Stochastic dissipative systems in Banach spaces driven by Lévy noise

In this paper, we are interested in the well-posedness of stochastic reaction diffusion equations like \begin{gather} \label{ciaoI} \eqsys{ \vspace{1pt} dX(t)(\xi)=\big[\Delta_\xi X(t)(\xi)-p(X(t)(\xi))\big]dt+RdW(t)+dL(t),\\ X(0)(\xi)=x(\xi)\in L^2(\mathcal{O}), } \end{gather} where $T>0$, $t\in [0,T]$, $\mathcal{O}$ is a bounded open domain of $\R^d$ with regular boundary, $d\in\N$, $\xi\in\mathcal{O}$, $p:\R\rightarrow\R$ is a polynomial of odd degree with positive leading coefficient, $R$ is a linear bounded operator on $L^2(\mathcal{O})$, $\{W(t)\}_{t\geq 0}$ is a $L^2(\mathcal{O})$-cylindrical Wiener process and $\{L(t)\}_{t\geq 0}$ is a pure-jump L\'evy process on $L^2(\mathcal{O})$. We complement the equation \eqref{ciaoI} with suitable boundary conditions on $\partial \mathcal{O}.$ In \cite[Chapter 10]{Pes-Zab2007} and \cite{Mar-Pre-Roc2010,Mar-Roc2010} the authors study existence and uniqueness of mild solutions for every $x\in L^p(\mathcal{O})$, for some suitable $p\geq 2$. The results of this paper allow to study reaction diffusion equations also on the space of continuous functions $C(\overline{O})$. This seems to be new in the L\'evy case (it is already done in the Wiener case in \cite[Chapter 6]{Cer2001}). We also discuss and review the previous cited works with the aim of unifying the different frameworks. We underline that, when $R=0$, for every $x\in C(\overline{O})$ (or $x\in L^p(\mathcal{O})$) the mild solution to \eqref{ciao} has a càdlàg modification in $C(\overline{O})$ (or $\in L^p(\mathcal{O})$), even if $\{L(t)\}_{t \geq 0}$ is not a L\'evy process taking values in $C(\overline{O})$ (or $\in L^p(\mathcal{O})$). This phenomenon for the linear problem (i.e., $F\equiv 0$ in the SPDE) has been investigated in \cite{Pes-Zab2013}; see also the references therein. \end{abstract}