Abstract
Since the seminal work by Chandrasekhar, the random walk scheme has been extensively developed both by expansion of new theoretical approaches and through applications to a vast deversity of experimental situations. In particular the original motivation for describing the transport by hopping conduction by Scher and Lax found a natural basement on the Montroll-Weiss continuous-time random walk on a lattice. Models of disorder are described by the properties of the distribution and specific symmetries in the underlying lattice. The aim of the theory is to find an approximation to calculate the mean value of the Green function of the problem. A unified description is studied, based on a diagrammatic technique, in order to compare the different approximations to tackle the problem of transport in disordered media. Important tools to solve this nonequilibrium problem are the effective medium approximation and non-Markovian random walk schemes. Particular emphasis is put on the calculation of the frequency dependence behavior of the electric conductivity, first passage times and residence times. The influence of boundary conditions on the Green function is put in correspondence with other interesting subjects like the first passage time problem and related ones. Studies of diffusion in pure and drifted lattices in presence of different kinds of disorder and anisotropies of the lattice are carried out.

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INTRODUCTION
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Clearly there are two different ways of doing physics, just to mention two pioneer papers let me point out Einstein's work and the Smoluchowski one on the theory of Brownian motion. Since then the random walk scheme has extensively been developed through a huge diversity of situations. In particular the seminal work on astrophysics by Chandrasekhar is a "modern" starting point to review the Markovian theory of diffusion processes.

An important chapter in the theory of diffusion process was written in 1965 by Montroll and Weiss when they formulate (deductive approach) the theory of non-Markovian processes to generalize the Brownian motion. Later on Scher and Lax (1973) extensively developed the theory of non Markovian processes to understand new experimental situations from the solid state physics. Since then the Montroll and Weiss random walk theory (CTRW) has been generalized in many different forms, for example to consider the diffusion of dimmer (introducing internal states in the theory), turbulence (introducing a coupled space-time structure), dynamical disorder (introducing a particular class of internal states), diffusion in discrete phase-space, diffusion-advection problems in random media, and in general to characterize anomalous diffusion in many different situations; interestingly the classical random walk approach has also been used to tackle quantum problems.

Another approximation to study the occurrence of anomalous diffusion comes from the stochastic transport theory and the theory of random matrix, in this case the main purpose of the theory was to evaluate the mean
value (over the disorder) of the propagator of the system, this deductive approach gave rise, in particular, to self consistent approximations, and scaling theories. This approach was an important point of view to formulate a general mobility formalism in classical disordered systems. Thus, models of disorder were described by the different properties of the distribution of random variables (disorder) and specific symmetries in the underlying system. Recently some generalizations of effective medium approximations considering "internal states'' have also been studied in order to understand the anomalous hydrodynamics dispersion in presence of multiple transport paths (fractures, dead ends, etc). In particular, emphasis was put on the characterization of the anomalous space-dispersion.

The present project is presented, with emphasis on comparisons between different theoretical approaches. Examples include studies of diffusion in pure and drifted situations away from equilibrium and in presence of different kind of disorder. The starting point is a random walk in a random media, typically in the form of a master equation with transition rates characterized by a given probability distribution. This fundamental equation is supplemented by specific symmetries that characterize the model of disorder and the presence of a bias and/or anisotropy in the lattice. Thus different theoretical approaches are presented in order to study the problem. The aim of the theory is to find an approximation to calculate the mean value (over the disorder) of the Green function of the problem. A unified description is studied, based on an exact diagrammatic technique in terms of projector operators, in order to compare the different approximations to tackle the problem. We studied that the specifics of each model of disorder enter through nonuniversal exponents that characterize the
nonequilibrium moments of the random walk. Then at long time different exponents are found depending of the strength of the disorder. The influence of boundary conditions on the Green function of the ordered system is put in correspondence with other interesting problems like the first passage time distribution and the mean residence time in random media. Thus the effects of finite-size and boundary conditions on the random walk domain are studied in connection with extreme phenomena which are experimentally accessible and enable to known the parameters of the stochastic dynamics. An important tool to solve this nonequilibrium problem is the occurrence of effective medium approximations like the self consistent approach (EMA) and the CTRW theory. In addition to the general features of the characterization of critical exponents in disordered systems discussed above, detailed reviews of theoretical and experimental works on many specific systems are made. These include electric conductivity in amorphous materials, i.e.: the generalized Einstein's law for the electric conductivity, first passage time and the residence time statistics in random media in a unified formalism, multifractality in the mean first passage time, etc.

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