Linear Langevin equation with time-dependent drift and multiplicative noise term: exact study

We investigate a linear Langevin equation with mulitplicative noise term and time-dependent drift. This system extends the system analyzed by Lillo and Mantegna [Phys. Rev. E 61 (2000) R4675]. We calculate the average of every moments and the variance of the system. We show that the system, with a particular choice of gamma(t), can describe different processes like diffusive, subdiffusive and superdiffusive. Other kinds of processes can also been obtained. We also show that the multiplicative noise term has the effect of splitting the peak of transition probability P(x, t\x', t'), at a fixed time, into two peaks in x coordinate. (C) 2002 Elsevier Science B.V. All rights reserved.