Levy processes and special functions

Special functions play an important role in the study of stochastic processes in general, but it is in the study of Levy processes (continuous time analogues of random walks) that we find some of the most interesting examples of their use. I will start this talk with a gentle introduction to Levy processes, after which we will discuss Bernstein, Stieltjes and Thorin functions and the role they play in the theory of Levy processes. Then we will discuss an example that involves the use of Mellin transform and Barnes double gamma function. All of these examples are similar in the sense that we are using known properties of special functions to derive new facts about Levy processes. At the end of this talk I will present an example with an opposite flow of information: here we will use known facts about Levy processes to derive new results about special functions (these new results can be considered as a generalisation of Lagrange Inversion Theorem).