Note: you may believe that life started in multiple places at multiple times throughout the universe, but the thread is concerned only about the strain of life that populated the earth.
Not to get off on a tangent, but this is why the study of the deep oceans is important; it's not just an obstacle so NASA has a smaller budget, it helps answer questions like these. But that goes two ways. If NASA finds life somewhere that isn't based on the same organic atoms we use, then that indicates that life can arise in places other than water worlds, and that would completely change how we look at biology.
Any hypothesis about shit that happened that long ago is just conjecture. We don't really have the ability to predict anything that far back. I'd say that what's here started here but I think we've had interaction to some extent with life from other places.