For the open manifold like $X\times \mathbb R$ or $X\times \mathbb R^+$, where $X$ is a closed manifold. 


Is there any decomposition like (Hodge Decomposition) of the Differential forms on it.



Hodge decomposition on open manifold
There certainly isn't a simple general statement like the Hodge decomposition in any category that contains, for example, complete noncompact manifolds with some pointwise condition on curvature, even if they are diffeomorphic to products of a compact factor and a real line. But you could try MR1815415 (2002j:58033) Ahmed, Zulfikar M.(1-CLMB); Stroock, Daniel W.(1-MIT) A Hodge theory for some non-compact manifolds. J. Differential Geom. 54 (2000), no. 1, 177–225. 58J05 (58A14 58J65)


