  In this paper, we prove the existence of martingale solutions to the
stochastic heat equation taking values in a Riemannian manifold, which admits
Wiener (Brownian bridge) measure on the Riemannian path (loop) space as an
invariant measure using a suitable Dirichlet form. Using the Andersson-Driver
approximation, we heuristically derive a form of the equation solved by the
process given by the Dirichlet form.
  Moreover, we establish the log-Sobolev inequality for the Dirichlet form in
the path space. In addition, some characterizations for the lower bounds of the
Ricci curvature are presented related to the stochastic heat equation.
