Martingale drift of Langevin dynamics and classical canonical spin statistics

Abstract

The martingale characterizes a kind of fairness or unbiased nature of the stochastic process which is associated with another stochastic process. If $x_t$ evolves according to the Langevin equation whose mean drift is $a_t$ as function of $x_t,$ and that $a_t$ as induced stochastic process is martingale in turn associated with the former process, then we show that the amplitude of $a_t$ is the Langevin function, which is originally the canonical response of a single classical Heisenberg spin under static field. Furthermore, the asymptotic limit of $x_t/t$ obeys the ensemble statistics of such Heisenberg spin.