Let $W$ be a standard Brownian Motion. Let $\epsilon>0$ be given. Let $X^\epsilon$ be the process which diffuses like $W$ on $(-\epsilon,\infty)$, but when $X^\epsilon$ reaches the level $-\epsilon$, it is immediately brought back to the value zero. It then diffuses again according to $W$ until hitting $-\epsilon$, and then is brought back to zero, and so forth. Let $X^0$ be a reflected Brownian Motion (reflected at zero). Then, as $\epsilon \rightarrow 0$, in what sense does $X^\epsilon \rightarrow X^0$ Are there any references for this? I'm also interested in when $W$ is a diffusion.
It looks like it should converge in distribution in Skorokhod space D. Martingale problem approach (see the book by Ethier & Kurtz on Markov processes) should work.