We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole, so that its asymptotic behavior is expected to depend on the density @math of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities @math except for at most two values @math . The asymptotic speed we obtain in our LLN is a monotone function of @math . Also, @math and @math are characterized as the two points at which the speed may jump to (or from) zero. Furthermore, for all the values of densities where the random walk experiences a non-zero speed, we can prove that it satisfies a functional central limit theorem (CLT). For the special case in which the density is @math and the jump distribution on an empty site and on an occupied site are symmetric to each other, we prove a LLN with zero limiting speed. Finally, we prove similar LLN and CLT results for a different environment, given by a family of independent simple symmetric random walks in equilibrium.
@cite_30: 
@cite_38: Abstract We consider random walks in dynamic random environments given by Markovian dynamics on Z d . We assume that the environment has a stationary distribution μ and satisfies the Poincare inequality w.r.t. μ . The random walk is a perturbation of another random walk (called “unperturbed”). We assume that also the environment viewed from the unperturbed random walk has stationary distribution μ . Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be finite-range. We derive a law of large numbers, an averaged invariance principle for the position of the walker and a series expansion for the asymptotic speed. We also provide a condition for non-degeneracy of the diffusion, and describe in some details equilibrium and convergence properties of the environment seen by the walker. All these results are based on a more general perturbative analysis of operators that we derive in the context of L 2 - bounded perturbations of Markov processes by means of the so-called Dyson–Phillips expansion.
@cite_18: 
@cite_11: 
@cite_37: In this paper we rigorously establish the existence of the mobility coefficient for a tagged particle in a simple symmetric exclusion process with adsorption desorption of particles, in a presence of an external force field interacting with the particle. The proof is obtained using a perturbative argument. In addition, we show that, for a constant external field, the mobility of a particle equals to the self-diffusivity coefficient, the so-called Einstein relation. The method can be applied to any system where the environment has a Markovian evolution with a fast convergence to equilibrium (spectral gap property). In this context we find a necessary relation between forward and backward velocity for the validity of the Einstein relation. This relation is always satisfied by reversible systems. We provide an example of a non-reversible system, where the Einstein relation is valid.
@cite_36: A proof is provided of a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof is based on a coupling argument that traces the space-time cones containing the infection clusters generated by single infections and uses that the random walk eventually gets trapped inside the union of these cones. For the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk eventually gets trapped inside a single cone. This in turn leads to the existence of regeneration times at which the random walk forgets its past. The latter are used to prove a functional central limit theorem and a large deviation principle. The qualitative dependence of the speed, the volatility and the rate function on the infection parameter is investigated, and some open problems are mentioned.
@cite_9: We consider a model of random walk on ℤν, ν≥2, in a dynamical random environment described by a field ξ= ξ t (x): (t,x)∈ℤν+1 . The random walk transition probabilities are taken as P(X t +1= y|X t = x,ξ t =η) =P0( y−x)+ c(y−x;η(x)). We assume that the variables ξ t (x):(t,x) ∈ℤν+1 are i.i.d., that both P0(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P0, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X t , and there is a corresponding random correction of order ( ) to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X t and a corresponding correction of order ( ) to the C.L.T.. Proofs are based on some new L p estimates for a class of functionals of the field.
@cite_29: We prove a law of large numbers for a class of ballistic, multidimensional random walks in random environments where the environment satisfies appropriate mixing conditions, which hold when the environment is a weak mixing field in the sense of Dobrushin and Shlosman. Our result holds if the mixing rate balances moments of some random times depending on the path. It applies in the nonnestling case, but we also provide examples of nestling walks that satisfy our assumptions. The derivation is based on an adaptation, using coupling, of the regeneration argument of Sznitman and Zerner.
@cite_27: 
@cite_20: In this paper we study random walks on dynamical random environments in @math dimensions. Assuming that the environment is invariant under space-time shifts and fulfills a mild mixing hypothesis, we establish a law of large numbers and a concentration inequality around the asymptotic speed. The mixing hypothesis imposes a polynomial decay rate of covariances on the environment with sufficiently high exponent but does not impose uniform mixing. Examples of environments for which our methods apply include the contact process and Markovian environments with a positive spectral gap, such as the East model. For the East model we also obtain that the distinguished zero satisfies a Law of Large Numbers with strictly positive speed.
@cite_17: 
The first works dedicated to the study of RWDRE focused on the case were the underlying medium exhibit fast mixing conditions. A broad range of such conditions have been considered such as: time independence @cite_9 @cite_17 , strong mixing conditions @cite_29 @cite_30 @cite_27 , exponential mixing rate @cite_36 @cite_18 @cite_11 @cite_37 , @cite_38 and fast decay of covariances @cite_20 . In all the above circumstances, one expects the random walk to exhibit Gaussian fluctuations and to satisfy a functional central limit theorem.