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We present a method to infer the arbitrary space-dependent drift anddiffusion of a nonlinear stochastic model driven by multiplicative fractionalGaussian noise from a single trajectory. Our method, fractional Onsager-Machlupoptimisation (fOMo), introduces a maximum likelihood estimator by minimising afield-theoretic action which we construct from the observed time series. Wesuccessfully test fOMo for a wide range of Hurst exponents using artificialdata with strong nonlinearities, and apply it to a data set of daily meantemperatures. We further highlight the significant systematic estimation errorswhen ignoring non-Markovianity, underlining the need for nonlinear fractionalinference methods when studying real-world long-range (anti-)correlatedsystems.

Inferring nonlinear fractional diffusion processes from single trajectories