Excursions and Lévy system of boundary process (Q1423921)

Let \({\mathbf M}=(X_t,P^x)\) and \(\hat{\mathbf M}=(\hat{X}_t,\hat{P}^x)\) be conservative diffusion processes. Assume that there exist a measure \(m\) and a continuous function \(p(t,x,y)\) such that \(p(t,x,y)m(dy)\) and \(p(t,y,x)m(dy)\) are the transition functions of \({\mathbf M}\) and \(\hat{\mathbf M}\), respectively. Fix a fine and cofine perfect set \(V\) and put \(D=V^c\), \(T=T_V\) and \(\hat{T}=\hat{T}_V\). Further assume that there exists a continuous function \(H_u(x,b)\) such that \(H_{u+v}(x,b)=\int P^0_v(x,dy)H_u(y,b)\) and \(h(x,b)=\int_0^\infty H_u(x,b) du\) is a density of the hitting distribution of \(V\) relative to a measure \(\mu\), where \(P^0_v\) is the transition function of the part process on \(D\). Assume the similar condition for the dual process \(\hat{\mathbf M}\) by the same measure \(\mu\). By using the explicit expression of \(P^x(f(X_T):T<s\mid X_t=y)\) by means of \(H\) and \(p\), the authors give the joint distribution of the endpoints of excursion from \(V\) straddling the fixed time \(t\). This result is used to characterize the Lévy system of the boundary process by means of the given data. Furthermore, the Douglas integral type Dirichlet form of the boundary process is given by a natural manner.