Abstract:
In this paper, we shall prove that the irreducibility in the sense of fine topology implies the uniqueness of invariant probability measures. It is also proven that this irreducibility is strictly weaker than the strong Feller property plus irreducibility in the sense of original topology, which is the usual uniqueness condition.

Abstract:
In this paper we explain that the natural filtration of a continuous Hunt process is continuous, and show that martingales over such a filtration are continuous. We further establish a martingale representation theorem for a class of continuous Hunt processes under certain technical conditions. In particular we establish the martingale representation theorem for the martingale parts of (reflecting) symmetric diffusions in a bounded domain with a continuous boundary. Together with an approach put forward in Lyons et al(2009), our martingale representation theorem is then applied to the study of initial and boundary problems for quasi-linear parabolic equations by using solutions to backward stochastic differential equations over the filtered probability space determined by reflecting diffusions in a bounded domain with only continuous boundary.

Abstract:
The main purpose of this paper is to explore the structure of regular subspaces of 1-dim Brownian motion. As outlined in \cite{FMG} every such regular subspace can be characterized by a measure-dense set $G$. When $G$ is open, $F=G^c$ is the boundary of $G$ and, before leaving $G$, the diffusion associated with the regular subspace is nothing but Brownian motion. Their traces on $F$ still inherit the inclusion relation, in other words, the trace Dirichlet form of regular subspace on $F$ is still a regular subspace of trace Dirichlet form of one-dimensional Brownian motion on $F$. Moreover we have proved that the trace of Brownian motion on $F$ may be decomposed into two part, one is the trace of the regular subspace on $F$, which has only the non-local part and the other comes from the orthogonal complement of the regular subspace, which has only the local part. Actually the orthogonal complement of regular subspace corresponds to a time-changed Brownian motion after a darning transform.

Abstract:
The regular subspaces of a Dirichlet form are the regular Dirichlet forms that inherit the original form but possess smaller domains. The two problems we are concerned are: (1) the existence of regular subspaces of a fixed Dirichlet form, (2) the characterization of the regular subspaces if exists. In this paper, we will first research the structure of regular subspaces for a fixed Dirichlet form. The main results indicate that the jumping and killing measures of each regular subspace are just equal to that of the original Dirichlet form. By using the independent coupling of Dirichlet forms and some celebrated probabilistic transformations, we will study the existence and characterization of the regular subspaces of local Dirichlet forms.

Abstract:
Roughly speaking, the regular subspace of a Dirichlet form is also a regular Dirichlet form on the same state space. It inherits the same form of original Dirichlet form but possesses a smaller domain. What we are concerned in this paper are the regular subspaces of associated Dirichlet forms of skew product diffusions. A skew product diffusion $X$ is a symmetric Markov process on the product state space $E_1\times E_2$ and expressed as \[ X_t=(X^1_t,X^2_{A_t}),\quad t\geq 0, \] where $X^i$ is a symmetric diffusion on $E_i$ for $i=1,2$, and $A$ is a positive continuous additive functional of $X^1$. One of our main results indicates that any skew product type regular subspace of $X$, say \[ Y_t=(Y^1_t,Y^2_{\tilde{A}_t}),\quad t\geq 0, \] can be characterized as follows: the associated smooth measure of $\tilde{A}$ is equal to that of $A$, and $Y^i$ corresponds to a regular subspace of $X^i$ for $i=1,2$. Furthermore, we shall make some discussions on rotationally invariant diffusions on $\mathbf{R}^d\setminus \{0\}$, which are special skew product diffusions on $(0,\infty)\times S^{d-1}$. Our main purpose is to extend a regular subspace of rotationally invariant diffusion on $\mathbf{R}^d\setminus \{0\}$ to a new regular Dirichlet form on $\mathbf{R}^d$.

Abstract:
In this paper, we shall first establish the theory of bivariate Revuz correspondence of positive additive functionals under a semi-Dirichlet form, which is associated with a right Markov process $X$ satisfying the sector condition but without duality. We extend most of the classical results about the bivariate Revuz measures under the duality assumptions to the case of semi-Dirichlet forms. As the main results of this paper, we prove that for any exact multiplicative functional $M$ of $X$, the subprocess $X^M$ of $X$ killed by $M$ also satisfies the sector condition and we then characterize the semi-Dirichlet form associated with $X^M$ by using the bivariate Revuz measure, which extends the classical Feynman-Kac formula.

Abstract:
In this paper, we shall consider the killing transform induced by a multiplicative functional on regular Dirichlet subspaces of a fixed Dirichlet form. Roughly speaking, a regular Dirichlet subspace is a closed subspace with Dirichlet and regular properties of fixed Dirichlet space. By using the killing transforms, our main results indicate that the big jump part of fixed Dirichlet form is not essential for discussing its regular Dirichlet subspaces. This fact is very similar to the status of killing measure when we consider the questions about regular Dirichlet subspaces in [6].

Abstract:
In this paper we shall prove the weak convergence of the associated diffusion processes of regular subspaces with monotone characteristic sets for a fixed Dirichlet form. More precisely, given a fixed 1-dimensional diffusion process and a sequence of its regular subspaces, if the characteristic sets of regular subspaces are decreasing or increasing, then their associated diffusion processes are weakly convergent to another diffusion process. This is an extended result of [13].

Abstract:
We extend the classical Douglas integral, which expresses the Dirichlet integral of a harmonic function on the unit disk in terms of its value on boundary, to the case of conservative symmetric diffusion in terms of Feller measure, by using the approach of time change of Markov processes.