I have a question about this paper BH, which is one of the basic literature on reflecting Brownian motions.
For any bounded open subset $D$, the semigroup $\{T_t\}_{t>0}$ on $L^{2}(D,dx)$ generated by Neumann laplacian admits an integral kernel. If $D$ is bounded Lipschitz domain, $\{T_t\}_{t>0}$ has the integral kernel $p_{t}(x,y)$ which is continuous on $(0,\infty) \times \bar{D} \times \bar{D}$.
The authors of this paper constructed reflecting Brownian motion on $\bar{D}$ in the following procedure:
- They proved $H^{1}(D)$ (1st order Sobolev space with Neumann boundary condition) is a regular Dirichlet form on $L^{2}(\bar{D},dx)$. Since, $H^{1}(D)$ is regular on $L^{2}(\bar{D},dx)$, there exists a Hunt process $X=(X_t,P_x)$ on $\bar{D}$ associated with $H^{1}(D)$.
- However, $X=(X_t,P_x)$ does not necessarily have transition density. So they introduced new probability measures $\{Q_x\}_{x \in \bar{D}}$. Each $Q_{x}$ is defined by \begin{equation*} Q_{x}(A \circ \theta_{t})=\int_{D}p_{t}(x,y)P_{y}(A)\,dy,\quad x\in\bar{D},\ t>0,\ A \in\mathcal{F}. \end{equation*} Here, $\theta_t$ is the shift operator of $X=(X_t,P_x)$ and $\mathcal{F}$ is the filtration generated by $X$.
My question
They claimed $Q_{x}$ uniquely determines a probability measure $Q_x$ on $\mathcal{F}$. However, I don't know why... For example, how do they define $Q_{x}(\{X_0=x\})$ ?