We consider a p-Laplace evolution problem with multiplicative noise on a bounded domain D ⊂ R d with homogeneous Dirichlet boundary conditions for 1 < p < ∞. The random initial data is merely integrable. Consequently, the key estimates are available with respect to truncations of the solution. We introduce the notion of renormalized solutions for multiplicative stochastic p-Laplace equations with L 1-initial data and study existence and uniqueness of solutions in this framework.

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<p style='text-indent:20px;'>We consider a <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplace evolution problem with multiplicative noise on a bounded domain <inline-formula><tex-math id="M4">\begin{document}$ D\subset\mathbb{R}^d $\end{document}</tex-math></inline-formula> with homogeneous Dirichlet boundary conditions for <inline-formula><tex-math id="M5">\begin{document}$ 1&lt;p&lt;\infty $\end{document}</tex-math></inline-formula>. The random initial data is merely integrable. Consequently, the key estimates are available with respect to truncations of the solution. We introduce the notion of renormalized solutions for multiplicative stochastic <inline-formula><tex-math id="M6">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplace equations with <inline-formula><tex-math id="M7">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula>-initial data and study existence and uniqueness of solutions in this framework.</p>