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$\left\{\begin{array}{l}d X(t)(\xi) = \left[\Delta_{\xi} X(t)(\xi)-p(X(t)(\xi))\right] d t+R d W(t)+d L(t), \\X(0)(\xi) = x(\xi) \in L^2(\mathcal{O}), \end{array}\right.\;\;\;\;(1)$where $ T>0 $, $ t\in [0, T] $, $ \mathcal{O} $ is a bounded open domain of $ {\mathbb R}^d $ with regular boundary, $ d\in {\mathbb N} $, $ \xi\in\mathcal{O} $, $ p: {\mathbb R}\rightarrow {\mathbb 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 Lévy process on $ L^2(\mathcal{O}) $. We complement equation (1) with suitable boundary conditions on $ \partial \mathcal{O}. $ In [33, Chapter 10] and [29,30], the authors studied 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 Lévy case (it is already done in the Wiener case in [12, Chapter 6]). 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 (17) has a càdlàg modification in $ C(\overline{O}) $ (or $ \in L^p(\mathcal{O}) $), even if $ \{L(t)\}_{t \geq 0} $ is not a Lévy 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 [34]; see also the references therein.