I would like to give a brief and intuitive introduction of my doctoral research without touching much technical jargon. Let us start with a basic example of a simple random walk on the integer number line, which is illustrated with the following figures:
(figures)
The random walk models fair bets by flipping a fair coin. For every flip, a gambler wins one dollar if the coin lands on heads, but loses one dollar if it lands on tails. The gambler wants to know what the probability of winning A dollars before losing B dollars is, given that (s)he has no initial wealth. The solution of this problem can be obtained as follows. Let $latex f(k)$ with $latex A\leq k\leq B$ denote the probability of winning A dollars before losing B dollars, given that the gambler initially has $latex k$ dollars. Some observations suggest that $latex f(k)$ satisfies the recurrence relation
$latex f(k) = \frac{1}{2}f(k-1) + \frac{1}{2}f(k+1),$
for any $latex A<k<B$, and
$latex f(A)=1$, $latex f(B)=0$. By solving these equations, we obtain a general formula for the probability of winning A dollars before losing B dollars, given that (s)he initially has $latex k$ dollars:
$latex f(k) = \frac{k+B}{A+B}$.

Random walks in cones, Discrete harmonic functions, Branching processes in random environments.