Reflected Spectrally Negative Stable Processes and their Governing Equations

Abstract

This paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one, reflected at its infimum. First we derive the forward equation using the theory of sun-dual semigroups. The resulting forward equation is a boundary value problem on the positive half-line that involves a negative Riemann-Liouville fractional derivative in space, and a fractional reflecting boundary condition at the origin. Then we apply numerical methods to explicitly compute the transition density of this space-inhomogeneous Markov process, for any starting point, to any desired degree of accuracy. Finally, we discuss an application to fractional Cauchy problems, which involve a positive Caputo fractional derivative in time.

References in corpus (3)

• Fractional Cauchy problems on bounded domains
• Brownian-Time Processes: The PDE Connection and the Half-Derivative Generator
• Semi-Markov approach to continuous time random walk limit processes

Cited by in corpus (9)

• Fractional Partial Differential Equations with Boundary Conditions
• Generalised Fractional Evolution Equations of Caputo Type
• Space-Time Duality and High-Order Fractional Diffusion
• Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker-Planck-Kolmogorov equations
• Variable Total Variation Regularization for Backward Time-Space Fractional Diffusion Problem
• Stochastic representation of solution to nonlocal-in-time diffusion
• A Note on Subexponential Rate of Convergence to Equilibrium for Processes on the Half-Line
• Spectral decomposition of fractional operators and a reflected stable semigroup
• Boundary conditions for nonlocal one-sided pseudo-differential operators and the associated stochastic processes II