Beyond the Triangle

The following sections are included:

• Dedication
• Preface
• Acknowledgments
• Contents

The following sections are included:

• Why fractional generalizations of the Fokker–Planck equation?
• The problem formulation

The following sections are included:

• Introduction
• Brownian motion
• Itô calculus
• FPK equations for stochastic processes driven by Brownian motion
  • FPK equation associated with Brownian motion
  • FPK equations associated with SDEs driven by Brownian motion
  • Connection with semigroup theory
  • Markovian processes and the Chapman–Kolmogorov equation
  • FPK equations associated with SDEs driven by Brownian motion in bounded domains

In this chapter we introduce and study properties of four different types of fractional order derivatives, namely the fractional derivatives in the senses of Riemann–Liouville, Caputo–Djrbashian, Liouville–Weyl and Riesz–Feller, and their distributed order generalizations. These derivatives are required in Chapter 7 to describe fractional Fokker–Planck–Kolmogorov equations. Fractional order derivatives in fact are defined as inverses of fractional order integration operators. Therefore this chapter starts with introducing the notion of fractional integral and studying some of its properties.

The following sections are included:

• Introduction
• Pseudo-differential operators
• Pseudo-differential operators with singular symbols
• Pseudo-differential operators associated with Lévy processes
• Some abstract facts on semigroups and linear operators
• Pseudo-differential operators on manifolds
• Pseudo-differential operators associated with stochastic processes in bounded domains

The following sections are included:

• Introduction
• The Skorokhod space and its relevant topologies
• Semimartingales and time-changes
• Lévy processes
• Subordinators and their inverses
• Gaussian processes

The following sections are included:

• Introduction
• Stochastic calculus for time-changed semimartingales
• SDEs driven by time-changed semimartingales
• CTRW approximations of time-changed processes in the Skorokhod spaces
• CTRW approximations of time-changed processes in the sense of finite-dimensional distributions
• Approximations of stochastic integrals driven by time-changed processes
• Numerical approximations of SDEs driven by a time-changed Brownian motion

The following sections are included:

• Introduction
• FPK and FPKΨ equations associated with SDEs driven by Brownian motion and Lévy processes
• TFFPKΨ/TDFPKΨ equations associated with SDEs driven by time-changed Lévy processes
  • Theory
  • Applications
• FPK equations associated with SDEs driven by fractional Brownian motion
  • An operator approach to derivation of fractional FPK equations
• Fractional FPK equations associated with stochastic processes which are time changes of solutions of SDEs driven by fractional Brownian motion
  • Theory
  • Applications
• FPK equations associated with general Gaussian processes
• Fractional FPK equations for time-changed Gaussian processes
  • Theory
  • Applications
• Fractional FPK equations associated with stochastic processes which are time changes of solutions of SDEs in bounded domains

The following sections are included:

• Bibliography
• Index