Stochastic tools in mathematics and science This introductory book on probability based modeling covers basic stochastic tools used in physics chemistry engineering and the life sciences. The topics include conditional expectations stochastic processes Brownian motion and its relation to partial differential equations Langevin equations the Liouville and Fokker Planck equations as well as Markov chain Monte Carlo algorithms renormalization and dimensional reduction and basic equilibrium and non equilibrium statistical mechanics. The applications include data assimilation prediction from partial data spectral analysis and turbulence. A noteworthy feature of the book is the systematic analysis of memory effects