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Particles perform independent continuous-time random walks until they meet, after which they share one path. A finite initial particle set remains finite and its cardinality cannot increase. They arise as the common ancestors of sites in a voter model. On the square lattice every pair eventually coalesces, because the difference walk is recurrent Markov chain.

If individual removal is absent, simultaneous removal of one molecule from each species leaves $X_1-X_2$ unchanged. Independent constant-rate births change it by plus or minus one, each at rate $\lambda$. The difference is therefore a symmetric continuous-time random walk on the integers with variance growing as displayed. It has no normalizable stationary law. This explains why a deterministic attracting line of equilibria does not provide a stationary stochastic distribution.

// Target: probability-and-statistics.bigb

In the voter model on ${\mathbb{Z}}^2$, each site has a rate-one Poisson process clock; when it rings, that site copies one of its four neighbors, chosen uniformly. States are zero or one. For a cylinder function $f$, the local infinitesimal generator iswhere ${\eta }^{x\leftarrow y}$ agrees with $\eta$ except that the voter model opinion at $x$ is replaced by the voter model opinion at $y$. Only sites in the support of $f$ contribute, so this infinitesimal generator sum is finite.

The voter-model duality comes from independent Poisson processes of copying arrows of rate $1/4$ on each ordered neighboring pair. Trace the ancestors of specified sites backwards from time $t$: each ancestral line is a rate-one continuous-time random walk with uniform nearest-neighbor steps, and lines coalesce when they meet. The voter model opinions at time $t$ are the initial voter model opinions at those ancestral locations. This construction also explains the general duality theorem being used.

Two ancestral walks on ${\mathbb{Z}}^2$ meet almost surely. Until their meeting, their difference is a rate-two symmetric nearest-neighbor walk, and simple random walk in two dimensions is recurrent. If ${\tau }_{xy}$ is their meeting time, this givesfor every initial law $\nu$. If $\nu$ is invariant, the left side is the fixed probability $\nu (\eta (x)\ne \eta (y))$, so it is zero. This holds for every pair. Countability then implies that $\nu$ is concentrated on configurations with all voter model opinions equal. Both constant configurations are absorbing state. Thus the invariant measures of the two-dimensional voter model are exactlyNo translation-invariance assumption on $\nu$ was used.

For the last assertion, let the initial set of one-valued opinions $F$ be finite. Single-line duality givesEach transition probability tends to zero. One direct verification is the Fourier transform for the rate-one walk:The integrand has modulus at most one and tends to zero except at the single point $k=0$; dominated convergence proves the claim. For any finite observation set $K$, the union bound now givesTherefore every finite voter model opinion pattern converges in law to the all-zero pattern. These cylinder distributions determine weak convergence of probability measures in the compact product topology, proving finite-seed local extinction in the voter model and

Write $q_{ij}=c{m}_{ij}$ for $i\ne j$, with a common positive constant $c$, and put $d_i={\sum }_{j\ne i}{m}_{ij}$. The infinitesimal generator matrix is symmetric because the graph is undirected. Thus the uniform distribution ${\pi }_i=1/|I|$ satisfies detailed balance, and the continuous-time random walk is reversible when started uniformly.

Its jump chain chooses a link uniformly from those incident to its current vertex:The flux ${\hat{\pi }}_i{\hat{P}}_{ij}=m_{ij}/{\sum }_j{d}_j$ is symmetric, so this random walk on a multigraph is also reversible. The continuous-time equilibrium is uniform because higher-degree vertices have shorter holding times; at jump times the equilibrium is degree-weighted. If the intended proportionality constant depends on the starting vertex, $q_{ij}=c_i{m}_{ij}$, reversibility still holds with ${\pi }_i\propto 1/c_i$, and the same jump chain results.

In the low-temperature regime, the particle rapidly equilibrates near a minimum and only rarely crosses a neighboring maximum. Applying Laplace's method to the exact current formula gives the Kramers escape ratesEach right or left escape changes position by $2\pi$, henceThe reduced continuous-time random walk on minima has off-diagonal transition ratesand diagonal generator entry $W(l|l)=-(k_{+}+k_{-})$. Its residence time in each well is exponentially distributed with rate $k_{+}+k_{-}$, and the next jump is right with probability $k_{+}/(k_{+}+k_{-})$.

Independent Poisson process copying arrows construct the voter model. A backward ancestral line from a site follows a continuous-time random walk; lines that meet then coincide. Thus a finite collection of Ising spins at time $t$ equals initial Ising spins at the positions of coalescing random walks. In particular $E_{\eta }{\xi }_t(x)={\sum }_z{p}_t(x,z)\eta (z)$, and the probability of disagreement at two sites is bounded by the probability their ancestral walks have not yet coalesced.