Trajectory of Probabilities, Probability on Trajectories, and the Stochastic-Quantum Correspondence

TL;DR

This paper systematically clarifies the relationship between probability trajectories and probability on trajectories, addressing conceptual issues in stochastic-quantum correspondence and introducing new notions like decomposability and implementation non-uniqueness.

Contribution

It provides a formal framework connecting probability dynamics and stochastic processes, clarifies misconceptions about transition matrices, and introduces decomposability as a key concept in probability evolution.

Findings

Every probability dynamics admits a Markovian implementation.

Implementations of probability dynamics are generally non-unique.

Decomposability is a broader concept than divisibility, especially in nonlinear cases.

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.