This is a question about a diffusion process on the unit ball.

In this article J.S, the author considered the following SDE in the **closed** unit ball $E subset mathbb{R}^n$:

begin{align*}

(1)quad dX_t=sqrt{2(1-|X_t|^2)},dB_t-cX_t,dt,

end{align*}

where ${B_t}_{t ge 0}$ is an $n$-dimensional Brownian motion, $|cdot|$ denotes the Euclidean norm on $mathbb{R}^n$ and $c$ is a nonnegative constant. We define an elliptic operator $(mathcal{A},text{Dom}(mathcal{A}))$ by $mathcal{A}=C^2(mathbb{R}^n)|_E$ and

begin{align*}

mathcal{A}f=sqrt{2(1-|x|^2)}Delta f-c xcdot nabla f,quad f in text{Dom}(mathcal{A}).

end{align*}

Then, standard results from martingale problems show that there exists a diffusion process ${X_t}_{t ge 0}$ on $E$ such that

$f(X_t)-f(x)-int_{0}^{t}mathcal{A}f(X_s),ds quad(t ge 0,, x in E)

$

is a martingale. Thus, the SDE $(1)$ possesses a solution. Furthermore, we can show that the solution is unique in the sense of distribution (by the way, the pathwise uniqueness for (1) is a very profound problem).

**My question is as follows.**

If $c=0$, then $mathcal{A}$ is a weighted Laplacian on $E$. However, we do not impose the Neumann boundary condition on $mathcal{A}$. Thus, the operator $mathcal{A}$ is not associated with a time-changed reflected Brownian motion on $E$, right? Indeed, there is no local time in the display of (1).

Even if $c=0$, is it difficult to describe the quadratic (Dirichlet) form of $X$? I am also interested in the fundamental solution of $X$. I’m not really sure that it exists…

I’m asking these questions to see what kind of diffusion process $X$ is.

Thank you in advance.