It\^o versus Stratonovich in a stochastic cosmological model

TL;DR

This paper compares Itô and Stratonovich interpretations of stochastic noise in a cosmological model, showing that the choice affects the solution's existence and long-term behavior of the Hubble parameter.

Contribution

It demonstrates that the interpretation of stochastic noise critically influences the solution's blow-up probability and asymptotic behavior in a stochastic cosmological model.

Findings

Stratonovich noise leads to finite time blow-up with positive probability.

Itô noise ensures global existence of solutions almost surely.

Expected asymptotic behavior is characterized under Itô interpretation.

Abstract

In this work we study a stochastic version of the Friedmann acceleration equation. This model has been proposed in the cosmology literature as a possible explanation of the uncertainty found in the experimental quantification of the Hubble parameter. Its noise has been tacitly interpreted in the Stratonovich sense. Herein we prove that this interpretation leads to a positive probability of finite time blow-up of the solution, that is, of the Hubble parameter. In contrast, if we just modify the noise interpretation to that of It\^o, then the solution globally exists almost surely. Moreover, the expected asymptotic behavior is found under this interpretation too.