Potential Well in Poincaré Recurrence.

Area of Science:

• Dynamical Systems and Probability Theory
• Stochastic Processes Analysis

Background:

• The potential well quantifies points escaping a neighborhood, crucial for recurrence time approximations in mixing processes.
• Existing literature highlights the potential well's significance in dynamical systems and stochastic processes.

Purpose of the Study:

• To review scaling factors for recurrence times.
• To introduce a novel exponential approximation for hitting and return times using the potential well.
• To analyze the uniform positivity of the potential well.

Main Methods:

• Utilizing the potential well as a scaling parameter for recurrence time distributions.
• Developing explicit and sharp error terms for the approximations.
• Investigating the uniform positivity of the potential well.

Main Results:

• A new exponential approximation for hitting and return times in ϕ-mixing and ψ-mixing processes is derived.
• Explicit and sharp error bounds for these approximations are established.
• The uniform positivity of the potential well is analyzed.

Conclusions:

• The potential well serves as an effective scaling parameter for approximating recurrence times.
• The findings offer precise estimations for hitting and return times in various mixing processes.
• The analysis extends to processes on countable alphabets without assuming complete grammar.