Fractional Brownian Motion with Drift: Theory and Numerical Validation

17 pages, 20 figuresWe study fractional Brownian motion of Hurst parameter $H$ with both a linear and a non-linear drift. The latter appears naturally when applying non-linear variable transformations. Using a perturbative expansion around Brownian motion, we analytically give the first-order corrections to Brownian motion, resulting in a significant change in the probability distribution functions. Introducing an adaptive bisection algorithm, which is about 1000 times faster and more memory efficient than the standard Davies-Harte algorithm, we test these predictions for effective grid sizes of up to $N_{\rm eff}=2^{28}\approx 2.7\times 10^{8}$ points. The agreement between theory and simulations is excellent, and by far exceeds in precision what can be obtained by scaling alone