# Research in Stochastic Processes

The stochastic processes Professor Doney is particularly interested in are random walks, Brownian motion, and Lévy processes which are the continuous time analogues of random walks. These classical processes are relatively simple to describe, and all have been studied intensively. Many of the problems he works on are also easy to state, but not usually simple to solve. Examples include

- Which random walks (S
_{n},n__>__0) are such that E(S_{N}) < infinity, where N = inf {n: s_{n}>0}? - What is the
probability that there
exists
some t > 0 with

t = B_{t}= sup_{0 < s < t}B_{s}, (B is Brownian motion)? - Which Lévy processes have points of increase?

Professor Doney is extremely active in collaborative work, mainly with the strong group of probabilists at the Lab. De Probabilite, Univ. Paris VI, whose members include Yor, Bertoin, Chaumont, Hu, Marchal and others.

He is also working with Jon Warren (Warwick), Ross Maller (Univ. of Western Australia) and Phil Griffin (Syracuse University, USA). Thus there is a stream of visiting probabilists which together with his research students, currently three in number, and the continued interest of Emeritus Professor Fredos Papangelou, makes this a stimulating environment for research.

## Members of staff involved:

Professor Ron Doney, Tel: +44 (0)161 275 5914, E-mail: Ron.Doney (@manchester.ac.uk)