Some Properties And Functional Inequalities For Finite Particle Systems On Polish Spaces

In this thesis, we study the regularity, quasi-regularity and functional inequalities of the Dirichlet forms for finite particle systems on Polish spaces, as well as the stochastic mono-tonicity and the preservation of positive correlations of the associated reaction process corresponding to the reaction particle system. To this end, we split the thesis into three parts.In the first part, we mainly consider the regularity and quasi-regularity of the Dirichlet forms corresponding to the finite particle systems. To this end, we generalize the results and get the sufficient and necessary conditions for the sum of two or countably many Dirichlet forms to be regular or quasi-regular.We consider in the second part the stochastic monotonicity and preservation of positive correlation of the process corresponding to the particle system when their is no diffusion (pure reaction). We get the sufficient and necessary conditions for the reaction process to be stochastic monotonic under certain conditions (e.g. in the birth-death case). Consequently, we confirm that under these conditions the associated measure-valued process preserves positive correlations.In part three, results in [29] are extended to a more general reaction-diffusion model. Correspondence between the functional inequalities for the particle system and those for the underlying based process as well as estimates on inequality constants are provided.