On nonlinear evolution equations for occupancies
For any natural number \(N\) subspaces of the direct product of \(N\) copies of a discrete probability space \(\Omega_0\), factored with respect to the group of permutation of indices, are defined by means of arbitrary linear constraints on occupancies. The direct sum of such quotient products over \(N\in \{1,2,\dots\}\) is considered. The formula for the characteristic functional of the occupation numbers is derived. Assuming that \(\Omega_0\) is a state space of the continuous time homogeneous Markov chain, the system of nonlinear evolution equations for the mean values of the occupation numbers is obtained. The solution of the Cauchy problem for these equations is represented as a composition of a solution of a linear Markov evolution equation and a solution of the defined first-order nonlinear ordinary differential equation. Some examples are discussed in detail.
scientific article; zbMATH DE number 1756285