Fisher information and trajectorial interpretation to the Itô--Langevin relative entropy dissipation

The dissipation phenomena of relative entropy from an Itô--Langevin dynamical system is a classic topic from stochastic analysis. Relying on the time-reversal of diffusions, a novel trajectorial approach investigates the pathwise behavior of relevant entropy processes, reveals more information from the delicate random structure, and eventually retrieves the known classical results. In essence, this approach provides novel insights and rederives the known results of the Itô--Langevin dynamics, as will be presented in this expository article. Another part is to view the stochastic time-evolution through the lens of the Wasserstein space, under which we observe the geometric feature of steepest descent of the entropy decay as well as its exponential rate of velocity.