Given a connected topological space $X$, its space of path $PX$ is again a topological space. On the other hand, for a simplicial set $K_{\bullet}$, its path space is given by $$ PK_{n}:=\operatorname{Map}_{sSet}(\Delta[1], K)_{n}. $$ Here my question: given a simplicial topological space $X_{\bullet}$, what is its "path space" $PX_{\bullet}$ (as a simplicicial topological space)? Do you know some candidates?

It is sensible to restrict to simplicial {\em based} spaces $X_*$ and then apply the functor $P$ levelwise. This is used, for example, to compare $|\Omega X_*|$ with $\Omega |X_*|$ in The Geometry of iterated loop spaces http://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf