In this note we introduce the notion of a Poisson smooth structure on a
symplectic stratified space. We show that under a mild condition many
properties of a symplectic manifold can be extended to a symplectic stratified
space, e.g. the existence and uniqueness of a Hamiltonian flow, the isomorphism
between the Brylinski-Poisson homology and the de Rham homology, the Hodge
structure on a symplectic stratified space. We give many examples of symplectic
stratified spaces satisfying these properties.