Small Steps at High Speed: The Diffusion Limit

We present and use some fundamental concepts that help us to understand heuristically how, and under which conditions, processes which are generically discrete eventually tend to diffusion processes which are continuous in time and space. This continuous framework permits us to cover, in a streamlined manner, processes that are more general than simple random walks. We indicate how the continuous treatment leads more directly to information about major characteristics of the specific process in question. The continuous framework does not rely on individual (albeit often ingenious) case-by-case arguments but instead on a more uniform approach that exploits standard mathematical tools such as differential equations and generating functions. These tools are explained in detail and permit us to investigate practically relevant properties of standard diffusion processes, such as their first-passage times or the maximum position they attain during a time period.