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

TL;DR

This paper establishes well-posedness for stochastic reaction diffusion equations driven by Lévy noise in Banach spaces, extending existing results to the space of continuous functions and unifying different stochastic frameworks.

Contribution

It introduces new well-posedness results for reaction diffusion equations with Lévy noise on $C(ar{O})$, a setting not previously addressed in the Lévy case, and unifies various existing frameworks.

Findings

Existence and uniqueness of mild solutions in $L^p$ and $C(ar{O})$ spaces.

Solutions have càdlàg modifications even with Lévy noise not taking values in the solution space.

Extension of linear problem analysis to nonlinear reaction diffusion equations with Lévy 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.