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Particles on a countable lattice attempt jumps at the rates $p(x,y)$, and an attempt into an occupied site is suppressed. Thus each site contains at most one particle and the number of particles is conserved when finite. Here ${\eta }^{xy}$ interchanges the occupations at $x$ and $y$. Bounded nearest-neighbour rates give a well-defined Markov process. A symmetric jump kernel gives the symmetric simple exclusion process; an unequal left-right kernel gives the asymmetric simple exclusion process.

Let the common vertex degree be $r>0$. For each unordered edge place an independent Poisson process of rate $1/r$, and at a mark interchange the two endpoint occupations. If exactly one endpoint is occupied, this moves its particle across that edge at rate $1/r$; if both or neither are occupied, the occupied set stays unchanged. Thus each particle proposes jumps at total rate one, uniformly among its neighbours, with occupied-target proposals suppressed. This is the stated symmetric simple exclusion process. An edgeless graph has the identity dynamics and both conclusions below hold trivially.

In the stirring representation of symmetric exclusion, put a distinct label at every vertex and interchange labels at every edge mark, even when the occupations agree. There are finitely many marks on every finite time interval. Their chronological composition is a random permutation ${\pi }_t$ of the vertex set, and simultaneously for every initial set,Each edge swap is its own inverse. Reflecting the edge-mark times in $[0,t]$ preserves their joint Poisson process law and reverses the order of the swaps. Hence ${\pi }_t^{-1}$ and ${\pi }_t$ have the same distribution. Pathwise,Taking probabilities and then using the inverse-permutation distribution proves the product self-duality of symmetric exclusion in set form:No assumption about disjointness or equal cardinality of $A$ and $B$ is required; impossible inclusions have probability zero on both sides.

For invariance, independently choose the initial occupied set $A$ under the product measure ${\mu }_{\rho }$. The occupations are independent identically distributed Bernoulli distribution variables, so permuting their sites by any deterministic permutation preserves their joint law. Conditional on the independent stirring permutation ${\pi }_t$, the output therefore still has law ${\mu }_{\rho }$. Averaging proves the Bernoulli invariant laws of finite symmetric exclusion:There is also a direct check from the duality. For every $B\subseteq V$,because the exclusion process conserves particle number. These are exactly the joint occupation moments of ${\mu }_{\rho }$; on a finite set they determine the law by the inclusion-exclusion principle. The endpoint densities give the absorbing empty and fully occupied configurations, and the empty-set moment is one also at $\rho =0$.

The duality requested below uses the symmetric simple exclusion process, so we take that standard convention here. A configuration $\eta \in \{0,1{\}}^{\mathbb{Z}}$ records a particle at $x$ when $\eta (x)=1$. Particles attempt nearest-neighbour jumps with equal left and right rates, and any attempt into an occupied site is suppressed. Thus there is at most one particle at each site.

A convenient normalization puts independent rate-one Poisson clocks on the unoriented edges $\{x,x+1\}$. At an edge mark, interchange its two endpoint occupations. If both endpoints have the same occupation, nothing changes; if exactly one is occupied, that particle jumps to the vacant endpoint. The generator on cylinder functions isOnly finitely many terms can be nonzero for a cylinder function. A convention with total jump-attempt rate one per particle uses edge clocks of rate $1/2$, merely rescaling time.

For a pathwise construction, label all sites and swap their labels at each clock mark, carrying the initial occupations along with the labels. Each label path has jump rate two and is nonexplosive on a bounded time interval. The same is true of a backward path from any fixed space-time point. This stirring representation of symmetric exclusion therefore defines the occupations for every initial configuration, even with infinitely many particles. It also conserves the number of particles when finite.

A general exclusion process may have asymmetric jump rates, but then the plain occupation-product self-duality in the next part need not hold. The symmetry convention is therefore mathematically necessary here, rather than an irrelevant choice of time normalization.

Give every site a distinct label and interchange labels at the edge marks defining the symmetric simple exclusion process. Each forward or backward label path makes only finitely many jumps in a bounded time interval, since its total rate is $2c$. Occupations are transported along these paths. The backward ancestors of a finite set have the law of the finite-particle symmetric exclusion process, by time reversal of the Poisson marks. This proves the product self-duality of symmetric exclusion.