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.

  • $\begingroup$ 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. $\endgroup$ Commented 4 hours ago


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