Diffusion processes, Feller semigroups and Wentzell boundary conditions

Area of Science:

• Mathematical Biology
• Genetics
• Probability Theory
• Partial Differential Equations

Background:

• Diffusion processes in genetics are studied using Probability, Functional Analysis, and Partial Differential Equations.
• Feller provided a unified treatment for the one-dimensional case of gene frequency changes.
• Taira demonstrated the equivalence of these approaches in the N-dimensional case for specific Markov processes.

Purpose of the Study:

• To establish the equivalence of different mathematical approaches for N-dimensional diffusion processes in genetics.
• To demonstrate that degenerate elliptic operators with Wentzell boundary conditions can generate Feller semigroups.
• To analyze the occurrence of viscosity diffusion at the boundary of a bounded domain.

Main Methods:

• Utilizing concepts from Probability, Functional Analysis, and Partial Differential Equations.
• Applying Taira's findings on the equivalence of mathematical approaches for Markov processes.
• Investigating degenerate elliptic operators with Wentzell boundary conditions.

Main Results:

• The generator of a Feller semigroup on C(D) can be identified with a Markov transition function.
• Certain degenerate elliptic operators with Wentzell boundary conditions generate Feller semigroups on C(D).
• Viscosity diffusion occurs at each boundary point under these conditions.

Conclusions:

• Different mathematical perspectives on genetic diffusion processes are unified, even in higher dimensions.
• Degenerate elliptic operators with specific boundary conditions are shown to generate Feller semigroups.
• The study provides a framework for understanding boundary diffusion phenomena in genetic models.