  We derive a posteriori error estimates for a fully discrete time-implicit
finite element approximation of the stochastic total variaton flow (STVF) with
additive space time noise. The estimates are first derived for an implementable
fully discrete approximation of a regularized stochastic total variation flow.
We then show that the derived a posteriori estimates remain valid for the
unregularized flow up to a perturbation term that can be controlled by the
regularization parameter. Based on the derived a posteriori estimates we
propose a pathwise algorithm for the adaptive space-time refinement and perform
numerical simulation for the regularized STVF to demonstrate the behavior of
the proposed algorithm.
