Some Properties Of The Two-parameter Semigroups Generated By Standard Para-transition Function

This paper mainly studies some properties of the two-parameter semigroupsgenerated by standard para-transtion function, including the continuity of the twosemigroups, their resolvents, infinitesimal operators and differential operators.To this end, we first research on one-parameter para-transition function andone-parameter semigroup. We study their continuity, differentiability, laplacechanged operator, the properties of resolvents, infinitesimal operators anddifferential operators, and the relationship between one-parameter para-transitionfunction and one-parameter semigroup, as well as some properties of the twoone-parameter semigroups.Furthermore, we introduce some properties of non-homogeneous para-transtionfunction. In addition to the continuity and differentiability, we also discuss theirstrong ergodicity. Simultaneously, we study the properties of resolvents,infinitesimal operators, differential operators and strong ergodicity of thetwo-parameter semigroups generated by non-homogeneous para-transtion function.Similarly, we also discuss the properties of hypo-homogeneous two-parametersemigroup, such as continuity and resolvents.Finally, we discuss the focus of this study-the properties of the two-parametersemigroups generated by standard para-transtion function. We first introduce adefinition of standard para-transtion function, as well as the two-parametersemigroups and their various kinds of operators. And we conclude the similarproperties to one-parameter para-transfer function and non-homogeneouspara-tranition function, such as continuity and quasi-homogeneity. Further, we studyoperators of the two-parameter semigroups, and get some properties, includingexchangeability of infinitesimal operators, equivalence of differential operatorsunder the condition of quasi-homogeneity, and equivalence relation with otheroperators.