A Modern Approach to Probability Theory by B. Fristedt and L. Gray. Topics Brownian motion and applications.
Brownian motion is surely the most important continuous time stochastic process.
It is, for example, the main building block for the theory of stochastic calculus. More general continuous time Markov processes. Using Brownian motion and pure jump processes as motivating examples, we will discuss the basics of more general Markov process theory, including semigroups and generators. If time permits, we will discuss some connections between Brownian motion and partial differential equations .