The Unequal Twins - Probability Distributions Aren't Everything

TL;DR

This paper demonstrates that identical probability distribution functions do not fully characterize stochastic models, as models with the same PDFs can differ significantly in other dynamic properties like FPT, autocorrelation, and aging.

Contribution

It reveals that models with identical PDFs can have different underlying dynamics, challenging the assumption that PDFs alone define stochastic behavior.

Findings

Two models with identical PDFs differ in FPT distributions

They exhibit different autocorrelation functions

They show distinct aging properties

Abstract

It is the common lore to assume that knowing the equation for the probability distribution function (PDF) of a stochastic model as a function of time tells the whole picture defining all other characteristics of the model. We show that this is not the case by comparing two exactly solvable models of anomalous diffusion due to geometric constraints: The comb model and the random walk on a random walk (RWRW). We show that though the two models have exactly the same PDFs, they differ in other respects, like their first passage time (FPT) distributions, their autocorrelation functions and their aging properties.