Analysis of a model for wealth redistribution
Daniel Matthes Giuseppe Toscani
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
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
Computing coherent sets using the Fokker-Planck equation
Andreas Denner Oliver Junge Daniel Matthes
We perform a numerical approximation of coherent sets in finite-dimensional smooth dynamical systems by computing singular vectors of the transfer operator for a stochastically perturbed flow. This operator is obtained by solution of a discretized Fokker-Planck equation. For numerical implementation, we employ spectral collocation methods and an exponential time differentiation scheme. We experimentally compare our approach with the more classical method by Ulam that is based on discretization of the transfer operator of the unperturbed flow.
keywords: transfer operator Coherent set Fokker-Planck equation.

Year of publication

Related Authors

Related Keywords

[Back to Top]