Brownian motion on manifolds with time-dependent metrics and stochastic completeness

The Brownian motion on a Riemannian manifold is a stochastic process such that the heat kernel is the density of the transition probability. If the total probability of the particle being found in the state space is constantly 1, then the Brownian motion is called stochastically complete. For manifolds with time-dependent metrics, the heat equation should be modified. With the modified heat equation, we study the Brownian motion on manifolds with time-dependent metrics and find conditions on metrics and the volume growth for stochastic completeness.