Nonlinear models of diffusion on a finite space (Q1087241)

The basic convergence theorems for finite state Markov chains are extended to the nonlinear case. An operator T in \(l_ 1\) of a finite space with counting measure is called nonexpansive if \(\| Tf-Tg\| _ 1\leq \| f-g\| _ 1\) holds for all f,g. It is shown that, for any f, there exists an integer \(p\geq 1\) such that \(T^{pn}f\) converges. Sufficient conditions for \(p=1\) are given. In the case of continuous parameter nonexpansive semigroups \(\{T_ t,t\geq 0\}\), \(T_ tf\) converges for \(t\to \infty.\) The main tool is a geometric theorem on isometries S of compact subsets of the above \(l_ 1:\) It is shown that any orbit under S is finite. The exponential speed of convergence does not extend from the Markov chain case to nonlinear T.