Spectral gaps for the O-U/Stochastic heat processes on path space over a Riemannian manifold with boundary

Fang-Wu\cite{FW17} presented a explicit spectral gap for the O-U process on path space over a Riemannian manifold without boundary under the bounded Ricci curvature conditions. In this paper, we will extend these results to the case of the Riemannian manifold with boundary. Moreover, we also derive the similar results for the stochastic heat process.