# Phenomenology of stochastic exponential growth

## TL;DR

This paper explores a broader class of models for stochastic exponential growth beyond the traditional Geometric Brownian Motion, emphasizing power law noise to better match experimental observations of biological and physical systems.

## Contribution

It introduces a generalized phenomenological model with power law multiplicative noise, providing analytical solutions and a method to infer parameters from experimental data.

## Key findings

- Mean-rescaled distributions are approximately stationary at long times.
- Power law noise models better fit experimental data than GBM.
- Analytical solutions and parameter inference methods are developed.

## Abstract

Stochastic exponential growth is observed in a variety of contexts, including molecular autocatalysis, nuclear fission, population growth, inflation of the universe, viral social media posts, and financial markets. Yet literature on modeling the phenomenology of these stochastic dynamics has predominantly focused on one model, Geometric Brownian Motion (GBM), which can be described as the solution of a Langevin equation with linear drift and linear multiplicative noise. Using recent experimental results on stochastic exponential growth of individual bacterial cell sizes, we motivate the need for a more general class of phenomenological models of stochastic exponential growth, which are consistent with the observation that the mean-rescaled distributions are approximately stationary at long times. We show that this behavior is not consistent with GBM, instead it is consistent with power law multiplicative noise with positive fractional powers. Therefore, we consider this general class of phenomenological models for stochastic exponential growth, provide analytical solutions, and identify the important dimensionless combination of model parameters which determines the shape of the mean-rescaled distribution. We also provide a prescription for robustly inferring model parameters from experimentally observed stochastic growth trajectories.