A Probability Space at Inception of Stochastic Process
Liteng Yang, Yuliang Liu, Jing Liu, Hongxuan Li, Wei Chen
TL;DR
The paper addresses how a deterministic, dissipative process subjected to resonance can spontaneously transition to a nondissipative stochastic dynamics and develops a rigorous probability-theoretic framework using signed measures and Borel σ-fields to model the inception of the stochastic process. It derives explicit probability density functions for forward and mirrored axes, including negative regions, and defines a stochastic process $X(t,\omega)$ on a common probability space comprising velocity and thermodynamic fields, proving that at a critical time $t_0$ the deterministic path and the stochastic path coexist with a formal transition mapping. The work shows that the random velocity admits tractable expressions for its mean and variance, reveals oscillatory and asymmetric pdfs with negative values, and discusses martingale properties, thereby enabling stochastic calculus tools to analyze nondissipative resonant dynamics. Overall, the framework provides a canonical bridge between deterministic and stochastic descriptions in continuum systems, with potential applications across turbulence, vibrations, and thermal transport where resonance-induced nondissipative dynamics emerge.
Abstract
Recently, progress has been made in the theory of turbulence, which provides a framework on how a deterministic process changes to a stochastic one owing to the change in thermodynamic states. It is well known that, in the framework of Newtonian mechanics, motions are dissipative; however, when subjected to periodic motion, a system can produce nondissipative motions intermittently and subject to resonance. It is in resonance that turbulence occurs in fluid flow, solid vibration, thermal transport, etc. In this, the findings from these physical systems are analyzed in the framework of statistics with their own probability space to establish their compliance to the stochastic process. In particular, a systematic alignment of the inception of the stochastic process with the signed measure theory, signed probability space, and stochastic process was investigated. It was found that the oscillatory load from the dissipative state excited the system and resulted in a quasi-periodic probability density function with the negative probability regimes. In addition, the vectorial nature of the random velocity splits the probability density function along both the positive and negative axes with slight asymmetricity. By assuming that a deterministic process has a probability of 1, we can express the inception of a stochastic process, and the subsequent benefit is that a dynamic fractal falls on the probability density function. Moreover, we leave some questions of inconsistency between the physical system and the measurement theory for future investigation. We believe that the establishment of the probability density function of resonance nondissipative dynamics in contemporary statistics should make many mathematical tools available and the analytical formulas for the random velocity and probability density function can provide a convenient platform for the development of statistics.