A nonlinear Markov evolution is a dynamical system generated by a measurevalued ordinary differential equation (ODE) with the specific feature that it preserves positivity. This feature distinguishes it from a general Banachspace-valued ODE and yields a natural link with probability theory, both in the interpretation of results and in the tools of analysis. However, nonlinear Markov evolution can be regarded as a particular case of measure-valued Markov processes. Even more important (and not so obvious) is the interpretation of nonlinear Markov dynamics as a dynamic law of large numbers (LLN) for general Markov models of interacting particles. Such an interpretation is both the main motivation for and the main theme of the present monograph.

The power of nonlinear Markov evolution as a modeling tool and its range of applications are immense, and include non-equilibrium statistical mechanics (e.g. the classical kinetic equations of Vlasov, Boltzmann, Smoluchovski and Landau), evolutionary biology (replicator dynamics), population and disease dynamics (Lotka–Volterra and epidemic models) and the dynamics of economic and social systems (replicator dynamics and games). With certain modifications nonlinear Markov evolution carries over to the models of quantum physics.