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 $p_{t}(x,y)$?


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 $\mathbf{P}(|X(t)| > K) \lesssim \exp(-c K^2)$.

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