Application of boundary functionals of random processes in statistical physics

TL;DR

This paper explores boundary functionals of random processes and their applications in modeling physical, chemical, and biological systems, providing definitions, characteristic functions, and examples across various stochastic models.

Contribution

It introduces a comprehensive framework for boundary functionals of random processes and demonstrates their applications in diverse physical and biological models.

Findings

Characteristic functions derived for exponential demand models

Applications demonstrated in Brownian motion and diffusion models

Generalizations of boundary functional limitations considered

Abstract

The potential applications of boundary functionals of random processes, such as the extreme values of these processes, the moment of first reaching a fixed level, the value of the process at the moment of reaching the level, the moment of reaching extreme values, the time the process stays above a fixed level, and other functionals, are considered for the description of physical, chemical, and biological problems. Definitions of these functionals are provided, and characteristic functions are presented for the model with an exponential distribution of incoming demands. A generalization of these limitations is also considered. The potential uses of boundary functionals are demonstrated through examples such as a unicyclic network with affinity A, an asymmetric random walk, nonlinear diffusion, two-level model, Brownian motion, and multiple diffusing particles with reversible target-binding kinetics.