Reflected Spectrally Negative Stable Processes and their Governing Equations
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In this article, the transition densities of a spectrally negative stable process with index greater than one, reflected at its infimum, were derived using the theory of sun-dual semigroups.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.read more
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References
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The random walk's guide to anomalous diffusion: a fractional dynamics approach
Ralf Metzler,Joseph Klafter +1 more
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One-Parameter Semigroups for Linear Evolution Equations
Klaus-Jochen Engel,Rainer Nagel +1 more
TL;DR: In this paper, Spectral Theory for Semigroups and Generators is used to describe the exponential function of a semigroup and its relation to generators and resolvents.
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