STOCHASTIC HEAT EQUATIONS WITH VALUES IN A MANIFOLD VIA DIRICHLET FORMS
In this paper, we prove the existence of martingale solutions to the stochastic heat equation taking values in a Riemannian manifold, which admits the 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 bound of the Ricci curvature are presented related to the stochastic heat equation.
52
3
2237-2274
2237-2274
Siam Publications