Abstract
We consider the dynamics of the particle influences by driving force, Gaussian and Poisson noises, which is a modification of the model by Friedman et al (2006 Phys. Rev. Lett. 97 168302). We study the relative integro-differential Fokker–Planck equation (IDFPE) for the probability density distribution and obtain the exact steady state solution of the IDFPE, which is consistent with both analytical and numerical solutions of the IDFPE in the limit of large time t. As in the case of zero Gaussian noise, there is a threshold value of the level of Poissonian noise, related to the intensity of the driving force: below the threshold the maximum of distribution is at the origin; nevertheless, even weak Gaussian noise prevents the formation of a singularity at the maximum point. Above the threshold value, the Gaussian noise cannot shift the maximum to the origin, but makes it less pronounced.