Probability that particle hits shaded region on random walk?

Question

Consider the following grid in the image, where the squares are equally spaced. Assume that the area of the entire image is $1$ and the black squares take up exactly $0 < k < 1$ of the region. Now suppose some point located in the white region begins a random walk at some constant speed in the image. (Further assume that the image is on a torus, so if the point travels right on the right edge it will appear on the left edge.) How do I determine the probability that the point enters one of the black regions?

I have no idea how to proceed since I am not familiar with the notation behind random walks. I only have general knowledge in elementary probability theory. How do I go about solving this problem?

Answer 1

There is an answer for the discrete time solution provided by Tomi. In the continuous time case, a random walk is a solution to the diffusion equation : $$ \frac{\partial}{\partial t}P(\vec{r}, t) = D \nabla^2 P(\vec{r},t) $$

where $D$ is the diffusion constant (provided it does not depend on time or space) and $P(\vec{r}, t)$ is the probability of finding your particle in the region $\vec{r}$ at time $t$. In the usual cartesian space, solving this equation with $P(\vec{r}, 0) = \delta(\vec{0})$ and $P(\vec{r} \rightarrow \infty, t) = 0$ yields the gaussian solution. In your case, you will want to solve it for 2D toroidal coordinates with $P(\vec{r}, 0) = \delta(\vec{r_0})$. For a given time, the probability of finding the particle on a black square would be given by $$\int_{B} P(\vec{r}, t) d\vec{r}$$ where $B$ is the black region.

EDIT : I'm not sure what you mean by use the assumptions. The solution $P$ does not have to follow the symmetry of your squares (the initial condition $P(\vec{r}, t=0) = \delta(\vec{r_0})$ does not). I would guess you're interested in getting the probability at some time $t$ of finding the particle in a black region given that the initial condition is some point in the white space. Since you can calculate $P(\vec{r} = B, t; \vec{r_0})$, the probability of finding the particle in the black region at time $t$ for some initial condition $\vec{r_0}$, you just have to sample all the possible initial points in the white region :

$$ P_B(t) = A^{-1}\int_{\vec{r_0} \in W} P(\vec{r}=B, t; \vec{r_0}) d\vec{r_0}$$

Where A is the white space area that you consider. Given your symmetry you only have to perform integration in the irreducible region, which should be one quarter of a single black/white square.

Answer 2

First you should try to identify a simpler "base tile" that is being used to create the overall shape. I suspect that you have drawn could be simplified to be sixteen square tiles, each with one dark square inside it.

Identify the number of pixels that your basic tile has. Each of these pixels is a different state. For each pixel, determine where you would end up if you were to go up, down, left or right. These are the possible destination states. You can set the dark pixels as a single sink state.

You can use this information to create a transition matrix. You can then work out the probability of being in the sink state after $n$ turns.