Stochastic dissipative systems in Banach spaces driven by L\'evy noise

Abstract

In this paper, we are interested in the well-posedness of stochastic reaction diffusion equations like \begin{equation} \begin{cases} dX(t)(\xi)=\big(\Delta_\xi X(t)(\xi)-p(X(t)(\xi))\big)dt+RdW(t)+dL(t) , \quad t\in [0,T];\\ X(0)=x\in L^2(\mathcal{O}) \end{cases} \end{equation} where $\mathcal{O}$ is a bounded open domain of $\mathbb{R}^d$ with regular boundary, $d\in\mathbb{N}$, $p:\mathbb{R}\rightarrow\mathbb{R}$ 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, $\{L(t)\}_{t\geq 0}$ is a pure-jump L\'evy process on $L^2(\mathcal{O})$. We complement the equation with suitable boundary conditions on $\partial \mathcal{O}.$ Some papers in literature analize 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 function $C(\overline{O})$. This seems to be new in the L\'evy case (it is already done in the Wiener case).\\ 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 the equation has a c\`adl\`ag modifications 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 other papers.