Stochastic Nonlinear Diffusion Equations with Singular Diffusivity
Viorel Barbu, Giuseppe Da Prato, Michael Röckner · SIAM Journal on Mathematical Analysis · 2009
In this paper we are concerned with the stochastic diffusion equation $dX(t)=\mathrm{div}[\mathrm{sgn}( abla(X(t)))]dt+\sqrt{Q}\;dW(t)$ 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 $\mathrm{sgn}( abla X)= abla X/| abla X|_d$ if $ abla X eq 0$ and sgn $(0)=\{v\in\mathbb{R}^d:|v|_d\le1\}$. The multivalued and highly singular diffusivity term $\mathrm{sgn}( abla X)$ describes interaction phenomena, and the solution $X=X(t)$ might be viewed as the stochastic flow generated by the gradient of the total variation $\|DX\|$. Our 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.