Random additive perturbation of a $k$ term recurrence relation

TL;DR

This paper investigates stochastic processes defined by nonlinear recurrence relations with additive noise, establishing conditions for the existence of an invariant measure, which is crucial for understanding their long-term behavior.

Contribution

The paper provides new sufficient conditions for the existence of an invariant measure in nonlinear recurrence relations with additive noise.

Findings

Existence of invariant measure established under specific conditions.

Applicable to a class of nonlinear stochastic recurrence relations.

Contributes to understanding the stability of such processes.

Abstract

We are interested in stochastic processes satisfying a nonlinear recurrence relation of the form $$X_{n + k} = \Phi_0 (X_n, ..., X_{n + k - 1}) + \Theta_n$$ where $\Theta$ is a noise term. We establish the existence of an invariant measure for this process under given sufficient conditions on $\Phi_0.$