A note on the antisymmetry in the speed of a random walk in reversible dynamic random environment

In this short note, we prove that $v(-\epsilon )=-v(\epsilon )$. Here, $v(\epsilon )$ is the speed of a one-dimensional random walk in a dynamic \emph{reversible} random environment, that jumps to the right (resp. to the left) with probability $1\mathrm{/}2+\epsilon$ (resp. $1\mathrm{/}2-\epsilon$) if it stands on an occupied site, and vice-versa on an empty site. We work in any setting where $v(\epsilon ),v(-\epsilon )$ are well-defined, i.e. a weak LLN holds. The proof relies on a simple coupling argument that holds only in the discrete setting.