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We provide a general construction scheme for $\mathcal L^p$-strong Fellerprocesses on locally compact separable metric spaces. Starting from a regularDirichlet form and specified regularity assumptions, we construct an associatedsemigroup and resolvents of kernels having the $\mathcal L^p$-strong Fellerproperty. They allow us to construct a process which solves the correspondingmartingale problem for all starting points from a known set, namely the setwhere the regularity assumptions hold. We apply this result to constructelliptic diffusions having locally Lipschitz matrix coefficients and singulardrifts on general open sets with absorption at the boundary. In thisapplication elliptic regularity results imply the desired regularityassumptions.

Construction of $\mathcal L^p$-strong Feller Processes via Dirichlet Forms and Applications to Elliptic Diffusions