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Article

On the Change of Measure for Brownian Processes

by
Francis J. Pinski
Department of Physics, University of Cincinnati, Cincinnati, OH 45221, USA
Entropy 2025, 27(6), 594; https://doi.org/10.3390/e27060594 (registering DOI)
Submission received: 5 February 2025 / Revised: 27 April 2025 / Accepted: 27 May 2025 / Published: 31 May 2025
(This article belongs to the Section Non-equilibrium Phenomena)

Abstract

Sometimes, limits can be singular, implying that they assume 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. For instance, 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.
Keywords: stochastic differential equations; Brownian dynamics; singular limit; change of measure stochastic differential equations; Brownian dynamics; singular limit; change of measure

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MDPI and ACS Style

Pinski, F.J. On the Change of Measure for Brownian Processes. Entropy 2025, 27, 594. https://doi.org/10.3390/e27060594

AMA Style

Pinski FJ. On the Change of Measure for Brownian Processes. Entropy. 2025; 27(6):594. https://doi.org/10.3390/e27060594

Chicago/Turabian Style

Pinski, Francis J. 2025. "On the Change of Measure for Brownian Processes" Entropy 27, no. 6: 594. https://doi.org/10.3390/e27060594

APA Style

Pinski, F. J. (2025). On the Change of Measure for Brownian Processes. Entropy, 27(6), 594. https://doi.org/10.3390/e27060594

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