Two-parameter semigroups, evolutions and their applications to Markov and diffusion fields on the plane. (English) Zbl 0866.60042

Mishura, Yu.; Tomilov, Yu.

J. Appl. Math. Stochastic Anal. 9, No. 3, 281-302 (1996).

Two-parameter coordinate-wise so-called \(C_0\)-semigroups are considered. Necessary and sufficient conditions for a linear operator to be a generator of such a semigroup are proved. Resolvents of semigroup generator are also considered. Two-parameter evolution and differential equations of the second order are studied for such a semigroup. The results are applied to obtain Hille-Yosida theorem for homogeneous Markov fields of the Feller type. Forward, backward and mixed Kolmogorov equations of the second order for the densities of non-homogeneous diffusion fields on the plane are presented.

Reviewer: V.Parkhomenko (Kiev)

MSC:

60G60	Random fields
60J25	Continuous-time Markov processes on general state spaces
60J35	Transition functions, generators and resolvents
60J60	Diffusion processes
47D06	One-parameter semigroups and linear evolution equations
47D07	Markov semigroups and applications to diffusion processes

Keywords:

two-parameter semigroup; two-parameter evolution operator; Markov field; diffusion field