Győző, Egri and Gomori, Marton and Gyenis, Balazs and Hofer-Szabó, Gábor (2026) Trajectory of Probabilities, Probability on Trajectories, and the Stochastic--Quantum Correspondence. [Preprint]

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Abstract

The probabilistic description of the time evolution of a physical system can take two conceptually distinct forms: a trajectory of probabilities, which specifies how probabilities evolve over time, and a probability on trajectories, which assigns probabilities to possible histories. A lack of a clear distinction between these two probabilistic descriptions has given rise to a number of conceptual difficulties, particularly in recent analyses of stochastic-quantum correspondence. This paper provides a systematic account of their relationship. We define probability dynamics and stochastic process families together with a precise notion of implementation that connects the two descriptions. We show that implementations are generically non-unique, that every probability dynamics admits a Markovian implementation, and characterize when non-Markovian implementations are possible. We expose fallacies in common arguments for the linearity of probability dynamics based on the law of total probability and clarify the proper interpretation of "transition matrices" by distinguishing dynamics-level maps from the conditional probability matrices of implementing processes. We further introduce decomposability as the appropriate general notion of stepwise evolution for (possibly nonlinear) probability dynamics, relate it to divisibility in the linear case - showing that the two can come apart - and disentangle both notions from Markovianity and time-homogeneity. Finally, we connect these results to what we call statistical dynamics, in which linearity is indeed physically motivated, and contrast the framework with quantum mechanics.

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Item Type:	Preprint
Creators:	Creators Email ORCID Győző, Egri egri.gyozo@gmail.com 0009-0000-1947-3850 Gomori, Marton gomorim@gmail.com Gyenis, Balazs gyepi@hps.elte.hu Hofer-Szabó, Gábor gsz@szig.hu
Keywords:	quantum mechanics, interpretations of quantum mechanics, stochastic-quantum correspondence, probabilistic modeling, stochastic process, stochastic process family, linearity, Markovianity, divisibility, statistical mixing, interpretation of probability
Subjects:	Specific Sciences > Probability/Statistics Specific Sciences > Physics > Quantum Mechanics
Depositing User:	Dr. Balázs Gyenis
Date Deposited:	27 Feb 2026 21:15
Last Modified:	27 Feb 2026 21:15
Item ID:	28368
Subjects:	Specific Sciences > Probability/Statistics Specific Sciences > Physics > Quantum Mechanics
Date:	26 February 2026
URI:	https://philsci-archive.pitt.edu/id/eprint/28368

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