Robert Brown's new thing -- Brownian motion as a Gaussian process -- Constructions of Brownian motion -- The canonical model -- Brownian motion as a martingale -- Brownian motion as a Markov process -- Brownian motion and transition semigroups -- The PDE connection -- The variation of Brownian paths -- Regularity of Brownian paths -- Strassen's functional law of the iterated logarithm -- Skorokhod representation -- Stochastic integrals: L²-theory -- Stochastic integrals: beyond -- Itô's formula -- Application of Itô's formula -- Stochastic differential equations -- On diffusions -- Simulation of Brownian motion / Björn Böttcher -- Appendixes: A.1. Kolmogorov's existence theorem -- A.2. A property of conditional expectations -- A.3. From discrete to continuous time martigales -- A.4. Stopping and sampling -- A.5. Remarks on Feller processes -- A.6. The Doob-Meyer decomposition -- A.7. BV functions and Riemann-Stieltjes integrals -- A.8. Some tools from analysis
