This chapter is devoted to the study of Langevin equations, first order in time differential equations, which depend on a random noise, and which belong to a class of stochastic differential equations that describe diffusion processes, or random motion. From a Langevin equation, a Fokker–Planck (FP) equation for the probability distribution of the solutions, at given time, of the Langevin equation can be derived. It is shown that observables averaged over the noise can also be calculated from path integrals, whose integrands define automatically positive measures. The path integrals involve dynamic actions that have automatically a Becchi–Rouet–Stora–Tyutin (BRST) symmetry and, when the driving force derives from a potential, exhibit the simplest form of supersymmetry. In some cases, like Brownian motion on Riemannian manifolds, difficulties appear in the precise definition of stochastic equations, quite similar to the quantization problem encountered in quantum mechanics (QM). Time discretization provides one possible solution to the problem.
Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of ℏ
