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Markov processes are well understood in the case when they take place in thewhole Euclidean space. However, the situation becomes much more complicated ifa Markov process is restricted to a domain with a boundary, and then asatisfactory theory only exists for processes with continuous trajectories.This research, into non-classical boundary value problems, is motivated by thestudy of stochastic processes, restricted to a domain, that can havediscontinuous trajectories.  To make this general problem more tractable, we consider a particularoperator, $\mathcal{A}$, which is chosen to be the generator of a certainstable L\'evy process restricted to the positive half-line. We are able torepresent $\mathcal{A}$ as a (hyper-) singular integral and, using thisrepresentation, deduce simple conditions for its boundedness, between Besselpotential spaces. Moreover, from energy estimates, we prove that, under certainconditions, $\mathcal{A}$ has a trivial kernel.  A central feature of this research is our use of Mellin operators to dealwith the leading singular terms that combine, and cancel, at the boundary.Indeed, after considerable analysis, the problem is reformulated in the contextof an algebra of multiplication, Wiener-Hopf and Mellin operators, acting on aLebesgue space. The resulting generalised symbol is examined and, it turns out,that a certain transcendental equation, involving gamma and trigonometricfunctions with complex arguments, plays a pivotal role. Following detailedconsideration of this transcendental equation, we are able to determine whenour operator is Fredholm and, in that case, calculate its index. Finally,combining information on the kernel with the Fredholm index, we establishprecise conditions for the invertibility of $\mathcal{A}$.

Mellin and Wiener-Hopf operators in a non-classical boundary value problem describing a L\'evy process