Nonlinear L\'evy and nonlinear Feller processes: an analytic introduction

The program of studying general nonlinear Markov processes was put forward in V. N. Kolokoltsov “Nonlinear Markov Semigroups and Interacting L'evy Type Processes” (Journ. Stat. Physics 126:3 (2007), 585-642), and was developed by the author in monograph “Nonlinear Markov processes and kinetic equations”. Cambridge University Press, 2010, where, in particular, nonlinear L'evy processes were introduced. The present paper is an invitation to the rapidly developing topic of noninear Markov processes. We provide a quick (and at the same time more abstract) introduction to the basic analytical aspects of the theory developed in Part II of the above mentioned book.