Wiener-Hopf factorization via indefinite inner products

• Main Author: Andrews, Shaun Leigh
• Published: Swansea University 2003
• Subjects: 519
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.635785

Most of the thesis is concerned with a seemingly simple example, referred to as the two-boundary problem. The problem illustrates (a) that the use of indefinite inner products can illuminate Probabilistic Wiener-Hopf Theory in symmetrizable cases, and (b) that half-winding probabilities should be thought of as branching measures for Ray processes. The use of the indefinite inner product provides us with an efficient way to tackle the traditionally difficult issue that it duality. A fully rigorous study of time reversal is always a problem for Probability Theory. The analytic approach reveals some results of considerable independent interest. The two-boundary problem is examined in Chapter 2. We begin with the necessary analysis and then confirm everything with the corresponding probability. The nature in which everything tallies is amazing, chiefly due to the crucial theorem that shows the equivalence of PDE and local martingale properties. Moreover, the way in which the analysis effortlessly provides us with the desired duality results cannot be underestimated. In Chapter 3, we look at a one-boundary problem with a drift component. The importance of the duality arguments in Chapter 2 is emphasised. In addition, unlike in the two-boundary problem, continuity of one of the underlying semigroups poses a rather serious problem. This provides motivation for part of Chapter 4. Of particular interest her is a certain mystifying ‘independence of drift’ result. Chapter 4 is concerned with (unorthodox) non-minimal, non-negative solutions of the Riccati equations for both the drift and two-boundary problems. The ‘continuity’ difficulty mentioned above in resolved in some generality.