Spaces of probability measures with positive density function on a compact Riemannian manifold are endowed with a closed 2-form associated with the Fisher information metric by using a divergence-free vector field. In this note we give a necessary and sufficient condition on the vector field that this 2-form is a strong symplectic structure.
Yuichi Shishido. "Strong symplectic structures on spaces of probability measures with positive density function." Proc. Japan Acad. Ser. A Math. Sci. 81 (7) 134 - 136, Sept. 2005. https://doi.org/10.3792/pjaa.81.134