Well-posedness of renormalized solutions for a stochastic $p$-Laplace equation with $L^1$-initial data

Abstract

We consider a $p$-Laplace evolution problem with stochastic forcing on a bounded domain $D\subset\mathbb{R}^d$ with homogeneous Dirichlet boundary conditions for $1<p<\infty$. The additive noise term is given by a stochastic integral in the sense of It\^{o}. The technical difficulties arise from the merely integrable random initial data $u_0$ under consideration. Due to the poor regularity of the initial data, estimates in $W^{1,p}_0(D)$ are available with respect to truncations of the solution only and therefore well-posedness results have to be formulated in the sense of generalized solutions. We extend the notion of renormalized solution for this type of SPDEs, show well-posedness in this setting and study the Markov properties of solutions.