We will see the basics of Ito calculus and Girsanov’s change of measure formula. Following Lehec, one can use these tools to construct a “minimum energy” coupling (Follmer’s drift) between the Gaussian measure and any other absolutely continuous probability measure on Rn. The log-Sobolev inequality and Talagrand’s Entropy-Transport inequality then fall out effortlessly. The same philosophy will then be applied to discrete spaces like the hypercube, the symmetric group (with transpositions as generators), and general discrete Markov chains. (Joint work with Ronen Eldan).