The probabilistic structure of controlled diffusion processes

This paper surveys those aspects of controlled diffusion processes wherein the control problem is treated as an optimization problem on a set of probability measures on the path space. This includes: (i) existence results for optimal admissible or Markov controls (both in nondegenerate and degenerate cases), (ii) a probabilistic treatment of the dynamic programming principle, (iii) the corresponding results for control under partial observations, (iv) a probabilistic approach to the ergodic control problem. The paper is expository in nature and aims at giving a unified treatment of several old and new results that evolve around certain central ideas.