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Large-deviation functions for nonlinear functionals of a Gaussian stationary Markov process.

Satya N Majumdar1, Alan J Bray

  • 1Laboratoire de Physique Quantique (UMR C5626 du CNRS), Université Paul Sabatier, 31062 Toulouse Cedex, France.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 13, 2002
PubMed
Summary

This study presents a quantum mechanics-based method to analyze the large-time behavior of a nonlinear functional for stationary Gaussian Markov processes. The findings reveal a large-deviation function describing the distribution

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Area of Science:

  • Statistical Physics
  • Quantum Mechanics
  • Stochastic Processes

Background:

  • Investigating the long-time behavior of stochastic processes is crucial in various scientific fields.
  • Nonlinear functionals of stationary Gaussian Markov processes appear in diverse applications, including finance and climate science.

Purpose of the Study:

  • To develop a general method for analyzing the large-time limit of the distribution of a nonlinear functional of a stationary Gaussian Markov process.
  • To derive a large-deviation function for this distribution.

Main Methods:

  • A novel approach mapping the problem onto quantum mechanics.
  • Analysis of the distribution P(r,T) for the nonlinear functional r[V]=(1/T)integral(T)(0)dT' V[X(T')].
  • Derivation of explicit results for special cases, including V(X)=XH(X).

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Main Results:

  • For large times (T→∞), the distribution P(r,T) approaches exp[-theta(r)T] at fixed r.
  • Identified theta(r) as a large-deviation function.
  • Obtained explicit results for the Heaviside function case, relevant to weather derivatives.

Conclusions:

  • The developed quantum mechanics-based method provides a powerful tool for studying the asymptotic behavior of stochastic processes.
  • The derived large-deviation function offers insights into the tail probabilities of the nonlinear functional.
  • The findings have potential applications in areas like financial modeling and climate data analysis.