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 provide a necessary and sufficient condition for the metastability of a Markov chain, expressed in terms of a property of the solutions of the resolvent equation. As an application of this result, we prove the metastability of reversible, critical
zero-range processes starting from a configuration.
We examine in this article the one-dimensional, non-local, singular SPDE begin{equation*} partial_t u ;=; -, (-Delta)^{1/2} u ,-, sinh(gamma u) ,+, xi;, end{equation*} where $gammain mathbb{R}$, $(-Delta)^{1/2}$ is the fractional Laplacian of order $
1/2$, $xi$ the space-time white noise in $mathbb{R} times mathbb{T}$, and $mathbb{T}$ the one-dimensional torus. We show that for $0<gamma^2<pi/7$ the Da Prato--Debussche method applies. One of the main difficulties lies in the derivation of a Schauder estimate for the semi-group associated to the fractional Laplacian due to the lack of smoothness resulting from the long range interaction.
We consider weakly asymmetric exclusion processes whose initial density profile is a small perturbation of a constant. We show that in the diffusive time-scale, in all dimensions, the density defect evolves as the solution of a viscous Burgers equation.
We review recent results on the metastable behavior of continuous-time Markov chains derived through the characterization of Markov chains as unique solutions of martingale problems.
Let $mathbb{T}^d_N$, $dge 2$, be the discrete $d$-dimensional torus with $N^d$ points. Place a particle at each site of $mathbb{T}^d_N$ and let them evolve as independent, nearest-neighbor, symmetric, continuous-time random walks. Each time two parti
cles meet, they coalesce into one. Denote by $C_N$ the first time the set of particles is reduced to a singleton. Cox [6] proved the existence of a time-scale $theta_N$ for which $C_N/theta_N$ converges to the sum of independent exponential random variables. Denote by $Z^N_t$ the total number of particles at time $t$. We prove that the sequence of Markov chains $(Z^N_{ttheta_N})_{tge 0}$ converges to the total number of partitions in Kingmans coalescent.
We review recent progress in potential theory of second-order elliptic operators and on the metastable behavior of Markov processes.
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.
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.
We present two variational formulae for the capacity in the context of non-selfadjoint elliptic operators. The minimizers of these variational problems are expressed as solutions of boundary-value elliptic equations. We use these principles to provid
e a sharp estimate for the transition times between two different wells for non-reversible diffusion processes. This estimate permits to describe the metastable behavior of the system.
