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

## TL;DR

This paper studies a non-classical boundary value problem for a Lévy process generator, using Mellin and Wiener-Hopf operators to analyze boundedness, kernel triviality, and invertibility conditions.

## Contribution

It introduces a novel approach combining Mellin and Wiener-Hopf operators to analyze boundary value problems for Lévy process generators with boundary singularities.

## Key findings

- Derived conditions for boundedness of the operator between Bessel potential spaces.
- Proved the operator has a trivial kernel under certain conditions.
- Determined when the operator is Fredholm and calculated its index.

## 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 L\'evy 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 $\mathcal{A}$.