Sharp metastability threshold for two-dimensional bootstrap percolation

We introduce Sobolev spaces and capacities on the path space Pm 0(M) over a compact Riemannian manifold M. We prove the smoothness of the Itô map and the stochastic anti-development map in the sense of stochastic calculus of variation. We establish a Sobolev norm comparison theorem and a capacity comparison theorem between the Wiener space and the path space Pm 0(M). Moreover, we prove the tightness of (r, p)-capacities on Pm 0(M), r in N, p>1, which generalises a result due to Airault-Malliavin and Sugita on the Wiener space. Finally, we extend our results to the fractional H?lder continuous path space P_{m 0}^{2m, alpha}(M), m in N, m geq 2, alpha in ({1/2m}, {1/2}).