Measure-valued limits of interacting particle systems with k-nary interactions II. Finite-dimensional limits

It is shown that Markov chains in describing k-nary interacting particles of d different types approximate (in the continuous state limit) Markov processes on having pseudo-differential generators with symbols p (x,ξ) depending polynomially (degree k) on x. This approximation can be used to prove existence and non-explosion results for the latter processes. Our general scheme of continuous state (or finite-dimensional measure-valued) limits to processes of k-nary interaction yields a unified description of these limits for a large variety of models that are intensively studied in different domains of natural science from interacting particles in statistical mechanics (e.g. coagulation-fragmentation processes) to evolutionary games and multidimensional birth and death processes from biology and social sciences.