Communications in Mathematical Sciences

Volume 16 (2018)

Number 2

Wealth distribution in presence of debts: A Fokker–Planck description

Pages: 537 – 560



Marco Torregrossa (Department of Mathematics, University of Pavia, Italy)

Giuseppe Toscani (Department of Mathematics, University of Pavia, Italy; and IMATI of CNR, Pavia, Italy)


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.


wealth distribution, Fokker–Planck equation, Fourier-based metrics, convergence to equilibrium

2010 Mathematics Subject Classification

35B40, 82C40, 91B60

This work has been written within the activities of the National Group of Mathematical Physics (GNFM) of INdAM (National Institute of High Mathematics), and partially supported by the MIUR-PRIN Grant 2015PA5MP7 “Calculus of Variations”.

Received 26 September 2017

Accepted 14 January 2018

Published 14 May 2018