The Stability of some stochastic processes

TL;DR

This paper introduces a new criterion for the stability of e-processes and demonstrates its application to specific shell models, establishing their asymptotic stability through theoretical proofs.

Contribution

It formulates a novel stability criterion for e-processes and applies it to shell models, proving their stability based on boundedness and concentration properties.

Findings

E-processes that are averagely bounded and concentrating are asymptotically stable.

The criterion is successfully applied to Goy and Sabra shell models.

Processes in these models satisfy the e-process property and stability conditions.

Abstract

We formulate and prove a new criterion for stability of e-processes. It says that any e-process which is averagely bounded and concentrating is asymptotically stable. In the second part, we show how this general result applies to some shell models (the Goy and the Sabra model). Indeed, we manage to prove that the processes corresponding to these models satisfy the e-process property. They are also averagely bounded and concentrating. Consequently, their stability follows.