# Backshift operator applied to a constant

This questions is two part:

1) What happens when you apply the backshift operator to a constant? For example, if I have the AR process $$(1-\phi B)(y_t-\mu)=\epsilon_t$$ does that equal $$y_t-\mu-\phi By_t-\phi B\mu = \epsilon_t$$ which (I believe reduces to) $$y_t-\mu-\phi y_{t-1}-\phi \mu = \epsilon_t\longrightarrow y_t=\mu+\phi y_{t-1}+\phi \mu+\epsilon_t$$

So am I correct in assuming that the backshift of a constant (in my example $\mu$) is just the constant?

2) If I assume that $\epsilon_t\sim N(0,v)$, then what is the likelihood of the above AR process in 1?

$$Y_i | Y_{i-1},\dots,Y_0 \sim \mathcal{N}\left((1+\phi) \mu+\phi Y_{i-1},v\right)$$
You need to specify what the distribution of $Y_0$ will be (will it contain the unknown parameters $\phi$, $v$? If not, it doesn't really matter.