12 Finite-state Feynman–Kac transfer
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Let \(K\) be a column-stochastic matrix, let \(V\) be a potential on the states, and let \(f\) be terminal data. Define
The backward Feynman–Kac value after \(n\) steps is
Theorem 12.1: Discrete Feynman–Kac path formula
For an initial state \(i\),
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Proof of Theorem 12.1
For \(n=1\),
which is the stated expectation. Iterating and expanding the matrix product sums over every path \((i=i_0,i_1,\ldots,i_n)\) with probability \(\prod_rK_{i_{r+1},i_r}\) and weight \(\prod_re^{-V(i_r)}\).
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