Dear Arjen,
thanks for your most valuable comment. The idea of the stochastic least action principle is to eliminate the need for an action principle coupled with an additional principle for quantum mechanics. As this theory is equivalent to quantum mechanics in its current setup I am not sure if it gives any new direct testable predictions (other than the ones in quantum mechanics). I don't think I have lost any quantum effects when S_i is finite. The Wick rotation on the quantum wave function creates an ensemble of actions. The other way around is that the stochastic least action principle on regular mechanics creates a fundamental uncertainty in the action of a system. Only upon observation the action revealed, before that it is fundamentally hidden. So in an interference situation (eg double slit experiment) the stochastic least action principle merely states that the particle has a weighted probability of travelling all paths (actions) through this slit. Under Wick rotation this is equivalent to Feynmann's approach to quantum mechanics and should thus have the same dynamics.
The expected action from the stochastic least action principle is indeed sort of an entropy with a different unit.
Perhaps the truly interesting stuff happens if one investigates the stochastic differential equation approach. That is an open problem as of now.
I hope I have given you some adequate information.
Feel free to post anything on your blog/twitter!
Best Wishes
/Joakim