Painting stone in entrance porch

[bixbycanyon]

Firstly, it would be helpful to know the technical term for the entrance bit by the front door you see on 1950's semi's... ?!
Like this one...(inside the arch)

The stonework inside the arch and all around the front door is painted white. Parts have started peeling off and it's looking tatty.

What is the best way to tackle this? The paint is very thick so even if painted over with white would still leave noticeable crevices (albeit white ones)

 

[dcdec]

Scrape off the loose stuff and spot prime. Then fill these areas with a decent masonary filler ( i use toupret ), rub down and two coats of bedec masonary paint.