An Introduction to L\'evy and Feller Processes. Advanced Courses in Mathematics - CRM Barcelona 2014

TL;DR

This paper provides an advanced introduction to Levy and Feller processes, covering theoretical foundations and extensions to space-inhomogeneous processes, aimed at graduate students and researchers in stochastic analysis.

Contribution

It extends the classical theory of Levy processes to Levy-type (Feller) processes, illustrating how local behavior resembles Levy processes and discussing their applications in stochastic analysis.

Findings

Comprehensive overview of Levy processes and Feller processes.

Extension of Levy process theory to space-inhomogeneous cases.

Connections to stochastic partial differential equations (SPDEs).

Abstract

These lecture notes are an extended version of my lectures on L\'evy and L\'evy-type (Feller) processes given at the "Second Barcelona Summer School on Stochastic Analysis" 2014 organized by the Centre de Recerca Matemaatica (CRM). The lectures are aimed at advanced graduate and PhD students. In order to read these notes, one should have sound knowledge of measure theoretic probability theory and some background in stochastic processes, as it is covered in my books "Measures, Integals and Martingales" (Cambridge University Press) and "Brownian Motion" (de Gruyter). My purpose in these lectures is to give an introduction to Levy processes, and to show how one can extend this approach to space inhomogeneous processes which behave locally like L\'evy processes: L\'evy-type or Feller processes. These course notes will be published, together Davar Khoshnevisan's notes on "Invariance and Comparison Principles for Parabolic SPDEs" as "From L\'evy-Type Processes to Parabolic SPDEs" by the CRM, Barcelona and Birk\"auser, Cham 2017 (ISBN: 978-3-319-34119-4). The arXiv-version and the published version may differ in layout, pagination and wording, but not in content