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Related Experiment Video

Updated: Dec 12, 2025

Measurement of Particle Size Distribution in Turbid Solutions by Dynamic Light Scattering Microscopy
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Dynamical large deviations of reflected diffusions.

Johan du Buisson1, Hugo Touchette2

  • 1Institute of Theoretical Physics, Department of Physics, Stellenbosch University, Stellenbosch 7600, South Africa.

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|August 16, 2020
PubMed
Summary
This summary is machine-generated.

We explore large deviations in Markov diffusions with reflecting boundaries. Modifying spectral methods with zero-current conditions reveals how these boundaries influence long-term behavior.

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Area of Science:

  • Stochastic Processes
  • Statistical Physics
  • Mathematical Analysis

Background:

  • Dynamical large deviations theory typically uses spectral methods.
  • Reflecting boundaries in Markov diffusions introduce complexities not handled by standard approaches.
  • Understanding system behavior under extreme conditions is crucial in various scientific fields.

Purpose of the Study:

  • To adapt spectral methods for large deviations in Markov diffusions with reflecting boundaries.
  • To analyze the impact of zero-current conditions (Neumann or Robin) on the driven process.
  • To provide a framework for studying extreme events in systems with reflecting boundaries.

Main Methods:

  • Modification of the standard spectral approach to incorporate reflecting boundary conditions.
  • Imposition of zero-current conditions to derive Neumann or Robin boundary conditions.
  • Analysis of the resulting driven process governing long-time behavior.

Main Results:

  • The spectral approach requires modification to handle reflecting boundaries, necessitating zero-current conditions.
  • These boundary conditions alter the driven process that describes the emergence of large deviations.
  • Illustrative examples include drifted Brownian motion and Ornstein-Uhlenbeck processes with reflection.

Conclusions:

  • Reflecting boundaries significantly impact the analysis of large deviations in Markov diffusions.
  • The modified spectral approach provides a robust method for studying such systems.
  • The findings have implications for diverse applications involving stochastic processes with boundaries.