Introduction to the Backward Fokker-Planck equation

The Fokker-Planck equation is a generalisation of the diffusion equation and is commonly used to understand the evolution and steady states of nonequilibrium systems. Less well-known is the backward Fokker-Planck equation where one typically conditions on some aspect of the future of the dynamics (for example, a random walker visiting a particular region of space) and asks how the probability this happens varies with the initial condition. It turns out to be a powerful tool for answering such problems, and I will use it to illustrate the different properties of diffusion in dimensions less than, equal to and greater than two.