Solving the generalized Langevin equation with the algebraically correlated noise

We solve the Langevin equation with the memory kernel. The stochastic force possesses algebraic correlations, proportional to $1/t.$ The velocity autocorrelation function and related quantities characterizing transport properties are calculated with the assumption that the system is in thermal equilibrium. Stochastic trajectories are simulated numerically, using the kangaroo process as a noise generator. Results of this simulation resemble L\'evy walks with divergent moments of the velocity distribution. We consider motion of a Brownian particle, both without any external potential and in the harmonic oscillator field, in particular the escape from a potential well. The results are compared with memory-free calculations for the Brownian particle.