# Local time of Levy random walks: a path integral approach

## TL;DR

This paper develops a path integral method to analyze the local time of Levy random walks, linking stochastic process properties with quantum-inspired Hamiltonian techniques for fractional diffusion.

## Contribution

It introduces a phase-space path integral approach to study local times of Levy flights, connecting stochastic analysis with Hamiltonian resolvent methods.

## Key findings

- Provides a framework for calculating local times of Levy walks.
- Links local time properties to Hamiltonian resolvent operators.
- Enhances understanding of fractional diffusion processes.

## Abstract

Local time of a stochastic process quantifies the amount of time that sample trajectories $x(\tau)$ spend in the vicinity of an arbitrary point $x$. For a generic Hamiltonian, we employ the phase-space path-integral representation of random walk transition probabilities in order to quantify the properties of the local time. For time-independent systems, the resolvent of the Hamiltonian operator proves to be a central tool for this purpose. In particular, we focus on local times of Levy random walks (or Levy flights), which correspond to fractional diffusion equations.