Brownian motion: definition and some properties
Max Planck Institute for Mathematics in Sciences, Leipzig.
We start with simple random walk on $\mathbb Z$ and using Donsker's invariance principle we show that the rescaled walk converges to the Brownian motion. We address some key properties. If time permits, we move on to the Markov semigroup associated to the process, its generator and the celebrated Feynman-Kac formula, which is the starting point of our second talk.