Consider the following SDE: $$dX_t=dW_t,\, t\in[s，T]; \quad X_s=x.$$
By the definition of Ito integral, we know the strong solution to the above SDE is given by $X_t=x+W_t-W_s$. However, I was wondering whether the stochastic process $Y_t=x+W_{t-s}$ for $t\ge s$, which has the same distribution as $X_t$, has any relation with the given SDE.
Moreover, it is OK to say $Y_t$ satisfy the SDE: $$dY_t=dW_{t-s},\, t\in[s，T]; \quad Y_s=x.$$