The solution represents the probability of finding our random walker at position x at time t if the initial distribution was placed at 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→∞ converges to the normal distribution. It is then possible to show that
W(il−jl,nϵ)→W(y,t+Δt|x,t)=1√4πDΔtexp[−((y−x)2/4DΔt)],and that it satisfies the normalization condition and is itself a solution to the diffusion equation.