SOME ASPECTS OF WIENER-HOPF FACTORIZATION

Wiener-Hopf factorization means many apparently different things, both in theory and in its wide variety of applications. This paper is designed so that almost all of it may be read by non-probabilists, though it makes demands on the reader's ability to use analogy. It is written in response to request from people in other fields to give some idea of what probabilists are doing. It gives some reformulations of the probabilistic Wiener-Hopf problem studied in London et al. One reformulation as a problem of simultaneous reduction of quadratic forms is used to motivate another as a Riemann-Hilbert problem. In addition to trying to synthesize various results, it answers affirmatively a question of McGregor as to whether a useful convolution formula which he obtained in a special case holds generally. Section 4 on examples, methods, and their interrelations is the liveliest part of the paper. Though algebra and complex analysis are successful and link perfectly with probability in much of what has so far been achieved, the scope of these methods is very severely limited, and much more challenging problems lie ahead. Motivation for this study derives originally from the practically important fact that integrals of Markov processes often provide better models than Markov processes themselves; but it has obvious pure-mathematical 'rightness' too.