# Coordinate transform to exploit symmetry

I have a stochastic process that can be described the following master-equation:

$$\partial_{t}P(x,y)=-\left(W_{12}(x,y)+W_{13}(x,y)+W_{21}(x,y)+W_{23}(x,y)+W_{31}(x,y)+W_{32}(x,y)\right)P(x,y)\\ +W_{12}(x+1,y-1)P(x+1,y-1)+W_{13}(x+1,y)P(x+1,y)\\ +W_{21}(x-1,y+1)P(x-1,y+1)+W_{23}(x,y+1)P(x,y+1)\\ +W_{31}(x-1,y)P(x-1,y)+W_{32}(x,y-1)P(x,y-1).$$

$P(x,y)$ is the probability mass function, and the state space can be described by two number $x,y$ with the constraints $0\le x\le N,0\le y\le N,x+y\le N$. $W_{\alpha,\beta}$ is transition probabilities. We can see that out of the 8 possible transitions, only 6 are allowed ($(x,y)\rightarrow(x+1,y+1),(x,y)\rightarrow(x-1,y-1)$ are not allowed), so the state space can be more naturally described on a triangular lattice. For this reason, this equation might benefit from a coordinate transform.

My question is, how to rewrite, through a coordinate transform, this equation so as to exploit the underlying symmetry?

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