Abstract
 
The Boltzmann equation and the Schroedinger equation are two examples of
evolution equations coming from mathematical physics that describe the
evolution of probabilities of experimental outcomes, but in which there
is nothing whatsoever random about the evolution of the state of the
underlying physical system: At least in the standard physical model, all
of the randomness in the Boltzmann equation setting comes from random
initial data, and in the Schroedinger equation setting, it comes form
randomness inherent in the measurement process. But the evolution of the
underlying state of the system itself is, in both cases, deterministic.
Nonetheless, in both cases, one can construct stochastic processes, in  
which the evolution itself is truly random, that provide solutions to the
deterministic evolution equations. This was done in the Boltzmann
equation setting by Mark Kac, and in the Schroedinger equation setting by
Nelson, with important contributions by Guerra and Morato. In this
lecture, I will introduce these random models of deterministic evolution,
describe some theorems  and results illustrating the advantages of
analyzing the evolution using the stochastic models, and discuss some
interesting open problems that are suggested by the results obtained so
far.