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Local time of Lévy random walks: A path integral approach.
1Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19 Praha 1, Czech Republic.
Physical Review. E
|June 17, 2017
Summary
This study quantifies local time properties for stochastic processes using phase-space path integrals. We analyze Lévy random walks, connecting local time to fractional diffusion equations.
Area of Science:
- Statistical Physics
- Stochastic Processes
- Mathematical Physics
Background:
- Local time measures trajectory proximity to points in stochastic processes.
- Understanding local time is crucial for analyzing complex system dynamics.
Purpose of the Study:
- To quantify the properties of local time for generic Hamiltonians.
- To investigate local times specifically for Lévy random walks.
Main Methods:
- Employed phase-space path-integral representation of random walk transition probabilities.
- Utilized the resolvent of the Hamiltonian operator for time-independent systems.
Main Results:
- Developed a method to quantify local time properties using path integrals.
- Identified the resolvent of the Hamiltonian as a key tool for time-independent cases.
- Characterized local times for Lévy random walks.
Conclusions:
- The phase-space path-integral approach effectively quantifies local time.
- Local times of Lévy random walks are linked to fractional diffusion equations.
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