Well-posedness of semilinear SPDEs with singular drift: a variational approach

We prove well-posedness for singular semilinear SPDEs on a smooth bounded domain $D$ in $\mathbb{R}^n$ of the form \[ dX(t) + AX(t)\,dt + \beta(X(t))\,dt \ni B(t,X(t))\,dW(t)\,, \qquad X(0)=X_0\,. \]

The linear part is associated to a linear coercive maximal monotone operator $A$ on $L^2(D)$, while $\beta$ is a (multivalued) maximal monotone graph everywhere defined on $\mathbb{R}$ on which no growth nor smoothness conditions are required. Moreover, the noise is given by a cylindrical Wiener process on a Hilbert space $U$, with a stochastic integrand $B$ taking values in the Hilbert-Schmidt operators from $U$ to $L^2(D)$: classical Lipschitz-continuity hypotheses for the diffusion coefficient are assumed. A comparison with the corresponding deterministic equation and possible generalizations are discussed.

This study is based on a joint work with Carlo Marinelli (University College London).