Steve Fitzgerald (University of Leeds) – Simple stochastic processes and complex classical mechanics

Joint work with Daniel Baldwin (Leeds) and Alan McKane (Manchester emeritus)

Traditionally, stochastic processes are modelled using either a Fokker-Planck PDE approach, or a Langevin SDE approach. There is also a third way: the functional or path integral. Originally developed by Wiener in the 1920s to model Brownian motion, path integrals were famously applied to quantum mechanics by Feynman in the 1950s. However, they also offer much to classical stochastic processes. In this talk I will introduce the formalism, focussing on the one-dimensional case when the noise is weak. There exists a remarkable correspondence between the most-probable stochastic paths and Hamiltonian mechanics in an effective potential [1,2,3]. It turns out that in some cases, the paths that dominate the integral, and hence quantities like the potential barrier crossing (Kramers) rate, depart from the real line. This is in some sense analogous to the way the residues at complex poles control ordinary integrals along the real line.

[1] Ge, Hao, and Hong Qian. Int. J. Mod. Phys. B 26.24 1230012 (2012)

[2] SPF et al. J. Chem. Phys. 158.12 (2023)

[3] Honour, Tom and SPF. J. Phys. A 57 175002 (2024)