Generative Modeling and Controllability

In generative AI, the approach called diffusion-based generative modeling has introduced the problem of modifying a probability through a diffusion process on a given interval of time, in order to obtain a target probability. If, instead of speaking of probability distribution one speaks of random variables, this problem looks very similar to a controllability problem for a stochastic dynamic system. It is thus meaningful to explore the connection between the two problems, which has not been considered in the literature so far. The approach of controllability can open new possibilities for generative modeling. At the same time, many differences occur. It is interesting to notice that the concept of probability distribution and that of random variable, although representing an equivalent uncertainty, are complementary and not identical. We survey in this work the two domains and how to make use of the two methodologies.