Baez' series on network theory and information geometry answer a couple of your questions in a very accessible way.

Here Baez and Fong prove a version of Noether's theorem for Markov processes.

http://math.ucr.edu/home/baez/networks/networks_11.html

Earlier Baez talked about conservation of total probability.

http://johncarlosbaez.wordpress.com/2011/09/09/network-theory-part-8/

Here he relates the distribution of existing species to a prior and how Bayes' rule says how the number of each species changes over time.

http://johncarlosbaez.wordpress.com/2011/05/26/information-geometry-part-8/