In this paper we consider a class of one-dimensional interacting particle systems in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied/vacant sites has a local drift to the right/left. We adapt a regeneration-time argument originally developed by Comets and Zeitouni [2] for static random environments to prove that, under a space-time mixing property for the dynamic random environment called conemixing, the random walk has an a.s. constant global speed. In addition, we show that if the dynamic random environment is exponentially mixing in space-time and the local drifts are small, then the global speed can be written as a power series in the size of the local drifts. From the first term in this series the sign of the global speed can be read off.The results can be easily extended to higher dimensions.Acknowledgment. The authors are grateful to R. dos Santos and V. Sidoravicius for fruitful discussions. MSC 2000. Primary 60H25, 82C44; Secondary 60F10, 35B40.

We use random spanning forests to find, for any Markov process on a finite set of size n and any positive integer m ≤ n, a probability law on the subsets of size m such that the mean hitting time of a random target that is drawn from this law does not depend on the starting point of the process. We use the same random forests to give probabilistic insights into the proof of an algebraic result due to Micchelli and Willoughby and used by Fill and by Miclo to study absorption times and convergence to equilibrium of reversible Markov chains. We also introduce a related coalescence and fragmentation process that leads to a number of open questions.

We propose a model of a one-dimensional random walk in dynamic random environment that interpolates between two classical settings: (I) the random environment is sampled at time zero only; (II) the random environment is resampled at every unit of time. In our model the random environment is resampled along an increasing sequence of deterministic times. We consider the annealed version of the model, and look at three growth regimes for the resampling times: (R1) linear; (R2) polynomial; (R3) exponential. We prove weak laws of large numbers and central limit theorems. We list some open problems and conjecture the presence of a crossover for the scaling behaviour in regimes (R2) and (R3).MSC 2010: 60F05, 60G50, 60K37.

We consider a one-dimensional continuous time random walk with transition rates depending on an underlying autonomous simple symmetric exclusion process starting out of equilibrium. This model represents an example of a random walk in a slowly nonuniform mixing dynamic random environment. Under a proper space-time rescaling in which the exclusion is speeded up compared to the random walk, we prove a hydrodynamic limit theorem for the exclusion as seen by this walk and we derive an ODE describing the macroscopic evolution of the walk. The main difficulty is the proof of a replacement lemma for the exclusion as seen from the walk without explicit knowledge of its invariant measures. We further discuss how to obtain similar results for several variants of this model.MSC 2010. Primary 60K37, 82C22; Secondary 82C44.

We propose a new method for performing multiscale analysis of functions defined on the vertices of a finite connected weighted graph. Our approach relies on a random spanning forest to downsample the set of vertices, and on approximate solutions of Markov intertwining relation to provide a subgraph structure and a filter bank leading to a wavelet basis of the set of functions. Our construction involves two parameters q and q . The first one controls the mean number of kept vertices in the downsampling, while the second one is a tuning parameter between space localization and frequency localization. We provide an explicit reconstruction formula, bounds on the reconstruction operator norm and on the error in the intertwining relation, and a Jackson-like inequality. These bounds lead to recommend a way to choose the parameters q and q . We illustrate the method by numerical experiments.2010 Mathematics Subject Classification. 94A12, 05C81, 05C85, 15A15, 60J20, 60J28.