Ramifications of generalized Feller theory

TL;DR

This paper extends generalized Feller theory to broader contexts, clarifies their properties, introduces extended Feller processes, and connects these concepts to differential equations on weighted spaces, with detailed proofs and counterexamples.

Contribution

It provides new insights into the construction, properties, and applications of generalized Feller processes, including the introduction of extended Feller processes and their relation to differential equations.

Findings

Extended several folklore results with detailed proofs.

Introduced the concept of extended Feller processes.

Connected generalized Feller semigroups to transport equations and semiflows.

Abstract

Generalized Feller theory provides an important analog to Feller theory beyond locally compact state spaces. This is very useful for solutions of certain stochastic partial differential equations, Markovian lifts of fractional processes, or infinite dimensional affine and polynomial processes which appear prominently in the theory of signature stochastic differential equations. We extend several folklore results related to generalized Feller processes, in particular on their construction and path properties, and provide the often quite sophisticated proofs in full detail. We also introduce the new concept of extended Feller processes and compare them with standard and generalized ones. A key example relates generalized Feller semigroups of algebra homomorphisms via the method of characteristics to transport equations and continuous semiflows on weighted spaces, i.e. a remarkably generic way to treat differential equations on weighted spaces. We also provide a counterexample, which shows that no condition of the basic definition of generalized Feller semigroups can be dropped.