Beurling-Deny Formula Of Non-Symmetric Dirichlet Forms And The Theory Of Semi-Dirichlet Forms

The theory of Dirichlet forms provides a bridge between classical potential theory and stochastic analysis. It has applications in potential theory, Markov processes, Malliavin analysis, quantum field theory, etc. The Beurling-Deny formula of symmetric Dirichlet forms plays an important role in the theory of symmtric Dirichlet forms and symmetric Markov processes. It provides a structure result of the forms which corresponds to the sample path behavior of the corresponding processes. Although, there have been some attempts for the extension of Beurling-Deny formula to non-symmetric case, but up to now it is still available only in symmetric case. This paper discuss the decompositions of non-symmetric Dirichlet forms and semi-Dirichlet forms. We give some structure results which can be regarded as an extension of the classical Beurling-Deny formula, and can also be regarded as an extension of Levy-Khintchine formula or more generally, an extension of Courrege's Theorem in Dirichlet forms or semi-Dirichlet forms setting. We also discuss some related problems.At first, we discuss the Beurling-Deny formula of regular (non-symmetric) Dirichlet forms. We construct a compatible metric p on E such that for any we have the following decomposition:In chapter 3 we prepare some potential tools for the discussions of the decomposition of semi-Dirichlet forms. In this chapter we mainly discuss the measures of finite energy integral with respect to a regular semi-Dirichelt forms.In chapter 4 we study futher some properties of regular semi-Dirichlet formsand its relation with associated Hunt process. We obtain the equivalence of exceptional sets and ε-exceptional sets, the equivalence of ε-quasi-continuous functions and q.e.-finely continuous functions. We prove that the part of a regular semi-Dirichlet form on an open set is also a regular semi-Dirichlet form, analogous to the case of regular Dirichlet forms.In chapter 5 we discuss the local compactification and quasi-homeomorphism of semi-Dirichlet forms , and get similar results as in Dirichlet forms setting.In final chapter of this paper we study the decompositions of quasi-regular semi-Dirichlet forms. In section 6.1 we establish an integral representation theorem for quasi-regular semi-Dirichlet spaces (cf, Theorem 6.1.1), which plays a key role in the discussion of the Beurling-Deny decompositions of quasi-regular semi-Dirichle forms. In section 6.2 we discuss the Beurling-Deny decompositions of regular semi-Dirichlet forms. By the results in section 6.1, 6.2 and quasi-homeomorphism of quasi-regular semi-Dirichlet forms discussed in chapter 5, we show that for any and we have (cf, Theorem 6.3.1)Moreover, we construct a quasi-compatible metric such that for all u, v in a special core and we have (cf, Theorem 6.3.6)...