In the following $W_t$ is a Brownian motion say in $\mathbb{R}$ or $\mathbb{R}^d$ and $y_0$ is deterministic.
If you have an SDE say $Y_t\in \mathbb{R}^d$ $$ dY_t=b(Y_t)dt+dW_t,~~Y_0=y_0$$
you can choose to recenter its initial condition without changing the dynamics much by defining $Z_t:=Y_t-y_0$ then $Z_0=0$, and $\mathbb{E}(Z_t)=\mathbb{E}(Y_t)-y_0$.
I would like to do the same but now with a reflected SDE.
Suppose we have a solution $(X_t,k_t)$ of the following Reflected SDE in some convex domain $D$.
$$ dX_t=b(X_t)dt+dW_t+dk_t,~~~X_0=x_0 $$
by solution I am referring to a solution to the Skorohod problem (see definition in https://projecteuclid.org/euclid.hmj/1206135203).
I would like to recenter the above reflected SDE, i.e define some $Z_t$ with $Z_0=0$ but with similar dynamics to $X_t$.
$\textbf{Question}$: Is this possible? Does $Z_t$ need to live in some `shifted/re-centred' domain $\tilde{D}=D-x_0$, and evolve as the solution of the Skorohod problem in this new domain. Wont its dynamics be very different to those of $X_t$? i.e we cant hope for a simple relation between their expected values, as in the case of the regular SDE.