I have a question about reflecting Brownian motion on an unbounded domain.

Let us consider the **reflecting** Brownian motion $\{X_t\}_{t \ge 0}$ on the following domain $\bar{D}$ of $\mathbb{R}^2$:
\begin{equation*}
D=\{(x,y)\in\mathbb{R}^{2}: |xy|<1\}.
\end{equation*}

**My research**

- We can construct reflecting Brownian motion $\{X_t\}_{t \ge 0}$ on $\bar{D}$ by using the Dirichlet form theory. That is, the following bilinear form (regular Dirichlet form on $L^{2}(\bar{D})$) generates $\{X_{t}\}_{t \ge 0}$: \begin{equation} \mathcal{E}(f,g)=\frac{1}{2}\int_{D}(\nabla f, \nabla g)\,dx,\quad f,g \in H^{1}(D), \end{equation} where $H^{1}(D)$ is the first order $L^2$-Sobolev space on $D$ with Neumann boundary condition.
- We can prove that for all $q>2$, the following Sobolev inequality
\begin{align*}
\|f\|_{L^{q}(D)} \le C \|f\|_{H^{1}(D)}
\end{align*}
**does not hold for all $f \in H^{1}(D)$.**

**My question**

Can we show that the transition probability density $p_{t}(x,y)$ of $\{X_{t}\}_{t \ge0}$ exists?

If $p_{t}(x,y)$ exists, can we have a upper/lower estimate of $p_{t}(x,y)$?