Expected Hitting Time for Simple Random Walk from origin to point (x,y) in 2D-Integer-Grid - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T10:06:43Z http://mathoverflow.net/feeds/question/112470 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112470/expected-hitting-time-for-simple-random-walk-from-origin-to-point-x-y-in-2d-int Expected Hitting Time for Simple Random Walk from origin to point (x,y) in 2D-Integer-Grid carnage 2012-11-15T11:15:23Z 2012-11-15T16:29:37Z <p>Consider a simple random walk on the lattice $\mathbb Z^2$ starting at the origin $(0,0)$ where in each step, one of the four adjacent vertices in chosen uniformly at random, i.e. with probability $1/4$. I want to know the distribution of the time that it takes to hit a certain vertex $(x,y)$. I have already done quite some research on related work but so far I was not able to come up with a paper that gives an answer to my question. I would appreciate any hint towards a paper that covers this problem or a hint on how I can calculate it myself.</p> http://mathoverflow.net/questions/112470/expected-hitting-time-for-simple-random-walk-from-origin-to-point-x-y-in-2d-int/112473#112473 Answer by Johan Wästlund for Expected Hitting Time for Simple Random Walk from origin to point (x,y) in 2D-Integer-Grid Johan Wästlund 2012-11-15T11:47:54Z 2012-11-15T11:47:54Z <p>Oops, too long for a comment: </p> <p>The expectation of the time until you reach a given point other than the origin is infinite already in one dimension. To see this, let $T$ be the expected time until you get from 0 to 1 (in the obvious 1-dimensional setting). The first step is either to the left or to the right, and if you go to $-1$, the expected remaining time is going to be $2T$ since you have to get back to the origin and then to 1. Consequently $T = 1 + 1/2\cdot (2T)$, from which we see that finite $T$ leads to contradiction. In higher dimensions it becomes even worse. Presumably you want to modify your question and ask about something like the distribution of the time (in dimensions 1 and 2 you actually get there!) or about the expected time to get from one point to another in a finite box.</p>