Dear Joakim,
I like how you shed light on one fundamental principle with different perspectives. This helps to gain new insight. I think I'll have to read your essay quietly three or four times in order to have a better understanding. At first read, I have the following questions: aren't you just counter Wick-rotating Feynmans least action principle and writing entropy in a new unit? When S_i goes to infinity you retrieve classical mechanics, but when S_i is of the order of a few h-bar, does your stochastic action principle have sense as you've lost the quantum interference effects of the complex exponential? I don't see for which experiments your principle can be of value in the quantum domain.
By the way, I promote the FQXi contest on my twitter profile and my blog. Would you mind if I post a quote of yours, linking to your essay?
Regards,
Arjen