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Title: Mellin and Wiener-Hopf operators in a non-classical boundary value problem describing a Levy process
Author: Hill, Tony
ISNI:       0000 0004 6497 9166
Awarding Body: King's College London
Current Institution: King's College London (University of London)
Date of Award: 2017
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This research, into non-classical boundary value problems, is motivated by the study of stochastic processes, restricted to a domain, that can have discontinuous trajectories. We demonstrate that the singularities, for example delta functions, that might be expected at the boundary, can be mitigated, using current probability theory, by what amounts to the inclusion of a carefully chosen potential. To make this general problem more tractable, we consider a particular operator, A, which is chosen to be the generator of a certain stable Levy process restricted to the positive half-line. We are able to represent A as a (hyper- ) singular integral and, using this representation and other methods, deduce simple conditions for its boundedness, between Bessel potential spaces. Moreover, from energy estimates, we prove that, under certain conditions, A has a trivial kernel. A central feature of this research is our use of Mellin operators to deal with the leading singular terms that combine, and cancel, at the boundary. Indeed, after considerable analysis, the problem is reformulated in the context of an algebra of multiplication, Wiener-Hopf and Mellin operators, acting on a Lebesgue space. The resulting generalised symbol is examined and, it turns out, that a certain transcendental equation, involving gamma and trigonometric functions with complex arguments, plays a pivotal role. Following detailed consideration of this transcendental equation, we are able to determine when our operator is Fredholm and, in that case, calculate its index. Finally, combining information on the kernel with the Fredholm index, we establish precise conditions for the invertibility of A.
Supervisor: Shargorodsky, Eugene ; Pushnitski, Alexander Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available