Ramifications of generalized Feller theory

Area of Science:

• Probability Theory
• Stochastic Processes
• Mathematical Analysis

Background:

• Feller theory is fundamental for analyzing Markov processes, typically in locally compact state spaces.
• Generalized Feller theory extends these concepts to broader state spaces, crucial for advanced stochastic modeling.
• Applications include stochastic partial differential equations and signature stochastic differential equations.

Purpose of the Study:

• To extend and rigorously prove folklore results concerning generalized Feller processes.
• To introduce and analyze the novel concept of extended Feller processes.
• To establish connections between generalized Feller semigroups and differential equations on weighted spaces.

Main Methods:

• Detailed mathematical proofs for construction and path properties of generalized Feller processes.
• Comparative analysis of generalized Feller processes with classical Feller processes and Doob's h-transform.
• Application of the method of characteristics to link semigroups with transport equations and semiflows.

Main Results:

• Extension and rigorous validation of several key results for generalized Feller processes.
• Introduction of extended Feller processes, offering a new framework for stochastic analysis.
• Demonstration of a generic method for solving differential equations on weighted spaces via Feller semigroups.

Conclusions:

• Generalized Feller theory is a powerful tool for analyzing complex stochastic systems.
• The newly introduced extended Feller processes warrant further investigation.
• A counterexample confirms the necessity of all conditions in the definition of generalized Feller semigroups.