In this thesis, we provide a comprehensive overview of classical fractional derivatives and collect results on mapping properties. In particular, we discuss mapping properties e.g. we prove that the 𝛼 order fractional derivative maps the Sobolev space W_0^(p,s) to the fractional Sobolev-Slobodeckij space W^(p,s-α) for all 𝛼 < 𝑠< 1.
Further, we present several definitions of “Bernstein fractional derivatives” using the Bernstein function and in particular, we study the Bernstein censored fractional derivative by using the Picard method to get its inverse Bernstein censored fractional integral. Moreover, we use analytic tools to get the existence and uniqueness of the solution of the corresponding resolvent equation.
Finally, we construct a stochastic process through Ikeda–Nagasawa–Watanabe (INW) piecing together procedure such that its generator is the Bernstein censored fractional derivative. Additionally, we show that this process gives a Feller semigroup.:Introduction
1 Basics
1.1 Some results in functional analysis
1.2 Fourier, Laplace and Mellin transforms
1.3 Regularly varying functions
1.4 Markov processes
1.5 LĂ©vy processes and subordinators
2 Fractional derivatives and integrals
2.1 Classical fractional integrals and derivatives
2.2 Mapping properties of fractional integrals and derivatives
2.3 Bernstein functions
2.4 Fractional derivatives based on Bernstein Functions
2.5 Probabilistic interpretation of fractional derivatives
2.6 Fractional Laplace operator
3 Censored Bernstein fractional derivative and integral
3.1 Sonine pairs
3.2 Examples of Sonine pairs
3.3 Mapping properties of general fractional derivatives
3.4 Censored Bernstein fractional derivative and integral
3.5 Linear censored initial value problem
4 Censored process
4.1 Construction
4.2 Probabilistic representation
5 Application
5.1 Censored subordinator for a regularly varying kernel
5.2 Linear censored initial value problem for regularly varying kernels
Bibliography
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:90178 |
| Date | 04 March 2024 |
| Creators | Li, Cailing |
| Contributors | Schilling, René L., Mishura, Yuliya, Technische Universität Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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