Stochastic Partial Differential Equations on Evolving Surfaces and   Evolving Riemannian Manifolds
  We formulate stochastic partial differential equations on Riemannian manifolds, moving surfaces, general evolving Riemannian manifolds (with appropriate assumptions) and Riemannian manifolds with random metrics, in the variational setting of the analysis to stochastic partial differential equations. Considering mainly linear stochastic partial differential equations, we establish various existence and uniqueness theorems. 
C. M. Elliott, M. Hairer, M. R. Scott
math.AP math.DG