Stochastic Nonlinear Diffusion Equations with Singular Diffusivity

Barbu V, Da Prato G, Röckner M. Stochastic Nonlinear Diffusion Equations with Singular Diffusivity . SIAM Journal on Mathematical Analysis. 2009;41(3):1106-1120.In this paper we are concerned with the stochastic diffusion equation dX(t) = div[sgn(del(X(t)))]dt + root Q dW(t) in (0, infinity) x O, where O is a bounded open subset of R-d, d = 1, 2, W(t) is a cylindrical Wiener process on L-2(O), and sgn(del X) = del X/vertical bar X vertical bar(d) if del X not equal 0 and sgn (0) = {v is an element of R-d : vertical bar v vertical bar(d) <= 1}. The multivalued and highly singular diffusivity term sgn(del 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 parallel to DX parallel to. Our main result says that this problem is well posed in the space of processes with bounded variation in the spatial variable.. The above equation is relevant for modeling crystal growth as well as for total variation based techniques in image restoration