The boundary value problem and the Jensen inequality for an entropy functional of a Markov diffusion process

The paper introduces the recent results related to an entropy functional on trajectories of a controlled diffusion process, expressed through an additive functional of the diffusion process, with a Lagrangian, determined by the parameters of a controlled stochastic equation. These results include a minimum condition for the entropy functional and the functionalʹs Jensen inequality, which both are useful for the solution of important mathematical and applied problems.