Understanding Markov Chains

As is frequently the case in the sciences, ideas seem to be hanging around in the air and are often discovered by several thinkers independently in a span of years or even months. Something similar took place at the turn of the 19th century when scientists increasingly became aware of the importance of stochastic processes such as random walks in the sciences. In a span of years, random walks popped up in the context of mosquito populations (where they were tied to the spread of disease), Brownian motion of molecules (part of Einstein's annus mirabilis), acoustics, and the financial market. To start formally, a random walk is a stochastic process that describes the path of a subject in a mathematical space, which can be constituted by something like the integers, but also a 2-dimensional or higher-dimensional Euclidian space. We can illustrate this with a simple intuitive example.