Lemma: Finite chain and normalization

For every finite $J$, (eq:generator) defines a nonexplosive irreducible continuous-time Markov chain. Its invariant law is $\pi$, and it is reversible.

If $K=n^{-1}{\sum }_i{K}_i$, then

In particular, $K$ counts uniformly chosen single-site attempts, whereas ${P_u}^J$ uses rate one at each site.

Proof. Each $K_i$ is a stochastic matrix. The exponential identity follows from $L_J=n(K-I)$. Its right-hand side is a Poisson mixture of stochastic matrices, so it defines the chain with rate-$n$ attempts and uniformly chosen sites. The number of attempts on a bounded interval is finite almost surely.

Each conditional spin probability is strictly positive when $J$ is finite. Single-coordinate changes connect the cube, which proves irreducibility. The operator $K_i$ is conditional expectation under $\pi$, given all coordinates except $i$. Therefore it is self-adjoint on $L^2(\pi )$, preserves $\pi$, and satisfies detailed balance. Summing gives reversibility of $L_J$ and ${P_u}^J$.