# Slow-roll versus stochastic slow-roll inflation

## TL;DR

This paper analyzes the stochastic dynamics of slow-roll inflation incorporating quantum and thermal noise, demonstrating conditions under which slow-roll can be probabilistically maintained and exploring the impact on inflationary variables.

## Contribution

It introduces a stochastic Einstein-Klein-Gordon framework with quantum and thermal noise, showing slow-roll conditions can hold probabilistically over long times and large field ranges.

## Key findings

- Slow-roll conditions can be satisfied probabilistically with noise.
- Quantum and thermal fluctuations affect inflationary variable expectations.
- Mean acceleration can be positive or negative depending on the model.

## Abstract

We consider the classical wave equation with a thermal and Starobinsky-Vilenkin noise which in a slow-roll and long wave approximation describes the quantum fluctuations of the graviton-inflaton system in an expanding metric. We investigate the resulting consistent stochastic Einstein-Klein-Gordon system in the slow-roll regime. We show in some models that the slow-roll requirements (of the negligence of $\partial_{t}^{2}\phi$) can be satisfied in the probabilistic sense for the stochastic system with quantum and thermal noise for arbitrarily large time and an infinite range of fields. We calculate expectation values of some inflationary variables taking into account quantum and thermal noise. We show that the mean acceleration $\langle \partial_{t}^{2}a\rangle$ can be negative or positive (depending on the model) when the random fields take values beyond the classical range of inflation.