Would oxygen pool in sink holes if the upper atmosphere was helium, hydrogen and methane?

Imagine a big planet, gravity quite strong, 1.5 times earth. Irregular surface, rocky etc... (though flora and fauna will come into it later). It has a very thick atmosphere, drained off by a twin planet, even bigger, which is now a small gas giant.

Is it possible that - the air that is left could be held in craters? The gravity holds it low, in a shallow dense layer. In pockets of holes etc? Could the atmosphere above maintain a helium/hydrogen/methane mix? Or would it most probably just thin out - as per our atmosphere and maintain the same mix but thinner?

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Would oxygen pool in sink holes if the upper atmosphere was helium, hydrogen and methane?

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