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Register a new userNoise reinforcement for L{e}vy processes
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Authors
Jean Bertoin
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In a step reinforced random walk, at each integer time and with a fixed probability p $in$ (0, 1), the walker repeats one of his previous steps chosen uniformly at random, and with complementary probability 1 -- p, the walker makes an independent new step with a given distribution. Examples in the literature include the so-called elephant random walk and the shark random swim. We consider here a continuous time analog, when the random walk is replaced by a L{e}vy process. For sub-critical (or admissible) memory parameters p < p c , where p c is related to the Blumenthal-Getoor index of the L{e}vy process, we construct a noise reinforced L{e}vy process. Our main result shows that the step-reinforced random walks corresponding to discrete time skeletons of the L{e}vy process, converge weakly to the noise reinforced L{e}vy process as the time-mesh goes to 0.
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We obtain the lower bounds for ergodic convergence rates, including spectral gaps and convergence rates in strong ergodicity for time-changed symmetric L{e}vy processes by using harmonic function and reversible measure. As direct applications, explicit sufficient conditions for exponential and strong ergodicity are given. Some examples are also presented.
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