There are many hypotheses but they are quite hard to test. The problem is that any such process is likely to have involved millions of years (at a minimum) of essentially random search among initial building blocks before something "takes."
How do you even try to replicate that? We could perhaps fast forward it in a lab but that is going to make it contrived.
The best we can probably do is to work backward from the simplest life to a credible model of prebiotic replicating molecular systems and then back from there to random organics plus energy. Then try to do isolated experiments to assign probability bounds and then try to estimate time and probability. This is not a true replication or a direct observation but it could at least provide a credible model.
I remember a coherent-sounding (to a somewhat smart layman) theory being presented in "The Vital Question". It involved alkaline vents (like these: [0]). As I remember it, the idea was that the nonviolent flow of matter and the porous rock of the vent created a natural chemical reactor, where gunk could accumulate in the pores, until it slowly chanced into a primitive pump, primitive membrane and primitive division; eventually, some of that gunk learned how to close a membrane around itself, with stuff from the alkaline vent on the inside, and could now survive out in the open, continuously pumping to keep the gradient. Evolution continued from there.
How do you even try to replicate that? We could perhaps fast forward it in a lab but that is going to make it contrived.
The best we can probably do is to work backward from the simplest life to a credible model of prebiotic replicating molecular systems and then back from there to random organics plus energy. Then try to do isolated experiments to assign probability bounds and then try to estimate time and probability. This is not a true replication or a direct observation but it could at least provide a credible model.