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On a degenerate SDE in the unit ball

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=(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 $E$ and the corresponding diffusion process $X$ is a time-changed absorbing Brownian motion on the interior of $E$. The boundary $partial E$ is the cemetery of $X$. If $c>0$, the corresponding diffusion process is a time-changed standard Brownian motion with drift?

Thank you in advance.

As for the above question, I may solved it myself.

MathOverflow Asked on February 2, 2021

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