# On the Change of Measure for Brownian Processes

## TL;DR

This paper shows that the physical limit of Brownian processes differs from the commonly accepted mathematical expression.

## Contribution

The paper derives the correct continuous-time limit of discrete-time Brownian dynamics and challenges existing concepts.

## Key findings

- The physical limit of Brownian processes differs from the commonly used mathematical expression.
- Concepts like 'most probable path' and 'minimum thermodynamic action' are shown to be unphysical artifacts.

## Abstract

Sometimes, limits can be singular, implying that they take on different values depending on the order of arithmetic operations. In other words, the limit map lacks commutativity. While all such limits are mathematically valid, only one can be the physical limit. The change of measure for Brownian processes illustrates this phenomenon. A substantial body of elegant mathematics centered around continuous-time Brownian processes has been embraced by the physics community to investigate the nonequilibrium and equilibrium thermodynamics of systems composed of atoms and molecules. In this paper, we derive the continuous-time limit of discrete-time Brownian dynamics, specifically focusing on the change of measure. We demonstrate that this result yields the physical limit that differs from the commonly used expression. Consequently, the concepts of “the most probable path”, “minimum thermodynamic action”, and “the small-noise limit” are unphysical mathematical artifacts.