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A martingale is a stochastic process that encodes a kind of fairness or unbiasedness, which is associated with a reference process. Here we show that, if the reference process xt evolves according to the Langevin equation with drift a(x) and if a(xt) is a martingale, then its amplitude is the Langevin function, which originally described the canonical response of a single classical Heisenberg spin under static field. Furthermore, the asymptotic limit of xt/t obeys the ensemble statistics of such a Heisenberg spin.
PHYSICAL REVIEW E
By: Ken Sekimoto
Phys. Rev. E 109, 014106 – Published 5 January 2024
DOI: https://journals.aps.org/pre/abstra...