This paper is concerned with the stochastic diffusion equation \(dX(t)=\text{div}[\text{sgn}(\nabla(X(t)))]dt+\sqrt{Q}dW(t)\) in \((0,\infty) \times \mathcal{O}\), where \(\mathcal{O}\) is a bounded open subset of \(\mathbb{R}^d, d=1,2, W(t)\) is a cylindrical Wiener process on \(L^2(\mathcal{O})\), and \(\text{sgn}(\nabla X)=\nabla X/|\nabla X|_d\) if \(\nabla X\neq 0\) and sgn \((0)=\{v\in \mathbb{R}^d:|v|_d\leq 1\}\). The multivalued and highly singular diffusivity term \(\text{sgn}(\nabla X)\) describes interaction phenomena, and the solution \(X=X(t)\) might be viewed as the stochastic flow generated by the gradient of the total variation \(\|DX\|\). The main result says that this problem is well posed in the space of processes with bounded variation in the spatial variable \(\xi \). The above equation is relevant for modeling crystal growth as well as for total variation based techniques in image restoration.