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Noise correction of large deviations with anomalous scaling
Daniel Nickelsen1, Hugo Touchette2
1African Institute for Mathematical Sciences (AIMS), Muizenberg 7950, South Africa.
Physical Review. E
|July 20, 2022
Summary
We calculated the probability distribution for time-integrated moments in the Ornstein-Uhlenbeck process. This reveals anomalous large deviations and provides insights into their origins, challenging standard large deviation theory methods.
Area of Science:
- Statistical Physics
- Stochastic Processes
- Nonlinear Dynamics
Background:
- The Ornstein-Uhlenbeck process is a fundamental model in stochastic calculus.
- Large deviation theory describes rare events in stochastic systems.
- Previous work identified an instanton term for low-noise limits of time-integrated moments.
Purpose of the Study:
- To compute the full probability distribution of time-integrated moments for the Ornstein-Uhlenbeck process.
- To incorporate Gaussian prefactor corrections to the low-noise approximation.
- To investigate the nature and origin of anomalous large deviations.
Main Methods:
- Path integral calculation
- Incorporation of Gaussian prefactor
- Comparison with importance sampling simulations
- Extension of direct Monte Carlo simulations
Main Results:
- The probability distribution includes both instanton and Gaussian prefactor terms.
- Anomalous scaling of large deviations with integration time is confirmed.
- Definition of an instanton variance provides insight into deviation generation.
- Simulations validate the theoretical calculations.
Conclusions:
- The Gaussian prefactor offers crucial corrections to low-noise approximations.
- Anomalous large deviations arise from nonlinear scaling with time.
- Standard large deviation theory methods are insufficient for these anomalous cases.
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