# 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?

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

MathOverflow Asked on February 2, 2021

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