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The martingale characterizes a kind of fairness or unbiased nature of thestochastic process which is associated with another stochastic process. If$x_t$ evolves according to the Langevin equation whose mean drift is $a_t$ asfunction of $x_t,$ and that $a_t$ as induced stochastic process is martingalein 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 asingle classical Heisenberg spin under static field. Furthermore, theasymptotic limit of $x_t/t$ obeys the ensemble statistics of such Heisenbergspin.

Martingale drift of Langevin dynamics and classical canonical spin statistics