Local time of random walks on graphs

Area of Science:

• Graph theory
• Probability theory
• Stochastic processes

Background:

• Random walks are fundamental models in various fields.
• Understanding the local time (occupation time) is crucial for analyzing random walk behavior.
• Existing methods for calculating local time averages can be computationally intensive or limited in scope.

Purpose of the Study:

• To present a general method for calculating sample-path averages of local time functionals for discrete-time random walks on generic graphs.
• To provide a computationally tractable approach for analyzing random walk behavior.

Main Methods:

• The study focuses on discrete-time random walks on generic graphs.
• A general method is presented for calculating sample-path averages of local time functionals.
• The method utilizes the resolvent of the transition matrix.

Main Results:

• A general formula is derived for computing sample-path averages of local time functionals.
• The method is applicable to any discrete-time random walk on a generic graph.
• The resolvent of the transition matrix is identified as a key component for calculation.

Conclusions:

• The developed method offers a powerful tool for analyzing the local time of random walks.
• This approach simplifies the calculation of occupation time averages, enhancing the study of graph-based stochastic processes.
• The findings have implications for network analysis, statistical physics, and computer science.