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Space-time-modulated stochastic processes.

Massimiliano Giona¹

¹Dipartimento di Ingegneria Chimica, Materiali e Ambiente, La Sapienza Università di Roma, Via Eudossiana 18, 00184 Roma, Italy.

Physical Review. E

|January 20, 2018

Summary

This study introduces space-time-modulated stochastic processes for systems with finite propagation velocity. These processes feature a unique feedback loop between the perturbation and the evolving physical system, enabling applications in curved space-times.

Area of Science:

Physics
Stochastic Processes
General Relativity

Background:

Lorentzian transformations of Poisson-Kac processes in inertial frames present physical challenges.
Stochastic processes with finite propagation velocity require novel modeling approaches.

Purpose of the Study:

Introduce space-time-modulated stochastic processes for systems with finite propagation velocity.
Establish a framework for understanding the interplay between stochastic perturbations and evolving physical observables.
Extend stochastic process modeling to curved space-time manifolds.

Main Methods:

Definition of space-time-modulated processes incorporating nonlinear amplitude and time-horizon modulation functions.
Derivation of balance equations for probability density functions of modulated Poisson-Kac processes.

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Analysis of feedback mechanisms between stochastic perturbations and physical observables.

Main Results:

Introduced space-time-modulated stochastic processes with two-way coupling.
Demonstrated that these processes possess a unique feedback property crucial for curved space-time extensions.
Derived balance equations for modulated Poisson-Kac processes.

Conclusions:

Space-time-modulated stochastic processes offer a powerful new class of models for physical systems with finite propagation velocity.
The inherent feedback mechanism is key to their applicability in complex space-time geometries.
Further analysis of specific examples reveals the unique characteristics of these processes.