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We derive a posteriori error estimates for a fully discrete time-implicitfinite element approximation of the stochastic total variaton flow (STVF) withadditive space time noise. The estimates are first derived for an implementablefully discrete approximation of a regularized stochastic total variation flow.We then show that the derived a posteriori estimates remain valid for theunregularized flow up to a perturbation term that can be controlled by theregularization parameter. Based on the derived a posteriori estimates wepropose a pathwise algorithm for the adaptive space-time refinement and performnumerical simulation for the regularized STVF to demonstrate the behavior ofthe proposed algorithm.

A posteriori estimates for the stochastic total variation flow