Stirring representation of symmetric exclusion

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.

For finite $A\subset \mathbb{Z}$ put $D(\eta ,A)={\prod }_{x\in A}\eta (x)$, with empty product one. Interchanging occupations gives $D({\eta }^{xy},A)=D(\eta ,A^{xy})$, where $A^{xy}$ interchanges membership of the two sites. The edge-swap generators therefore agree on $D$. The stirring representation of symmetric exclusion proves the displayed identity at all times by tracing the ancestors of $A$ backwards. A general asymmetric jump kernel does not satisfy this product self-duality.

If the probability of occupation at every site in a finite set depends only on its cardinality, the law is exchangeable. The product self-duality of symmetric exclusion and conservation of the finite dual particle count imply that all these occupation-product moments are unchanged by evolution. Inclusion-exclusion principle reconstructs every finite cylinder probability from them, proving invariance. This direct argument does not require a representation theorem for exchangeable laws.