In log-linearized New Keynesian model, what do $Y$, $Y_t$, $y_t$ actually mean? This may be a weird question, but I am unfortunately confused by terms. Let us suppose log-linearized New Keynesian model as suggested by Gali here: http://crei.cat/people/gali/pdf_files/monograph/slides-ch3.pdf

My first question is, apparently steady value $Y$ is assumed to do log-linearization of $Y_t$, output, but is this $Y$ a constant value, or the whole steady output path? Equivalently, is $Y$ about how output will evolve if $Y_t$ evolves without stochastic factors and errors according to long-run natural rate?

My second question, related to the first question, is whether $Y_t$ refers to total output or normalized output. That is, if economy has positive output growth rate, will $Y_t$ grow? Or is it normalized output that does not change without stochastic elements? 

My third question is, what $y_t$ actually mean. As I understand it, it is just $\log Y_t$. Is this correct? 

The fact that consumption euler equation exists seems to support the intuition that $Y$ is steady output path, not constant steady value, as real interest rate is often positive for economy. My confusion all rises from here, and I am not sure if this is correct understanding.
The following post explains in a somewhat easier way what exactly is happening when we log linearize a model. 

http://economictheoryblog.com/2012/06/22/latexgx_t/

Going through the provided example should make it clear what the single steps are.