Fluctuations in Markov Processes : Time Symmetry and Martingale Approximation Diffusive phenomena in statistical mechanics and in other fields arise from markovian modeling and their study requires sophisticated mathematical tools. In infinite dimensional situations time symmetry properties can be exploited in order to make martingale approximations along the lines of the seminal work of Kipnis and Varadhan. The present volume contains the most advanced theories on the martingale approach to central limit theorems. Using the time symmetry properties of the Markov processes the book develops the techniques that allow us to deal with infinite dimensional models that appear in statistical mechanics and engineering (interacting particle systems homogenization in random environments and diffusion in turbulent flows to mention just a few applications). The first part contains a detailed exposition of the method and can be used as a text for graduate courses. The second concerns application to exclusion processes in which the duality methods are fully exploited. The third part is about the homogenization of diffusions in random fields including passive tracers in turbulent flows (including the superdiffusive behavior). There are no other books in the mathematical literature that deal with this kind of approach to the problem of the central limit theorem. Hence this volume meets the demand for a monograph on this powerful approach now widely used in many areas of probability and mathematical physics. The book also covers the connections with and application to hydrodynamic limits and homogenization theory so besides probability researchers it will also be of interest to mathematical physicists and analysts. Preface Part I: General Theory 1.A Warming up Example 2.Central Limit Theorems 3.RandomWalks in Random Environment 4.Bounds and Variational Principles for the Asymptotic Variance Part II: Simple Exclusion Processes 5.The Simple Exclusion Process 6.Self Diffusion 7.Equilibrium Fluctuations of the Density Field 8.Regularity of the Asymptotic Variance Part III: Diffusions in Random Environments 10.Variational Principles for the Limiting Variance 11.Diffusions with Divergence Free Drifts 12.Diffusions with Gaussian Drifts 13.Ornstein Uhlenbeck Process with a Random Potential 14.Analytic Methods in Homogenization Theory References Notation Subject Index.