= Product self-duality of symmetric exclusion
{title2=$\mathbb E^\eta D(\eta_t,A)=\mathbb E^A D(\eta,A_t)$}
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.
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