Regularity of solutions of the Dyson equation and applications

Abstract

The goal of this short paper is to investigate the regularity of the solutions of the Dyson equation. In the work of Bertucci and al. [3, 4, 5], a new notion of solutions for the Dyson equation has been introduced using the viscosity solutions theory and they proved a regularization of solutions in L^{$\infty$} . According to the work of Biane [6] we should expect a regularization in C^{1/3} of solutions (and not better). In the spirit of the works of Bertucci and al., we shall prove using PDE methods that for almost all time t \ge 0 the solution is as expected in C^{1/3} . Our approach allows us to extend this result in addition to a drift term in the Dyson equation for which the semi explicit solutions is not necessarily known. We shall also give an application of this result, proving that the solutions of the periodic Dyson equation converge in long-time toward the uniform distribution on the circle in L^{$\infty$} norm which was an open question in [5].