The solution represents the probability of finding our random walker at position \( \mathbf{x} \) at time \( t \) if the initial distribution was placed at \( \mathbf{x}=0 \) at \( t=0 \).
There is another interesting feature worth observing. The discrete transition probability \( W \) itself is given by a binomial distribution. The results from the central limit theorem state that transition probability in the limit \( n\rightarrow \infty \) converges to the normal distribution. It is then possible to show that
$$ W(il-jl,n\epsilon)\rightarrow W(\mathbf{y},t+\Delta t|\mathbf{x},t)= \frac{1}{\sqrt{4\pi D\Delta t}}\exp{\left[-((\mathbf{y}-\mathbf{x})^2/4D\Delta t)\right]}, $$and that it satisfies the normalization condition and is itself a solution to the diffusion equation.