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In a step reinforced random walk, at each integer time and with a fixedprobability p $\in$ (0, 1), the walker repeats one of his previous steps chosenuniformly at random, and with complementary probability 1 -- p, the walkermakes an independent new step with a given distribution. Examples in theliterature include the so-called elephant random walk and the shark randomswim. We consider here a continuous time analog, when the random walk isreplaced by a L{\'e}vy process. For sub-critical (or admissible) memoryparameters p < p c , where p c is related to the Blumenthal-Getoor index of theL{\'e}vy process, we construct a noise reinforced L{\'e}vy process. Our mainresult shows that the step-reinforced random walks corresponding to discretetime skeletons of the L{\'e}vy process, converge weakly to the noise reinforcedL{\'e}vy process as the time-mesh goes to 0.

Noise reinforcement for L{\'e}vy processes