly in the macroscopic time scale. We derive an equation which describes the evolution of the density up to the order $epsilon_N$.
We consider small perturbations of a dynamical system on the one-dimensional torus. We derive sharp estimates for the pre-factor of the stationary state, we examine the asymptotic behavior of the solutions of the Hamilton-Jacobi equation for the pre-
factor, we compute the capacities between disjoint sets, and we prove the metastable behavior of the process among the deepest wells following the martingale approach. We also present a bound for the probability that a Markov process hits a set before some fixed time in terms of the capacity of an enlarged process.
Consider the symmetric exclusion process evolving on an interval and weakly interacting at the end-points with reservoirs. Denote by $I_{[0,T]} (cdot)$ its dynamical large deviations functional and by $V(cdot)$ the associated quasi-potential, defined
as $V(gamma) = inf_{T>0} inf_u I_{[0,T]} (u)$, where the infimum is carried over all trajectories $u$ such that $u(0) = barrho$, $u(T) = gamma$, and $barrho$ is the stationary density profile. We derive the partial differential equation which describes the evolution of the optimal trajectory, and deduce from this result the formula obtained by Derrida, Hirschberg and Sadhu cite{DHS2021} for the quasi-potential through the representation of the steady state as a product of matrices.
We describe how to analyze the wide class of non stationary processes with stationary centered increments using Shannon information theory. To do so, we use a practical viewpoint and define ersatz quantities from time-averaged probability distributions. These ersa