Stochastic dissipative systems in Banach spaces driven by Lévy noise
Davide A. Bignamini, Enrico Priola
TL;DR
This work establishes well-posedness for stochastic dissipative systems in infinite dimensions driven by Lévy noise, focusing on stochastic reaction-diffusion-type equations. By decomposing the solution as $X(t)=Y(t)+W_A(t)+L_A(t)$ and analyzing a random PDE with dissipative drift, the authors prove existence and uniqueness of generalized mild solutions on a Banach space $E$ (e.g., $C(\overline{\mathcal{O}})$ or $L^p$ spaces) and extend these results to $H$ via density, under two sets of hypotheses handling different ambient spaces. The analysis leverages Yosida approximations, dissipativity, and careful regularity estimates for the stochastic convolutions, including cadlag properties for both Wiener and pure-jump Lévy noise, and yields stability bounds with respect to initial data. The results unify and extend prior Wiener- and Lévy-driven frameworks and enable well-posedness for reaction-diffusion SPDEs in spaces relevant to continuous-function and $L^p$-type initial data, with potential applications in dissipative dynamics under jump noise. Overall, the paper provides a rigorous pathwise approach to existence, uniqueness, and regularity of semilinear SPDEs driven by Lévy noise in Banach/Hilbert settings, offering a versatile toolkit for infinite-dimensional stochastic dissipative systems.
Abstract
In this paper, we are interested in the well-posedness of stochastic reaction diffusion equations like \begin{equation} \begin{cases} dX(t)(ξ)=\big(Δ_ξX(t)(ξ)-p(X(t)(ξ))\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 Lévy 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 Lévy 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 càdlàg modifications 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 other papers.