Stochastic partial differential equations of the form \[ dX_t=\operatorname{div}(\phi(X_t))\,dt+B(X_t)\,dW_t,\quad X_0=x_0,\tag{1} \] with the homogeneous Dirichlet boundary condition on a bounded convex domain with smooth boundary \(\mathcal O\subseteq\mathbb R^d\) for \(d\leq 6\) are studied. Under fairly general assumptions on \(\phi\) and \(B\) – covering the mean curvature flow in one dimension, the minimal surface/image denoising and the additive or the linear multiplicative diffusion terms – the equation (1) can be rewritten in a relaxed form \[ dX_t\in-\partial\varphi(X_t)\,dt+B(X_t)\,dW_t,\quad X_0=x_0,\tag{2} \] for a suitably defined \(\varphi\), and it is shown that it has a unique generalized solution if \(x_0\in L^2(\Omega;L^2(\mathcal O))\). If, in addition, \(\mathbb E\,\varphi(x_0)<\infty\) then the unique generalized solution is also a strong solution (in the PDE sense defined in the paper). In both cases, the authors prove an energy estimate that implies a better regularity of the unique solutions.
The above result, however, does not cover the mean curvature flow with a homogeneous normal noise, i.e., \[ dX_t=\frac{\partial^2_xX_t}{1+(\partial _xX_t)^2}\,dt+\alpha\sqrt{1+(\partial_xX_t)^2}\circ\,d\beta_t,\quad X_0=x_0,\tag{3} \] on \(\mathcal O=(0,1)\) with the periodic boundary condition, where \(\beta\) is a real-valued Brownian motion. The equation (3) is rewritten as \[ dX_t\in\frac{\alpha^2}{2}\partial^2_xX_t\,dt-\partial\varphi(X_t)\,dt+B(X_t)\,d\beta_t,\quad X_0=x_0,\tag{4} \] for a suitably defined \(\varphi\). In this case, the existence and uniqueness of solutions is also proved for every \(x_0\in L^2(\Omega;L^2(\mathcal O))\) but the solutions must be understood in a weaker PDE sense as a stochastic variational inequality (which is a weaker notion than the generalized strong solution). The authors also prove an energy estimate that implies a better regularity of the unique solution, and an \(L^2\)-contraction principle \[ \mathbb E\,\|X^{x_0}_t-X^{x_1}_t\|^2_{L^2(\mathcal O)}\leq\mathbb E\,\|x_0-x_1\|^2_{L^2(\mathcal O)}. \]