Abstract

In this thesis, we discuss three topics on Dirichlet forms and non-symmetric Markov processes.

First, we explore the analytic structure of non-symmetric Markov processes. Let U be an open set of \mathbf{R}^n, m a positive Radon measure on U, and (P_t)_{t>0} a strongly continuous contraction sub-Markovian semigroup on L^2(U;m). We give an explicit Levy-Khintchine type representation of the generator A of (P_t)_{t>0}. If (P_t)_{t>0} is an analytic semigroup, we give an explicit characterization of the semi-Dirichlet form {\cal E} associated with (P_t)_{t>0}.

Second, we consider the Dirichlet boundary value problem
for a general class of second order non-symmetric elliptic operators L with singular coefficients. We show that there exists a unique, bounded continuous solution by using the theory of Dirichlet forms and heat kernel estimates. Also, we give a probabilistic representation of the non-symmetric
semigroup generated by L.

Finally, we present new results on Hunt's hypothesis (H) for Levy processes. These include a comparison result on Levy processes which implies that big jumps have no effect on the validity of (H), a new necessary and sufficient condition for (H), and an extended Kanda-Forst-Rao theorem.