- Aims, interface of probability with PDEs; - Review of continuous-time Markov processes - Feller semigroups & Hille-Yosida: transition semigroups; finite-state CTMCs and $Q$-matrices; Poisson/pseudo-Poisson examples; the Hille–Yosida theorem linking generators ↔ semigroups. - Martingale problems: setup and link to Markov (pre)generators; recap of SDEs; diffusion processes via martingale problems and equivalence to SDEs. - Stroock–Varadhan diffusion approximation: weak convergence notions; approximating diffusions by Markov chains. - Method of duality: using dual processes to analyze distributions. - One-dimensional diffusions: speed & scale: scale function, speed measure; boundary classification (Feller); Green’s functions; stationary/reversible laws. - PDEs via Brownian motion/diffusions: probabilistic solutions to Dirichlet/Poisson problems, Feynman–Kac, and semilinear PDEs (branching).