# Polynomial behavior in mean of stochastic skew-evolution semiflows

## TL;DR

This paper introduces a generalized concept of polynomial stability in mean for stochastic skew-evolution semiflows, extending classical deterministic results using probabilistic and functional analysis techniques.

## Contribution

It develops a new framework for polynomial stability in mean in stochastic settings, generalizing classical stability concepts and extending Datko's theorem.

## Key findings

- Established variants of Datko's theorem for polynomial stability in mean.
- Extended deterministic techniques to stochastic skew-evolution semiflows.
- Provided conditions for polynomial (in)stability in mean in a probabilistic context.

## Abstract

In this paper, we are interested in the more general concept of a polynomial (in)stability in mean in which the polynomial behaviour in the classical sense is replaced by a weaker requirement with respect to some probability measure. This concept includes the classical concepts of a polynomial (in)stability as particular cases. Extending techniques employed in the deterministic case, we obtain variants of a well-known theorem of Datko for a polynomial (in)stability in mean. This is done by using the techniques of stochastic skew-evolution semiflows and Banach spaces of functions or sequences.