Multi-valued, singular stochastic evolution inclusions

We provide an abstract variational existence and uniqueness result for multi-valued, monotone, non-coercive stochastic evolution inclusions in Hilbert spaces with general additive and Wiener multiplicative noise. As examples we discuss certain singular diffusion equations such as the stochastic 1-Laplacian evolution (total variation flow) in all space dimensions and the stochastic singular fast diffusion equation. In case of additive Wiener noise we prove the existence of a unique weak-* mean ergodic invariant measure.Comment: 39 pages, in press: J. Math. Pures Appl. (2013

Well-posedness of SVI solutions to singular-degenerate stochastic porous media equations arising in self-organised criticality

We consider a class of generalised stochastic porous media equations with multiplicative Lipschitz continuous noise. These equations can be related to physical models exhibiting self-organised criticality. We show that these SPDEs have unique SVI solutions which depend continuously on the initial value. In order to formulate this notion of solution and to prove uniqueness in the case of a slowly growing nonlinearity, the arising energy functional is analysed in detail.Comment: Version submitted to journa

On a stochastic singular diffusion equation in Rd

AbstractWe establish the existence and uniqueness of a strong solution to the Cauchy problem for a singular diffusion equation with random noise in Rd with initial data in L2(Rd) with bounded variation or in H1(Rd). We also prove the existence of an invariant measure and extinction of a solution in finite time

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