Stochastic equation for a jumping process with long-time correlations

A jumping process, defined in terms of the jump size and waiting time distributions, is presented. The jumping rate depends on the process value. The process, which is Markovian and stationary, relaxes to an equilibrium and is characterized by a power-law autocorrelation function. Therefore, it can serve as a model of 1/f noise as well as of the stochastic force in the generalized Langevin equation. This equation is solved for noise correlations approximately 1/t ; the resulting velocity distribution has sharply falling tails. The system preserves memory about the initial condition for a very long time.