Consider a two-dimensional random walk, but this time the probabilities are not 1/4, but some values p_1, p_2, p_3, p_4 with $\sum_{i=1}^4 p_i=1$. For example, from (0,0), it goes to (1,0) with p_1, goes to (0,1) with p_2 etc.
The question is how to compute the probability x of going back to (0,0), starting from (0,0). In general, this probability is not 1.
Thanks.
Best Answer
Since the number of visits has geometric distribution, it's enough to find its expected value.
Here is one approach: since you can solve the one-dimensional case, treat the two-dimensional case as two "interleaved" one-dimensional walks, one north-south, the other east-west. Let's say $p_1$ is the probability of north, $p_2$ is the probability of south. Then at each step, we do a step from the north-south walk with probability $p_1+p_2$ and a step from the east-west walk with probability $p_3+p_4$; each step of the north-south walk is north with probability $p_1/(p_1+p_2)$ and south with probability $p_2/(p_1+p_2)$.
So then the probability of being back at the origin at time $N$ is the sum over $m$ of
$P(X=m) P(\text{north-south walk at 0 after $m$ steps})P(\text{east-west walk at 0 after $N-m$ steps})$
where $X$ is Binomial$(N, p_1+p_2)$.
Sum over $N$ to get the total expected number of visits.
I don't see that this will give you a closed-form answer, but it may at least make it easier to compute and/or obtain bounds.