Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on   metric measure spaces
  Starting with a regular symmetric Dirichlet form on a locally compact separable metric space $X$, our paper studies elements of vector analysis, $L_p$-spaces of vector fields and related Sobolev spaces. These tools are then employed to obtain existence and uniqueness results for some quasilinear elliptic PDE and SPDE in variational form on $X$ by standard methods. For many of our results locality is not assumed, but most interesting applications involve local regular Dirichlet forms on fractal spaces such as nested fractals and Sierpinski carpets. 
Michael Hinz, Michael R\"ockner, Alexander Teplyaev
math.FA math.PR