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

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.

### 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
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