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.