Taking a walk on $\mathbb{N}$, starting at 1, I need to find out how many steps I expect to take before returning to the origin, as a fraction. For each step, I either walk forward, backward, or stay still (+1, -1, or 0 respectively) with probabilities $a, b, c$. Walking forward and staying still have a small probability, while walking backwards has a high probability. Is there a general method I can use?
[Math] Random walk with weighted probabilities
discrete mathematicsmarkov chainsprobabilityrandom walk
Best Answer
I'm not sure I got the 'as a fraction' part, but the rest can be handled by standard MC machinery. Since $a+b+c=1$ you need to solve a recurrent equation: $$ m_{k,1}=1+a m_{k,1}+b m_{k,1} + c m_{k+1,1} $$
Can you handle from here?