This textbook is the result of the enhancement of several courses on non-equilibrium statistics, stochastic processes, stochastic differential equations, anomalous diffusion and disorder. The target audience includes students of physics, mathematics, biology, chemistry, and engineering at undergraduate and graduate level with a grasp of the basic elements of mathematics and physics of the fourth year of a typical undergraduate course. The little-known physical and mathematical concepts are described in sections and specific exercises throughout the text, as well as in appendices. Physical-mathematical motivation is the main driving force for the development of this text.It presents the academic topics of probability theory and stochastic processes as well as new educational aspects in the presentation of non-equilibrium statistical theory and stochastic differential equations.. In particular it discusses the problem of irreversibility in that context and the dynamics of Fokker-Planck. An introduction on fluctuations around metastable and unstable points are given. It also describes relaxation theory of non-stationary Markov periodic in time systems. The theory of finite and infinite transport in disordered networks, with a discussion of the issue of anomalous diffusion is introduced. Further, it provides the basis for establishing the relationship between quantum aspects of the theory of linear response and the calculation of diffusion coefficients in amorphous systems.

Fills the gap in the literature for an introductory and didactical book on stochastic processes with different applications in physics and biology

Elements of probability theory.- 2. Fluctuations close to thermodynamic equilibrium.- 3. Elements of Stochastic processes.-  4. Irreversibility, the Fokker-Planek equation.- 5. Irreversibility, linear response.- 6. Introduction to diffusive transport.- Diffusion in disordered media.- 9. Non-equilibrium statistical mechanics.- 9. Metastable and unstable states.- Appendix A. Thermodynamic variables in statistical mechanics.- Appendix B. Relaxation to the stationary state.- Appendix C. The Green function of the problem of an impurity.- Appendix D. The waiting-time function of the CTRW.- Appendix E. Non-Markovian effects against irreversibility.- Appendix F. The density matrix.- Appendix G. Kubo's formula for the susceptibility.- Appendix H. Fractals.- Appendix I. Quantum open systems.