Title: Mellin and Wiener-Hopf operators in a non-classical boundary value problem describing a Lévy process,
Abstract: Markov processes are well understood in the case when they take place in the
whole Euclidean space. However, the situation becomes much more complicated if
a Markov process is restricted to a domain with a boundary, and then a
satisfactory theory only exists for processes with continuous trajectories.
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.
To make this general problem more tractable, we consider a particular
operator, $\mathcal{A}$, which is chosen to be the generator of a certain
stable Lévy process restricted to the positive half-line. We are able to
represent $\mathcal{A}$ as a (hyper-) singular integral and, using this
representation, deduce simple conditions for its boundedness, between Bessel
potential spaces. Moreover, from energy estimates, we prove that, under certain
conditions, $\mathcal{A}$ 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 $\mathcal{A}$.