Multidimensional advection and fractional dispersion.

Area of Science:

• Mathematical Physics
• Stochastic Processes
• Partial Differential Equations

Background:

• Extending fractional diffusion equations to multiple dimensions presents challenges not seen in second-order equations.
• Standard methods for generating multidimensional probability distributions, like Gaussian, are insufficient for general stable vectors.

Purpose of the Study:

• To develop a method for constructing general multidimensional stable vectors for fractional diffusion.
• To define a multidimensional fractional differential operator.

Main Methods:

• Utilizing random combinations of maximally skewed stable variables on the unit sphere.
• Analyzing the properties of generated stable vectors and their relation to diffusing particle models.

Main Results:

• Demonstrated that general stable vectors, unlike Gaussians, require non-atomistic measures on coordinate axes.
• Showcased that random combinations of skewed stable variables generate a general model for diffusing particles.
• Identified subsets of these vectors as previously known symmetric stable vectors and multidimensional Brownian motion.

Conclusions:

• The proposed method successfully generates general multidimensional stable vectors for fractional diffusion.
• A novel multidimensional fractional differential operator is defined through this construction.