Remarks on Random Evolutions in Hamiltonian Representation

Time evolution of random processes differs in one essential respect from evolution of conservative systems in general and Hamiltonian systems in particular. As a great number of various limit theorems attest, the final states loose all the information about the initial conditions. Thus, there is nothing to be conserved, no constants of motion can exist, and no use can be made of the powerful machine of the Hamiltonian formalism. Or so it seems. Sometimes there are things which do not change during time evolution, such as rates of decay and other universal exponents. This suggests that one may hope to find constants of motion and even Hamiltonian forms in at least some probabilistic systems provided one is willing to make time-dependent rescalings of the basic dynamical variables. Besides, there is something like a precendent in the history of attempts to quantize dissipative systems, a close relative of random processes. The simplest of such systems is a particle moving on a line under the influence of a harmonic force and a friction: