This has been a world wind tour of the
wonderful world of random walks,
diffusion and first passage phenomena.
These are well studied topics and many
resources are available to delve more
deeply. Just use some of the key words
from this tutorial in your favorite search
tool. Even though they have been
investigated for more than a century,many
aspects of diffusion and firstpassage are
still hot topics. One example is
diffusion in realistic geometries, such as
complex networks or disordered
environments. Here we are still learning
how heterogeneity affects the long time
properties of random walks.
There are many other open questions that
are too numerous to list.
Random walks are merely a specific
realization of general stochastic
processes in which a state of a many
body system changes incrementally due
to interactions of all types. This
evolution is captured by the master
equation which quantifies how the
probability distribution of the system
changes due to these interactions.
We already encountered the master equation
for the one dimensional random walk. So a
firm background on random walks will help
you in your understanding of general
stochastic processes.
Finally, I gave a brief glimpse into the
fascinating topic of first passage processes.
An amazing feature of first passage in
one dimension is that a random walk is
certain to return to its starting point,
but the average time for this return is
infinite. This dichotomy has many
intriguing consequences. First passage
processes also underlie many real
examples, such as chemical reactions,
financial transactions, and physiological
processes where the system responds when
a random walk first reaches a given
threshold. I encourage you to explore
these intriguing first passage phenomena
and their wide range of applications.