Dr. Steve Fitzgerald - Path integral approach to stochastic processes

Traditionally, most investigations into stochastic processes follow one of two routes: the stochastic (Langevin) differential equation approach, where the noise is included explicitly in the equation of motion, and the PDE (Fokker-Planck) approach, where the deterministic equation for the probability density is considered instead. A third approach using functional integrals is also possible, and in some cases can prove powerful and convenient. The idea of expressing a probability as a sum over paths goes back to Wiener in the 1920s, and was later extended and popularized by Feynman for quantum mechanics and field theory. Further progress in its application to classical stochastic processes was made by Onsager and Machlup in the 1950s, and later by Graham, Martin-Siggia-Rose, McKane and others in the 1970s and more recently. In this talk I will introduce the formalism at a physicist's level of rigour, and discuss some new results and applications. In particular, much of the previous work focussed on infinite-time average quantities such as barrier-crossing rates. I will describe how the finite-time dynamical information can be retained, and how quantities of interest such as first-passage densities may be obtained.