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Fractal dimensionality of Lévy processes
1Center for Studies of Nonlinear Dynamics, La Jolla Institute, P. O. Box 1434, La Jolla, California 92038.
Summary
We determined the fractal dimensionality of trajectories for translationally invariant Markov processes. Simple operational measures are introduced to estimate this fractal dimension.
Area of Science:
- * Statistical Physics
- * Stochastic Processes
- * Dynamical Systems
Background:
- * Markov processes are fundamental in modeling systems with memoryless transitions.
- * Understanding the geometric properties of stochastic trajectories is crucial in various scientific fields.
Purpose of the Study:
- * To determine the fractal dimensionality (D) of trajectories generated by translationally invariant Markov processes.
- * To develop practical methods for estimating the fractal dimension of these trajectories.
Main Methods:
- * Analysis of the geometric properties of trajectories.
- * Theoretical determination of fractal dimensionality for a specific class of Markov processes.
Main Results:
- * The fractal dimensionality (D) was successfully determined for the studied class of processes.
- * Two straightforward operational measures were developed to estimate D.
Conclusions:
- * The study provides a method to quantify the complexity of Markov process trajectories using fractal dimension.
- * The introduced measures offer practical tools for estimating fractal dimensionality in relevant systems.