In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on $\mathbb{R}^+$ or $\mathbb{R}$ with values in a general Riemannian maifold, which is only assumed to be complete and stochastic complete. This work is an extension of the previous paper \cite{RWZZ17} on finite volume case. Moveover, we also obtain some functional inequalities associated to these Markov processes. This implies that on infinite volume case, the exponential ergodicity of the solution if the Ricci curvature is strictly positive and the non-ergodicity of the process if the sectional curvature is negative.