Tail behaviour of stationary densities for one-dimensional random diffeomorphisms

TL;DR

This paper investigates how the stationary densities of one-dimensional random diffeomorphisms behave near their support boundaries, revealing dependence on noise distribution and local dynamics at fixed points.

Contribution

It provides a detailed analysis of boundary behavior of stationary densities, linking it to noise characteristics and local fixed point dynamics of the diffeomorphisms.

Findings

Stationary density behavior depends on noise distribution.

Boundary behavior linked to fixed point linearisation or nonlinear terms.

Results differentiate hyperbolic and non-hyperbolic fixed points.

Abstract

We study the asymptotic behaviour of stationary densities of one-dimensional random diffeomorphisms, at the boundaries of their support, which correspond to deterministic fixed points of extremal diffeomorphisms. In particular, we show how this stationary density at a boundary depends on the underlying noise distribution, as well as the linearisation of the extremal diffeomorphism at the boundary point (in case the corresponding fixed point is hyperbolic), or the leading nonlinear term of the extremal diffeomorphism (in case the corresponding fixed point is not hyperbolic).