What is the trigger of pollen tube formation when the pollen is on the stigma?
Germination of the pollen tube begins when the pollen grain becomes hydrated on the tip of the stigma. The pollen grain absorbs water and also exchanges signals with the stigma. Microchannels form between the pollen grain and the tip of the stigma papillae and water is transferred from the stigma to the pollen grain (Taylor 1997). cAMP, which is produced by the protein PSiP, and cytosolic free calcium can be used as signals to orient the pollen tube to grow in the proper direction (Moutinho et al 2001).

Pollen tubes can also germinate in vitro even without contacting a stigma. If you place a pollen grain in germination medium, which contains calcium and boron, among other things, the tube will germinate but it will not grow to its full length.

Often when pollen of a different species lands on the stigma, the pollen will germinate but the pollen tube growth will stop before it reaches the ovary (Ascher and Peloquin 1968). Mating barriers are very complicated, however, and there are many different mechanisms for species recognition and arrest of pollen tube growth. In some cases, the tube is halted in the style while in other cases the tube grows, but slowly and inefficiently (Swanson et al. 2004). Species recognition in some species (for example, tobacco) is related to self-incompatibility and the S-locus. Species recognition can also be unilateral: that is, the pollen of species A can fertilize the ovule of species B, but the pollen of species B cannot fertilize the ovule of species A.

I would recommend that Swanson paper if you want to read more.