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 $p\_{t}(x,y)$?



Reflecting Brownian motion and its transition probability density
Since building reflected Brownian motion in smooth bounded domains is not a problem, the only potential obstruction to the existence of the transition probabilities is that it escapes to infinity in finite time. This can be ruled out by checking that the function $(x^2-y^2)^2$, which is in the domain of the generator, is a Lyapunov function.


Regarding estimates on the transition probabilities, it is relatively easy to get tail estimates in a similar way, by considering the martingale problem for functions of the type $f(x^2-y^2)$. For example, by choosing for $f$ a smooth approximation to the absolute value function (for example $f(n) = \sqrt{1+n^2}$), one can find that the law of $X$ has Gaussian tails in the sense that $\mathbf{P}(|X(t)| > K) \lesssim \exp(-c K^2)$. 


