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A step reinforced random walk is a discrete time process with memory suchthat at each time step, with fixed probability $p \in (0,1)$, it repeats apreviously performed step chosen uniformly at random while with complementaryprobability $1-p$, it performs an independent step with fixed law. In thecontinuum, the main result of Bertoin in [7] states that the random walkconstructed from the discrete-time skeleton of a L\'evy process for a timepartition of mesh-size $1/n$ converges, as $n \uparrow \infty$ in the sense offinite dimensional distributions, to a process $\hat{\xi}$ referred to as anoise reinforced L\'evy process. Our first main result states that a noisereinforced L\'evy processes has rcll paths and satisfies a $\textit{noisereinforced}$ L\'evy It\^o decomposition in terms of the $\textit{noisereinforced}$ Poisson point process of its jumps. We introduce the jointdistribution of a L\'evy process and its reinforced version $(\xi, \hat{\xi})$and show that the pair, conformed by the skeleton of the L\'evy process and itsstep reinforced version, converge towards $(\xi, \hat{\xi})$ as the mesh sizetend to $0$. As an application, we analyse the rate of growth of $\hat{\xi}$ atthe origin and identify its main features as an infinitely divisible process.

Noise Reinforced L\'evy Processes: L\'evy-It\^o Decomposition and Applications