This course is meant to serve as an introduction to the theory of Markov processes on finite or countable state spaces. We will discuss what a Markov process is along with associated concepts such as transition probabilities, recurrence, transience, ergodicity, reversibility etc. We will motivate various abstract notions introduced with concrete examples from physics and statistics.

We will study two approaches to treating singular SPDEs. The first approach is the theory of energy solutions, while the second is the so-called paracontrolled calculus. We will use these two techniques to study two concrete examples: the parabolic Anderson model and the stochastic Burgers equation.