Joint probability distributions and multipoint correlations of the continuous-time random walk

Area of Science:

• Statistical physics
• Probability theory
• Stochastic processes

Background:

• Continuous-time random walks (CTRWs) are fundamental models in various scientific fields.
• Calculating multipoint correlation functions in CTRWs can be computationally intensive.
• Existing methods often require explicit determination of joint probability distributions.

Purpose of the Study:

• To present an efficient method for determining the Fourier-Laplace transform of CTRW n-point probability distributions.
• To develop a recursive procedure for calculating Laplace transforms of multipoint correlation functions.
• To demonstrate the applicability of these methods to various waiting time and step size distributions.

Main Methods:

• Derivation of an efficient method for the Fourier-Laplace transform of joint n-point probability distributions.
• Development of a recursive procedure to compute Laplace transforms of multipoint correlation functions.
• Application of the methods to examples with independent and dependent distributions.

Main Results:

• An efficient analytical method for the Fourier-Laplace transform of joint n-point distributions was established.
• A recursive procedure was devised, bypassing the need to compute joint probability distributions first.
• Successful application to diverse examples, including dependent waiting times and step sizes.

Conclusions:

• The presented methods offer an efficient and versatile approach to analyzing continuous-time random walks.
• The recursive procedure simplifies the calculation of multipoint correlation functions.
• These techniques are valuable for studying complex stochastic systems.