Let $X$ be a connected topological space. Then its based loops space $\Omega X$ is an examples of $A_{\infty}$ space. Let $PX$ be the free paths space. Which type of homotopical structures are naturally defined on $PX$?
Edit: The multiplication of loops on the based loop spaces is defined via the ordinary composition followed by a reparametrization, this construction is associative only up to homotopy (the resulting objects in the associative diagram differ by a parametrization) and give to $\Omega X$ the structure of an $A_{\infty}$ spaces. Why this structure is not well defined on $PX$? It seems to me that the composition of two paths is well defined on $PX$ and it is again associative up to homotopy...