Analysis of a model for wealth redistribution
Daniel Matthes Giuseppe Toscani
Kinetic & Related Models 2008, 1(1): 1-27 doi: 10.3934/krm.2008.1.1
A recent application of the kinetic theory for many particle systems is the description of the redistribution of wealth among trading agents in a simple market economy. This paper provides an analytical investigation of the particular model with quenched saving propensities, which has been introduced by Chakrabarti, Chatterjee and Manna [11]. We prove uniqueness and dynamical stability of the stationary solution to the underlying Boltzmann equation, and provide estimates on the rate of equilibration. As one main result, we obtain that realistic steady wealth distributions with Pareto tail are only algebraically stable in this framework.
keywords: Boltzmann equation Pareto distribution. Econophysics
A gradient flow scheme for nonlinear fourth order equations
Bertram Düring Daniel Matthes Josipa Pina Milišić
Discrete & Continuous Dynamical Systems - B 2010, 14(3): 935-959 doi: 10.3934/dcdsb.2010.14.935
We propose a method for numerical integration of Wasserstein gradient flows based on the classical minimizing movement scheme. In each time step, the discrete approximation is obtained as the solution of a constrained quadratic minimization problem on a finite-dimensional function space. Our method is applied to the nonlinear fourth-order Derrida-Lebowitz-Speer-Spohn equation, which arises in quantum semiconductor theory. We prove well-posedness of the scheme and derive a priori estimates on the discrete solution. Furthermore, we present numerical results which indicate second-order convergence and unconditional stability of our scheme. Finally, we compare these results to those obtained from different semi- and fully implicit finite difference discretizations.
keywords: Wasserstein gradient flow numerical solution. Higher-order diffusion equation

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