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 $D$ 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 $D$ by using the Dirichlet form theory. That is, the following bilinear form (regular Dirichlet form on $L^{2}(D)$) 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 Gaussian upper/lower estimate of $p_{t}(x,y)$? \begin{align*} C_{1}t^{-d/2}\exp(-C_{2}|x-y|^2/t) \le p_{t}(x,y)\le C_{3}t^{-d/2}\exp(-C_{4}|x-y|^2/t) \end{align*}