When happens when pollen grains land on the a part other than the stigma of a flower of the same species?
Sometimes, the morphology and the topology of the flower structure is such that the pollen grains falling on certain parts are gravitationally (or hydrostatically) drawn towards the stigma.
I remember having read that the waxy streamlined stigma of submerged hydrophytes (Vallisneria, Hydrilla)  creates mini vortices around it during steady water flow. These vortices then pull nearby pollen, (which may be sitting on nearby flower parts) towards itself. Certain hydrophyte morphologies also maximise the water flow towards stigma, which in turn gathers pollen incident elsewhere on the plant and bringing in towards the stigma.(For example, the streamlined and bladder shaped hydrophyte flowers have stigmas at their centre) 

Another way to do this is via the common sloping surface of the petals or the perianth, which on gathering pollen, gravitationally push them to the centre of the flower where the stigma usually resides. The entire inflorescence stalk might be modified into specific shapes to do this. The common pyramidal arrangement of the inflorescence (broad at the base, tapering towards the apex) helps in gathering the pollen incident on non-stigma parts of the apex region at the bottom.   

Apart from this, pollen falling elsewhere, may again be taken up by the agents of pollination. But I believe the formation of pollen tube never occurs at any place other than stigma. Nor does any re-absorption take place.  

It is also worth mentioning that just because pollen falls on the correct stigma does not ensure fertilization. Absence of stigma maturity/receptivity, synergid abnormality, inviable pollen etc could still hinder fertilization.  

[I am not perfectly aware about studies on whether any chemical reaction occurs if pollen lands on places other than stigma. But I doubt it, since the chemical specificity required to begin the pollen tube formation is very high in most species.]