29, Khodzhaeva st., 700143, Tashkent, Uzbekistan
Romanovskii Mathematical Institute of the National Academy of Sciences of the Uzbek Republic
Ganikhodzhaev N. N., Rozikov U. A.
On quadratic stochastic operators generated by Gibbs distributions
2006, vol. 11, no. 4, pp. 467-473
We give a constructive description of quadratic stochastic operators which act to the set of all probability measures on some measurable space. Our construction depends on a probability measure $\mu$ and cardinality of a set of cells (configurations) which here can be finite or continual. We study behavior of trajectories of such operators for a given probability measure $\mu$ which coincides with a Gibbs measure. For the continual case we compare the quadratic operators which correspond to well-known Gibbs measures of the Potts model on $Z^d$. These investigations allows a natural introduction of thermodynamics in studying some models of heredity. In particular, we show that any trajectory of the quadratic stochastic operator generated by a Gibbs measure $\mu$ of the Potts model converges to this measure