# Relations between boundary and inner probabilities for 2D finite state Markov chains

In calculus the Green's theorem relates an integral around a curve to a double integral over the plane region bounded by that curve.

Does there exist an analogy of this theorem for 2-dimensional Markov chains with a finite discrete state space, which relates boundary and inner probabilities?

For example.

Consider a 2-dimensional Markov chain with the state space is $$(i,j), 0 \le i,j \le N$$, $$N$$ is a positive integer. Denote $$p_{ij}$$ the steady state probability of the state $$(i,j)$$.

Then the possible theorem might say: if one knows distribution on the boundary (i.e. $$p_{00},p_{10},\dots,p_{N0}$$, $$p_{01},\dots,p_{0N}$$, $$p_{1,N},\dots,p_{N-1,N}$$, $$p_{N,1},\dots,p_{N,N-1}$$), then the whole distribution $$p_{ij}$$ can be computed.

Would be grateful for any references and thoughts.

• It depends on how many closed communicating classes the Markov chain has. If there's at least two such classes then the distribution is not unique. Commented 4 hours ago