Wealth distribution in presence of debts. A Fokker--Planck description

We consider here a Fokker–Planck equation with variable coefficient of diffusion which appears in the modeling of the wealth distribution in a multi-agent society. At difference with previous studies, to describe a society in which agents can have debts, we allow the wealth variable to be negative. It is shown that, even starting with debts, if the initial mean wealth is assumed positive, the solution of the Fokker–Planck equation is such that debts are absorbed in time, and a unique equilibrium density located in the positive part of the real axis will be reached.