Product self-duality of symmetric exclusion (${\mathbb{E}}^{\eta }D({\eta }_t,A)={\mathbb{E}}^{A}D(\eta ,A_t)$)

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.