Well-posedness and existence of an invariant measure for the linearly-damped KdV equations driven by a jump noise

In this paper, we investigate the linearly damped KdV equation on the one-dimensional torus $\mathbb{T}$, perturbed by a multiplicative Lévy noise. For any damping coefficient $\gamma >0$, we establish the existence and uniqueness of a pathwise weak solution with values in $H^2(\mathbb{T})$. In the second part of the paper, we analyze the long-time behavior of these solutions. This study is particularly subtle as the presence of jumps in time can significantly influence the asymptotics. We show, using the techniques of Maslowski and Seidler, that, provided the frictional damping coefficient $\gamma >0$ is sufficiently large, the system influenced by square-integrable jumps admits an invariant measure in $H^2(\mathbb{T})$.